Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers Vol 1 (Studies in Logic and the Foundations of Mathematics Vol 125)

CLASSICAL RECURSION THEORY The Theory of Functions and Sets of Natural Numbers

STUDIES I N LOGIC AND THE F O U N D A T I O N S OF MATHEMATICS V O L U M E 125

Editors.

ELSEVIER AMSTERDAM

LALJSANNE N E W Y O R K

OXFORD SHANNON

SINGAPORE TOKYO

CLASSICAL RECURSION THEORY The Theory of Functions and Sets of Natural Numbers Piergiorgio ODIFREDDI Uiii\v?rsirj, of' Tirriti Tir rin, Itcil!

ELSEV IHR AMSTbKDAM

LAUSANNE

N E W YORK

OXF*ORD SHANNON

SINGAPORE

TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 21 I , I000 AE Amsterdam. The Netherlands

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To Lidia W h a t thou lovest well remains, t h e rest is dross W h a t t h o u lov’st well shall not be reft from t h e e W h a t thou lov’st well is t h y t r u e heritage. (Pound, Cantos, LXXXI)

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Foreword Odifreddi has written a delightful yet scholarly treatise on recursion theory. Where else can one read about mezoic sets? His book constitutes his answer t o the central question of recursion theory: what is recursion theory? His answer, I am pleased t o note, is idiosyncratic. He makes numerous references t o set theory, for example Baire’s category theorem, the analytical hierarchy, the constructible hierarchy, and the axiom of determinateness. To my mind a n understanding of recursion theory, even a t the level of Turing degrees and recursively enumerable sets, is incomplete until the connection t o higher levels is made via set theory. If recursion theory is about computations, then the familiar finite case allows only a shallow view of the matter. Infinitely long computations, as in Kleene’s account of finite type objects, or as in Takeuti’s version of recursive functions of ordinals, permit a deeper insight into the nature of computation. This is borne out by t h e work of Slaman and others on fragments of arithmetic and polynomial reducibility, in which ideas from high up are applied low down. The author’s use of ‘classical’ in his title is partially meant, in this volume, t o date the material he covers. He concentrates on the early days of recursion theory. Perhaps those were the glory days. Perhaps only the early results will survive. The author makes the set theoretic connection but does not pursue it fully here. Let us hope he writes his next volumes on ‘modern’ recursion theory. His sparkling first volume proves him worthy of the task.

G.E. Sacks Harvard University and M.1.T November 1987

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Preface T h e origins of this book go back to fourteen years ago when, having done my studies in a country t hat , as Kreisel later remarked t o me, was ‘logically underdeveloped’, I thought I could learn Recursion Theory by writing it. There were at t h e time a few textbooks, prominent among t hem Kleene [1952] a n d Rogers [1967], but I was unsatisfied with them because papers I was interested in, on progressions of formal systems, seemingly required many results t h a t they did not cover. Thus I sat down, read a lot, and wrote a first version in Italian. Fortunately, I did not publish it. Meanwhile, I had gotten in touch with some recursion theorists, a n d decided I would go to t h e United States t o study some more. T h e Italian Center of Researches (C.N.R.) provided support, and in 1978 I landed in UrbanaChampaign, where t h e world opened up t o me. I found there a very sensitive a n d kind teacher, Carl Jockusch, who taught me in one pleasant year more t h a n I could have taught myself in a lifetime. And I found a friend in Dick Epstein, from whom I learned how t o write mathematics. Then, having read some of their papers, I went t o t he U.C.L.A. people for a year, and I’m afraid I tried their patience with my many questions. There I learned what I know of Set Theory an d Generalized Recursion Theory, through t h e teaching and help of Tony Martin, Yannis Moschovakis and John Steel. Back in Italy, I rewrote t h e whole book, this time in English. In t h e meantime, I had grown aware of t he fact t h a t mathematics was not t h e universal science t hat I had once thought it was: not only personal, but also social a n d historical influences shape the work of t h e researchers. More specifically, I had learned t h a t t h e Soviets were doing, in Recursion Theory, work t h a t t h e Westerners did not know much about, and they themselves were largely unaware of what people did in t h e West. I found this a n odd situation, a n d decided I would go t o t h e Soviet Union t o bridge t h e gap, at least in my knowledge. T h e Italian and Soviet State Departments provided support, and I stayed in Novosibirsk for one and a half years, in 1982-83, again learning a lot, in both mathematical and human terms. In particular, great help was provided by Marat Arslanov, Sergei Denisov, Yuri Ershov, and Victor Selivanov. Despite

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some difficulties there, which cost me a marriage among other things, I came back with more experience, and the book was now ready. A final and unexpected touch was added by Anil Nerode and Richard Shore, who invited me t o Cornell for a year in 1985, a n d in the following summers. With them I started a (for me) very fruitful collaboration, partly financed by a joint N.S.F.-C.N.R. grant. In particular, Richard Shore has relentlessly proved theorems t h a t covered blank spots in t h e book. In cornell I also met Juris Hartmanis, who changed my perspective in Complexity Theory. In addition to all t h e people mentioned above, I was greatly helped by those who have read, and commented upon, substantial parts of t h e manuscript, or have taught me different things, including Klaus Ambos-Spies, Felice Cardone, Alexander Degtev, Leo Harrington, Georg Kreisel, Georgi Kobzev, Manny Lerman, Jim Lipton, Gabriele Lolli, Flavio Previale, Mark Simpson, a n d Bob Soare. Many other people have provided various kinds of help a n d corrections, in particular those who attended classes and seminars on various parts of t h e book in Torino and Siena (Italy), Urbana, U.C.L.A. and Cornell (United States), Novosibirsk and Kazan (Soviet Union). I t would take t o o much space t o mention them all, but t o everybody go my sincerest thanks. Since Gutenberg, books have usually been written to be printed. In my case this was made possible by Solomon Feferman a n d Richard Shore, who introduced t h e book t o different editors. Michael Morley convinced me t h a t I could type it myself in UT)-$, a t a time when I did not even know how t o t u r n a computer on, and he and Anil Nerode helped afterwards with t h e machines, in many ways. While I was preparing t h e typescript, t h e amazing Bill Gasarch and Richard Shore provided overall corrections in real time. Finally, support for typesetting was provided by the C.N.R. Thanks to all of them, too. Different and very special thanks go t o Lidia. She was there before it all, saw the book taking shape, and heard about it more than anybody else. She followed me on my pilgrimages, and I could perceive how great a toll this was taking on her only when it was too late. She is not here anymore, to see t h e end of it, and this is most sad. T h e immense amount of time stolen from her and devoted t o this work is partly responsible for her absence. No doubt it was a stupid trade, b u t now, after fourteen years, here is t h e book: devoted t o her as a partial, late compensation for what she deserved, a n d I was unable t o give. Torino - Urbana - Los Angeles Novosibirsk - Ithaca 1974-1988

Preface to the Second Edit ion After its first appearance in 1989, Classical Recursion Theory has seen a second printing and a paperback edition in 1992, all sold out. With the appearance of Volume I1 in 1999, the time has come to publish a second edition of Volume I, revised and corrected of all the misprints and mistakes that have been noticed through the years. I seize this opportunity t o thank all the readers who have kindly pointed out imperfections in the text, thus helping me t o make it better. Recursion Theory has seen many exciting developments in the last decade, but most of them concern advanced areas which are treated in Volume 11. Since Volume I has been left largely untouched by these developments, I have not attempted any major rewriting of the original text. Instead, I have inserted in t h e two volumes a number of forward and backward pointers (to pages, results and chapters) t o make the whole book more interconnected, following the new Gospel of the Net. Torino, June 1999.

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Contents Foreword by G.E. Sacks

vii

Preface

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Preface to the Second Edition

xi

Introduction W h a t is ‘Classical’ . . . . . . . . . . . . . . . . . . . . W h a t is in t h e Book . . . . . . . . . . . . . . . . . . . Applications of Recursion Theory . . . . . . . . . . . . How to Use t h e Book . . . . . . . . . . . . . . . . . . Notations a n d Conventions . . . . . . . . . . . . . . . I

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RECURSIVENESS AND COMPUTABILITY 1.1 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions by inductions . . . . . . . . . . . . . . . . . . . . . . Proofs by induction . . . . . . . . . . . . . . . . . . . . . . . . . Recursiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical roots of Recursion Theory * . . . . . . . . . . . . . . Formal Arithmetic * . . . . . . . . . . . . . . . . . . . . . . . . Some primitive recursive functions and predicates . . . . . . . . Codings of t h e plane . . . . . . . . . . . . . . . . . . . . . . . . Elimination of primitive recursion . . . . . . . . . . . . . . . . . 1.2 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . T h e formalism of equations . . . . . . . . . . . . . . . . . . . . Definability by systems of equations . . . . . . . . . . . . . . . Derivability from systems of equations . . . . . . . . . . . . . . A logical programming language * . . . . . . . . . . . . . . . . I .3 Arithmetical Formal Systems . . . . . . . . . . . . . . . . . Notions of representability . . . . . . . . . . . . . . . . . . . . . xiii

1 2 5 11 13

17 18 20 20 21 22 23 24 26 28 31 31 33 36 38 39 39

Contents

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1.4

1.5

1.6

1.7

1.8

Formal systems representing Invariant definability * . . . Definability of functions * . Turing Machines . . . . .

t h e recursive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Variations of t he Turing machine model . . . . . Physical Turing machines * . . . . . . . . . . . . . Finite automata * . . . . . . . . . . . . . . . . . . Turing machine computability . . . . . . . . . . . Machinedependent programming languages * . . Flowcharts . . . . . . . . . . . . . . . . . . . . . . Unstructured programming languages * . . . . . Unlimited register, random access machines * . . Flowchart computability . . . . . . . . . . . . . . . Structured programming languages * . . . . . . . P r o g r a m for primitive Petri nets * . . . . . . Functions as Rules A-calculus . . . . . . .

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recursion * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other formulations of t h e A-calculus * . . . . . . A-definability . . . . . . . . . . . . . . . . . . . . . Functional programming languages * . . . . . . . Arithmetization . . . . . . . . . . . . . . . . . . Historical remarks * . . . . . . . . . . . . . . . . . Numerical tools for arithmetization . . . . . . . . T h e Normal Form Theorem . . . . . . . . . . . . .

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Equivalence of t h e various approaches t o recursiveness . . . . . T h e basic result of t he foundations of Recursion Theory . . . . Church’s Thesis * . . . . . . . . . . . . . . . . . . . . . . . . Introduction t o Church’s Thesis . . . . . . . . . . . . . . . . . . Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . Computers and physics . . . . . . . . . . . . . . . . . . . . . . . Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic physics . . . . . . . . . . . . . . . . . . . . . . . . computers and thought . . . . . . . . . . . . . . . . . . . . . . T h e brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructivism . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 44 45 46 49 51 52 53 59 61 63 64 65 68 70 74 75 76 82 83 86 87 87 88 90 97 100 101 102 105 106 107 109 113 115 118 122

Contents I1 BASIC RECURSION THEORY 11.1 Partial Recursive Functions . . . . . . . . . . . The notion of partial function . . . . . . . . . . . . Partial recursive functions . . . . . . . . . . . . . . Universal Turing machines and computers * . . . .

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Recursively enumerable sets . . . . . . . . . . . . . . . . . . . . R.e. sets as foundation of Recursion Theory * . . . . . . . . . . A programming language based on r.e. sets * . . . . . . . . . .

11.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . The essence of diagonalization . . . . . . . . . . . . . . . . . . . Recursive undecidability results . . . . . . . . . . . . . . . . . . Limitations of mechanisms * . . . . . . . . . . . . . . . . . . . . Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . Limitations of formalism * . . . . . . . . . . . . . . . . . . . . . Self-reference * . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-reproduction and cellular automata * . . . . . . . . . . . . 11.3 Partial Recursive Functionals . . . . . . . . . . . . . . . . . Oracle computations and Turing degrees . . . . . . . . . . . . . The notion of functional . . . . . . . . . . . . . . . . . . . . . . Partial recursive functionals . . . . . . . . . . . . . . . . . . . . First Recursion Theorem . . . . . . . . . . . . . . . . . . . . . . Recursive programs * . . . . . . . . . . . . . . . . . . . . . . . Topological digression . . . . . . . . . . . . . . . . . . . . . . . Iteration and fixed-points * . . . . . . . . . . . . . . . . . . . . Models of A-calculus (part I) * . . . . . . . . . . . . . . . . . . Different notions of recursive functionals * . . . . . . . . . . . . Higher Types Recursion Theory * . . . . . . . . . . . . . . . . . Computability on abstract structures * . . . . . . . . . . . . . . 11.4 Effective Operations . . . . . . . . . . . . . . . . . . . . . . Effective operations on partial recursive functions . . . . . . . . Effective operations on total recursive functions . . . . . . . . . Effective operations in general * . . . . . . . . . . . . . . . . . . Recursive analysis * . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Indices and Enumerations * . . . . . . . . . . . . . . . . . . Acceptable systems of indices . . . . . . . . . . . . . . . . . . . Axiomatic Recursion Theory * . . . . . . . . . . . . . . . . . . Models of A-calculus (part 11) * . . . . . . . . . . . . . . . . . . Indices for recursive and finite sets . . . . . . . . . . . . . . . . Enumerations of classes of r.e. sets . . . . . . . . . . . . . . . . The Theory of Enumerations * . . . . . . . . . . . . . . . . . . 11.6 Retraceable and Regressive Sets * . . . . . . . . . . . . . . Retraceable versus recursive . . . . . . . . . . . . . . . . . . . .

125 126 127 127 132 134 143 144 145 145 146 149 152 158 165 170 174 175 177 178 181 185 186 192 194 196 199 202 205 205 208 210 213 214 215 221 223 225 228 236 238 239

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Contents Regressive versus r.e. . . . . . . . . . . . . . . . . . . . . . . . . Existence theorems and nondeficiency sets . . . . . . . . . . . . Regressive versus retraceable . . . . . . . . . . . . . . . . . . .

IIIPOST’S PROBLEM AND STRONG REDUCIBILITIES 111.1 Post’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . Origins of Post’s Problem * . . . . . . . . . . . . . . . . . . . . Turing reducibility on r.e. sets . . . . . . . . . . . . . . . . . . 111.2 Simple Sets and Many-One Degrees . . . . . . . . . . . Many-one degrees . . . . . . . . . . . . . . . . . . . . . . . . . . Simple sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effectively simple sets * . . . . . . . . . . . . . . . . . . . . . . 111.3 Hypersimple Sets and lkuth-Table Degrees . . . . . . . Truth-table degrees . . . . . . . . . . . . . . . . . . . . . . . . . Hypersimple sets . . . . . . . . . . . . . . . . . . . . . . . . . . The permitting method * . . . . . . . . . . . . . . . . . . . . . 111.4 Hyperhypersimple Sets and Q-Degrees . . . . . . . . . Q-reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperhypersimple sets . . . . . . . . . . . . . . . . . . . . . . .

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Maximal sets * . . . . . . . . . . . . . . . . . . . . . . . . . . .

111.5 A Solution to Post’s Problem . . . . . . . . . . . . . . . . . Semirecursive sets . . . . . . . . . . . . . . . . . . . . . . . . . 7-hyperhypersimple sets . . . . . . . . . . . . . . . . . . . . . . 111.6 Creative Sets and Completeness . . . . . . . . . . . . . . . Effectively nonrecursive sets . . . . . . . . . . . . . . . . . . . . Creative sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasicreative sets * . . . . . . . . . . . . . . . . . . . . . . . . Subcreative sets * . . . . . . . . . . . . . . . . . . . . . . . . . Effectively inseparable pairs of r.e. sets . . . . . . . . . . . . . . 111.7 Recursive Isomorphism Types . . . . . . . . . . . . . . . . Mezoic sets and l-degrees . . . . . . . . . . . . . . . . . . . . . Recursive isomorphism types . . . . . . . . . . . . . . . . . . . Recursive equivalence types and isols * . . . . . . . . . . . . . . 111.8 Variations of Truth-Table Reducibility * . . . . . . . . . . Bounded truth-table degrees . . . . . . . . . . . . . . . . . . . . Weak truth-table degrees . . . . . . . . . . . . . . . . . . . . . Other notions of reducibility * . . . . . . . . . . . . . . . . . . 111.9 The World of Complete Sets * . . . . . . . . . . . . . . . . Relationships among completeness notions . . . . . . . . . . . . Structural properties and completeness . . . . . . . . . . . . . . III.10Formal Systems and R.E. Sets * . . . . . . . . . . . . . . Formal systems and r.e. sets * . . . . . . . . . . . . . . . . . . .

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251 252 253 254 256 257 259 263 267 269 272 277 280 281 282 288 294 294 299 304 304 306 311 314 316 319 320 324 328 330 331 337 339 341 341 348 349 350

Contents

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Undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . Essential undecidability . . . . . . . . . . . . . . . . . . . . . . Independent axiomatizability . . . . . . . . . . . . . . . . . . .

352 353 357

IV HIERARCHIES AND WEAK REDUCIBILITIES IV.l The Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . T h e definition of t r u t h * . . . . . . . . . . . . . . . . . . . . . . T r u t h in First-Order Arithmetic . . . . . . . . . . . . . . . . . T h e Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Arithmetical Hierarchy . . . . . . . . . . . . . A; sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativizations * . . . . . . . . . . . . . . . . . . . . . . . . . . IV.2 The Analytical Hierarchy . . . . . . . . . . . . . . . . . . . Truth in Second-Order Arithmetic . . . . . . . . . . . . . . . . T h e Analytical Hierarchy . . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Analytical Hierarchy . . . . . . . . . . . . . . II: sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 363 363 363 365

A: sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 392 394

Descriptive Set Theory * . . . . . . . . . . . . . . . . . . . . . . Relativizations * . . . . . . . . . . . . . . . . . . . . . . . . . .

Post's Theorem in t h e Analytical Hierarchy * . . . IV.3 The Set-Theoretical Hierarchy . . . . . . . . . Truth in Set Theory . . . . . . . . . . . . . . . . . . Standard structures . . . . . . . . . . . . . . . . . . T h e Set-Theoretical Hierarchy . . . . . . . . . . . . AFKp functions . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Set-Theoretical Hierarchy . . . . W F a n d t h e Arithmetical Hierarchy . . . . . . . .

367 373 374 375 376 377 380 381

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395 . . . . . . . 397 . . . . . . 397 . . . . . . 401 . . . . . . . 405 . . . . . . 406 . . . . . . . 411 . . . . . . . 414

Absoluteness a n d t h e Analytical Hierarchy . . . . . . . . . . . . 418 Admissible sets * . . . . . . . . . . . . . . . . . . . . . . . . . . 421 IV.4 The Constructible Hierarchy . . . . . . . . . . . . . . . . T h e Constructible Hierarchy . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Constructible Hierarchy . . . . . . . . . . . T h e structure of L . . . . . . . . . . . . . . . . . . . . . . . . . Constructible sets of natural numbers . . . . . . . . . . . . . Cksets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1C a n d t h e Analytical Hierarchy . . . . . . . . . . . . . . . . Recursion Theory on t h e ordinals * . . . . . . . . . . . . . . . Relativizations * . . . . . . . . . . . . . . . . . . . . . . . . . .

. 422 422 . 424 425 . 432 437

. 441 . 443 444

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Contents

V TURING DEGREES V . l The Language of Degree Theory . . . . . . . . . . . . . . . T h e join operator . . . . . . . . . . . . . . . . . . . . . . . . . . T h e jump operator . . . . . . . . . . . . . . . . . . . . . . . . . First properties of degrees . . . . . . . . . . . . . . . . . . . . . T h e Axiom of Determinacy * . . . . . . . . . . . . . . . . . . . V.2 The Finite Extension Method . . . . . . . . . . . . . . . . Incomparable degrees . . . . . . . . . . . . . . . . . . . . . . . Embeddability results . . . . . . . . . . . . . . . . . . . . . . . T h e splitting method . . . . . . . . . . . . . . . . . . . . . . . . Forcing t h e jump . . . . . . . . . . . . . . . . . . . . . . . . . . V.3 Babe Category * . . . . . . . . . . . . . . . . . . . . . . . . . Topologies on total functions . . . . . . . . . . . . . . . . . . . Comeager sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bake Category a n d Degree Theory . . . . . . . . . . . . . . . . Meager sets of degrees . . . . . . . . . . . . . . . . . . . . . . . Measure Theory and Degree Theory * . . . . . . . . . . . . . . V.4 The Coinfinite Extension Method . . . . . . . . . . . . . . Exact pairs and ideals . . . . . . . . . . . . . . . . . . . . . . . Greatest lower bounds and least upper bounds . . . . . . . . . Extensions of embeddings . . . . . . . . . . . . . . . . . . . . . V.5 The n e e Method . . . . . . . . . . . . . . . . . . . . . . . . Hyperimmune-free degrees . . . . . . . . . . . . . . . . . . . . . Minimal degrees . . . . . . . . . . . . . . . . . . . . . . . . . . Minimal upper bounds * . . . . . . . . . . . . . . . . . . . . . . Konig's Lemma and @ classes * . . . . . . . . . . . . . . . . . Complete extensions of Peano Arithmetic * . . . . . . . . . . .

V.6 Initial Segments * . . . . . . . . . . . . . . . . . . . . . . . . Uniform trees . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447 448 448 450 451 453 456 457 459 463 467 471 472 473 477 481 484 484 485 488 490 493 495 498 502 505 510 516 516

Minimal degrees by recursive coinfnite extensions . . . . . . . . 520 T h e threeelement chain . . . . . . . . . . . . . . . . . . . . . . 523 T h e initial segments of the degrees * . . . . . . . . . . . . . . . 528 530 V.7 Global Properties . . . . . . . . . . . . . . . . . . . . . . . . Definability from parameters . . . . . . . . . . . . . . . . . . . 530 T h e complexity of t h e theory of degrees . . . . . . . . . . . . . 536 Absolute definability . . . . . . . . . . . . . . . . . . . . . . . . 540 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 V.8 Degree Theory with Jump * . . . . . . . . . . . . . . . . . 550

Contents VI MANY-ONE AND OTHER DEGREES VI.l Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . .

XiX

555

555

Distributive upperseniilattices . . . . . . . . . . . . . . . . . . . 556 Ideals of distributive uppersemilat. tices . . . . . . . . . . . . . . 558

V1.2 Countable Initial Segments . . . . . . . . . . . . . . . . . . Finite initial segments . . . . . . . . . . . . . . . . . . . . . . . Countable initial segments . . . . . . . . . . . . . . . . . . . . . VI.3 Uncountable Initial Segments . . . . . . . . . . . . . . . . Strong minimal covers . . . . . . . . . . . . . . . . . . . . . . . Uncountable linear orderings . . . . . . . . . . . . . . . . . . . Uncountable initial segments . . . . . . . . . . . . . . . . . . . VI.4 Global Properties . . . . . . . . . . . . . . . . . . . . . . . . Characterization of t h e structure of many-one degrees . . . . . Definability, homogeneity, a n d automorphisms . . . . . . . . . . T h e complexity of t h e theory of many-one degrees . . . . . . . VI.5 Comparison of Degree Theories * . . . . . . . . . . . . . . 1-degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truth-table degrees a n d weak truth-table degrees . . . . . . . . Elementary inequivalences . . . . . . . . . . . . . . . . . . . . . VI.6 Structure Inside Degrees * . . . . . . . . . . . . . . . . . . Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inside many-one degrees . . . . . . . . . . . . . . . . . . . . . . Inside truth-table degrees . . . . . . . . . . . . . . . . . . . . . Inside Turing degrees . . . . . . . . . . . . . . . . . . . . . . . .

561 562 566 569 569 570 571 574 575 575 577 582 582 584 590 591 591 594 598 600

Bibliography

603

Notation Index

643

Subject Index

649

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Introduction Classical Recursion Theory is the study of real numbers or, equivalently, functions over the natural numbers. As such it has a long history, a n d a number of notions a n d results t h a t were originally proved in different fields a n d for different purposes are incorporated, unified a n d extended in a systematic study. We are thinking here, for example, of t h e different equivalent definitions of real number, of Cantor’s theorem t h a t the real numbers a r e uncountable, of Gadel’s class of constructible real numbers, and so 011. All of these are now part of Recursion Theory and of our study, but t h e theory also provides new tools of its own, the origins of which can be traced back to Dedekind [1888]: he introduced t h e study of functions definable over t h e set w of t h e natural numbers by recurrence using the well-ordered structure of w , whence t h e name Recursion Theory. T h e power of recursion as a tool for defining functions was analyzed in detail by Skolem [1923], Peter [1934], and Hilbert a n d Bernays [1934], but its limitations were also pointed out. Gradually the collective work of Post [1922], Church [1933], Godel [1934], Kleene [1936], and Turing [1936], led to t h e identification of t h e most general form of the recursion principle and to what we now call recursive functions. In a bold philosophical abstraction Church [1936] proposed t o identify the notion of ‘effectively computable function’ of natural numbers with t h a t of recursive function, thus providing a feeling of absoluteness to t h e notion. With Post [I9441 Recursion Theory became a n independent branch of mathematics, studied for its own sake.

What is ‘Classical’ In more recent decades Recursion Theory has been generalized in various ways t o different domains: ordinals bigger than w ,functionals of higher order, abstract sets. All these subjects belong t o what we call Generalized Recursion Theory. We use the word ‘classical’t o emphasize the fact t h a t we confine our treatment t o t h e original setting, and we will deal with notions of Generalized 1

2

Introduction

Rccursion Theory only when t h e theory provides results for t h e case we a r e interested in. If we see classical mathematics as t h e s t u d y of concrete structures, like t h e set of natural numbers in Number Theory, or t h e set of functions over t h e real or complex numbers in Analysis (as opposed to modern mathematics, where t h e emphasis is on abstract structures, like algebraic or topological ones), t h e n Classical Recursion Theory is p a r t of classical mathematics, a n d s i t s between Number Theory a n d Analysis. T h i s provides another reason for t h e word ‘classical’ in our subject. Mathematics is usually formalized in well-established systems of Set Theory such as Z F C ( t h e Zermelo-Fraenkel system, together with t h e Axiom of Choice). Our final use of t h e word ‘classical’ emphasizes t h e fact t h a t we will be working mostly in Z F C . I t is not surprising, d u e to t h e well-known independence results of Gijdel [1938] a n d Cohen [1963], t h a t only a p a r t of t h e s t u d y of real numbers can b e carried o u t in Z F C a n d we will point o u t t h e limits of our approach, together with possible extensions of Z F C suitable for Recursion Theory, at t h e e n d of t h e book.

What is in the Book T h e basic methods of analysis of t h e real numbers t h a t we a r e going to use a r e two:

Hierarchies. A hierarchy is a stratification of a class of reals built from below, starting from a subclass t h a t is taken as primitive (either because well understood, or because already previously analyzed), a n d obtained by iteration of a n operation of class construction. Degrees. Degrees a r e equivalence classes of reals under given equivalence relations, t h a t identify reals with similar properties. O n c e a class of reals has been studied a n d understood, degrees a r e usually defined b y identifying reals t h a t look t h e s a m e from t h a t class point of view. Degrees were used for t h e purpose of a classification of r e d s already in Euclid’s Book X (w.r.t. a geometrical equivalence relation, between rational a n d algebraic dependence). See Knorr [1983] for a survey. As might be imagined t h e two methods a r e complementary: first a class is analyzed in terms of intrinsic properties, for example by appropriately stratifying it in hierarchies, a n d t h e n t h e whole structure of real numbers is studied modulo t h a t analysis with t h e appropriate notion of degrees induced by t h e given class. T h e two methods also have a different flavor: t h e first is essentially definitional, t h e second essentially computational.

Introduction

3

To give t h e reader a n idea of what (s)he will find in the book we outline its bare skeleton, referring t o the introductions of the various chapters for more detailed outlines. T h e starting point of our study is t h e class of recursive functions i n t r e duced in Chapter I. T h e idea of its definition is simple: we t r y to isolate t h e functions over w t h a t are 'computable' in ways appealing both to t h e mathematician a n d t o t h e computer scientist. Having many different approaches available, and various different intuitions of t h e notion of computability, we t r y them all, a n d discover t h a t they all produce, once appropriately formalized, t h e same class of functions (and sets, through characteristic functions). Chapter I1 considers two fundamental generalizations of t h e notion of recursiveness. Partial recursive functions are the natural formalization of algorithms: these, in t h e common use of t h e term, do not necessarily define total functions but only provide for specifications t h a t allow t h e computation of values if particular conditions are satisfied. Partial recursive functionals take care of a different aspect of computations, namely t h e interactive procedure according t o which a machine can be piloted, in its behavior, by a human agent. This can be formalized by the use of oracles t h a t help t h e computation when requested by the machine. A set is recursive if membership in it is effectively computable. T h e next level of complexity is reached when a set is effectively generated. I n this case membership still can be effectively determined by waiting long enough in t h e generation of t h e set until the given element appears, but nonmembership r e quires waiting forever, a n d thus does not have effective content. Such sets are called recursively enumerable, and are the subject of Chapter 111. But t h e emphasis of t h e study here is on the relative difficulty of computation. In other words, we identify sets which are equally difficult t o compute. Then we attack t h e problem of whether t h e only relevant distinction among recursively enumerable sets, from a computational point of view, is between recursive a n d nonrecursive. T h e answer is t h a t the world of recursively enumerable sets is a variegated one, in which different nonrecursive effectively generated sets may have different computational difficulty. Chapter IV introduces t h e first hierarchies, by building on t h e fact t h a t t h e recursively enumerable sets are exactly those definable in t h e language of FirstOrder Arithmetic with exactly one existential quantifier (coding t h e fact t h a t a n element is in a given recursively enumerable set if a n d only if there is a stage of t h e enumeration in which it appears). A natural hierarchy is thus obtained by looking at t h e arithmetical sets as those sets which are definable in FirstOrder Arithmetic, counting the number of alternations of quantifiers. Other hierarchies in t h e same vein are possible: counting alternations of function quantifiers in Second-Order Arithmetic which stratifies the analytical sets; or measuring t h e complexity of t h e definition of a set of natural numbers in

4

Introduction

t h e language of Set Theory in terms of previously defined sets which defines t h e constructible sets of integers. Hierarchies are, by their nature, only partial tools of analysis. T h e notion of degree is instead a global one, classifying all sets modulo some equivalence relation. Chapters V and VI study t he structure of t h e continuum with respect t o two notions of relative computability, Turing degrees a n d m-degrees, a n d obtain two structural results. T h e first equates t h e complexities of t h e decision problem for t h e theories of Turing and m-degrees with t h a t of SecondOrder Arithmetic, t he second gives a complete algebraic characterization of t h e continuum in terms of t h e structure of m-degrees. T h e Baire Category method, in both its original version and generalized forms, is t h e basic method of proof. This completes Volume I, which introduces t h e fundamental notions a n d methods. Volumes I1 a n d I11 are a deeper and more sophisticated study of t h e same topics, in which t he structures already introduced are revisited a n d analyzed more carefully and thoroughly. Volume I1 deals with sets of t h e arithmetical hierarchy, Volume 111 with t he rest. Chapters VII and VIII resume t he analysis of t h e fundamental objects in Recursion Theory, t he recursive sets and functions, a n d provide a microscopic picture of them. We start in Chapter VII with a n abstract st udy of t h e complexity of computation of recursive functions. T hen in Chapter VIII we will attempt t o build from below t h e world of recursive sets a n d functions t h a t was previously introduced in just one go. A number of subclasses of interest from a computational point of view are introduced a n d discussed, among them: th e polynomial time (or space) computable functions which provide an upper bound for t h e class of feasibly computable functions (as opposed t o th e abstractly computable ones); t h e elementary functions, which are t h e smallest known class of functions closed under time (deterministic or not) and space computations; t h e primitive recursive functions, which are those computable by t h e ‘for’ instruction of programming languages like PASCAL, i.e. with a preassigned number of iterations (as opposed to t h e recursive functions, computable by t h e ‘while’ instruction, which permits a n unlimited number of iterations). Chapters IX and X return to t h e treatment of recursively enumerable sets. In Chapter 111 a good deal of information on their structure had been gathered, but here a systematic study of t h e structures of both t h e lattice of recursively enumerable sets and of t h e partial ordering of recursively enumerable degrees is undertaken. Special tools for their treatment are introduced, most prominent among them being t he priority method, a constructive variation of t h e Baire Category method. Chapter XI deals with limit sets, also known as A! sets, which are limits of recursive functions. They are a natural formalization of t h e notion of sets for which membership can be determined by effective trials a n d errors, unlike

Introduction

5

recursive sets (for which membership can be effectively determined), and recursively enumerable sets (for which membership can be determined with at most one mistake, by first guessing that a n element is not in the set, and then changing opinion if it shows up during the generation of the set). T h e following chapters produce an analysis of the sets introduced, and only touched upon, in Chapter IV, in particular arithmetical, hyperarithmetical, A;, and constructible sets, and various other classes. In all these chapters the study proceeds by first analyzing the classes themselves, and then looking at t h e notions of degree associated with them (respectively: arithmetical degrees, hyperdegrees, A;-degrees, constructibility degrees, as well as degrees with respect to appropriate admissible ordinals). T h e final chapter deals with nonclassical set-theoretical worlds in order t o point out t h e limitations of the classical approach, t o exactly establish its limits, and t o reach beyond it by adding appropriate axioms (prominent among them the Axiom of Projective Determinacy). Starred subsections deal with topics related t o t h e ones at hand thought sometimes quite far away from the immediate concern. They provide those connections of Recursion Theory t o the rest of mathematics and computer science which make our subject part of a more articulate and vast scientific experience. Limitations of our knowledge and expertise in these fields make our treatment of t h e connections rather limited, but we feel they add important motivation and direct the reader to more detailed references. Particular themes on which continuous commentary is made throughout t h e book are relationships with computers, logic, and t h e theory of f o r m a l systems, in particular the results known as Gadel’s Theorems. As our development becomes more technical, connections t o fields outside logic in general, a n d other branches of Recursion Theory in particular, become less important. As will be clear by now, we have opted for breadth rather than depth, a n d have provided rudiments of many branches of Classical Recursion Theory, rather than complete and detailed expositions of a small number of topics. In this respect our book is in t h e tradition of Kleene [1952] and Rogers [1967], and differs from recent texts like Hinman [1978], Epstein [1979],Moschovakis [1980], Lerman [1983], and Soare [1987], which can be used as useful complements and advanced textbooks in their specialized areas.

Applications of Recursion Theory No sound mathematical theory is self-contained or detached from t h e rest of mathematics or science. It takes inspiration from, and provides matter of reflection to other branches of knowledge. Recursion Theory is no exception and, despite this being a book on the pure theory, we will touch on applications

6

Introduction

a n d connections whenever possible. Here we give a n idea of t h e applications t h a t our subject can have in other branches of science some of which will be taken u p again in more detail in t h e book.

Philosophy If one of t h e main goals of Philosophy of Science is t h e conceptual analysis of epistemological notions, t h e n t h e foundations of Recursion T h e o r y provide sonie astounding successes for it. O n e of t h e original concerns of Recursion Theory had been t h e analysis of t h e notion of effective computability a n d of t h e related concept of algorithm. T h e isolation of t h e technical notion of recursiveness as a formal proposal intended to capture t h e essence of computability on natural numbers (see Chapter I) is a first success of t h e philosophical side of t h e theory, b u t by no means t h e only one. After all, computability o n natural numbers is j u s t one p a r t of t h e whole story. A great deal of work has been spent on axiomatizing t h e a b s t r a c t notion of computability (see p. 22l), a n d on analyzing t h e role of t h e special properties of natural numbers in computations. Decent notions of elementary computability have been proposed for abstract domains (see p. 202), a n d deeper properties have been shown to extend to a variety of domains more general t h a n w (such as c?dmissible ordinals, see p. 443). T h i s has required a n analysis of t h e role of finiteness in computations, a n d a n isolation of its essential properties. T h e familiarity of t h e notion involved, which is usually used unconsciously, magnifies t h e success obtained. T h e concern of Recursion Theory with predicativity predates even i t s concern with computability (see p. 22), a n d it is reflected in i t s widespread use of hierarchies as a mean of building classes of functions from below. O n e of these hierarchies ( t h e hyperarithmetical, see p. 391) has t u r n e d o u t to be particularly interesting a n d to provide for a n upper bound to t h e notion of A predicatively defined set of natural number (Kreisel [1960]). Related work h a s sulisequently been able to isolate a precise analogue of this notion (Feferman [1964]), t h u s doubling t h e success obtained with computability,

Computer Science T h e area of Recursion Theory t h a t deals with recursiveness is p a r t of Theoretical Computer Science. Turing’s analysis of computability i n t e r m s of machines provided t h e conceptual basis for t h e construction of phystcal computers in t h e late Forties: in t h e United States through Von Neumann, who knew Turing’s work, a n d in t h e United Kingdom through Turing himself ( s e e p . 132). Different approaches to recursiveness generate different types of programming languages, a n d we discuss (Chapter I a n d Section 11.1) how t h e computational core of

Introduction

7

PASCAL, LISP, PROLOG, and SNOBOL can easily be obtained from t h e appropriate versions of recursiveness. Finally, a good deal of Recursion Theory is devoted t o t h e analysis of t he complexity of algorithms and t o a classification of recursive functions according t o t h e tools needed t o compute them. This is rapidly becoming a field of its own, called Complexity Theory, with methods a n d results strongly influenced by other parts of Recursion Theory (see Chapters VII a n d VIII).

Number Theory T h e very origins of Recursion Theory place it close t o Number Theory: t h e m G tivation of Dedekind [1888] was t he analysis of t h e concept of natural number (see p. 22), while Skolem [1923] wanted t o present a formulation of Arithmetic t h a t avoided t h e difficulties of t h e common solutions t o t h e paradoxes. But perhaps t h e most striking application of Recursion Theory t o Number Theory is t h e solution of Hilbert’s Tenth Problem (see p. 135) which asked for a d e cision procedure t o determine the existence of solutions of given diophantine equations. Matiyasevitch [1970] proved a representation theorem, showing t h a t t h e sets of (non-negative) solutions of diophantine equations are exactly t h e re cursively enumerable sets. A negative solution t o Hilbert’s Tenth Problem then follows from t h e existence of a recursively enumerable, nonrecursive set.

Algebra Until t h e second half of t h e last century, including t he work of Lagrange, Gauss, Abel, and Galois, algebra had been developed in a strictly constructive way. T h e dichotomy between constructive and nonconstructive methods arose with t h e notion of prime ideal, which both Kronecker and Dedekind discovered from th e usual constructive approach, but which Dedekind published in t h e now coninion set-theoretical framework. After t hat , nonconstructive methods which may produce less informative b u t more easily graspable arguments have become standard (see Metakides and Nerode [1982] for more historical background). Recursion Theory makes t h e analysis of t h e constructive content of classical results possible, as t h e following typical case illustrates. Steinitz Theorem shows t h a t a field has a n algebraic closure which is unique u p to isomorphism. I ts original proof does not constructivize: this is an accident for t h e existence part, but necessary for t he uniqueness. T h e former follows from Rabin [1960] who, using a different existence proof, showed t hat a recursively presented field (i.e. a field with recursive set of elements and field operations, including equality) always has a recursively presented algebraic closure. T h e latter comes from Metakides and Nerode [1979], who showed t hat uniqueness

8

Introduction

(up to recursive isomorphism) of the recursively presented algebraic closure is equivalent to t h e existence of a splitting algorithm (to determine whether a polynomial is irreducible or not), a result t h a t uses t h e priority method introduced in Chapter X. T h e analysis of t h e effective content of classical algebra has been thoroughly pursued: see Ershov [1980], Crossley [1981], Nerode a n d Remmel [1985] for references. T h e usefulness of Recursion Theory in the analysis of constructivity in algebra is plausible. But there are unexpected uses too, such as in Higman (19611 who shows t h a t t h e finitely generated groups embeddable in a finitely presented group are exactly t h e recursively presented ones (i.e. those for which t h e set of words equal t o 1 is recursively enumerable), thus linking a purely algebraic notion with t h e notion of recursiveness. Higman’s representation theorem easily implies t h e undecidability of t h e word problem (to determine whether two words are equal) for finitely presented groups, proposed by Dehn in 1911 and solved by Novikov (19541 a n d Boone [1959]. T h e undecidability of the easier word problem for semigroups, proposed by Thue [1914] and solved by Post [1944] and Markov [1947], is historically important, being t h e first undecidability result of a problem from classical mathematics. These results started a whole area of research, devoted to t h e determination of which properties of algebraic structures a r e (un)decidable. See Tarski, Mostowski a n d Robinson [1953], Ershov, Lavrov, Taimanov a n d Taislin [1965] and Ershov [1980] for detailed treatments a n d references.

Analysis Bore1 [1912] introduced t h e notion of computable real number, using t h e intuitive notion of computability. T h e very paper in which Turing introduced his influential approach to computability was motivated by t h e search for a formal definition of computable reals, and was thus t h e beginning of recursive analysis. Turing isolated a class of recursive reds t h a t is independent of t h e proposed constructivization (in the sense t h a t all classically equivalent definitions of real number remain equivalent when appropriately constructivized), contains all commonly used reals, and is algebraically closed. Subsequent work extending the notion of recursive functional (Section 11.4) defined t h e notion of a recursive function of a real variable as a function defined on all reals, not only on the recursive ones. This provided the needed tools t o analyze t h e effective content of analysis: a result is constructive if whenever it has recursive data it provides us with recursive solutions. As a typical example, Weierstrass proof of t h e existence of a maximum for a continuous real function on a closed interval is constructive; b u t a n argument at which t h e maximum is attained cannot be constructively found

Introduction

9

(Lacombe [1957],Specker [1959]). Another example is provided by the ordinary differential equation y' = f ( z , y ) : the original proof of Picard that i f f satisfies a Lipschitz condition t h e solution exists and is unique is constructive, but Aberth [1971] and Pour El and Richards [1979] showed that even t h e existence alone is not constructive if f is only uniformly continuous. See p. 213 for more on the subject.

As for algebra, one can look for undecidability results as well, some of which have been obtained by Richardson [1968], Adler [1969] and Wang [1974]. As an example, the latter proves that there is no recursive procedure t o decide whether a real elementary function has a zero.

Set Theory Recursion Theory and Set Theory have a large overlap in the study of sets of integers of high complexity: the material dealt with in Volume I11 could hardly be classified as solely belonging to one of them; it is rather a new field sprung from their marriage. But Recursion Theory does have successful applications t o pure Set Theory in areas were the latter seems to be classically impotent. The best-developed applications have been two theories about cardinals: recursive equivalence types, and admissible ordinals. The former deals with sets that, in a constructive sense, are infinite but Dedekind-finite, i.e. can be one-one mapped neither t o a proper initial segment of w, nor t o a proper subset of themselves. Classically such sets do not exist in the presence of the Axiom of Choice, but their recursive versions have generated a rich theory that provides new insights into the notion of finiteness (see p. 328). Another branch of Set Theory which is classically unmanageable is the theory of large cardinals: even the inaccessible ones, the smallest proposed type, cannot be proved to exist i n classical Set Theory. The lack of examples different from w forces one to resort to trivial cases, such as considering 1 as weakly but not strongly inaccessible because ' 0 = 1 (Godel [1964]). Recursion Thcory provides a well-developed analogue of the theory of large cardinals, in which the role of the first regular cardinal is taken by the first ordinal which is not the order type o f a recursive well ordering of w (see p. 385). The notion of adtnissible ordinal (11. 443) takes care of the analogue of regular cardinal i n general as an ordinal closed under recursive operations on ordinals, and analogues of a great variety of large cardinals can already be seeii to exist among the countable ordinals. The existence of analogues of Ramsey cardinals can be disproved which might prompt some reflection on the role of very large cardinals in Set Theory (see Volume I11 for details).

10

Introduction

Descriptive Set Theory Cantor's Set Theory, a n d in particular t h e unlimited use of t h e power set, provoked various reactions at t h e turn of t h e century, one of which produced Descriptive Set Theory as a study of larger a n d larger classes of sets of reals which were explicitly defined (see p. 392). This approach, in which hierarchies a r e one of t h e main tools, is obviously a forerunner a n d a n analogue of various recursion theoretical hierarchies (see Chapter IV), t h e main difference being one level of complexity: sets of reals a r e considered in t h e first case, sets of integers in t h e second. B u t Addison [1954], [1959] discovered t h a t not only a r e there analogies: t h e full classical theory can be obtained by relativization of t h e recursive hierarchy theory by substituting continuous functions a n d open sets for recursive functions a n d recursively enumerable sets (see p . 392). T h i s implies t h a t all classical theorems have recursive versions of which they a r e consequences (but not conversely). This allows a unified approach, with recursion theoretical methods applicable to t h e classical case, a n d t h e theory has been resurrected from t h e state of lethargy in which it had fallen in t h e Forties.

Constructive Mat hemat ics T h e use of constructivism in classical mathematical theories is conservative: nonconstructive methods a r e accepted, a n d t h e issue is only whether given proofs a r e constructive as they stand, or can be replaced by constructive ones, a negative answer being interesting a n d acceptable. B u t constructivism can be taken more seriously as a philosophy of mathematics t h a t would simply banish nonconstructive notions a n d proofs from practice. One possible approach to constructive mathematics consists of using t h e notion of recursiveness as a substitute for t h e notion of constructivity. This can be taken literally, as in Markov's school (see p. 214), which considers only those mathematical objects a n d operations on them t h a t can be effectively described by recursive p r o c o dures as existing. But it can also be taken as a tool of analysis to compare different approaches. For example, in Kolmogorov [1932] intuitionism is seen as a logic of problems: a V P means to solve one of cr a n d p, cr --+ p to reduce t h e problem of solving /3 to t h a t of solving a , 3za(z) to solve a(.) for some z, a n d so on. Kleene [ 19451 then introduced t h e notion of recursive realizability for Intuitionistic Number Theory: numbers realize formulas if they code, inductively, recursive procedures t h a t prove t h e formula according to t h e constructive meaning of t h e logical operations. Realizability has been extended t o Intuitionistic Set Theory by Kreisel a n d 'Ikoelstra [1970] and, even if not accepted as t h e only possible way of interpreting intuitionistic provability, it has become a common tool of analysis since it provides for constructive models of theories. See Troelstra

Introduction

11

[1973] and Beeson [1985] for detailed treatments of t h e subject.

Logic After t h e first fifty years in which Recursion Theory was mainly motivated by mathematical problems about Arithmetic, t h e logicians took over. Their main interest was still in Arithmetic, but their point of view was metamathematical. In their hands t h e theory obtained its most astonishing and revolutionary results which a r e also the best known applications of t h e subject and one of t h e main impulses to its growth. By a balanced use of two of t h e most fundamental methods of proof of Recursion Theory, arithmetization and diagonalization, a complete characterization of the expressiveness of formal systems was obtained, t h e result being that (as in the case of diophantine equations) exactly t h e recursively enumerable sets are (weakly) representable in any consistent formal system having a minimal arithmetical strength. T h e existence of a recursively enumerable, nonrecursive set then implies t h e undecidability and incompleteness of any such system (see Section 11.2), thus showing t h e inadequacy of t h e concept of formal system. These are t h e highlights of t h e extensional analysis of formal systems provided by recursion theoretical methods, b u t by no means t h e only ones (see p. 349). A result of Myhill [1955] (111.7.13) points out t h e limits of this analysis and shows that, from a n extensional point of view, all formal systems of common use look alike in the sense of being all recursively isomorphic.

How to Use the Book This book has been written with two opposite, and somewhat irreconcilable, goals: t o provide for both an adequate textbook, and a reference manual. Supposedly, t h e audiences in t h e two cases are different, consisting mainly of students in t h e former, and researchers in the latter. This has resulted in different styles of exposition, reflecting different primary goals: self-containment and detailed explanations for textbooks, and completeness of treatment for manuals. We have tried to solve the dilemma by giving a detailed treatment of t h e main topics in t h e text, and sketches of t h e remaining arguments in t h e exercises a n d in t h e starred parts. T h e exercises usually cover material directly connected to t h e subject just treated and provide hints of proofs in the majority of cases, in various degrees of detail. In a few cases, for completeness of treatment a n d easiness of reference, some of t h e exercises use notions or methods of proof introduced later in t h e book. T h e starred chapters and sections treat topics t h a t can be omitted on a

12

Introduction

first reading. T h e starred subsections deal with side material, usually giving broad overviews of subjects t h a t a r e more or less related to t h e main flow of thought, b u t which we believe provide interesting connections of Recursion Theory with other branches of Logic or Mathematics. T h e style is mostly suggestive: we t r y to convey t h e spirit of t h e subject by quoting t h e main results a n d , sometimes, t h e general ideas of their proofs. Detailed references a r e usually given, both for t h e original sources a n d for appropriate u p d a t e d treatments. T h e general prerequisite for this book is a working knowledge of first year undergraduate mathematics. When dealing with applications, knowledge of t h e subject will be assumed b u t , since t h e treatment is kept separate from t h e main text, there will be no loss in skipping t h e relative parts. T h e chapters have been kept self-contained as far as possible. We have d o n e our best to keep t h e style informal a n d devoid of technicalities, a n d we have resorted to technical details only when we have not been able to avoid t h e m , no doubt because of our inadequacy. Instead of t h e usual complicated diagrams of dependencies, we give suggestions on how t h e first two volumes of t h e book can be used as a textbook for classes in which Recursion Theory is t h e main ingredient.

Elementary Recursion Theory Chapters I a n d I1 provide a number of alternative approaches to recursiveness a n d t h e baqic development of t h e theory. Scctions 2 to 6 of C h a p t e r I a r e independent a n d can be chosen according to t h e audience in t h e class. More precisely, mathematicians can concentrate on Sections 2 a n d 3 a n d cover also t h e Incompleteness a n d Undecidability Results, treated in Section 11.2. On t h e other hand, computer scientists will find inore interest in Sections 4 to 6 of Chapter I a n d Section 11.1, where t h e foundations of a number of prograniniing languages a r e laid, a n d can also cover self-reproducing machines, touched upon in Section 11.2, a n d t h e tools needed to build models of A-calculus a n d combinatory logic, covered i n Sections 11.3 a n d 11.5. Section 1.0 t r e a t s Church’s Thesis in a less simpleminded way t h a n usual (i.e. facing t h e problems, instead of sweeping them under t h e rug), a n d i t is perhaps more appropriate for philosophers.

Recursively Enumerable Sets T h e elementary theory of r.e. sets a n d degrees is contained in C h a p t e r 111 which requires only sonic background in elementary Recursion Theory. T h e chapt,er goes u p to t h e solution t o Post’s problem (Sections 1 to 5 ) a n d t h e basic classes of r.e. sets. It can be used either as a final section of a course

Introduction

13

on elementary Recursion Theory (not dealing with alternative definitions of recursiveness), or as t h e initial segment of a n advanced course on r.e. sets. In t h e latter case, it should be followed by Chapter IX, dealing with t h e lattice of r.e. sets, an d a choice of material from Chapter X, in which priority arguments are introduced. Some of t h e material here, e.g. t he theory of r.e. m-degrees, is not standard, b u t is useful in various respects: intrinsically, this structure is much better behaved than t h e schizoid one of r.e. T-degrees, and it reflects t h e global structure of degrees, which t he latter does not; moreover, arguments on T-degrees (such as t h e coding method) ar e better understood in their simpler versions for m-degrees.

Degree Theory Elementary degree theory is treated in Chapter V which, with some background in elementary Recursion Theory, can be read autonomously. We develop t h e theory u p to a point where it is possible t o prove a number of global results. This forms t h e nucleus of a course, a n d it can be followed by a number of advanced topics including a choice of results from Chapters XI a n d XII, on degrees of A! a n d arithmetical sets. Chapter VI,on m-degrees, is often unjustly neglected, b u t it does provide for t he only existing example of global characterization of a structure of degrees. It can be read independently of Chapter

V.

Complexity Theory Chapters VII an d VIII deal with abstract complexity theory and complexity classes, an d do not require any background, except for a working knowledge of recursiveness and Turing machines (like Sections 1 and 4 of Chapter I). T h e treatment is fairly complete but , going beyond t h e usual unbalanced confinement t o polynomial time and space computable functions, it also covers unjustly neglected classes of recursive functions, such as elementary, primitive recursive, a n d €0-recursive ones which are of interest t o t he computer scientist.

Not at ions and Conventions w = {0,1,. . . } is t h e set of natural numbers, with t he usual operations of plus (+) a n d times ( X or .), a n d t h e order relation 5. P(w)is t h e power set of w , i.e. t h e set of all subsets of w. ww and p are, respectively, t h e sets of total a n d partial functions from w t o itself. We reserve certain lower or upper case letters t o denote special objects: 0

a, b , c, . . . ,~ , yz ,, . . . for natural numbers

14

Introduction 0

f,g, h, . . . for total functions of any number of variables

0

a,@,r,. . . ,(p, $,

0

x,. . . for partial functions of any number of variables

F, G, H , . . . for functionals, i.e. functions with some variables ranging over numbers, an d some over functions

0

A , B , C , .. . , X , Y ,2 , ., . for sets of natural numbers

0

P, Q, R , . . . for predicates of any number of variables

. . for strings, i.e. partial functions with finite domain a n d values in {0,1).

0 CT,~,.

Regarding sets: 0 2

E A means t hat z is an element of

A

0

(A1 is t h e cardinality of A , i.e. t h e number of its elements

0

A

0

0

0

0

0

0

0

CB

and A

c B ar e t he relations of inclusion a n d strict inclusion

is t h e complement of A , and t he prefix ‘cc-’ in front of a property of a set means th a t the complement has this property (i.e. a set is c e i m m u n e if its complement is immune)

A U B is th e union of A and B , i.e. t h e set of elements belonging t o at least one of A and B

A

@ B is the disjoint union of A a n d B , i.e. t h e set of elements of t h e form 2 2 if 2 E A , and 2 2 + 1 if 5 E B

A n B is th e intersection of A and B , i.e. t h e set of elements belonging t o both A an d B A x B is th e Cartesian product of A and B , i.e. t h e set of pairs whose first and second components are, respectively, in A a n d B A . B is the recursive product of A and B , i.e. t h e set of codes pairs ( 2 , ~E)A x B (see p. 26 for codings)

(2,~)

(2, y)

of

is t h e characteristic function of A , with value 1 if t h e given argument is in t h e set, arid 0 otherwise.

CA

Regarding predicates:

Introduction 0

15

-

- P , P A Q, P V Q, P --+ Q , P Q, VxP, 3xP are t h e usual logical operations of negation, conjunction, disjunction, implication, equivalence, universal an d existential quantification. T h e symbols -+ and H will be used in a formal way, t o build new properties from given ones. T h e symbols =+ and will be used informally, as abbreviations for ‘if . . .then’, and ‘if and only if’.

*

We use bounded quantifiers as abbreviations:

(32 I Y)P(4 for (Vx 5 y ) P ( x ) for 0

(3x)[x I Y A q x ) ] (Vx)[x 5 y + P(x)].

cp is t h e characteristic function of P , with value 1 if P holds for t h e given argument and 0 otherwise.

Regarding binary relations on a set A , R is: 0

reflexive if x R z for every x E A

0

antireflexive if ~ ( x R z )for , every x E A

0

symmetric if xRy

0

transitive if xRy A yRz =+ xRz for every x , y, z E A

3

yRa: for every x , y E A

0

a (weak) partial ordering if it is reflexive and transitive (weak partial orderings are indicated by 2,5 , or C)

0

a (strict) partial ordering if it is antireflexive and transitive (strong partial orderings ar e indicated by <, 4 , or C)

0

0

0

a total ordering if it is a partial ordering, and xRy V yRa: V (x = y) for every x, y E A a n equivalence relation if it is reflexive, transitive, and symmetric; in this case t h e set A is partitioned into equivalence classes (each consisting of t h e elements t h a t ar e in t he relation R with each other) a n uppersemilattice if any pair of elements of A has a I.u.b., a n d a lattice if any pair of elements of A has both 1.u.b. and g.1.b. (given two elements 2 a n d y , their least upper bound (1.u.b.) an d greatest lower bound (g.1.b.) are, respectively, t h e smallest element of A greater than bot h x and y, a n d t h e greatest element of A smaller than both x and y).

Regarding functions: 0

f o g or f g denote t h e composition of f and g

Introduction

16 0

f(") denotes the result of n iterations o f f , i.e. n successive applications o f f (by convention, f(')(x) = x)

0

cp(x)1 means that cp is defined on x

0

cp(x)T means that cp is undefined on

0

0

0

0

2

the set of elements on which cp is defined is called its domain, a n d t h e set of elements which are values of cp for some argument is called its range cp N II, means that cp and II, are equal as partial functions, i.e. on each argument they are either both undefined, or both defined and equal

the set of pairs (x,y) such that cp(x)N Y is called t h e graph of cp

a p means that as partial functions p extends a, i.e. if a is defined on a n argument, then p is too and has t h e same value.

Each chapter is divided into numbered sections, a n d each section is divided into unnumbered subsections. There is a unique progressive numbering inside sections, including definitions, results, and exercises. Internal references in a given chapter may omit t h e chapter number. T h e bibliography only includes papers quoted in t h e book. We have done our best to attribute results and quote t h e original sources. In case of unpublished results, when an attribution has been possible through personal communication or other sources we have attached names without references, and the mistakes that may have occurred are unintentional. We are, of course, well aware of the fact t h a t simply quoting original sources is only a ghost of history, and it barely hints at the growth and interaction of ideas. But at least it provides the bare facts.

It is now time to plunge into the real work. We hope you will find t h e book readable, despite the difficulties imposed partly by t h e subject, but mostly by our limitations. Try t o be patient, and remember patience is the great thing, and above all things else we must avoid anything like being or becoming out of patience. (Joyce, Finnegans Wake)

Chapter I

Recursiveness and Computability This chapter attacks t h e problem of characterizing t h e notion of effective computability, by isolating various different proposals. T h e methods i n t r e duced in Section 7 show them all t o be equivalent, thus demonstrating that we have certainly found a natural and fundamental class of functions. In Section 8 we discuss whether we have reached a satisfactory solution, and t o which extent it is possible t o believe that the class of functions so isolated coincides with the class of effectively computable functions. T h e various approaches we introduce can be roughly classified into two groups:

Mathematical. We start in Section 1 with a class of functions defined by mimicking t h e basic arithmetical notions, the principle of induction among them. We note that the functions of this class are naturally defined by means of equations, and thus undertake in Section 2 a general study of systems of equations. We then discover that by adopting special formal rules we can derive the values of a function from a system of equations defining it. In Section 3 we thus investigate the functions whose values can be derived by any logical means in current formal systems suitable for arithmetic.

Computational. By analyzing the human process of routine calculation, we set up in Section 4 a machine-like model of computation and programs for it. In Section 5 we then consider the purely algorithmical skeleton of programs, by abstracting from t h e specific implementation of the machine. In a final generalization we then set up, in Section 6, a theory of 17

I. Recursiveness and Computability

18

functions as abstract programs.

Of course different classifications a r e possible. A particularly relevant one, from a computational point of view, would make a distinction between deterministic a n d nondeterministic notions. So, e.g., Herbrand-Godel computability, representability in formal arithmetical systems, A-definability a n d , more generally, all notions of derivability in suitable formal systems (the most comprehensive formulation in this direction being Post canonical systems, introduced in Section 11.1) a r e nondeterministic, since they provide rules which can be applied in certain situations, b u t do not establish t h e order of application when multiple choices are available. It is however possible t o introduce restrictions in nondeterministic approaches to turn t h e m into deterministic ones, usually without affecting their power (and this is actually done when these approaches a r e taken as basis for programming languages). It is important to stress t h a t for much of the later development of Recursion Theory, alternative characterizations of recursiveness, as well as its relation with effective computability, are not needed. T h e various sections have been kept mostly independent from each other, so t h a t they can be read separately. T h e reader not interested in foundational aspects of Recursion Theory can even skip t h e whole chapter, except for Sections 1 a n d t h e first part of Section 7, in which t h e recursive functions a n d t h e fundamental method of arithmetization a r e respectively introduced. Arithmetization is a basic technical tool, which is here applied to produce a normal form for t h e recursive functions, a n d to show the equivalence of t h e various approaches to recursiveness. As a whole, this introductory chapter (and t h e first two sections of t h e next one) may b e thought of as a technical version of what Webb [1980] does philosophically a n d Hofstadter (19791 pyrotechnically. These books may offer various (and sometimes unexpected) complements to t h e matters here discussed (especially so for those we just hint a t ) . T h e y a r e recommended reading.

I. 1

Induct ion

T h e subject of this book is a close look at functions from natural numbers to natural numbers. T h e interest of our study is evident: t h e natural numbers a r e one of t h e most natural type of mathematical objects, a n d thus our functions a r e among t h e most natural mathematical functions. B u t to even understand what such functions are, we must first of all have a good grasp of t h e objects they relate. We then s t a r t by analyzing t h e intuitive picture of t h e natural numbers, trying t o characterize their structure. Something is clear: t h e natural numbers a r e all in a single discrete row, with a first b u t no last element. Since what matters to us is just their mutual relationship a n d not their ultimate

1.1 Induction

19

individual nature, we may imagine them as obtained from a first element (the number 0), by iteration of a generation procedure (the successor operation S). T h u s t h e numbers ar e

0 S(0) S ( S ( 0 ) ) . .. or (by using now t h e natural numbers metalinguistically, t o indicate t h e number of iterations of S) 0 S q o ) S y o ) .. .

We simply write n for s ( ~ ) ( o ) . Three axioms t h a t we take for granted from our intuitive picture above are t h e following (in a first-order logic with equality):

Axioms 1.1.1 (Dedekind [lSS8]) A1 S(z) = S(y)

+z

=y

A3 z # 0 --+ ( 3 ~ ) ( = z S(y)). T h e y say t h a t t h e successor induces a n isomorphism between w (t he set of natural numbers) and (3 - ( 0 ) . Also, they rule out some unwanted pictures of t h e natural numbers, like ones with cycles, or with two infinite sequences of elements like a0 a1 a2

... bo bl

b2

...

Unfortunately, they leave space for structures like

a n d t o be able t o isolate just t he initial part of these structures we need t o say t h a t every element can be reached from 0 by a finite number of applications of S. This seems to involve t he very notion of integer t h a t we are trying to characterize, an d might seem t o be circular (It is actually impossible to do this in a first-order way. See t h e related remarks on p. 24). We then take a n operational stand, and begin to study how we can deal with functions a n d properties of natural numbers. There are basically two ways: we may want to define something new, or t o check properties of something we already have. Necessarily (according t o our intuition) we have t o proceed in both cases by induction, i.e. starting from 0 and going on by means of t h e successor operation.

20

I. Recursiveness and Computability

Definitions by inductions A typical example is given in the following definitions of sum a n d product, that reduce each of these two binary functions to an infinite family of unary functions (obtained by fixing t h e first argument). Definition 1.1.2 (Grassmann [lSSl]) A4 A5 A6 A7

x+O=X ”+S(y) =S(X+Y) x . 0 ~ 0

x.S(Y)=X-Y + X.

In both cases a new function is defined, first for 0 and then for a generic S(y), using the work already done for y. A general formulation of this process (with parameters) is the following, where we write y + 1 for S(y), as usual.

Definition 1.1.3 (Dedekind [ISSS]) A f i n c t i o n f is defined f r o m g and h by primitive recursion if

I(.’,O) f(CY

=

+ 1)

= h(Z,Y,f(lc’>Y)).

Proofs by induction Suppose we have a property cp of natural numbers a n d we wish to check t h a t it holds for every number. From the way the numbers are generated, this follows if the property holds for 0 and it propagates through t h e successor operation, since every number is obtained from 0 by a finite iteration of S. This is expressed by the Axiom of Induction:

Axiom 1.1.4 (Dedekind [ISSS])If cp is a formula with one free variable then A8 cp(0) A ( W c p ( X )

+

cp(S(4)l

+

PY)cp(Y).

In terms of sets this means t h a t any set containing 0 and closed under successor contains w , or t h a t the numerals Scn)(0) exhaust t h e natural numbers. This is thus a tentative t o restrict the possible models of t h e axioms Al-A3. For our purposes it is better t o express this principle in the equivalent form of Complete Induction, which refers t o the natural ordering of t h e natural numbers, that can be introduced for example as:

x5y x
(3z)(x+z=y) e x
@

21

1.1 Induction We then have the following equivalent form of A8: W‘.”X

< 4cp(4

+

cp(4l

+

(~Y)cp(Y).

By writing ?c, in place of l c p and taking the contrapositive, Complete Induction is equivalent t o the following Least Number Principle:

(3Y)$J(Y)

+

.I).(+-).

(34[$J(4A (V’. <

Its content is simply that if we know that a number with a certain property exists, then we also know t h a t there is the least number satisfying that property. A general formulation of this principle (with parameters) in terms of functions is:

Definition 1.1.5 (Kleene [1936]) A function f is defined from a relation R b y p-recursion’ i f 1. R is a regular predicate, i.e. (VZ)(3y)R(Z,y). 2. f(2)= p y R ( Z ,y), where p y R ( Z ,y) is the least number y such that R ( 5 , y )

holds. Similarly, f is defined fromg b y p-recursion if 1. 2

PWkM3c’,Y)= 0) f (3= PY(g(?cY) = 0).

Note that the Least Number Principle can be simply written in p-notation as (3y)?c,(Y) ( 3 4 ( 2 = P9$J(Y)>. The name recursion for both the processes above (primitive recursion and p-recursion) is justified by the fact that they are both defined by recurrence on the natural numbers. +

Recursiveness We are now ready for our first attack on the notion of effective computability. The idea is simple: the two processes just introduced certainly produce effectively computable functions when applied to effectively computable functions and predicates. We just have t o take an inductive approach, by starting from the effectively computable functions corresponding t o 0 and S and by successively building up new functions using primitive recursion and p-recursion. We will also permit a rudimentary logical intuition t o contribute t o the class, both in initial functions (identities or projections) and in building rules (composition of known functions), t o allow for useful manipulations. We are thus led t o the following notion. p is the Greek equivalent of the first letter of ‘minimum’.

22

I. Recursiveness and Computability

Definition 1.1.6 (Dedekind [1888], Skolem [1923], Godel (19311) The class of primitive recursive functions is the smallest class of functions 1. containing the initial functions

O(s)

= 0

S(s) = s + l r ( s 1 ..., , s ), = xi (1 I

i

I n )

2. closed under composition, i.e. the schema that given 91,. . . ,gm,h produces

3. closed under primitive recursion.

A predicate is primitive recursive i f its characteristic function is. Definition 1.1.7 (Kleene [1936]) The class of recursive functions is the smallest class of functions 1. containing the initial functions

2. closed under cornposition, primitive recursion and p-recursion.

A predicate i s recursive i f its characteristic function is.

Historical roots of Recursion Theory

*

Dedekind [1888], improving the work of Grassmann [1861], was t h e first to succeed in t h e analysis of the concept of natural number. He was able to isolate t h e axioms for 0 a n d S , and t h e principle of second-order induction. He was immediately faced with the problem of justifying his formal theory as adequately describing t h e informal notion of number. For this purpose he offered a characterization theorem: the theory described, up to isomorphism, only one structure (in modern terms, it was categorical). T h e basic idea for the proof was t o see t h a t a n isomorphism of any two structures satisfying t h e axioms can be immediately defined by (primitive) recursion. T h e main task for Dedekind was, thus, t h e justification of t h e existence of functions defined by such a principle. By doing this he pulled t h e trigger of Recursion Theory. A great impetus for t h e early work on t h e field was set by t h e suggestion of considering only effectively defined functions: this underlay various constructive approaches t o mathematics, among which two have been particularly relevant t o our subject. Semi-intuitionists (like Kronecker [1887], Poincar6 [1903], [1913], and Lebesgue [1905]) were interested, on t h e positive side, in

1.1 Induction

23

effective solutions to mat hematical (especially algebraic) problems. They were also reacting, on the negative side, t o Cantor’s Set Theory and his use of the power set (which, they thought, should be taken as consisting of just those sets of natural numbers which are somehow explicitly definable). Finitists (Hilbert [1904] and his school), stimulated by t h e discovery of paradoxes, were trying t o constructivize Dedekind’s second-order result on Arithmetic by bringing it into the realm of first-order logic: one of their main interests was a consistency proof for Arithmetic done by finitary means. This soon led t o general problems of characterizing finitistic arithmetical methods and their relationships with primitive recursion.

Formal Arithmetic

*

T h e simplest partial formalization of arithmetic that can be extracted from our treatment is the Robinson Arithmetic Q (R.M. Robinson [1950]). It consists, in t h e language of first-order logic with equality, of a constant 0, functional symbols S, and . , and the axioms Al-A7. Local variations are possible, e.g. both t h e constants 0 and S can be defined (and thus eliminated) in t h e following way:

+

z = o @ z+z=z ~ = 1@ x . x = x A ~ # O S(z) = y

@

y=z+l.

Also, axiom A3 can be replaced by t h e following:

z=ovo
< is t h e predicate so defined: z


@

(3z)(z

+ S ( z ) = y).

This system is quite weak, but nevertheless sufficient to represent every recursive function (in a precise sense introduced in definition 1.3.1). Robinson Arithmetic adds the defining equations of plus and times to the axioms for successor. Primitive Recursive Arithmetic (Skolem [1923]) actually adds axioms corresponding t o the definitions of all the primitive recursive functions. Equivalently, one could add a general schema of primitive recursion:

R(f,z,O) = R(f,z,S(y)) =

2

f(W,~,Y),lI).

T h e next step would be t o add the equation for p-recursion (or equivalently, as we know, t h e induction principle), and this would result in Peano

I. Recursiveness and Computability

24

Arithmetic Pd (a first-order version of Peano [l889]), which can be defined as the extension of Q obtained by adding t o it the axiom A8. Note t h a t A8 is actually a schema of axioms, one for each formula of t h e language with one free variable. Axiom A3 now becomes derivable from the others, and it is usually dropped. But dramatic simplifications are not possible: Pd,although independently axiomatizable, is not finitely axiomatizable (Ryll-Nardzewski [1952], Montague and Tarski [1957]). Leaving aside formal systems, we can consider t h e structure (w,+,-) underlying our intuition of natural numbers and take a semantic approach t o arithmetic. This pertains to the following chapters of our study and we quote here two extremal cases: First-Order Arithmetic, that is, the set of formulas of the first-order language with equality true in the structure; and SecondOrder Arithmetic, obtained similarly by considering the second-order language. Note that First-Order Arithmetic, though completely determining the first-order sentences true in the standard model (i.e. determining the structure up to elementary equivalence), admits countable models not isomorphic to it (Skolem [1934]). This shows that no purely first-order version of Dedekind’s isomorphism theorem exists. But, quite appropriately, t h e notion of recursiveness does provide the missing ingredient: a n y recursive model of Peano Arithmetic is isomorphic t o the standard one (Tennenbaum [1959]).

Some primitive recursive functions and predicates We begin our treatment by noting (after Skolem [1923]) t h a t many interesting arithmetical functions and predicates in common use are primitive recursive. To illustrate the use of identities and composition, we show t h a t addition is primitive recursive. Recall its defining equations

2+0

2

= S(z+y).

2+S(y)

They define a function f ( z , y ) = z

=

+ y such that

f(X,O)

f b , Y + 1)

=

=

2

w.7 Y)).

This can be put in the form allowed by Definition 1.3 by letting (by composition of initial functions)

= S(mz,Y,z)) f(z,O) = I;(.) f(z,!/+ 1) = h(WAf(2,Y)). +,Y,Z)

1.1 Induction

25

In t h e following we do not attempt to put definitions in the allowed forms, which can be easily done as a n exercise by the reader. It should be clear however t h a t , due t o t h e existence of the identity functions TF and the possibility of substitutions (compositions), the schemata of primitive recursion and p-recursion may be applied quite freely b y interchanging, identifying and intro-

ducing variables when needed. T h e following are primitive recursive:

Predecessor

pd(0) = 0 p d ( x + 1) = 2. Integer difference2 x-0 2 - (y+

a

= X

1) = p d ( 2 - y ) .

Bounded sums

Bounded products, as above. This shows that the primitive recursive predicates are closed under logical connectives and bounded quantifiers, since e.g. the characteristic functions of Y P , PA&a n d (Vy 5 z ) R ( Z , y )are respectively 1-cp, C P - C Q and c~(.‘,y).

nYrZ

This allows us t o translate usual arithmetical definitions into logical terms, a n d to often find out that they are primitive recursive. For example:

x divides y

x I y e (32 5 y)(x. z = y).

x is a prime

P r ( z ) @ x 2 2 A (Vy 5 x)(y I x

-+

y = 1 V y = x).

2Since we do not consider negative integers, the differenceof two numbers is set equal to 0 when it would give a negative value.

I. Recursiveness and Computability

26:

To define the sequence of prime numbers p,, t h e natural guess would be: Po pz+l

= 2 = the smallest prime number greater than p , = P Y ( W Y ) A Y > P5).

This is a permissible application of the p-operator since t h e predicate is regular (there a r e infinitely many prime tiutribers), b u t we a r e not allowed to use t h e 11-operator i n primitive recursion. However p,+l 5 p5! 1 (by Euclid's proof), i.e. we have a primitive recursive bound (the factorial is primitive recursive because z! = n,,,(y I)). And the primztzve recursive fimctzons are closexi under bounded p-recurszon

+

+

Indeed, if

g(Z, y) = cliaracteristic function of R( Z ,y) A (b'z

Y<

< y ) y R ( Z ,2 )

2

We can also decortipose a number in a primitive recursive way: the exponent of k in thc decomposition of y

e z P ( Y ,k ) = Pz
++l

I Y)l.

Here we stop our first taste of primitive recursion: t h e unsatiated reader can have a bellyful by turning t o Chapter VIII.

Codings of the plane It is possible (after Cantor [1874]) t o p u t t h e plane (i.e. t h e set w x w of ordered pairs of natural numbers) into a oneone, onto correspondence with t h e line (i.e. t h e set w of natural numbers) in a primitive recursive way. T h i s can be achieved by dovetailing: enumerating the pairs of t h e first row in t h e picturc, a n d inserting after each of them all t h e pairs connected to it by arrows.

... ...

1.1 Induction

27

Note t h a t all pairs on a same arrow-connection have t h e same sum. We thus order t h e pairs by their sum, and t h e pairs with t h e same sum in a lexicographical way (i.e. by first component):

Since there are z order will be

+ 1 pairs with sum z , t he position (x + y)2

of t he pair (z,y) in t h e

+ 3z + y 2

i
a n d J is primitive recursive. It also has primitive recursive inverses:

L .I(. L ( z ) = PL”
R ( z ) = PY&a:

= J(Z1Y)) =J(z,d).

It is easy t o check t hat t h e following properties hold: zI J ( X , Y ) and Y 0

I J(5,Y)

+ y > 1 3 x < J ( z ,y ) A y < J ( z ,y ) z < z’ * J ( z , y ) < J(z’,!/) Y < Y’ * J ( Z , Y ) < J ( Z 1 Y ’ )

z

R ( J ( z , y )= ) Y,L ( J ( z , y ) )=

) z. and J ( L ( z ) , R ( z )=

Slightly different codings are:

1. By noting t h a t each natural number different from 0 can be uniquely written as t h e product of a n even (22) and an odd (2y 1) number, t h e map (5,y) H 22(2y 1) - 1.

+

+

2. By considering t h e 1’s in t he binary expansion of natural numbers as markers, th e map (z, y)

H

1 0 . . . 0 1 0...o = 2y

v

z times

w

+ 2z+y+1.

y times

This is not onto as a coding of pairs. But since every natural number can be thought of as coding a finite sequence of natural numbers this way, this actually provides an onto coding of all finite sequences (Minsky

[1967]). Other codings will be considered in t h e next theorem (Giidel’s function a n d in Section 7.

p)

I. Recursiveness and Computability

28

Elimination of primitive recursion We have introduced two definitional schemata based o n induction: primitive recursion a n d p-recursion. T h e next result shows t h a t t h e former is subsumed under t,he latter, modulo appropriate initial functions. T h e intuition comes from Peano Arithmetic, which is built only o n plus, times a n d t h e induction principle. Theorem 1.1.8 (Godel [1931], Kleene [1936a]) T h e class of recursive functions is the smallest class 1. containing sum, product, identities

ZT and the characteristic function 6

of equality

2. closed under composition 3. closed under p-recursion. Proof. Let C be t h e smallest class satisfying t h e conditions of t h e theorem: clearly every function in C is recursive, a n d for t h e converse we only have to show t h a t t h e constant function 0 a n d t h e successor S a r e in C, a n d t h a t C is closed under primitive recursion. Since 0 is t h e least number equal to 0,

Also, since 0 ifz#y 1 otherwise a n d 1 is t h e least number different from 0,

l(z) = ~LY(S(02(Z,Y),2;2(2,Y)) = 0) = py(S(0,y) = 0) = 1 S(z) = T;(z)+l(z). T h e whole problem is t h u s to show t h e closure of C under primitive reciirsion. T h e idea is t h e following: we will show t h a t it is possible to define in C a function p such that for every finite sequence ao, . . . , a, of natural numbers there is one natural number a coding the sequence via 0,i.e. such t h a t (VZ

5 n)(P(a,i)= a,).

T h e n , if f is defined from g a n d h in C b y primitive recursion as

f(.’,O)

=

d.’)

f ( Z , Y + 1) = h(KY,f(KY))

29

1.1 Induction we can code t h e sequence of values of f from 0 t o y by using t(5,y) = p z p ( z , O ) = g ( 5 ) A (Vz

p:

< y)(p(z,z + 1 ) = h(Z,i,@(z,z)))].

Since then

f(5,Y) = P(@,Y),Y), f will be in C if t is, and this is t he case if C is closed under logical operations (including universal bounded quantifier). We thus have three steps t o perform:

1. Existence o f p This is a purely number-theoretical argument. We want p s.t. for any sequence ao,. . . ,a, there is a s.t. for i 5 n, p(a, i) = a,. We will t ry to get t h e ai’s as remainders of t he division of a number c by given numbers &, . . . ,d,. T h a t is, we want a , = r m ( c , d , ) . Then c = Qidi ai, a, < di. Suppose this holds for another c’: c’ = qidi a i . By subtracting we get c - c’ = (q, - qri)d,, i.e. t he difference c - c’ is also divisible by d i . If t h e di are relatively prime, then c - c‘ is also divisible by their product p = d o . . . d,. This means t h a t for different c, c’ < p we get different sequences of remainders when dividing by di (otherwise c - c’ 2 p ) . Moreover, any sequence of n 1 numbers less than t he di’s is obtained (since t h e number of possible sequences is p , and different c < p give different sequences, i.e. we get them all). Then one of these sequences is t h e given one ao, . . . ,a,. 3

+

+

+

We now have t o obtain t he di’s, with the stated conditions t h a t they be relatively prime and ai < di. T h e easiest way t o get them relatively prime is t o consider t h e sequence

l+d

1+2d

... l + ( ? ~ + l ) d

+

+

for any d = s!, s 2 n. (1 r d , 1 r’d are relatively prime since if q divides both of them, then it also divides their difference ( r - r’)d, a n d hence it divides d because r - r’ 5 n is a factor of d = s ! , since s 2 n. But then q divides d and 1 r d , i.e. it divides 1.) We also want these numbers t o be greater t han t he respective ai’s: for this is enough to have d 2 a,, hence s 2 a,.

+

+ +

We can then let d , = 1 (i l ) d with d = s!, for any s greater t han n an d t h e ai’s. By coding c and d into a = J ( c , d ) we can actually define

p(~,i) =

rm(c,di) =T

= rTTL(C(U),1

+ (i + 1 ) d )

~ ( c1,

+ ( 2 -k 1)R(a))

3This is the Chinese Remainder Theorem, so called because it was known to the Chinese already in the 6th Century B.C.

I. Recursiveness and Computability

30

2 C’losure of C undcr logical op~rittioris For t h e proposit,ional connectives: 0

negation We need a function interclrangirig 0 arid I . Note that,

S(z,O)=

0 1

ifzfO otherwise

arid thus c ~ , n ( z= ) 6(cR(z),O). 0

disjunction We need a fiirictioii which is 0 exactly when both the arguiiieiits are 0, i.e. wheii z + y = 0. Then d ( z ,y) = 6(6(z y,O), 0) satisfies the need, and cIzVs(:c) = d ( c l t ( z ) c.y(z)). ,

+

Closure under negatioii iinplies that, i i i C we can apply the p-operator tlirtx:t,ly to predic.at,es:

0

bounded poperator We use, in this proof only, the f o r m

which is expressiblc in C as

0

0

bounded existent i d quantifier By t tie part i c x l a r fornr o f I)oiindetl p-operator defined above w r h a w

Thc other operations arc o b t a i n d by compcx;il,ion as

iisiial,

e.g

1.2 Systems of Equations

J

31

is defined as

Since in C we have sum, product a n d coniposilion, we only need t h e function f(z) = z / 2 : f(2)

R

and

= py(2y = z

v 2y + 1 = z )

C a r e in C by their very definitions, e.g. R ( z ) = px(3y < z + l ) ( z = J ( X , Y ) )

t h e remainder of t h e division of x by

r m ( z , y ) = pz<$q

JJ

is defined as

< x + 1)(.

= qy

+

2).

0

T h e use of Godel’s function p in t h e proof above can be avoided by using different codings. Of course t h e usual coding by prime numbers (see Section 7) would not work here, since t h e relevant functions a n d predicates a r e d o fined by primitive recursion, which is exactly what we have to avoid. For very elementary coding techniques, see Quine [1946] a n d Smullyan [1961]. Complementary results have been obtained by .J. Robinson [1950], [1968], o n e of t h e m being t h a t the recursive functions of any number of variables can be obtained, b y composition alone, from the recursive functions ofjust one variable, plus s u m and identities. And the recursive functions of &st one variable can be defined independently, without using functions of more variables, by t h e operations of composition a n d inversion of onto functions ( t h e latter clearly corresponding to p-recursion applied to regular predicates), from t w o ( b u t not from only one) suitable initial functions.

1.2

Systems of Equations

From a mathematical point of view a function is usually defined from a set of equations, a n d we have used this fact informally in t h e previous section. We now undertake a n analysis of t h e expressive power of systems of equations, bearing in mind t h a t our concern is effective computability, a n d t h a t we therefore require not only a definition of a function, b u t also some way to compute it.

The formalism of equations Definitions a r e linguistic objects, a n d we t h e n need, first of all, a n appropriate language t o express them. Also, since we a r e dealing with numerical functions,

I. Recursiveness and Computability

32

we need linguistic analogues of t h e numbers, which we c a n generate from 0 by t h e sucwssor operation. Our fornialism t h u s cwnsists of 1. symbols 0

equality '='

0

coristant,s 0 (for t h e number 0) arid

0

parentheses '(' a n d ')'

0

comma ',' (to s e p a r a k variablw)

0

variables n:o, 5 1 , . . . for nuinbers

0

c o n s t a n t s f;, f;', . . . for w a r y fitnctions (for each n ) .

S (for t h e s~~c:cessor operation)

2 . numerals

0

0 is a nunic:ral i f a is a numeral, so is S ( a )

0

nothing else is a nunieral.

0

wc will write

w

for tile nuniera~S ( ~ ~ )representing (O) tIie nunilwr n

3. terms 0

0 is a terni

0

variables a n d 0-ary functional lett,c:rs a r e terms

0

i f t is a t e r n i s then

0

if

0

nothing else is a t e r m .

t l , . ..

so

is S ( t )

,t,, a r e ternis t h e n so is

f:'(tl,.

.. ,t T l )

Note, in particular, that, the ~iurneralsa r e tcrnls. 4. equations If t and s a r e terms, and t is of t h e form fF(t1,. . . , t,,) for sonic' 71, i ant1 1 1 , . . . , t , not, c:ontaining any functional letter except possibly S , t h e n 1 = s is a n q u a t , i o n . T h e iclc'a is t h a t cxluwtions define functions b y t h e i r right-hand sides, while left-hand sides j u s t tell which functions a r e defined. 5 . systems of equations A system of eqiiations is simply a finite set of e q u a t i o ~ i s .

We will write & ( f l , . . . , f,,; Z) for a system of e q u a t i o n s whose functional letters a n d nunicrical v a r i a l h a r e all, respectively, aniong f l , . . . , f,,, a n d

-

9

Y .

1.2 Systems of Equations

33

Definability by systems of equations T h e first thought t h a t comes to mind is to consider systems of equations t h a t define functions uniquely w.r.t. a given letter (which we may always suppose to be f1):

Definition 1.2.1 (Herbrand [1931]) A function f i s definable by a system of equations i f there is a system & such that: 1. there is a solution to the system:

(i.e. there are functions satisfying every equation of &) 2. any solution determines uniquely

fl

as f: for every f l ,. . . ,f,,

Note t h a t t h e existence condition 1) is necessary, otherwise any system of equations & without solutions would define any function f , since t h e n t h e clause (V.i?)&(fl,. . . , fn; 2') would be false, a n d t h e implication of condition 2) would be vacuously true. For our purposes of identifying computable functions this notion of definability is however unsatisfactory, because each value f(2) is determined (independently of 2) by a global infinitary condition involving every Z. Herbrand probably intended t h e uniqueness condition 2) to be proved constructively, a n d such a proof would probably give a n explicit computation procedure for f. B u t , we do not have a precise notion of constructiveness (after all, we a r e precisely trying to characterize t h e related notion of effectiveness). A n d from a classical standpoint t h e class of functions definable by systems of equations is too comprehensive (although mathematically very interesting), and i t very much transcends t h e class of recursive functions (Kalrnar [1955]): it coincides with t h e class of hyperarithmetical functions (Grzegorczyck, Mostowski a n d Ryll-Nardzewski [1958]), which will be studied in Volume 111 (see also p. 391). Computations, whatever they might be, a r e certainly finite objects a n d t h u s each value f(Z) should be uniquely determined by j u s t a finite a m o u n t of information. W e a r e t h u s led to t h e following modification of the notion introduced above.

Definition 1.2.2 (Kreisel and Tait [1961]) A function f is finitely definable by a system of equations if there is a system & such that:

I. Recursiveness and Computability

34 1.

(3f1)...(3f,)~")&(fl,... ,fn;.3

2. for every 5 there is a finite set 21,. . . ,ZP (p depending o n 5) such that

the substitution instances of E by them determines the value off such that for every f 1 , . .. ,f,

E(fl,. . . ,f,;

6 )A . . . A &(f1,. .. ,f,; g )+-fl(S)=

(Z),i.e.

m.

For example, the system of equations (written informally)

f (0)= 0

f .(

+ 1) = f (4+ 2

g ( 4 = f ( g ( x+ 1 ) )

defines (for every x) f(x) = 2 x and g ( x ) = 0. Here f is finitely defined (by t h e first two equations), while g is not: indeed g is uniquely determined by the infinite set of equations

g ( 0 ) = 2 . g ( l ) = 4 . g ( 2 ) = - .. b u t there are infinitely many solutions to any finite subset of these equations.

Theorem 1.2.3 (Herbrand [1931],Giidel [1934],Kleene [1935])Every recursive function is finitely definable. Proof. 1. initial functions 0,sand are, respectively, finitely defined by:

= i3 fO(x0) = S ( x 0 ) fo"(21,... ,x,) = xi.

f,'(.o)

2. composition Suppose gi(z1,. . . ,x,) (i = 1,. .. ,m) and h a r e finitely defined, respectively, by &, and & w.r.t. f? and f g + l (by changing t h e letters if needed, we can always reduce to this case). Then

f ( % . .. ,En)= h ( g l ( z 1 , . .. ,4,. . . , g m ( G , . . . ,GJ)

-

is finitely defined by El U . . U Em U & together with fOn(Zl,...

,%) = fm"+l(fin(21,...,2,),... , f ; ( Z l , . - .

,GL)).

Of course we must suppose (here and in t h e following) that there are no conflicts of functional letters in E l , . ..,&m,& (which can always be achieved by possible changes of letters).

1.2 Systems of Equations

35

3. primitive recursion If g a n d h a r e finitely defined by El a n d &z w.r.t. f(Z1,.

f(Z1,.

= S(%, . . . ,%) 1) = h(z1,. . . , %,Y,

fg

and

f;+'

then

.. , % , O )

. . ,%,Y

+

f(% . . . >%,Y)).

is finitely defined by El U &2 together with f;+l(z1,.

. . , zn,Q =

fo"(%.

w.r.t.

and

. . ,%)

and

4. p-recursion If g is finitely defined by

&I

f;+l

t h e n let & be El plus t h e equations defining s u m a n d product (which a r e primitive recursive, a n d for which we t h e n already have finite definability), together with

f;(O)

=G

f;@) = i

f , j ( s (=~i~ ) ) f;(s(z0)) =S

a n d t h e formal translations of t h e following: f ; + l ( Z l , . . . r ~ n r ~ n + l= ) f,(f;+l(%.

..,Zn,%+l))

. fin+l(ZI,.

.. ,%,S(G+l))

+fi'(f;+'(%-. . ,Zn,Zn+I))-. %+1 fo"(x1,. . . , Z n )= f3"+'(%. . . ,Zn,O). T h e n & finitely defines f w.r.t. following facts: i t defines h('c',') =

fg, because t h e system j u s t translates t h e

{ $ 2 , ~+

if g(Z,y) = O 1) otherwise

w.r.t. f;L+l (by using two auxiliary functions distinction), a n d t h e n it defines

f(Z)= h(Z,O)

f; a n d fi

to allow for case

I. Recursiveness and Computability

36

w.r.t,. f;.

1ritlt:td

h ( Z , O ) = py(g(Z,y) = 0) hraiise if py[!g(3C,y) = 01 = 2 theii /I(:?,

0) = h ( Z , 1 ) = . . . = h ( Z ’ , z )= 3.

fr”,

N o k t,liat. & has soliitioiis: t h e only t,rouMe might c o m e from but here the natural iiiterpretation applies: either for a given Z there are irifinitc~y~ r i a n yy ’ s siic~it,Iiat g ( ~y), = 0, ancl t,Iieii tIie clefiiiit,ioii o f f;”’ is the n a t u r a l step function, o r t h e r e tire only finitely rriaiiy suc.11 y’s, arid then a n y function constant from t h e last o n c of such y ’ s would work. 0

Derivability from systems of equations Kivirig sliown that (:very recursive filnction is finitely definable b y systeriis o f wlliiit,iolis (in logical t,erriis), wc would also like t,o concoc:t,explicit r u l c s to oblain thc vahies o f a hinctioii f r o m a systerri o f equations finitely defining it, thus detcrininirig a syntartical counterpart to the seriiaritical riotion o f finite definability. Since t h i s a m o u n t s to finding an a p p r o p r i a t c formal systcin ( c a p t u r i n g a notion of validity for a given interpretation) we have as n a t u r a l c a n d i d a t e s t h e aiialogues of well-known rilles iised for logical fornial systems: s u b s t i t u t i o n a11d cut.

Definition 1.2.4 (Godel [1934]) A n n-ary functioii f a s Herbrand-Godel computable if them is a fiiiite system of e4unlioi~s& such that i f f,” is the kftiriost lctter in thv last cquat.ion oft‘, thcn

holds if and only ,if the tquntioii

can be derived f r o m & b y means of the folloviing two rules: R1 Substitution o f a nuineral f o r every occwrenc(: of a particular variable in an equatioii.

R2 Replacement in thc right-hand-side of an q u a t i o n of a term- of the f o r m f;”(Z1,. . . ,E7,,)with a nuineral d , prorrideclJ,”’(T., , . . . ,C,,,) = d has alrtmdy been derivrd.

1.2 Systems of Equations

37

Theorem 1.2.5 (Herbrand (19311, Godel [1934], Church [1936], Kleene [1936]) Every recursive function is Herbrand-Godel computable.

Proof. W e consider here t h e systems of equations defined in t h e proof of Theorem 1.2.3. T h e r e a r e two things to prove: 1. Completeness property ( t h e values can be deduced from t h e appropriate systems of equations). T h i s is quite evident from t h e informal discussion of 1.2.3.

2. Consistency property (no other value can be deduced, i.e. t h e valutzj a r e uniquely determined). T h i s follows by inductlion on t h e construction of t h e systems. As a n example, we show t h e case of primitive recursion (with notations as iri 1.2.3). Suppose n+l l o ( X i , . . . , z n , g )= b has been derived. T h e r e a r c two cascj: -

0

g=o

T h e n , since It2 only allows for replacement on t h e right-hand side, a n d since there is only one equation with (51,. . . ,Z, 0) on t h e left-hand side, i t follows t h a t

font'

-

n+l -

fo

= fg(zi,...,xn) fg(T1,. . . ,zn) = b (xl,...,zn,Q

have been previously derived. But t h e consistency property liolds for ft by intluction hypothesis. 0

jj = S(Z) Theii, as in t h e above case, a n equation

(z, , . . . ,z, S ( 2 ) )= n+2

lo

-

(Xl,..

. r%,~,Ign+i(z,,..,z,,q>

must have been derived. Note t h a t by definition there c a n be no equation with t,he left-hand side like t h e right-hand side of t h e above equation. Also, R2 only allows for replacement of ternis of t h e form: functional letter followed by nunierak. T h e n equations like -

-

f;+'(zl,.. . ,xn,z ) = Ti

fg+2(z1,.. . ,zn,z,z) =6

must, have been derived. B u t then again t h e induction hypothesis applies. 0

I. Recursiveness and Computability

38

In Section 7 we will prove t h a t t h e syntactical notion of Herbrand-Gadel computability a n d t h e semantical notion of finite definability a r e globally equivalent, by showing them equivalent to recursiveness. Also, by t h e proof just given, every recursive function is Herbrand-Godel computable f r o m a system of equations finitely defining it. But the two notions are not locally equivalent, in t h e sense t h a t given a system & t h e following may happen: 1. g can be finitely defined by & without being Herbrand-Godel computable f r o m it, as t h e system

f(z) = 0

f(z)= h ( z ) g ( z ) = h(z).

a n d t h e function g ( z ) = 0 show. This simply results from t h e fact t h a t t h e rules R1 and R2 are of a very specific form, a n d do not even allow for full logical substitution of equal entities.

2. g can be Herbrand-Godel computable f r o m & without being finitely defined by it, as t h e system f(0) = 0

f(S(z))= S(f(S(z)))

g ( 4 = f(0)

and t h e function g ( z ) = 0 show. Here Herbrand-Gadel computability follows because only f(0) is used among t h e values o f f , but finite definability fails because there is no total function f satisfying every equation (since f(z) = f(z) 1 for z > 0).

+

Kreisel and Tait [I9611 isolate a notion of derivability from systems of equations, which is locally equivalent t o finite definability. Basically, t h e rules correspond t o t h e logical axioms for equality a n d successor. See Statman [1977] for a proof-theoretical analysis. Herbrand-Gadel computability has t h e advantage of using simple rules, and t h e disadvantage of not being complete, in t h e sense of not allowing t h e derivation of everything which is logically derivable.

A logical programming language * Note t h a t a n equation can be put into a normal form of t h e kind R1

A*-.AR,,

+

Q

with R,, Q atomic equations of t h e form fy(z1,.. . ,z), = y a n d xi, y variables or constants (the interpretation of variables being t h a t they are all universally quantified). For example, f(4= g ( h ( 4 )

1.3 Arithmetical Formal Systems

39

can be written as

h ( z )= y A g(y) = z

+

f(z)= Z.

T h en t h e previous results show t hat t he values of recursive functions can be logically deduced from axioms of t he described kind. T h e programming language PROLOG (Programming in Logic, Colmerauer, Kanoui, Pasero and Roussel [1972], Kowalski and Van Emden [1976]) is based on logical deductions from clauses of t he form above, with R,, Q atomic relations holding of terms. These are called Horn clauses, and are especially interesting because proof procedures for them are particularly manageable. They can be thought of as conditions breaking up a goal Q into a series of subgoals R,. T h e results of this section show t hat PROLOG, although concocted to handle deductive more than computational problems, has nevertheless the power of computing all the recursive functions.

1.3

Arithmetical Formal Systems

T h e general trend of this century’s mathematics has been to work in formal systems which are supposed t o capture, more or less accurately, some aspects of t h e objects in which we are interested. From a formalistic point of view we can thus consider a function as computable when we have a consistent formal system representing it, i.e. allowing us t o prove for t h e appropriate numbers (and for nothing else) t hat they ar e t h e function’s values for given arguments. Certainly th e approach of Herbrand-Godel computability falls in this trend, b u t t h e formal system involved there, concocted for different goals, is somewhat unnatural from a purely arithmetical point of view. T h e same will be t rue of t he approach of A-definability, see Section 6 . Since we are considering arithmetical functions, it is natural t o investigate which functions are representable in t he usual logical systems for arithmetic, for example in Peano Arithmetic (see p. 24). We attack th e problem in a general way, by isolating minimal conditions (which will tu r n o ut t o be very weak) sufficient t o represent every recursive function. I n this section, ‘formal system’ will always mean ‘formal system

extending first-order logic with equality, and having constants terms ii, called numerals, for each n ’. For more details on formal systems, see p. 350.

Notions of representability Definition 1.3.1 (Tarski [1931], Godel [1931], [1934], Tarski, Mostowski and Robinson [1953]) Given a formal system F and a function f , we say that:

I. Recursiveness and Computability

40

1. f is weakly representable ,in F if, f o r some f o n n u l a 'p of t h e language

of

F, f ( 5 1 ,... ,%)

= ?J

FF

@

-

2. f i s representable in .F 2% f o r some f o r m u l a

=+

f(:c,,. . . ,Zn) =y

f(XI,. ..,Xll)

# y =+

FF

FF

-

' ~ ( TI .I. . , %,Y) 'p,

Cp(S1,. . . ,:n,u)

-

y'p(z*,. . . , x n , p )

if f o r some fonriida 'p, f is repre3. f is strongly representable in sentable hy cp, and inoreoiler t h e following uniqueiiess c o n d i t i o n holds:

'I'he relationships arnoiig t,he various notions are: if f is strongly representable then it is rel)re;eiitabk, and i f f k represenl.&k in a consistent formal syst,eni then it is wcakly rc:present,al,le (becausc if .F is c.orisisf,ent, and k~ "p the11

y>-cp).

Exercises 1.3.2 a) ?'he two cmiditioiw /(ZI , . - . , ~ n ) ~ ?/: *

~p(-Cl,...,+n,S)

FF

-

F F (VY)(vz)['P(il,...r~n,~)ACp(~lr ...,Zn,Z) nrr cquivnlerit to the unique condition

F.F (VY)l'P(i,,. . . , Z , , Y ) b) If .F zs such that I

# y =+

FI

-

y

~

f(.l,,

..

+

Y

--I

.Tn)l.

l(Z = 5 )

t,Iwii strung repi-eseiitability off iii .F is iquiiralent to the unique condition

Proposition 1.3.3 (Tarski, Mostowski and Robinson [1953])I f .F is a coiisisteiit. foriiial system with a predicate < satisfying thc axiom s c h e m a t a 1.

2.

T(Z

< 0) -

x<11+1

-

u

z = o V ~ ~ ~ v 2 : = 7 1

9. x < n v x = E v E < : 1 : thrri a r q f u n r t i o n ruprcscntablc in

F i s strongly representablc in it..

41

1.3 Arithmetical Formal Systems

Proof, Suppose @ represents f in

F.Then the formula

. . ,zn,Y) @ @ ( E l , . . . , z n ,Y) A (vz < ~ ) - 4 ' ( 2 1 , . . ., z n ,z )

~p(z1,.

strongly represents f . Indeed: 0

If f(z1,.. . ,zn)= y then sentability of f via @

#

f(z1,.. . ,z,)

Z, for

every z A

Axioms 1 and 2 (depending on whether y = 0 or y t h e z < in t h e first part of t h e formula. Thus

$ ( % ., .. ,z n ,Y) A (vz< y)-+(.1,

> 0) take care of all

.. .,

&I 2) ,

(if y = 0 t h e second part is vacuously true, since there is no z t'3 (p(Z*, . . . ,Zn,Y). 0

< y), a n d

If f(z1,.. . ,zn)# y then, by representability, I-F +J(Z~,. .. ,Tn,v) and so

0

By repre-

x)@(Ti ,. . . ,Zn,3).

t - ~- I @ ( Z .~.,. ,Zn,Ti) A . . . A T$J(Z~, . . . ,Z n ,

tF

< y.

. . ,z,,y).

I-F T(p(51,.

To show the uniqueness condition we prove (see 1.3.2.a)t h a t -

t3 (try>[~p(~l,. . . ,z n , y )

++

Y = f(z1,.. zn)1.

We have I-F (p(Z1,.. . ,Z,,!(XI,. . . , 5 , ) ) from t h e first part of t h e proof. Suppose now FF 'p(z1,. . . ,zn,y).By axiom 3 t h e only possibilities are y

< f(z1,.. . ,xn) v Y = f ( 2 1 , . . . , G I ) v

. . ,Xn) < Y.

f(x1,.

T h e first one is ruled out, since from l - ~cp(T1,. . . ,Tn,f(zl,. . . ,zn))we have t3 +J(Z~,. . . ,Zn,y),while from I-F (p(T1,. . . ,Z,,y) (assumed by hypothesis) we have I-F @(?El,.. . ,Z,,y), and F is consistent. Similarly we can rule out f(z1,.. . ,zn)< 9. Then y = f ( ~ 1 ,... ,zn). 0 T h e notion of representability makes sense for predicates as well:

Definition 1.3.4 Given a formal system F and a relation R, we say that: 1. R is weakly representable iJ for some

R ( ~ i , . - - , z n@ ) FF 2. R is representable if, for some

(p,

(p,

(~(%,...,2n)

I. Recursiveness and Computability

42

Note that i f tlic charac:terist,ic function CIz of I< is (weakly) rcprcsetited by z),t,lic:n 11’ is (weakly) reprcsetit,tible l)y cp(z’:l ,. . . rxL’ll, 1). Also, if .T is such that, 2,’# y =$ t :,- 7(z = 5 ) ‘p(z1,.. . , x,,

atid R is represetitotl by ‘ p ( z l , . . . , : i z T L )then , A h =

‘112

is (strongly) represetitable by

-

I ) v (7cp(z,, . . . ,&)

A 3 =Ti)

(tliis follows from 1.3.2.1)). Note t.liat the axionis a r e tiecxldetl evcti for simplr rel)txsetiiabiIity of (:R.I)ctc.aitsc: w l i w c ~ ~ ( z l ., . : I ’ ~ ~#) 2 and 2 # O, 1 wc nctxl to ktiow 3 # t,o be able to i t i f e r t l i w t , thc formula i i i t , r : n t l e t l t.o r e p r c x n t C R

o,r

is tiot prova1)Ie.

Formal systems representing the recursive functions N o t c that if j(.t.,. . . . , n.,,) = pyR(.rl , . . . ,.T,,, !/) then

Proposition 1.3.5 (Tarski, Mostowski and Robinson [1953]) 1s F jOl.lllol a‘lJ.5tClIL S U ( h 11101

Proof. Suppose

j(.q = y ( l t l ( T ) ,. . . ,h,n(.?))

LI

(I

1.3 Arithmetical Formal Systems

43

if kF cp(??,y) let yl, . ~.,ym be such that

0

x (Y1, . . . Ym,Y). By strong representability it must be yi = hi(Z)and thus l-F $1

(2,y1) A . . . A $m (2,Y m

A

y = g(h1(Z),. . . , h,(Z))

= f(?).

1

0

We are now ready t o conclude our search for axioms which allow representability of every recursive function.

Theorem 1.3.6 (Godel [1936], Mostowski [1947], Tarski, Mostowski and Robinson [1953]) I n any formal system 3 with a predicate < and functions and . satisfying the following axiom schemata, any recursive function is (strongly) representable (and thus, i f the system is consistent, also weakly representable):

+

B1 B2 B3 B4 B5 B6

y(T=jj), f o r x f y V A < x

x
<Ti)

-

x
-

++

x=OV...Vx=A

Z+jj== Z . j j = 2-y

Proof. We refer t o the characterization of recursive functions given in T h e e rem 1.1.8. We have just proved that closure under composition is implied by B1, and closure under p-recursion follows from B2-B4, as in 1.3.3. It then suffices t o note that: 0

0

Equality is representable because if x = y then obviously l-7 ?C = jj, and if x # y then k~ T(T = jj) by B1. Then its characteristic function is representable too.

S u m is representable by cp(x,y,z)

+

-3 x

+y=z

+

Indeed, if x y = z then k~ s+y = 2 and (by B5) l - ~Z jj = 2, i.e. FF cp(Z,jj,Z). And if 2 y # z then l - ~T(= E) by B1 and kF - ~ ( T + j j= 2) by B5, i.e. l - -cp(T,jj,Z). ~ 0

+

Product is similarly represented by

cp(x,y,z) w 2 . y = z.

I. Recursiveness and Computability

44 0

Identitics 1 : a r e obviously represented by

Thc axioms U l - R f i define a theory R with infinitely many arioms. Tarski, Mostowski and Robinson [I 9531 show t h a t R is not finitely axioinntizable, since any finite subset of t h e axioms (and thus, by compactness, ariy finite set o f theorems) a d m i t s a natural finite model consisting (for n sufficiently big) of t h e numbers {0,1,.. . , n } nat,itrally ordered, and having t h e operations restricted to value equal to n wlieii t h e original value would exceed it. Of course 77, canriot, have a finite model (by R I ) and thus it is not reducible t o a finite set of t,heorenis. T h i s also provcs that, any closed theorem of R is t r u e in some finite riiodel, a n d t h u s t h e power of R is very limited. Theorem 1.3.6 is, thus, very general. Local iniproveuierits o f Theorem 1.3.6 a r e possible however, as Cobham (see Vaught, [ 19601) and Jones and Shephcrdsoii [ 19831 have shown. 'The cxmclusion still holds if B2 arid B5 a r c dropped. Moreover, by changing t h e language a n d iiit,rotlitc.ing a predicate 5 in p1ac.e o f <, B 2 - ~ H 4niay be rcducctl to t.wo axioiii schenlata

1tol)insoii Arithmetic Q (see p. 2 3 ) is a n extension of R,and t h u s provides a n example o f a finitely a x i o n i a t i z d t h w r y iri which every recursive fiinction is reprweiita1)le. Robinson has proved that, if ariy o n e of t h e axioms of Q is r e n i o v d , then some rec:iirsive f'itiictioii is not strongly rcpresentable. 'Thus Q is u iriiiiinial finitely axiornatizible theory in which every recursive function is ,strori.ql?j representable. For details arid more information, see Tarski, Mostowski a n d R o b i i i s o i i [1953].

Invariant definability

*

Every model A of R has a sitl)niotlel isomorphic to w , which is called t h e standard part of t h e rriodel arid is still deiioted by w. Given a forrriula 'p arid a subset A of t h e universe of A, we say that, A is defined by 'p in A if

A = (2: : A

+ cp(z)}

+

(where is t h e usual notion of sat,isfactioii in a structure, see I V . l . l ) . A is defined on w by p in A if

A = w n(3: : A

\ cp(x)}.

1.3 Arithmetical Formal Systems

45

We can also introduce uniform versions of these notions, by saying t h a t A is invariantly defined (on w ) by cp if A is defined (on w) by t h e same cp in every model J1 of R. T h e Compactness Theorem implies t h a t a set is invariantly definable i f and only i f i t is a finite subset of w ,while the proof of 1.3.6 (together with t h e results of Section 7) shows t h a t invariant definability on w and recursiveness coincide (Kreisel [1965a]). Thus we have a purely model-theoretical reformulation of recursiveness, and by changing the class of models one gets natural generalizations of it (see Mostowski [1962] and Kreisel [1965a]), some of which will be considered in Volume 111. Note t h a t t h e reformulation of finiteness explains t h e need of considering t h e relative notion of invariant definability: w is not invariantly definable in models of R. For models of PA a stronger fact is true: w is not even definable in a single nonstandard model of PA. Indeed, given a formula cp defining a set A, consider -y.If there is no element satisfying it then A is t h e whole universe, which is nonstandard. If there is a n element satisfying l c p then, by t h e Axiom of Induction, there is a minimal one 2 , which is not in A because -cp(x) holds. If z is 0 or t h e successor of a standard element then A does not contain all standard elements. If z is t h e successor of a nonstandard element y then, by minimality of 2 , cp(y) holds and A contains a nonstandard element. In any case, A is not w.

Definability of functions

* fo.

We say t h a t f is definable in .F if for some term t( Z ) , l - ~t(%)= If 3 satisfies B l then definability implies strong representability (by t h e formula t ( Z ) = z , see 1.3.2). But definability is not an absolute notion, since t h e fact t h a t a function is definable is quite accidental a n d simply depends on t h e functional constants of the theory language. For example, in Robinson Arithmetic only polynomials are definable, because t h e language has only t h e function symbols S , and . . By extending a theory with axioms for a formal analogue of t h e p-operator, namely

+

every function which is strongly representable by cp(Z,y ) becomes definable by pycp(Z, y ) . In particular, there are formal systems in which every recursive

function is definable.

46

I. Recursiveness and Computability

In any theory with a predicate <, a functional symbol f can be introduced in a conservative way (i.e. not producing new theorems in t h e old language) for any function which is strongly represented by cp(,‘. y), together with t h e axioms

In any theory with the Axiom of Induction (e.g. Peano Arithmetic and its extensions) t h e same can be done even more generally, in t h e sense t h a t only (3y)p(3,y) is required, instead of strong representability. T h e provable existence of pycp(3,y) then follows by t h e Least Number Principle. From 1.7.7 it follows t h a t functions definable in consistent formal systems satisfying B1 are recursive, since in this case definability implies representability, as noted above.

1.4 Turing Machines We have been analyzing t h e notion of effectiveness by taking our inspiration from mathematical activity. We now switch our point of view a n d begin a n analysis based on computational activity, by providing a model of certain aspects of human behavior during routine computations. T h e final goal is to understand what we are doing when we compute, in such a way as to be able to simulate this activity by a machine. In this section we present t h e analysis of Turing [1936] and Post [1936]. In doing a computation we are given an input and then produce a n output, usually after a n appropriate amount of written calculations. Fortunately, for t h e pure writing activity we already have a mechanical device: t h e typewriter. O n it we base our model. We may imagine our abstract machine as being capable of printing symbols on paper. In real life writing is usually done on planar sheets of paper, but since we write in consecutive (horizontal or vertical) lines we may simplify, and just think of our machines as writing on a linear tape. Since we d o not put a priori bounds on t h e amount of scratch work needed for a computation, we think of our tape as potentially infinite in both directions (i.e. we suppose t h a t we are always able t o glue more tape at either end, when needed). Also, since symbols a r e actually (or can be) written one at a time, we think of our tape as consisting of cells, each of which can contain a single symbol. Although we might, in general, need symbols for infinitely many entities (e.g. one for each number), we can certainly use complex symbols (like words) built up from a finite stock of atomic ones (like letters). T h u s our machine will simply be capable of printing symbols from a finite alphabet, as in real life.

1.4 Turing Machines

47

And as in concrete typewriters, we allow for the possibility of two directions of movement along t h e tape, one cell at a time. Movement is necessary to print symbols in different cells, and t h e two directions allow for a recall of t h e work already done. It does not matter whether we picture t h e situation as a n actual movement of the tape, or of the machine, or even just of a n extendible telescopic arm examining t h e tape and transmitting the information to t h e machine. Since we can come back to cells containing a printed symbol, we allow for t h e possibility of erasing it (as modern typewriters can do). An abstract human being can carry on his computation by using just such a n abstract machine for his explicit writing activity, b u t his assistance is at this point obviously still necessary (as it would be for writing something by a typewriter). We now take advantage of our restriction to routine work: no step must require ingenuity, and thus each move must be automatically carried on on t h e basis of t h e previous work. In particular, the machine needs a memory, i.e. a way to recall t h e crucial features of what it has already done. One possible way to implement this requirement is to think of t h e machine as being at each moment in some particular physical state, determined by t h e previous action u p to this point, and determining t h e successive action. As a simple example, consider once again the typewriter: one possible routine decision t h a t is done when we write something in most western languages is to begin a word coming after a dot with a capital letter. We might thus build a typewriter that automatically steps, after printing a dot, from its current writing state to t h e state for printing capital letters (in actual machines this change of state is obtained by pushing an appropriate key). If we allow for sufficiently (possibly infinitely) many states we can certainly record any action: it is enough to consider the tree of all possibilities, associate states to its nodes and, by representing possible actions as paths of the tree, to change states by following a given path. But infinitely many states would contradict t h e intuition both of routine work (as the implementation of an effectively describable task) a n d of machine (as a device of limited complexity). We thus impose the limitation of having only finitely many states. To be able to work automatically, t h e machine must perform its elementary operations according to given instructions, telling it what to do on the basis of given information. Some of this information reaches the machine through its internal memory (codifled by t h e current state), but other (in particular t h e input) might be fed from outside. T h e connections with the outer world (the tape) consists not only in getting, but also in providing information (the output among others) and it may be pictured (following the parallel with the typewriter) as happening through a head, which is both able to read and write. Since the tape consists of cells, t h e head’s range of action will just be one single cell, both statically (when reading and printing) and dynamically (when moving).

I. Recursiveness and Computability

48

I t is clear at this point, t h a t only finitely many instructions are needed, specifying w h a t to do in a given situation. Indeed, there are only finitely m a n y possible local situalioris (determined b y state a n d r e a d symbol) a n d actions (consisting in printing o r erasing a symbol, possibly moving t h e h e a d o n e cell left o r right, a n d possi1)ly changing state). We new1 not analyze the a c t u a l physical irnplementation of t h e rriachine work. Just imagine t h e existence of a black box, which sornehow k n o w s t h e inst,riic:tions a n d t h e way to carry t h e m o u t (for more o n t h i s see t h e r e m a r k s o n pp. 52 a n d 116). T h u s t h e rriachine hardware consists of a t a p e , a head ancl a black box. I t c a n 1)e pictured as follows:

T h e physical intuition o f a machine as a generalized a u t o m a t i c t y p e w r i t e r is u : r h i n l y useful, but for o u r a1)stract p u r p m e s not,hirig rnore t h a n i t s software is net:ded:

Definition 1.4.1 (Turing [1936], Post [1936]) A Turing machine M co11sists of: 1. a f i n i t e alphabet { s g , , s n } , with two distiiigwishecl syrribols (blank) and sl = 1 (tally)

SO

=

2. a fiiiite set of stalcs {(lo, . . . , qlrL},among which distinguished states (initial state) and qJ ('iiial slate)

* (10

3. a f i n i t e set of consislcnl iii,striictions { I , , . . . , l p } , ench one of the following lhr(re basic types:

qa s b s, it and

'Id

: tht: inuihine r n state (la a n d rcndiiiq the symbol s b erases s, in its plnctr, then chaiiges r b state t o yd

priiits

qa s b R yd : 11w iriach%iicin state qa and rmding the symbol one cell t o thc r i g h t and changes its state bo q d q,,

sb

s b Illo'lies

c

I, (id : siinilar, irroiriiig one cell t o the k$.

Consistency mtmis that n o pair of instructions is contradictory, i.e. with blw s a m e prcmise q,, s b but with digerelit coiLcluTions.

1.4 Turing Machines

49

Note t h a t instructions are local: their behavior is completely determined by t h e part of t h e tape immediately adjacent t o t h e head. Except for this crucial feature, t h e form of instructions could be different. We could further analyze t h e first type of instructions into pure erasing and pure writing actions. Note t h a t pure erasing is a special kind of printing, with S, = *. Or we could slightly complicate t h e basic actions into a unique type qa sb 5, q d j , telling t h e machine in state qa a n d reading sb t o print s, in place of s b , change its state t o qd a n d (depending on whether j = 0, 1,2) stay still, move one cell to t h e right or move one cell to t h e left. An instantaneous configuration of the machine is a complete recording of all t h e relevant data in a given instant. It can be represented by a sequence o1

... oj qa U j + l .. . uz

in which all t h e consecutive symbols o; (including t h e blanks) written in t h e relevant portion of t h e tape (at least t h e one between t h e two outermost nonblank symbols) are indicated, together with t h e state and t h e head position (shown by t h e position of t h e state symbol among the alphabet symbols). Thus t h e sequence above records the following situation:

blanks

I

I

blanks

~

T h e machine behavior can be represented by a sequence of instantaneous configurations, each obtained from the previous one by application of a necessarily unique instruction, a n d starting from a n initial configuration (coding t h e input a n d t h e initial state). A final configuration is reached when t h e machine finds itself in the final state, and we say figuratively t h a t t h e machine stops. An alternative formulation, not using final states, simply states: t h e machine stops when no instruction is applicable. We will show in 1.4.2 how a Turing machine can compute a function, which is our real concern. Before t h a t we discuss Turing machines in more detail, but t h e interested reader can skip this, and go directly to p. 53.

Variations of the Turing machine model T h e particular model of a Turing machine introduced in definition 1.4.1 is absolutely unimportant, as long as we are only concerned with computational power and not with efficiency. Since in this book we d o not deal with machine theory (although we will prove some scattered results in Chapter VIII), we will

50

I. Recursiveness and Computability

only quote some facts, and refer t o Minsky [1967], Arbib [1969] and Hopcroft a n d Ullman [1979] for their proofs and for further information.

States. Although only one state is not in general enough to compute every recursive function (Shannon [1956], Wang [1957]: basically, a one-state Turing machine must behave in t h e same way on every cell outside the input), two states are (Shannon (19561). Thus it is not relevant whether we restrict our model t o machines with only a fixed number n 2 2 of states, or we allow any number of states. Symbols. Clearly we must have at least two symbols, since we consider blank as a symbol. Two symbols are enough t o compute any recursive function (Shannon [1956]), since it is possible t o economize on t h e number of symbols by increasing t h e number of states. See Priese [1979] for results concerning t h e simultaneous economy of states a n d symbols. Erasing. This is dispensable (Wang [1957]), in t h e sense t h a t all the recursive functions can be appropriately computed by machines that never erase. Basically, given a Turing machine it is possible t o simulate it by a new nonerasing machine, that writes on its tape codes of successive configurations of the original machine. T h e result shows t h a t in principle we d o not need erasable material, like magnetic tapes or disks, for t h e external memory of computers. see also p. 51. Tapes and heads. Here t h e freedom is practically absolute. We synthesize it in the following result. A mring machine with finitely m a n y tapes, each with its own (finite, or even countable) dimension and its own finite number of heads simultaneously scanning it, can be simulated by a Turing machine with only one linear tape, infinite in just one direction, and scanned b y a single head (Hartmanis and Stearns [1965]). However, we d o need the two directions of movement, since to restrict it t o one would be compatible only with finite or periodical behavior on cells outside the inputs (Wang [1957]). Determinism. Our model of a Turing machine is deterministic, in t h e sense that the instructions are required t o be consistent (at most one of them is applicable in any given situation). Randomizing elements in computing devices were introduced early on by Shannon [1948] a n d De Leeuw, Moore, Shannon and Shapiro [1956]. There a r e basically two models. Nondeterministic Thing machines behave, in a n ambiguous situation where conflicting instructions might be applicable, by randomly choosing one of them: their computational power, at least for 0,l-valued functions (sets), does not exceed the power of deterministic ones. Probabilistic

1.4 Turing Machines

51

machines differ from nondeterministic ones in t h a t t h e next state has a probability, and thus conflicting instructions do not have t h e sam e chance of being chosen by t h e machine.

Physical Turing machines * Turing machines are theoretical devices, but have been designed with a n eye to physical limitations. In particular, we have incorporated in our model restrictions coming from: (a) atomism, by ensuring t hat t h e amount of information t h a t can be coded in any configuration of t he machine (as a finite system) is bounded; a n d (b) relativity, by excluding actions at a distance, a n d making causal effect propagate through local interactions. Gandy [1980] has shown t h a t t h e notion of a Turing machine is sufficiently general t o subsume, in a precise sense, any computing device satisfying similar limitations. However, there is at least one aspect which has been neglected in our discussion, namely energy consumption. This is a central problem especially for actual modern computers, whose sizes are getting increasingly bigger. T h e point is th a t in devices with packed components, energy dissipation is proportional t o volume, while heat removal is proportional only to surface. To provide ,adequate cooling, it would thus seem necessary t o expand machines only in two dimensions, a n d keep them flat and thin. But this would mean a spread of components in space and a n increase in t h e time needed t o transmit information, thus a decrease in speed. I t is then crucial t o limit energy consumption in computations, not only for costs limitation, but also for physical realization. At a macroscopical level, energy seems t o be needed in at least two different ways. First of all, physical computations involve data handling (like storing a n d transmitting), an d thus measurements: this implies energy consumption. Secondly, Turing machines a n d real computers are irreversible devices (many sets of inputs may produce the Same output, and it is usually impossible t o invert computations): this implies energy dissipation. Theoretical bounds on energy requirements have been computed by various authors (e.g. from t h e uncertainty principle AE . At N h, by arguing that if At represents a switching time, then AE must represent energy dissipation). See Landauer [1961], Bremerniann [1962], [1982], and Mundici [1981]. At a microscopical level, dynamical physical laws are reversible: it is thus natural t o look for reversible models of Turing machines. Logical reversibility (of rules) is easy t o achieve (Bennett (19731). First note t h a t a computation can be trivially simulated in a reversible way, by having a n extra tape t h a t successively records th e quadruples of the Turing machine used, so t h a t t h e present configuration, an d t he last record on t he history tape, allow for a recovery of t h e previous configuration. Then note t h a t erasing a computation done by

52

I. Recursiveness and Computability

reversible rules is also reversible, and that it can be tlone after the outpiit has been copied (to l)e saved) on a separate blank tape: no record is needed a t this stage, since copying onto a blank tape is a one-one operat,ion. Then the whole computation, including the erasure of scratch work, c a n be tlone by reversible rules. T h e next, step is to look at energetic reversibility. Rennet [1973] and liaiidaiicr [ 19761 provide Brownian niotlels for any finite cornput,ation on Turing niachines, dissipating arbitrarily little energy when proceeding slowly enough. The filial step is to look for plainly conservative models of Turing niachineS, not, dissipating any criergy a t all, whilc conipiit,irig at finite s p e d . For classical nimliaiiics, Fredkin and 'l'offoli [ 19821 give a ballistic model consist,ing o f hard spheres (whose presence or aI)sencr: in a constant flow code digits 1 or 0) that collide elastically wit,h each other arid with fixed barriers placed inside the coinputer (see also Landauer [1981] aiid 'Ihffoli [1981]). For quantum mtrhaiiit:s, niotlels have been given by Benioff [1980], [1981], [1982]. It should be noted that, all these niotlels are mathematical. They only show that dissipationless computatims are i i o t contrary to current physical laws, not. that t h y are physicAly realizable. Moreover, energy dissipation does occur in these systems, in interactioiis with external o1)servet-s (t,o receive inpiits and give oiitputs): it, is not, needd for the internal system evoIut,ion, i.e. for the conipiitation itself, but it. is needed in the end to c:apt,iireand communicate the information that, is wandering inside the process, devoid of itieaning. For more OII physics o f coniputations, see tlie issue devoted to the subjcct by the In2ernationul Journal of Theorelicnl Physics (vol. 21 (1982) nos. 3,4), as well as IJennet, [ 19821, Landauer [ 19851 and Feynman [ 19961.

Finite automata * The control box of a Turing machine exeniplifies the notion of finite automaton (McCiil1oc:h antl Pitts (1943]), as a machine with finite sets of states and o f inputs and oiitpnts, hgether with fiinctioiis for next state antl for output Idiavior (both depending on input and current. state): this is a rnntherri;itic:ally rendition of finite state system. IIaving only a finite number of possible inputs aiid outputs, the ptfisil)le behavior of a finite aiitomaton is a logical function (0, l}" + ( 0 , I}", and it, is thus representable by a switching circuit (Sliannon [1938]) or a neuronic net (McCulloch arid Pitts [1943], sec also p. 1 I G ) , by writing a truth-table representation for it in disjiinc:tive normal form, and hy using chips for the logical connectives. A finite automaton is basically a passive device, producing only an output froni an input by a series of internal changes. Moreover, it has only a finite short-term memory, since the only way it is able to record is by changing it,s state. A Turing mac:liine is a more coniplex device, and it may be seen as a fiiiit,e aiitoniaton supplied with an external, potentially infinite long-term

1.4 Turing Machines

53

memory, a n d with t h e ability of interacting with t h e outer world ( t h e tape) in a dynamical, active way (through t h e head). Actual modern computers lie somewhere between finite a u t o m a t a a n d Turing machines. T h e theory of finite a u t o m a t a is highly developed, a n d we will only touch on it in Chapter VIII. For more information see Hartmanis a n d Stearns [1966], Minsky [1967], Arbib [1969], Trakhtenbrot a n d Bardzin [1973], Eilenberg [1974], a n d Hopcroft a n d Ullman [1979].

Turing machine computability W e have introduced Turing machines as computing devices, b u t we have not yet spelled o u t how they compute functions.

Definition 1.4.2 (Turing [1936],Post [1936])A function f(x1,.. . ,x,) is Turing machine computable i f there is a mring machine that, when starting in the initial configuration

(with the integers represented in unary notations b y tally sequences, and nothing else on the tape except the input representation) and when following its instructions, reaches a final configuration of the kind

(with f ( X I , . . . ,x,) represented in unary notation b y a tally sequence, and possibly something else o n the tape). It should be noted t h a t t h e details of t h e definition a r e arbitrary. T h e important t h i n g is t h a t t h e machine carries configurations coding in some way t h e inputs, to configurations coding in some (not necessarily t h e same) way t h e o u t p u t . T h e point is t h a t , as we have already noted, t h e class of functions computable by Turing machines is widely independent of t h e details of t h e definition. T h e next result shows t h e computational power of Turing machines. T h e proof is not difficult in outline b u t cumbersome in details, a n d we a r e going t o give a sketch only: for more details, see e.g. Davis [1958] or Hermes [1965].

I. Recursiveness and Computability

54

Theorem 1.4.3 (Turing [1936]) Every recursive function is Turing machine coinputable.

Proof. We proceed b y iritludion on t h e d c h i i t i o n of rcxursive function. To have a sufficiently s t r o n g inductive hypothesis, we prove t h a t every recursive function is c:ompiital)lc by Turing machines t h a t : may have initial tape rionerript,y at t h e left of t h e inputs, work only o n t h e half o f t h e tape c:ontwining t h e i n p u t a n d at t h e right o f it,, halt i n a haltiiig state, a n d p r i n t t h e value of t h e f u n d i o n inimediately t,o t h e right of t h e original i n p u t s . 1. co~istarilzerci We s t a r t in t h e initial configurat,ion

and w a n t

t,o

reach t,hc following final one:

The following instrwtioiis will do ( n o t e t.hat 42 is t h e final state)

2. identit irs 2; is corriputed by a n y machine t,hat, copics t h e input,. T h e flowchart o f Figure 1 i1idicates t h e sequelice o f ac:t,ionsto perforrri. Figure 2 exhibits a 1)rograni written along these l i r m (the inst,rilctions a r c written in boxes only to iridic,atleexplicitly to which piirt,s of the flowchart, they correspond). N o t e t,liat, is t h e final st,ate.

A program for Tr c a n I)e written in tlic sariie way, b y corisic1t:ring z,as t h iiipiit,, and by moving ac~tr;szL+l* * z,, 1iitc.k aiid fortti (whic.11 ( ' i t i i bc casily t h e by using i1iorc chiiiiges of s t a t c s , wit,h t h e only function o f rriakiiig the head pass t,lrroiigli t h i s portion of t,he t a p e ) .

1.4 Turing Machines

Move one c d l to t,hc, right and p r i i i t a tally.

1as1,?

t

I riglitiiiost, Go t,o t,he I tally o f

56

I. Recursiveness and Computability

L 92 s1 L 92 92 so I, 9 3 93 so L 93

q 3 s1

s o 44

Figure 1.2: Program for 2:

input on the right.

and move one cell to

Figure 1.3: Flowchart for S

1.4 Turing Machines

57

Recopy t h e inputs on t h e right. Simulate M1 and get o u t p u t 2, Eraye t h e copy of t h e inputs arid move z1 to t h e right of t h e inputs.

Recopy t h e inputs to t h e right of z1. Simulate Mz a n d get o u t p u t 2 2 . I h s e t h e copy of tlie inputs and move 22 to t h e right of z1.

:

Itwopy tlhe inputs to t h e right of Sirnulate M,,, a n d get, o u t p u t zT,. Erase t h e copy of t,he inputs and move z.r,Lt o t h e right. o f z,,-

-nln-1.

1

Simulate M,,L+I a n d get o u t p u t Erase 21,. . . , z,, and move z near t h e inputs

2

Figure 1.4: Flowchart, for coniposition

58

I. Recursiveness and Computability 3.

s11 (:cessoI'

'I'he flowchart, is given i i i Figure 3. It is then enoiigh t o add, t,o tlie program for 1;giveii a h v c , t,Iic foIIowiiig t,wo instruct,ioiis (correspotiditig t,o the Iast box of the floudiart): 910 *%I *$1 (113

yi:c

913 7 1

As we can sec f r o m t,liese exatriplo;, t,o writ,e prograriis is q i i i t ~routine, otice w(: have t h e apI)ropriat,e flowchartas. I n tlic following we will then restrict oiirselvcs to t,hc la1 r, and lmvc iw an cxercise their trarislatioti into prograrris.

4. c:oiripo,si/ ion IA,

f(2)= h ( g l ( Z ) , . . , g n 1 ( Z ) ) , arid siipptae gl , . . . , g 7 1 fh, are coniputcd, in t,hc sense explained a t , tjlie begintiiiig oft l i e proof, by ttiacliitirs M I , . . . , M,,, , A 7 , 1 + 1 . 'I'hen f is C O I I I piited by a r r i x h i t i e iiriplcniciititig t,he flowchart of Figurc: 4 , where sitri111at~ioii of a given iiiacliiiie means cliangiiig t,hc iianic of its states in the appropriate way: by renaming itjs initial state by t,hc: riattie of t,lit: statc in which the new rtiachiric I)cgiris its sitiiulat,ioii, rctiarnitig the rctnainiiig stat,- by new iiatiies not yet. used, and cotititiiiing the work o f t l i e siniulatirig ritachitie from tlie halting st,at,e of the siirnilated one.

5 . prirnitivc rrr:iirsiori I,ct,

f(:?,O)

f(Z,:y

=

+ 1)

g(Z) h(:?,y,f(?,g)),

atid supposc ,q aritl I , are cwtiiputcd,

i i i t . 1 setise ~ explaincxl i i t tile begiiio f the proof, by tiiac.liities A f t and M 2 . Theti f is coiiipiit.tu1 by a niachiric: implenienting the flowcliatt of Iigure 5 . The idea is t o c ~ o n i p i i t , c : successive valiic; of f , using thr: previous otic each t,ime, iiiit,il we reach t,lie one we are interc:stecl i n . To keep t m c k o f the tiiitiib~rof it,(mt,ioiis st.ill to be done wc irit,rotliic:e a counter s, iriit,ially set up t o thc: v a l ~ i e y, and decreased by one at each step. We do differout siriiiilations, dcpending o n whetlicr y = 0 o r i i o t , . I n t,lie cotnputatiori we also ired ari additional input,, wliicli is not present ai. t,he first step (corti~~i~tat,ion of g ) , a i d i n c r e a s e I)y o i i e at, each step aftjerwartls: we theti introduce a second coiintcr 1 , which is itiitially etiipty. T h e t,apc: then codes i i sitilitfion of tlic kitid:

tiitig

51

* ' ..*

* y * s * XI * . . * '

* t * f(Xl,.

. . ,&,, t )

1.4 Turing Machines where s

59

+ t = y. Write, to t h e right of t h e inputs, a copy s of y, a copy of t h e inputs a n d a n enipty string t . Simulate MI a n d get output, 2.

Increase t a n d decrease s by one tally each. Simulate M z , get a new o u t p u t z

Erase everything between t h e inputs a n d 2, a n d iiiove z near t h e inputs.

Figure 1.5: Flowchart for primitive recursion

6. precursion

Let

f(.')

= P Y [S(Z,Y) = 01, a n d suppose 9 is computed, in t h e sense explained at t h e beginning of t h e proof, by a niachinc A4. T h e n f is computed by a machine implernenting t h e flowchart o f Figure 6. T h e itlea is t o compute successive values of y(Z,y), starting with y = 0, until one with output, 0 is reached. T h e n y is t h e value of f . 0 Exercise 1.4.4 Fill in the detaib of the proof of 'Ihemxm 1.4.3, by specifying Turing machine programs implementing the flowcharts given there.

Machine-dependent programming languages

*

To show t h a t a certain function is Turing machine roinputable we have t o write a coinplete program, using only t h e cleinen1,ary instructions of defiiii-

I. Recursiveness and Computability

60

write, on t h e right of t h e iiiputs, a copy of t,hem a n d a string y of j u s t one tally

simulate M a n d

erase t h e copy of the input,s and 2 , a n d move y tiear t h e inputs

erase t h e o u t p u t z and add o n e tally

Figure 1.6: 1Iowc:Iiart for p-recursion tion 1.4.1. Since they spell o u t explicitly t h e elementary physical operations t h a t tlic machine lias t,o perform, such progmms a r e said to be wri chine language. Exarnplcs have been given for 0,Z: a n d S in t h c proof of 1.4.3. T h e rest of t h e proof of 1.4.3hay been given i t 1 a different format, at a slightly niorc abstract, level: we still descril)ed the way to conrbine sonic basic physical operations t h a t t h e machirie has to perform, b u t left out, t,he task of specifying how t o translate these operations into sets of instruciions (to retliice t,heni to clctnentary operations t h a t t h e machine rail directly execute). T h i s is a typical situation of assembly programs, whtwe illstructions code blocks o f instructions in niacliine laiiguage, b u t still refer directly to machine implemcntatim . By pursuing this trend toward greater abstraction, we c a n break t h e prograniniing task into different assignments: first t h e algorithm is translated into it machine-independent (or high-level) language, whose instructions r c fcr to lmsic abstract operations, arid then these operations a r e p r o g r a n i n i d in machinr: languagc:. This has various advantages:

1.5 Flowcharts 0

61

It takes into account t h e difference between man and machine, and t h e fact t h a t a language suitable for one might not be suitable for t he other: machine languages a r e based on physical commands, high-level languages are closer to ordinary language.

0

T h e translation of t h e basic abstract operations into machine language can be done once and for all, by means of fixed programs called compilers or interpreters. T h e distinction between t he two is, at least in origin, th at the compiler translates t he given high-level program into a program in machine language, which is then executed; t h e interpreter instead proceeds directly t o t he execution of t h e given high-level program, by translating a n instruction only if and when it needs it in t h e computation.

0

T h e abstract analysis of t he algorithm is more easily carried out in terms of basic abstract operations, without paying attention t o concrete problems of implementation. T h e advantage is somewhat reflected by t h e compiler’s complexity, since whatever is done automatically does not have t o be taken care of directly in t he program.

0

T h e same abstract analysis can be implemented by niany different types of machines, each one using its own compiler or interpreter.

T h e general skeleton of programming thus takes t he following form. On one side th e given function is analyzed into a n algorithm, which is then translated into a machineindependent program. On the other side t he behavior of t h e given machine is structured into a set of basic instructions in machine (or assembly) language. A compiler or interpreter then relates t he two parts a n d produces a machine-dependent program t hat can be executed by t h e machine.

1.5

Flowcharts

Every program written in a machine-dependent language has a n abstract core t h a t can be written in a machine-independent language. It is not clear, at this stage of our development, whether t h e opposite also holds, i.e. whether t h e most general programs can be implemented on abstract machines (if not step by step at least globally, in t h e sense of having a machine computing t h e function defined by t h e program). Since, however, p r o g r a m do provide formalizations of algorithms, an d our concern is effective computability, we feel compelled t o analyze th e notion of program as a natural approximation t o it. T h e analysis will make t h e notion precise and, as a by-product, it will be possible t o clarify t h e relationships between programs and machines.

I. Recursiveness and Computability

62

To define t h e most general notion of arithmetical program we rely on the intuition given by t h e experience with Turing machines. It seems t h a t the constituents of abstract programs are reducible to actions of two kinds: perform some basic operations, and ask some basic questions. Both are clearly needed, since without operations we cannot compute, a n d without conditional actions we cannot exercise judgments or make choices, a n d then we could only have fixed behaviors. To picture programs we thus need to distinguish t h e two types of action, which we do figuratively by using (as in t h e proof of Theorem 1.4.3) boxes for basic operations, and diamonds for basic questions. We are then faced with t h e usual problem of choosing the basic constituents of our programs. For reasons already discussed for the other approaches, we stick to t h e most elementary arithmetical functions and predicates. As usual in Computer Science, we adopt in this section the convention of writing variables in capital letters.

*,

Definition 1.5.1 (Goldstine and Von Neumann [1947])A flowchart program is a diagram having exactly one entry and a finite number (possibly zero) of exit points, and built up b y connecting (through the outward edges) parts of the following kinds:

"

1. assignment statements of one of the following forms:

p+

T h e sign ':=' means: set the left-hand side equal to the right-hand one. Thus the three assignments correspond, respectively, to instructions of the form: set X equal to 0, increase X by 1, and decrease X by 1.

6

2. conditional statements of the form:

,yes

x = O?

no

In a flowchart, edges can split only when passing through conditional state ments, b u t they are allowed t o merge in a point (thus, even if formally many exit points a r e permissible, in practice they can be connected a n d reduced to at

1.5 Flowcharts

63

most one). Also, t h e outward edgcj of conditional statements must exit from t h e diamond b u t can go anywhere. In particular, they c a n re-enter t h e diagram a n d t h u s produce closed paths.

+

Exercises 1.6.2 Choices of primitives. a ) h m the assignments X := X 1 and X := X - 1 and the conditional ‘X = 0 ?’ one cnn derive the assignments X := 0 and X := Y , and the conditional ‘X = Y ? ’ . (Hint: decreaw X and Y simultaneously by 1, until one or both vanish.) b) h m the assignments X := X 1 and X :== 0 and the conditional ‘ X = Y ? ’ one can derive the assignment X := X - 1. (Hint: use two variables, set initially to 0 and 1, and increased simultaneously by 1 until the greatest reaches X . ) c) Both choices of primitives above are mininaal. (Hint: without conditionals the arguments cannot influence the form of the computation; without predecessor in a),

+

the only property that can influence a computation is whether an argument is zero or not; without successor there would be no way to write down numbers greater than the arguments.)

Unstructured programming languages T h e motivation t h a t led to t h e introduction of t h e planar structure of flowcharts is best, explained by its own inventor: T h e r e is reason to suspect t h a t our predilection for linear codes, which have a simple, alniost temporal sequence, is chiefly a literary habit, corresponding to our not particularly high level of combinatorial cleverness, a n d t h a t a very efficient language would probably depart from linearity. (Von Neumann [ 19661) T h e description of algorithms by means of flowcharts has led to t h e programming language GPSS (General Purpose Simulation System, Gordon [1961]; see Sanimett [1969] a n d Wexelblat [1981] for history a n d references), which uses block diagram notation directly. T h i s is useful for t h e simulation of discrete systems, since it allows for a direct representation of t h e system s t r u c t u r e (with blocks s t a n d i n g for operations performed in t h e system, a n d edges corresponding to possible sequences of events). However, in practice, most programming languages represent algorithms as sequences of instructions. We thus have to devise a way to unwind flowcharts, a n d lay t h e m o u t (which does not mean eliminating their planar structure, b u t only representing t h e m in a different way). A simple method is to label statements (e.g. by natural numbers). T h e n a flowchart can be represented as a juxtaposition of labeled statements (the order in t h e sequence, not, t h e order of labels, indicating t h e progressive order of execution), by translating conditional statements via conditional jumps (Post[1936]) of t h e kind if X = 0 go to

11

I. Recursiveness and Computability

64

(with th e meaning: if X = 0 then jump t o t he statement with label n, otherwise continue with th e next statement in t he list). Note t h a t it is also possible to define unconditional jumps go t o n as: if 0 = 0 then go to n . This permits t h e treatment of merging edges of a

flowchart. A ‘go to’ program is a finite sequence of labeled statements of t h e kind

X:=O

X:=X+1 X:=X-1 if X = 0 go to n,

with th e conditions t hat different statements have different labels, a n d every number mentioned after a ‘go to’ is t h e label of a statement. ‘Go to’ programs

and flowcharts programs are mutually translatable, and hence equivalent. ‘Go to’ statements are the natural solution t o flowchart unwinding: they are typical of unstructured programming languages, like FORTRAN (Formula Translator, Backus et al. [1957]; see Sammett [1969] a n d Wexelblat [1981] for history an d references). However, ‘go to’ statements are currently out of fashion (see p. 68).

Unlimited register, random access machines * Turing-like machines implementing ‘go t o’ programs have been proposed by Post [1936] (independently of Turing [1936]) and Wang [1957]. A different model is th e unlimited register, random access machine (Melzak [1961], Lambck [1961], Shepherdson and Sturgis [1963], Peter [1963], Elgot a n d Robinson [1964]). I t consists of a finite number of registers of unbounded capacity, each able t o contain a n integer, and labeled (by a number). T h e machine is able t o perform th e following operations: clear a register (set its content equal to zero), and increase or decrease by one t h e content of a register. T h e machine performs t h e operations by following t he instructions of a ‘go to’ program, with t h e convention th at variables represent t h e content of associated registers. This model is somewhat closer t o real computers t h a n Turing machines, since t h e menlory consists of labeled registers whose content can be made available by a direct call t o t h e label. T h e advantage of this kind of random access memory, versus a sequential access memory like t h e tape of a Turing machine, is subdivision (and consequent greater accessibility) of information. In real computers both kinds o f memory ar e present: t he random access (chips) is internal an d limited, while the sequential access (e.g. magnetic tapes or disks) is external and potentially (in theory, at least) unlimited. Of course, unlike random access machines, real computers’ registers have only a finite capacity.

1.5 Flowcharts

65

Flowchart computability Flowcharts have a natural semantics:

Definition 1.5.3 A function f ( 2 1 , . .. ,z,) is flowchart computable i f there is a flowchart program with an output variable Z, and possibly some of the input variables XI,. . . , X,, such that whenever at the entry point X I , . . . , X, are set equal t o xl,.. . ,x,, and all the other variables are set equal to 0, and their values are then modified according to the iwtructions of the program, then the exit point is reached with the variable Z having value f (XI,.. . ,x,). Theorem 1.5.4 (Wang [1957], Peter [1958], Ershov [1960]) Every recursive function is flowchart computable. Proof. We proceed by induction on t h e definition of recursive function. 1. initial functions 0 is computed by:

+

From I.5.2.a we have t h a t t he assignment X := Y is flowchart computable. Then Zr is computed by:

z:= xi

S is computed by:

0 z:= x

Z:=Z+l

Here a n d in t h e following, we use a single box with many lines to indicate t h e concatenation of many boxes with single lines, with instructions read in t h e order provided by t h e arrow.

I. Recursiveness and Computability

66

2. cornposit ioii Suppose 91,. . . g,,,, h a r e coinputed by flowcharts with appropriately distinct variables. We write st,aternents ol t h e kind

as shorthand for t h e program computing g with respect to input variables

X1,. . . , X,, a n d o u t p u t variable Y , a n d such t h a t t h e input variables have, in exit, t h e same values they h a d in entry (which c a n be d o n e b y storing their original values by using new variables, a n d restoring t h e m at t h e end). T h e n if .f(zl,.

* . rxTL)

= h(91(z11.. . ? z 7 1 ) ,

. ..

,g77L(x1,.

.. ~

~ n ) ) ,

.f is computed by t h e program of Figure 7.

Figure 1.7: Flowchart for composit,ion

3. primitivp recursion Let

.. ,%,O)

f(z1,.

f(..cl,

=

..., %,?4 +1) =

S(Zl 1 . .

. ,xn)

~(~,,...,..cn,Y,f(..cl,... ?ZnrY)),

a n d suppose g a n d h a r e computed by programs with appropriately distinct variables. T h e n f is computed by t h e program shown in Figure 8.

1.5 Flowcharts

67

Figure 1.8: Flowchart for primitive recursion

T := y ( X 1 , . . . ,X,, , Y ) Y:=Y+1

9 Z := Y

Figure 1.9: Flowchart for p-recursion

I. Recursiveness and Computability

68 4. p-recursion Let

.,

~ ( E I. ., En)

= p g ( g ( z l , . . ., G,Y) = 01,

a n d suppose g is computed by a program. T hen so is f , by t h e program shown in Figure 9. 0

Structured programming languages * It has been argued by many (e.g. Dijkstra [1968]) t h a t unstructured flowcharts present definite disadvantages: they may be excessively complicated a n d difficult t o visualize, even for their authors, and errors are consequently easy t o make and difficult to debug. T h e proof of 1.5.4 clearly shows t h a t there is no need of complicated flowcharts, with intricate interlacements of edges, at least for t h e computation of all t h e recursive functions. T h e current trend - with a departure from planarity (see p. 63) a n d a return t o linearity) - is, thus, t o stick t o structured flowcharts (Dahl, Dijkstra a n d Hoare [1972]), inductively made up from t h e assignments a n d conditional statements of blocks with one entry and one exit line, connected only in t h e ways shown by Figure 10. One crucial advantage of structured flowcharts is t h e possibility of a topdown approach, in which t h e algorithm is successively approximated and expanded, from general t o specific diagrams. Related t o this is t h e fact t h a t , while unstructured programs cannot, in general, be thought of as more t han sets of isolated instructions, structured programs are really built u p from subprograms. This brings in a consideration of operations on programs a n d t h e need (stressed in Rackus [1981]) of a n algebraic study of their properties.

Definition 1.5.5 Consider the programming language whose statements are the following: 1. assignment statements (X := 0, X := X 2. ‘while’ statements (while X

#Y

+ 1 and X

:= X - 1)

d o S , with S arbitrary statement)

3. compound statements (begin S1,. . . , Sn end, with Si arbitrary statements).

A ‘while’ program is any compound statement. ‘While’ programs generate all the recursive functions, as shown by t h e proof of 1.5.4. However, although ‘while’ programs ar e sufficient for recursion theory, t o simplify t h e overall picture of programs it is useful in practice t o have at hand more instructions than just those which are really needed. T h e common use instructions of structured programming languages considered in Figure 10

69

1.5 Flowcharts

Stqucncing (begin

5'1,.

. . , S,,end)

Forward conditionals ( if C then S, else

&, if C' then S )

Backward coiiditioiials (while C do S, repeat S until Figure 1.10: Structured flowcharts

c)

I. Recursiveness and Computability

70

can easily be defined as shorthand expressions for appropriate ‘while’ programs (for any Boolean combination C of atomic expressions of t h e kind a = b a n d a < b, with a , b variables o r numbers). T h e language obtained by enriching ‘while’ programs by these definit,ions is essentially PASCAL ( W i r t h [1971]) without declarations (variables being directly available) a n d t h e special features necessary to handle non-numerical data. PASCAL is a modern derivative of t h e ALGOL family (see Sammett [1969] a n d Wexelblat [1981] for t h e saga of its collective gestation, birth a n d development). Structured and unstructured flowcharts are equivalent (Bohm a n d Jacopini [1966]) in t h e sense t h a t they compute t h e s a m e functions. T h i s follows indirectly from t h e fact, to be proved in Section 7, t h a t unstructured flowcharts only compute recursive h i d o r i s , b u t this can easily be shown directly by an analysis of unstructured flowcharts. Equivalence is t h e best result t h a t can be achieved, since we cannot expect, in general, to be able to decompose unstruct u r e d flowcharts into p a r t s with only a fixed number of p a t t e r n s (because we can build unstructured flowcharts with a n y number of conditional nestings). Note t h a t our treatment of flowchart a n d ‘while’ programs is independent of Turing machines. A different proof, based on 1.5.4, of Theorem 1.4.3 would then consist of showing how t o compute t h e assignment s t a t e m e n t s a n d how to compose Turing machine programs by ‘sequencing’ a n d ‘while’. In other words, we would j u s t need a compiler (see p. 61). T h i s c a n be d o n e as a useful exercise (see Davis [1974] for details). Exercises 1.5.8 a) ’If then eke’and ‘repeat’can be defined as ‘while’pmgrams. b) ‘While’ can be defined using sequencing, ‘if then else’ and ‘repeat’.

Programs for primitive recursion

*

Ail examination of t h e proof of Theorem 1.5.4 shows t h a t primitive recursion a n d p-recursion a r e defined by similar flowcharts programs, with o n e basic difference: t h e number of iterations is fixed in advance in t h e first case, arid iiriknown i n t h e second. W e thus isolate, in t h e class of ‘while’ programs defined i n 1.5.5, t h e following class of structured programs:

Definition 1.5.7 (Meyer and Ritchie [1967])Consider the programming language whose statements are the following: 1. assignment statements 2. ‘for’ statements (?or Y do S

’, with S arbitrary statement)

3. compound statements (‘begin S,, . . . , S,, end’, with S, arbitrary state711e71t.s).

A ‘for’ program is a n y compound statement.

1.5 Flowcharts

71

T h e statement ‘for Y do S’means ‘iterate S for Y times’, a n d it is obviously definable as a ‘while’ program as follows: (begin T := 0, (while T

#Y

do (begin T := T

+ 1, S end))end)

(with T variable not appearing in S ) .

Proposition 1.5.8 (Meyer and Ritchie [1967]) A function is ‘for’ computable i f and only af at is primitive recursive.

Proof. Every primitive recursive function is ‘for’ computable, by the proof of Theorem 1.5.4. T h e opposite is shown by induction o n t h e d e p t h of nesting of ‘for’ statements in a program. It is convenient, to have a stronger inductive hypothesis, to show t h a t each variable of t h e program has, at t h e e n d of t h e program execution, a value which is a primitive recursive function of all t h e variables (we would need t h i s only for t h e o u t p u t variable, as a function of t h e input variables). W h e n t h e r e is no ‘for’ (i.e. when t h e depth of nesting is 0) t h e program consists of t h e sequencing of a finite number of assignment statements, a n d each variable h a s a primitive recursive value (obtained by composition of 0 , S a n d t h e predecessor operation). Consider now ‘for Y d o S’, a n d suppose S satisfies t h e inductive hypothesis: if t h e variables of t h e program S a r e a m o n g X I , . . . , X,, there a r e primitive recursive functions 91,. . . ,gTL such t h a t X, = g,(X1,. . . , X,). For 1 5 i 5 n, let f , ( X l , . . . , X,,2 ) be t h e value of X, after 2 executions of S . T h e n :

‘This is a simultaneous recursion on primitive recursive functions, which is easily seen (by coding, see 1.7.2) t o define primitive recursive functions. After t h e cxecution of ‘for Y do S’, t h e variable X, will have value f,(X,,. . . , X,, Y ) . In particular, t h e output, variable will be a primitive recursive function of t h e input variables. 0 ‘The result suggests a natural way of classifying t h e priniitive recursive functions, by rneasuring t h e smallest depth of nesting of ‘for’ statements in programs

I. Recursiveness and Computability

72

computing a given function. This is essentially equivalent to t h e Grzegorczyck hierarchy (Grzegorczyck [1953]), which will be studied in detail in Chapter VIII. A consequence of 1.5.8 is that primitive recursion in t h e form of definition 1.1.3 is not necessary: a simpler operation of simultaneous iteration is sufficient. This result can be further refined, and we now show t h a t , at t h e cost of a slight increase of t h e stock of initial functions, even simple iteration alone is sufficient.

Definition 1.5.9 A function f is defined b y iteration f r o m a function t if

where t(")(x)denotes the result of n successive applications of t (by convention, t(O)(x)= x ) .

Proposition 1.5.10 (Robinson [1947], Bernays) T h e class of primitive recursive functions is the smallest class of functions 1. containing the initial functions, together with coding and decoding functions for pairs

2. closed under composition 3. closed under iteration. Proof. Let C be t h e smallest class of functions satisfying t h e conditions just stated: every function in C is primitive recursive, because t h e coding a n d decoding functions for pairs are primitive recursive, a n d t h e class of primitive recursive functions is closed under composition a n d iteration, t h e former by the definition, and the latter because iteration is a special case of primitive recursion:

For the converse, we only have to show t h a t primitive recursion can be reduced t o iteration. First of all note t h a t , since C has coding a n d decoding functions for pairs and is closed under composition, it also has coding a n d decoding functions for n-tuples, for any fixed n. We continue to use for them t h e standard notations for coding and decoding functions. Let g a n d h be functions in C, and

1.6 Flowcharts

73

We will show t h a t t h e function

,.('

). = (2,n,f(Z,4)

is in C. T h e n so is f, by composition, because

(where m is t h e number of elements in t h e vector 2). Note t h a t

s(Z, 0) = (2,0, f(Z, 0)) =

(Z,O,g(q)

a n d t h u s s(Z,O) is in C, as a function of Z. Moreover,

s(Z,n

+ 1)

= =

+

+

( 2 , . l , f ( Z , n 1)) ( Z , n + l,h(Z,n,f(Z,n))).

Since

).,'(. s(Z, n

.I),

= (2,n, f(2,

+ 1) c a n be obtained from s(2,n) by one application of t h e function t((Z, = (2,n + 1, h ( Z ,n , z ) ) , 7 1 , ~ ) )

which is in C by composition, since g, h, t h e successor, a n d t h e coding functions a r e in C. Finally, s ( 2 , n )= t(n)(s(Z,O)), a n d s(2,n) is t h u s t h e composition of s(Z,O) a n d t h e iteration o f t . Since these a r e b o t h in C, so is s. 0 T h e result j u s t proved can be variously improved. First of all, Gladstone [1967], [1971] shows t h a t the introduction of new initial functions can be avoided: t h e class of primitive recursive functions is t h e smallest class containing t h e initial functions, a n d closed under composition a n d iteration. Second, t h e iteration schema can be further weakened in t h e following schema of pure iteration: f ( n ) = t(")(O) (Robinson [1947]). Some new initial functions a r e needed here, since b y composition a n d p u r e iteration we never get, from t h e initial functions, a n y function depending on t w o variables, like z+y. Choices of initial functions t h a t generate t h e primitive recursive functions by pure iteration, a n d a s t u d y of t h e algebraic

I. Recursiveness and Computability

74

structure of t h e class of primitive recursive functions, a r e in Robinson [1947], Poliakov [1964], [1964a), Lavrov [1967], Kozmnikh [1968], a n d Gladstone [1971]. R . Robinson [1947], 119551, a n d J. Robinson [1955],also show t h a t t h e p u r e iteration schema is enough to generate t h e unary primitive recursive functions by composition, starting from two ( b u t not from only one) appropriate unary functions. T h u s we do not need to use nonunary functions to get any unary primitive recursive function. See also Peter [1951] for a t r e a t m e n t of t h i s topic, a n d Georgieva [ 19761 for further simplifications.

Petri nets

*

Systeriis of separate, interacting components (like finite a u t o m a t a or computer hardware, flowcharts or computer software, physical systems, a n d so o n ) c a n be modeled by Petri nets (Petri [1962]), which consist of bipartite, directed multigraphs whose vertices can be either places or transitions (represented, respectively, by circles a n d bars), a n d whose arcs connect places to transitions, or transitions to places. Tokens can be assigned to (and c a n be thought to reside in) t h e places of a Petri net (which becomes marked), a n d their number in a place may change during t h e execution. T h e execution is controlled by t h e number a n d distribution of tokens in t h e net, a n d consists of transition firings, which remove tokens from their input places, a n d deposit new ones in their o u t p u t places. A transition fires when enabled, i.e. when each of its input places has at least as many tokens in it as there a r e arcs from t h e place to t h e transition (that, is, a r c s a r e seen as conductors of capacity one). Firing continues as long as t h e r e exist enabled transitions, then it halts. T h u s , nets model systems, a n d executions model t h e flow of information in them. Some of t h e places may be singled o n t as inputs or outputs, to model interactive behavior of a system with t h e outer world, a n d t h u s allowing a coniputational interpretation of t h e behavior of a net. Petri nets a r e q u i t e general devices, apt to model concurrence, d u e to their inherent parallelism, asynchronicity and nondeterminism. Events which a r e b o t h enabled a n d do not, interact may occur independently, there is no inherent measnre of t i m e flow in t h e execution, and no order is placed on transition firings when many transitions a r e enabled. An extension of Petri nets (Agerwala (19741, Hack [1975)) allows for inhibitor arcs froni places to transitions (w1iic.h a r e pictiired differently, as arrows with t h e arrowhead substituted by a srriall circle). T h e new firing rule is t h a t a transition is enabled if it is in t h e previous sense, a n d moreover t h e input places corresponding to inhibitor a r c s a r e empty. T h u s inhibitor arcs permit zero testing. T h e interest of this extension is t h a t Petri nets with inhibitor arcs compute all the recursive functions. T h i s is easily seen by modeling 'go to'

1.6 Functions as Rules

75

programs (p. 64) as Petri nets, with different places corresponding t o variables and statements, and transitions corresponding t o instructions.

in

Xi

Figure 1.11: Petri net for a ‘go to’ instruction For example, given a ‘go to’ program with a list of labeled instructions i n , the part of t h e corresponding Petri net relative to a ‘go to’ instruction

in : if Xi = 0 go to m is the one shown in Figure 11, whose places are labeled in the natural way. The program works in the following way: if X i = 0 then the inhibitor arc works, and the instruction ,i is activated, while if X i # 0 then the normal arc works, decreases X i by one, thus allowing the activation of the next instruction in+l , and then a new arc reintegrates the original value of X i , which is not supposed t o change. T h e parts corresponding to the other instructions (increasing or decreasing by one the value of a variable) are similar, but easier. For an introduction t o both theory and applications of Petri nets, and an annotated bibliography on the subject, see Peterson [1981].

1.6

Functions as Rules

The post-Dirichelet practice has been t o identify functions and their graphs, thus giving a prominent importance t o the values, independently of t h e way they are obtained. The rise of Computer Science has forced people t o look at functions in the same way they were originally considered, as intensional objects. We thus set up in this section a theory of functions as explicit rules.

I. Recursiveness and Computability

76

A-calculus Rules must be expressed somehow, and they can thus be identified with terms in a n appropriate language. A term t ( z )can be seen as t h e value of a function for th e argument 2. We introduce a n abstraction symbol ‘A’ t o indicate t h e step from t h e value t ( z )t o t h e function z H t ( z ) ,which we indicate by Xz. t ( z ) . To begin with, we work in full generality and consider a language having merely th e essential ingredients.

1. Symbols. 0

variables zo, z1,

0

t h e abstraction symbol ‘A’

0

parentheses ‘(’ a n d ‘)’

0

dot &.’ .

Note t h a t there is only one kind of variable, and no theoretical distinction is made between functions and arguments. T h e common practice in mathematics, following Russell [1908], is t o consider a function as being a n object of a t ype higher than its arguments and values. This was introduced as one possible way out of Russell’s paradox (see p. 81), which arose from t h e consideration of sets belonging to themselves. Since self-membership corresponds, in functional terms, t o application of a function to itself, t he practice of A-calculus (of dealing with only one kind of object, t hat can be function or argument at will) sounds at least suspicious, and will have t o be justified. Appropriately, Recursion Theory provides us with various ways of doing this, see pp. 194 a n d 223.

2. Terms. 0 0

a variable z alone is a term, and z occurs free in it if M , N are terms then t h e application ( M N ) of M to N is a t erm , an d free or bound occurrences of variables in M or N remain so in

(MN) 0

if M is a term and z is a variable then t he A-abstraction (Ax. M ) of M w.r.t. z is a term, z is bound in it, a n d t h e other variables are free or bound in it according t o what they were in M .

By convention,

77

1.6 Functions as Rules

T h e latter just avoids t h e need of many parentheses, but t h e former is theoretically important, since it defines functions of several arguments as composition

of functions of a single argument. As we see, notations in A-calculus ar e quite different from t h e common practice, a n d they might be confusing at first sight, both for t h e treatment of application (which is just written as juxtaposition) a n d of multiple arguments (which are not treated simultaneously, but one at a time). E.g., what is usually written f ( t 1 ,t2) is here rendered as (ft1)tz or simply, by t h e conventions just stated, ftlt2. Terms a r e names for t h e objects of A-calculus (which, as already noted, may be thought of as both functions and arguments). We now set u p a theory of transformations for them, and introduce rules t hat allow us t o manipulate terms. T h e idea is t o compute values of functions by purely syntactical transformations of th e terms representing t he functions and their arguments.

3. Reduction rules.

a-rule. We can rename bound variables: Xz. M 2 Xy.M[x/y]

provided y does not occur in M I where M[z/y] indicates t h e result of t h e substitution of x by y everywhere.

@rule. We can apply a function Ax. M t o a n argument N : (Xz. M ) N

3 M[x/N]

provided whatever was free in N before t h e substitution remains free afterwards. This proviso can always be fulfilled, by a possible application of t h e a-rule. For example,

(AZY.XY)Y

+

XY.YY

is a n illegal application of t he p-rule, a nd indeed would not preserve t h e intended meaning , but it can be replaced by

(Xzy.xy)y

+ Xz.

yz,

which can be derived by first changing t h e bound variable y into z by th e a-rule, a n d then applying t h e /3-rule.

B We write tl = t2, and say t hat tl a n d t2 are equal (modulo a or p reductions), if t l an d t2 can be reduced t o t he same term by a finite number of

78

I. Recursiveness and Computability

applications of t h e cy or @ rules. Also, a t e r m is said to be in normal form if it cannot be P-reduced (i.e. t h e p-rule is not applicable t o a n y of i t s s u b t e r m ) , a n d it is reducible otherwise. A t e r m is normalizable if it has a normal form, in t h e sense of being equal to a t e r m in normal form. Normal forms of terms c a n be considered as their ‘values’, a n d it is natural t o investigate when they exist a n d how to c o m p u t e t h e m . Unfortunately, a normal f o r m does not necessarily exist ( t h u s , in s o m e sense, t h e functions of A-calculus a r e partial ones, see p. 127). To sce this, we can simply produce a circular example of a t e r m reducing to itself via t h e P @-rule, say M N -+ M N : M must then have a A-abstraction (to allow for a n application of t h e @-rule), a n d produce a n application. For example,

applies a t e r m to itself: P At = It,

a n d hence it self-reproduces:

A second negative point is t h a t even i f a nonnal forin exists, not all sequences of reductions can produce i t , e.g. a t e r m might be reduced to normal form by one reduction a n d enter a loop by another: (Azy.y) (AA) a

= ’ a

by using t h e outer X

(Axy.y) (AA) a

P =

by using t h e A in A .

(Axy. y) (AA) a

Terms in which every subterm has a normal form a r e called strongly normalizable, a n d t h e example just given shows t h a t a t e r m can be noriiializable without being strongly so. On t,he positlive side, Church a n d Rosser [1936] have shown (by a dificnlt proof, outside t h e scope o f this book) t h a t i,erniinating reductions of a s a m e t e r m always give t h e same result, u p to renaming of b o u n d variables. In particular, the normal form is unique when it exists. Moreover, there are reduction strategies that produce the normal fonn, whenever it exists. O n e such strategy is t o always reduce t h e lckmost reducible A. T h e n s u b t e r m s a r e e v a l u a t d exactly as many times as needed, which means t,hat t h e y may be evaluated more t h a n once in some cases, b u t also t h a t t h e y a r e not evaluated if not useful. A different strategy would be to evaluate M a n d N before doing M N . T h i s does not work in general (as t h e exaniple above shows), b u t each t e r m is evaluated only once, which rneans t h a t t h e strategy is fast when i t works,

79

1.6 Functions as Rules

although some unnecessary terms may happen t o be evaluated (and this may be fatal if one of these is a nonterminating one). We have already noted a peculiar aspect of A-calculus: t h e absence of distinction between functions and arguments, which may be seen as a collapse of t h e concept of type. In Recursion Theory a cumbersome technique (the method of arithmetization of Section 7) is used for t h e related purpose of assimilating functions with numbers. This is partly achieved by identifying t h e recursive functions with their finite descriptions, e.g. programs, a n d by coding these as numbers. Being indirectly obtained, this assimilation does not avoid t h e need of continuously going back and forth between functions a n d numbers representing them, and some arguments may at times become darkened. This is t h e case of t h e Fixed-Point Theorem 11.2.10 and its proof, which are s o m e times considered quite mysterious. When recast in terms of X-calculus, both t h e statement and the proof of this result (based on methods fully explored in Section 11.2) become more transparent.

Theorem 1.6.1 (Kleene [1936b], Turing [1937a], Curry [1942], Rosenbloom [1950]) There is a fixed-point operator y which produces, for every term M , a @ed-point for it:

yM

E M(yM)

Proof. We give two different proofs. 1. We start with a n informal argument. Recall that we have defined a term

A = AX. xx such t h a t , for any term t , P At = tt.

By applying A t o itself, we then have

AA

E

AA. P

Given a term M , we want a term z such that z = M z : then z reproduces not itself, but t h e application of M to itself. It is then enough to generalize t h e definition of A, which gives what we want when M is t h e identity operator. Let AM = AX. M(xx). Then, for any term t ,

AM^

P

= M(tt).

I. Recursiveness and Computability

80

By applying A M to itself we then have P

A M A M= M ( A M A M ) , a n d a fixed-point of M is giver1 by A M A M =

(AX. M(zz))(Az. M ( z 5 ) ) .

Sirice this definition is unifornl in M , we can actually a b s t r a c t M from it, arid obtain y = Ay. (Ax. y(m))(Xx.y(zx)). Then

P

P

P

Y M = AMAM = M ( A M A M ) = M ( Y M ) . 2. We now give a proof h s ~onl t h e (contrapositive of the) diagonal method P

of Section 11.2. Consider M : a fixed-point z for it is such t h a t z = Mz, i.e. it must be of t h e forni t,t, for soiiie terms t , , t,. Also, M must be thought of as t h e result o f a trailsformation of t e r m s of t h i s kind. Consider a possible enumeration of all pairs of A-terrrrs:

a n d t h e effect of M

This is

ii

011

t h e diagonal:

hequeiice of X-t,ernis, of t h e form P t,,t, = M(t,t,)

for sonic

71.

But then it. is jiiht, t,he

7 ~ - t , I i row

of t h e matrix, in particular

is a fixed-point, o f M . Explirit,ly,

t,, = Az.M(zz) tnt, = B (Xz.M(:cz))(X:c. M ( z x ) ) ,

1.6 Functions as Rules

81

a n d t h i s is uniform in M , i.e. it produces a fixed-point operator

y

= Ay. (As. y(zz))(Az. y(m)).

Since all this might look quite mysterious, we can check t h a t gives u s a fixed-point for M :

yM

y

really

= (Ay. (Ax. y(zz))(As. y(sz)))M

(1.1)

!!

(1.2)

(As. M(zs))(As.M ( z s ) ) = P M((Az. M(sz)(Az.M ( z z ) ) \ 4 "

(1.3)

Y M

M(YM)

(1.4)

where ( 1 ) is obtained by definition of y , (2) by p-reducing t h e Ay, a n d (3) by &reducing t h e outer As. Note t h a t (3) is still reducible, using t h e outer A s . T h i s is a typical property of t h e terms of t h e form y M , and somewhat illustrates t h e circular aspect of fixed-point definitions. 0 Exercise 1.6.2 Y is a fixed-point operator if and only zf it is itself a fied-point of G = Xym. m(ym). (Bohm, Van Der Mey) (Hint: identify Am. Y m and Y . )

y is sometimes called t h e paradoxical combinator, because it embodies t h e argument used in Russell's paradox (Russell [1903]). T h e connection between Set Theory a n d A-calculus can be established by t h e following correspondences: element argument set function membership application set formation A-abstraction set equality t e r m equality. Russell's paradox is obtained by considering t h e set

A =

{Z

: z @ z}.

Theu

X E Ae x@x, a n d thus

AEA e A@A, contradiction.

I. Recursiveness and Computability

82

111 terms o f A-cAculus, thc negation operator can b e considered as a teriii N that, is never thc ideiitity. Since ineinbership corresponds t,o application, sclfiticiiilmdiip corresponds t o self-al)l'licatioii, and then the sct A cwrresponds to tlie term Az. N ( z z ) .

By p-rccluctiori, B

(AZ. N ( z z ) ) z = N(:c:c),

alld thns (AT. N(ZX))(XZ. N ( 3 - J ) )

i

il

N((A.1;.N(zz))(Az.N ( T T ) ) ) ,

(t

yN

N(YN).

Notc however that liere there I? no paradox. f r o i i i t,he last, asscrtion, which is N that is never the identity.

triic, we j u s t detliice that tliere ih no term

Other formulations of the A-calculus

*

O u r approach (Church [ 19331) has been t o define (by means of /\-abstractions) c d e t l coriibinators, rcRec.t,iiigthe way functional letters can b e effectively c.oitil)inetl to produce nanics for iritensionally presented functions. Our t.eriii foriliation rules allow Az. A1 to bc a term wlimever M is. 'I'his version is callcd tlie XK-calculus. If a rcstxiction wcre imposed on M , requiring that 2; occur free i n it, we would o l > t i i i i i a version callccl thc XI-calculus, i n which fii iict ions with fictitious argii 11ICIit,s are excluded. This rcst r ict,io11 has 1 )ccii int rotliiced t o avoid soltie patliologies, like t,he existence o f noriiializable, i i o t , st roiigly noriiializable t>ernis,as S C Y : ~ above. The: N arid j j rules define a system called the calculus of p-conversion. Varioiis riiotlific;it,ioiis o f it are possible, e.g. by adding aii extensionality law tcritis,

for ,\-t.cmns:

77-rule. \Vc can itlcntify evcry

t ~ r i i iwitli

a fnwt,ion:

1)rovidctl z is not frec i n 121. No1.c that., I)y t,lie rules of term forniaLioii, for a n y term A1 t,lie expr"?isiori Xz. A1:c is also a teriii, a i i t l thus it r q ) r e scnts a fiinctioii: (,lie r n l e wsiires that, this is exactly thc fiiriction t,h;it, is rc:presentetl by the term A1 itself.

1.6 Functions as Rules

83

T h e reason we call this a n extensionality law is that it implies that if M and N behave extensionally in t h e same way, i.e. Ax. M x and Ax. N x are equal, then so are M and N . An equivalent but different approach to the X-calculus (called the theory of combinators, Schonfinkel [1924], Curry [1930]) consists in postulating some, actually very few, of t h e combinators as primitive, and t o deduce all t h e others from these. This produces a kind of synthetical (bottom-up) analysis of t h e global concept of X-definability. In particular, it can be shown t h a t only two combinators are needed:

S = Xzyz. z z ( y z )

K = Xxy. z.

S can be seen as a kind of interpreted application, where x and y are first interpreted in t h e environment z, and then applied one t o the other. Since t h e identity I can be defined as S K K : I X= S K K X= K x ( K 2 )=x , the X operator can be defined by induction on the terms obtained from t h e variables by application, as follows:

Xx.x = I Ax. c = K c if c is K , S or a variable y A X . M N = S(Xx.M ) ( X z .N ) .

#x

We can then prove t h e P-reduction rule, by induction. Standard references on t h e subject are Church [1941], Curry and Feys [1958], Curry, Hindley and Seldin [1972], Barendreght [1981], Hindley and Seldin [1986]. Historical accounts on t h e origins of X-calculus and its interaction with Recursion Theory are in Kleene I19811 and Rosser [1984].

X-definability We are interested in numerical functions, but it would seem that until now we have just set up a logical basis, and that we still need to add t o t h e language of X-calculus numerical terms and some basic numerical functions. But then we would face the problem of not knowing exactly what t o add, and we would have to turn back to different approaches, with the X-calculus relegated to a mere role of convenient notation. Instead, and this is a most interesting aspect of this approach, it turns out that there is no need of additional notions. T h e natural numbers appear obliquely in this general setting, when we consider the number of iterations of a function application. We can thus define X-terms T i that, applied t o a function f and an argument x , give t h e result

I. Recursiveness and Computability

81

f(")(z)o f R iterations of the function t l i r iiit,rgers. Since we want

f

o n z, and take them as representing

wc ('a11just let:

Definition 1.6.3 (Peano [1891], Wittgenstein [1921], Church [1933]) vw iiiiiriera/ ii is the ,\-tr:mri ~f.. f(")(~). We write f and z to help the intuitive underst,aiiding, but note that we just, have one type of variable (with no distinctiori I)et,weeti functions and argunierits), arid t h u s we slioultl just write -

71

= Azy. z(747,

Not,c that the teriiis TE and ii are different if m # n, by the theorem o f ('1inrc:h aiid Itosser quoted on 1). 78, I)ecaiise they are in normal form arid tlist,iiict. Iiidiictively we have, by tlcfinit,ion of iteration, -

0 = Afx.2:

m

= Af:c.f(7if2:)

I)ccaiise f ( " ) ( ~ ) = z ant1 f(lt+')(:c) = f(f("'(2;)). Since these terms rcprcseiit. i n sonie way the constarit, function 0 and the sii( s o r operation S,we gel. froni t,liis t,he idea of reprc;cnt,ing iiunierical f i i r i c t i o i i s :

Definition 1.6.4 (Church [1933], Kleene [1935]) A I L?t,-ary f u n c t i o n f is A-definable if there i s a A - t e r m 1' such that

.

f ( ~ , l , .. ,a,)

=b

holds if m d only if lG, . . . ?in

5.

Theorem 1.6.5 (Church [1933], Rosser [1935], Kleene [1935], [1936b]) I < i w r y i r ~ c ~ ~ ~ i :fuiictioii siiie i s X-deJinahle. Proof. We proceed by iiidiictioii o n the definition of recursive function. '1'0 simplify tlic technical clctnils, wc rely o n t,he alt.eriiatjivec,liaract,erizationof thc class o f primitive recursive functions given in 1.5.10. 1. initid functioris 0 , S arid Tp are, respectively, A-defined by: AX.

ii

Azfz. f ( z f z ) A z 1 ' . ' X 7 & .2 ,

1.6 Functions as Rules

85

2. coding and decoding functions for pairs This is obtained, in analogy with the representation of natural numbers, as: (n,m)= Afgx. f'"'(g'"'(x)). Thus t h e pairing function is

AY zfgx. Y f ( W ) . Decoding is then immediate: if Z is the representation of T i , i.e. of the identity function, then

Thus t h e decoding functions are Xy f x . y f I x and Aygx. ylgx. 3. composition Suppose h, gi are A-defined by H , Gi. Then

is A-defined by

4. iteration This is a n immediate consequence of the representation of natural numbers: if T is a term representing the function t, then the iteration

is represented by

Axy. yTx. 5 . p-recursion This is obtained as in the proof of 1.2.3. Recall that if

I. Recursiveness and Computability

86 where

h(S’ y, =

{ :(?,

y

+

if g(5,y) = 0 1) otherwise

By t h e first part of the proof case definition, successor, a n d composition are all A-representable, because primitive recursive. By induction hypothesis, so is g. Thus there is a term M representing Ah-y. F , where

F(Z,Y,h) =

{ :(?,

y

+ 1)

if g(5,y) = 0 otherwise.

Then t h e function h defined above, being a fixed-point of F , is represented by y M . And f is finally represented by A?. ( y M ) & . To conclude t h e proof we note that: 0

T h e completeness property, stating t h a t t h e values can be deduced from the appropriate A-terms, is quite evident from t h e informal discussion just given.

0

T h e consistency property, stating t h a t no other value can be deduced, follows from t h e theorem of Church and Rosser quoted on p. 78, which ensures t h a t if the process of P-reduction of a t e r m produces a term in normal form (like T i ) , then this term is uniquely determined (up to renaming of t h e bound variables). 0

Exercises 1.6.6 a) Alternative coding and decoding functions are, respectively, Xxyu.uxy, Xz.zT and Xz.zF, where T = Xxy.x and F = Axy.y. Being distinct Xterms, T and F can be taken as representation of the truth values ‘true’ and ‘false’. b) 6 = Xz.z(Xu. F)T represents a function that, when its argument is a numeral, is T i f z is and F otherwise. Then Axyz. ( 6 z ) x y represents definition by cases, i.e. a function that returns the first or the second argument, according t o whether the -third is 0 or not. (Hint: a numeral applied to two terms produces the second if it is 0, and applies the first at least once otherwise.) c) The predecessor function can be directly represented, using only repwsentations of successor, coding, and decoding functions. (Kleene [ 19351) (Hint: the predecessor of n is the second component of the n-th iteration of the function on pairs defined as t ( ( z y, ) ) = (z 1, z), started on (0,O).) By 11.2.15 this provides an alternative proof of 1.6.5, not using 1.5.10.

a,

+

Functional programming languages * T h e programming languages discussed in Sections 4 a n d 5 a r e imperative in t h e sense t h a t they specify a sequence of instructions a n d a n order of execution to be followed t o produce a n output f(5).T h e functional approach, suggested by A-calculus, defines f directly, and computes t h e required values by

1.7 Arithmetization

87

P-reductions. T h e name ‘functional’ reflects the emphasis put on t h e functions themselves as intensional objects, as opposed t o extensional emphasis on t h e values. Relying on the characterization of recursive functions given in 11.2.15, McCarthy [I9601 notes t h a t a function is recursive if and only i f 0

there is a definition of it from identities, successor, and predecessor by means of A-abstraction, composition and the conditional operator

if t(5)= 0 then g(Z) else h(Z). 0

t h e function is computable from this definition by a call by value procedure, i.e. a computation which evaluates first the innermost (and leftmost, if there is more than one) occurrence of the letter defining t h e function.

Clearly t h e conditional operator replaces the definition by cases, and t h e fixedpoint operator is eliminated, in favor of a computational approach t h a t just produces it. This formulation of recursive functions can be extended from numerical functions t o functions on words of a given alphabet and, as such, it has furnished t h e computational basis for t h e programming language LISP (List Processing, McCarthy [1960]; see Sammett [1969] a n d Wexelblat [I9811 for history a n d references). Since there is no reason t h a t functional programming languages should be forced to run on machines designed for imperative ones, work has been done t o design hardware directly inspired by t h e functional approach, based on A-calculus ( t h e SECD machine, Landin [1963], implementing P-reductions) or on t h e theory of combinators (the SK machine, Turner [1979],implementing reductions of graphs t h a t represent t h e definition of a function, in terms of t h e combinators S and K , see p. 83).

I.?

Arithmetization

Arithmetization simply means translation into the language of arithmetic. We will give one detailed example of t h e method, and show how to code the machinery of computation of recursive functions in a primitive recursive way. T h e details are quite cumbersome, and in t h e rest of our work we will content ourselves t o sketch similar arguments, leaving the details t o the reader.

Historical remarks

*

T h e first attempt t o find number-like connections between propositions of various sorts probably goes back t,o Lullus’ Ars Mugna, b u t it was Leibniz [1666]

I. Recursiveness and Computability

88

who dreamt of arithmetization as a general method to replace reasoning in natural language by arithmetical propositions, with t h e goal of substituting arguments with computations. Leibniz went further than just this oneiric activity, and devised (see [1903]) a precise coding method, by first assigning numbers t o primitive notions, and then showing how t o associate composite numbers (e.g. by multiplication) t o composite notions. However, his tentative work was left unpublished until 1903, and did not influence modern developments. Hilbert [1904] again envisaged arithmetization in his idea of formalizing consistency proofs into Arithmetic, but it was Godel [1931] who first used it explicitly and formally t o translate t h e concepts relative to formal systems into a n arithmetical language. Tarski [1936] independently arrived at t h e method in his investigations of t h e concept of truth. Contrary to Leibniz's dreams, the effect of arithmetization was, ironically, not t o shield the language against its oddities, but rather to spring a leak in Arithmetic, through which the linguistic paradoxes poured only to reveal t h e inadequacies of formalism (see 11.2.17).

Numerical tools for arithmetization Primitive recursive functions and predicates are more than enough to carry out arithmetizations. We will use prime numbers a n d factorizations (recall, see p. 26, t h a t the sequence { p s } , E w of prime numbers is primitive recursive), because this coding is particularly simple. It should however be noted t h a t we could use functions from much smaller classes: this will be done in Chapter VIII, when t h e necessity for more efficient codings will arise. To code the sequence (50,.. . ,zm), t h e simplest way would be t o use t h e number pgo - .. p 2 . But then we could not uniquely decode a number, since we would not know whether a prime in the decomposition has exponent 0 accidentally or meaningfully (in t h e sense that it is coding t h e number 0). Thus, either we rule out 0 as a meaningful exponent, a n d let (20,.. . ,z),

= p;0+1

*

. . pEn+',

or we tell in advance how many numbers we are coding, and let

..., zn)= p ; * p ; 1 . . . p?.

(XI,

Since we have t o make a choice for the following, we decide t o use t h e second proposal. T h e decoding system is given by t h e following functions a n d predicates, all primitive recursive (see p. 26):

89

1.7 Arithmetization

We call h ( z ) t h e length of 2,and ( x ) the ~ n-th component of z. If S e q ( z ) holds then we say t h a t z is a sequence number. In this case, z = ((XI1 9 .

*.

7

(.)In(,)>.

We will also need a concatenation operation (21,.

*, such t h a t

. . , z n )* (?/I,.. . ,Ym)= h,.. . ,zn,Yl,.* . ,Y,>.

if S e q ( z ) A S e q ( y ) holds, a n d 0 otherwise. Finally, we will need t h e notion of initial subsequence, defined as

As a first use of sequence numbers we get t h e following useful result. Proposition 1.7.1 Course-of-values recursion (Skolem [1923],Peter [1934])The class of primitive recursive functions is closed under recursions in which the definition of f (2,y f 1) may involve not only the last value f (2,y), but any number of (and possibly all) the values { f (2,z ) } ,-< ~ already obtained. Formally, let f be the history function o f f , defined as: f(.',Y)

= (f(.',O),...,f(.'>!4).

Then, i f f is defined as

and g , h are primitive recursive, so is f . Proof. It is enough to show that

f is primitive recursive, since then also f

is:

I. Recursiveness and Computability

90

But this is imrriediat,e. since

+

1) only uses t h e last previous valuc f (.',y), Thus f(?,y recursive because so a r e coding arid coricatenation. 0

arid

f is primitive

Exercise 1.7.2 Simultaneous primitive recursion. The clnss of primit,ive w cursive functions is c l o d under simultaneous recursion on mom than one function. (Ililbert arid Rernays [ 19341) (Hint: reduce simultaneous primitive recursion of, say, f l (i) arid f2(z+),to primitive recursion of the single function (fl(?), f'~(.t)))), 1Jy coding.)

The Normal Form Theorem We are now in position t o give our first a n d only complete example of a r i t h n i o tization, by reducing t h e reciirsivc runc:t,ions to a normal form. Any approach to recursiveiicss would produce similar results, a n d we will skct,c:li t,he versions relative t o t h e approadies o f Sections 2--6 later in this sect,ioii, but, here we give a self-cont,ained treatment based or1 recursiveness alone.

Theorem 1.7.3 Normal Form Theorem (Kleene [1936]) Th,ere ,is a primitive recursive f u n c t i o i i U a n d (for each n I) primitiiie recursive predicates such that for every recursive f u n c t i o n f o f 71 variables there is n miirrber e (called index o f f ) for which the following hold:

>

zl,

1.

VXl

...Vxn3y?;l(f:,xl, . . . ,xn,g)

2. f ( X h , .

. ,%) '

= ~ ( P Y T 1 ( c l x l.>. ,.G L l Y ) ) .

Proof. The idea of t h e proof is to associate Iiuinbers to fuiictioiis a n d m n i putations i n such a way t h a t t h e predicate ~ l ( e l x.~. ,lx T . L l yt)r a n s l a t e t h e assertion: ?j is t h e tiuriiber of a computation of t h e value of t h e function with . this, pyTn(e,xl, . . . , xTL,y) associated nurriber e , on input,s z1,. . . ,z,,Ilaving will give t h e riurriber of one such coriipiitat~ion,arid t h e fiiiiction U will extract t h e value of t h e o u t p u t from it. T h i s is more easily said than done, and to achieve it we need to carry out a riuml~erof steps.

1.7 Arit hmetization

91

1. associate numbers t o recursive functions

This is done according to the inductive procedure that generates the recursive functions. T h e details are obviously irrelevant, and we just give one possible assignment: 0

(0) t o 0

0

(1) to

0

(2,n,i)t o Tin, for 15 i 5 n

0

0

0

s

( 3 , b l ,... ,b,,a) to f(Z)= g(hl(Z),..., h,(Z)), where b l , ... ,b, and a are numbers respectively associated t o hl , . . . ,h,, and g (4, a, b) to f(Z, y) defined by primitive recursion from g and h, where a and b are respectively associated t o g and h ( 5 , a ) t o f(2) = py(g(Z,y) = 0), if VZ3y(g(P,y) = 0) and a is associated to g.

Any number associated t o a recursive function is called a n index of this function. Since there are many ways t o define the same function, there will be many indices for each recursive function (see 11.1.6). Also, many numbers are not indices of recursive functions, either because they are not sequence numbers of t h e right form, or because some of their relevant components are not indices of recursive functions. We will see (p. 146) that in general there is no effective way t o tell whether a number is indeed the index of a recursive function, because basically there is no way t o tell whether VZ3y(g(Z,y) = 0).

2. put computations in a canonical form A natural way t o organize a computation for t h e values of a given recursive function, is by way of computation trees. Each node of such a tree will tell how a value needed in t h e computation can be inductively obtained. Of course, t h e only possibilities are those given by t h e permissible schemata of definition I. 1.7, namely: 0

nodes without predecessors:

f (4= 0 iff=U if f = S f(z)= z 1 f ( z 1 , . . . ,zn) = zi if f = Tr

+

0

composition If f(Z)= g(hl(Z),. . . ,h,(Z)) predecessors:

then the node f(2) = z has r n

+1

I. Recursiveness and Computability

92

f(2)= z

hl(Z)= z1 0

...

h,(Z) = z , g ( z * , . . . , z,,,) = z

primitive recursion I f f(Z,y) is dcfiried by primitive recursion from g a n d h t h c r c a r e two cases, respfxtively with one o r two predecessors:

p-recursion If f(2) = py(g(S,?y) = 0 ) t h e n there is r i o fixed p a t t e r n to t h e predecessors of f(Z)= y. T h e general situation is:

f(2)= z

where l o , . . . , t,-

I

a r e all different from 0.

3. associate numbers to cornputatjoris This is done by induction on t h e c.onstriic.l,ion of t h e computn1,iori tree. First of all, we assign numbers to nodes: since t h e y a r e expressions of t h e kind f(xl,. . . , z T L=) z,we give t h e m numbers

where e is a given index of f . T h u s a tiode is represented by t,hree numbers, correspotiding respectively to t h e firtiction, t h e inputs i i r i t l t h e olll,put.

We then assign niiinl)ers t o trees: each tree T consists of a vertex v with associated numl>er u,a n d of a certain number (finite, arid possibly equal to zero) of orderrd predecessors, each o n e being a siibtree T,. By

93

1.7 Arit hmetization V

Figure 1.12: A tree T with vertex v and subtrees Ti's induction, we assign to this tree t h e number

5

is t h e number assigned t o the subtree Ti. In particular, if t h e where vertex with number 21 does not have predecessors, then it has number (v) as a tree.

4. translate in a primitive recursive predicate 7 ( y ) the property that y is a number coding a computation tree To increase readability we use commas instead of nested parentheses, and write e.g. ( a ) i , j , k in place of ( ( ( a ) ; ) j ) k . To keep track of what we are doing, check Figure 13 and recall t h a t h

Y = (V,Tl,...,Trn), a n d hence:

(Yh (y)l,l (Y)1,2 (Y)1,3

(Y)i+l (Y);+~,]

= (e, (21 ,. . . ,%), 4 = various types, depending on e

=

(21 1 . .

. ,GJ

z

= = Ti = number of the vertex of T;. A

First we let:

This expresses t h e most trivial properties of y. We then have four cases, corresponding to t h e possible situations spelled o u t in Part 2 above.

I. Recursiveness and Computability ((3, b , , . . . , ( I , , , , a),(x,,. . ,:c,J, f

2)

a) composition

( 0 ,( X I , .

I

. . ,&),

2)

I,) primitive recursion

c ) p-recursion

1.7 Arithmetization

For composition we let:

For primitive recursion, recall that there are two possible cases:

For p-recursion we let:

95

I. Recursiveness and Computability 'I'liese cwiditions t a k e c a r e o f all possible cases, a n d thus we may define

iiitliictively:

'I'lien T is priniitive recursive because it is definccl by using only pririiit,ive rwiirsive clauses arid values (of its cliaracteristic function) for prcvious iwgumerits because, by definit,ioii of cwling, (y), < y . T h a t is, 7 is defined by coursc of value rtxmrsioii, wliicli is a priiiiitive recursive o p e r a t i o n by

1.7.1.

z,

5 . drfine arid U We a r e now rciidy to concliiclc: o u r work. Namely, for each n

2

1 we let:

ittid

U(Y) = ( Y ) l , 3 r 1

I lime are obviously pritnitivc: r m u r s i v e

Lct. iiow f 11c a recursive n-ary function with index e . Since Is total, for every . I : ~, . . . , x l I there is ii c u i n p i i t i i t i o i i t r e e for j ( x l , .. . , x1,)relative to t h e c o m p u t a t i o n procculure c:otlctl by c . This is forinally expressed by:

Moreover, from any coriipitatiori ( , r e (in par1,iciiIar from t.he one wit.li t h e sinallrst code iiiiniber) w c c a n extract thc value of t l i c funct,ioii by lookiiig at t h e t h i r d coniponeiit of its vertex. ThLs is foriiially expressed by:

Exercise 1.7.4 There is a rrcirrsivc Jwictioia munieratirq the uriary primitiiic rrarrsive fuirctions, i.e. a recursive function j ( e ,T ) such that: for every c the function Ax. f ( e , s) is prirriitive recursive, arid every primitive recursive function is cqual to As. J ( c ,s) for snmc e . (Petrr [ 19351) (Hint: let

J(e,x)

~

{

1A(,u~y?; ( e , 5 , y))

if e is a primitive recursive index o th erw is?,

wherc k i n g a primitive rcc:iirsive iridcx mcnns to define n recursive fiincticm from a r i d prirriitive recursion alone, without using the the iriitial functions by corrip~~+itioii

/i-operator.)

97

1.7 Arit hmetization

Equivalence of the various approaches to recursiveness By using t h e method of arithmetization we can get a cascade of results relative to t h e various approaches introduced in Sections 2-6. We will not go beyond sketches because we believe t h a t , once t h e method is understood, t h e translation of these into formal proofs should not present theoretical difficulties. I t is, however, a very useful exercise to t r y to fill in t h e cumbersome details of some of these sketches. We begin by dealing with t h e notions of Section 2.

Proposition 1.7.5 (Kreisel and Tait [196l]) Every finitely definable function is recursive. Proof. Suppose f is finitely definable by & w.r.t. t h e n fl(Z) = Z is a logical consequence of 4

f1.

We know t h a t if f(2)= z

for some Zl, . . . , Zp. By t h e completeness of t h e predicate calculus, a n d t h e fact t h a t whenever we have a model we also have a n w-model (i.e. one with domain t h e integers, a n d with 0 a n d S interpreted as zero a n d successor), t h i s is equivalent of saying t h a t if f(2)= z then f1 ( T ) = Z is derivable in a n y complete formalization of t h e predicate calculus with equality, from t h e premises 4

a n d t h e axioms for t h e successor operation:

(with t h e f l , . . . ,fm held fixed in t h e derivation). By arithmetization we can define a primitive recursive predicate 7,(e,Z,y) (where n is t h e number of components of t h e vector )’. meaning: 4

y codes a derivation of a n equation of t h e form fi(Z)= Z,in t h e predicate calculus with equality, from t h e axioms for successor a n d a finite conjunction of substitution instances of t h e system of equations coded by e. Let U be a primitive recursive function such t h a t whenever y codes a derivation, t h e n U ( y ) gives t h e value of t h e numeral on t h e right-hand side of t h e layt equation coded by y.

98

I. Recursiveness and Computability

Then we have that f is recursive, because

fF)= u(PY%(e,%Y)).

0

Proposition 1.7.6 (Kleene [1936]) Every Herbrand-Godel computable function is recursive. Proof. By arithmetization we can define 7,(e,z1,. sive, meaning:

. . ,xCn,y)primitive recur-

y codes a derivation, by means of the rules R1 a n d R2 and from the system of equations coded by e , of a n equation of t h e form f,'"(Zl,. . . ,z), = Z, where f? is t h e leftmost letter in t h e last equation of t h e system coded by e. Let

U be a primitive recursive function such that if y codes a derivation then U(y) is t h e value of t h e numeral in the right-hand side of the last equation coded by y.

If f is Herbrand-Gdel computable from the system of equations coded by e then f is recursive, because f ( x 1 , .. . ,x,) = U(pLy%(e,zl,. . . ,x , , ~ ) ) .

0

We turn now t o t h e representability approach of Section 3.

Proposition 1.7.7 (Godel [1936], Church[1936]) Every function weakly representable in a consistent formal system (with recursive sets of axioms and of recursive rules) is recursive. Proof. By arithmetization we can define 7,(e,x1,. . . ,z,,y) primitive recursive, meaning: y codes a derivation of a sentence of t h e form $ ( T I , . .. ,Z, Z), where $ is the formula coded by e, from the axioms of t h e given system and by means of its rules. Let U be a primitive recursive function such t h a t if y codes a derivation then U(y) is t h e value of t h e numeral which instantiates the last variable of the last formula of the derivation coded by y.

If f is weakly representable by the formula coded by e then f is recursive, because 0 f ( z 1 , .. . ,x,) = U ( P Y z ( e , x l , .. . ,Z,,Y)).

99

1.7 Arit hmetization

Corollary 1.7.8 For any consistent formal system extending R,the following arc equivalent: 1. f is weakly representable

2. f is representable 3. f is strongly representable.

Proof. Indeed, recursive =+ strongly representable =+ representable =+ weakly representable recursive

(by (by (by (by

1.3.6) definition) consistency) t h e proposition).

0

W e now t u r n to t h e computational approaches of Sections 4 a n d 5 .

Proposition 1.7.9 (Turing [1936], [1937]) Every Turing machine computable function is recursive. Proof. By arithmetization we can define 7 , ( e , z l , . sive, meaning:

. . ,z,,y)

primitive recur-

y codes a computation carried o u t by t h e Turing machine coded by

el on inputs

51,.

. . , 2,.

Let U be a primitive recursive function such t h a t if y codes a computation then U ( y ) is t h e value of t h e number written on t h e t a p e to t h e left of t h e head, in t h e last configuration of t h e computation coded by y.

I f f is computed by t h e Turing machine coded by e t h e n f is recursive, because .f(x11..

. ,G)

= U ( W ; T , ( ~ , X..I ~ x n , ~ ) ) . 0

Proposition 1.7.10 (Wang [1957], Peter [1959]) Every flowchart computable function is recursive. Proof. To simplify t h e details we can make t h e following conventions: any program has only variables named X O ,X I , . . .; t h e inputs a r e indicated by X I , .. . ,X,, a n d t h e o u t p u t by X O . By arithmetization we can define a primitive recursive predicate 7 , ( e , 5 1 , . . . ,x,, y), meaning: y codes a computation of t h e program coded by e when, at t h e beginning, t h e input variables a r e set equal to 5 1 , . . . ,z, a n d all t h e remaining variables a r e set equal to 0.

I. Recursiveness and Computability

lo0

Let, U be a priiiiitivc rccursive function siich t h a t

if y codrs a c.onil~iit.iitioii,t , I i c i r U(,y) Ls the value of tlic. varinble, in t,lie last step of the compiitatioii c.od~xlby y .

I f f is coniputecl

I)y t h e program c d e d by f(.l,.

I‘

oiit,piit

then J is recursive, becaiisc:

0 . . ,:En) = U(IL!/75I(P,.Tl,~. . ,GI ,!I)).

Wc filially t u r n to the X-defiiial)ility approach o f Sedioii G .

Proposition 1.7.11 (Church [1936], Kleene [1936b]) Ehcry A-dcJfiiiahlr: /iinc!ioii % s recursive.

Proof. I3y arit,Iiirrctizatioii we sive, rricmiiiiig:

caii

define 7 ; , ( c , z l , . . . , x T L , y primitivc ) recur-

y cotles a rculiictiori t,o a i i i i i i w r a l , via cr or /jr i i l t s , o f t h e t,erin codctl hy c , applictl to Z1, . . . ,x,.

I d , U I)(! a primitive rccursive

fiiric.t.ioir

such that

if y (.odes a rcxluctioii theii U ( y ) is the valiie of the iiii~ncralobtairietl in its last stc:p.

I f f is A-tlefinable by t h e tcrrir cotltxl 1)y e

t,heii

f is recursivc, brcausc

The basic result of the foiindations of Recursion Theory I’lie rc:siilt,s provctl in this first part, o f the Iwok iiirply t,hat all tlic a p p r o d i e s t,o offertive c.i)nipiital)ility irit,rotliic.etl SO far are equivalent, and thus sliow that /.he iiotioii of rf:ciirsiiii:ii~ss is absolute and very stable. ‘I‘his is a stxiking fact,, and t o s t r c w i1.s irriportance we isolate it in a thcorern o f its o w n , which c a p t u r e s thc esseii(~:o f t,liis chapter: anew tlirb, perch6 tii veggi pura

la vrritk rhe Ih giu si corifonde, fatta I e t t u r a (Dantc, Paradzso, XXIX)

q i i i v o ( w i ( 1 o i i i si

Theorem 1.7.12 Basic result. 7’hc folloiiriiig arc: cqiiiimlmt: ‘ I shall say more, so that you may s w clear ~y the truth that, t l i w p lwlow, has lwei1 cr)rifiised by tcachirrg that rriay I w arnbigiicius.

‘

1.8 Church’s Thesis

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101

1. f is recursive 2. f is finitely definable

3. f is Herbrand-Godel computable

4. f

is representable in a consistent formal system extending R

5. f is %ring computable

6. f is flowchart (or ‘while’) computable

7. f is A-definable.

Proof. T h e direction showing t h a t all these notions a r e not more extensive t h a n recursiveness is given by t h e results j u s t proved in t h i s section. T h e opposite direction, showing t h a t all these notions a r e at least as extensive as recursiveness, is given by Theorems 1.2.3, 1.2.5, 1.3.6, 1.4.3, 1.5.4a n d 1.6.5.

0

I t should be noted t h a t the equivalence proofs among different notions of computability are effective and eficient. Effectiveness means t h a t for a n y pair of notions t h e r e is a recursive function t h a t , given t h e code of a recursive function relative to one notion, produces a code of t h e s a m e recursive function relative to t h e other notion. T h i s is t h e basis of t h e definition of acceptable system of indices, see 11.5.2. A precise statement of efficiency requires concepts introduced in Chapter VII, a n d will be given there. T h e intuitive idea is t h a t t h e code of a function not only defines t h e function, b u t also shows a method t o compute it, a n d t h e translation roughly preserves t h e computational efficiency of t h e methods.

1.8

Church’s Thesis *

In t h i s section we discuss t h e assertion t h a t every effectively computable function is recursive, by considering physical a n d biological computers. For t h e former we rely on physical theory, a n d t r y first to determine how far t h e particular model of determinism provided by recursiveness accounts for t h e general model of determinism, a n d then to establish t h e extent of determinism itself. For t h e latter, d u e to lack of theory, we pursue t h e synthetical a n d analytical approaches, b y analyzing t h e brain structure a n d formulations of constructive reasoning. D u e to t h e generality of t h e discussion, we will freely quote results which either will be proved later in t h e book, or will not be proved at all, being outside t h e scope of our work. However, appropriate references will be given, whenever needed.

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Introduction to Church’s Thesis T h e work done so far shows t h a t t h e class of recursive functions is certainly a very basic one, since it arises in fields as varied as mathematics, logic, computer science a n d linguistics, with quite independent approaches ( a n d each one interesting in its own right), t h a t t u r n o u t to be equivalent a posteriori (by t h e Basic Result 1.7.12). T h e generality of t h e method of arithrnetization, t h a t allows for these equivalence results, also leads us to believe t h a t o t h e r possible approaches to t h e notion of computability a r e likely to produce notions not more extensive t h a n recursiveness, if not outright equivalent. T h e fact t h a t many variations in t h e details of t h e various approaches do not produce changes in t h e defined class (see e.g. t h e discussion on p. 49), shows t h a t t h e notion of recursiveness is very stable. By Theorem 1.7.12,t h e class of recursive functions is not sensitive to changes in t h e formal systems considered to represent its functions: t h a t is, t h e s a m e functions a r e representable in any consistent formal system having a least minimal power, independently of t h e system strength. A n d even more is true: Kreisel [1972] shows t h a t riot only in formal systems, b u t even in vast classes of recursive transfinite progressions of formal systems, only recursive functions a r e representable. T h u s t h e notion is absolute in a certainly astonishing way, with few (if any) analogues among other logical notions. To q u o t e Godel [1946]: With this concept o n e has for t h e first t i m e succeeded in giving a n absolute definition of a n interesting epistemological notion, i.e. o n e not depending on t h e formalism chosen. In all other cases treated previously, such as demonstrability or definability, o n e h a s been able to define t h e m only relative to a given language, a n d for each individual language i t is clear t h a t t h e one t h u s obtained is not t h e one looked for. For t h e concept of computability however, although it is merely a special kind of demonstrability or definability, t h e situation is different. Giidel referred t o t h e situatioii as ‘a kind of miracle’. These facts point out t h e exceptional importance of t h e class of recursive functions, a n d have led (see t h e ncxt subsection for historical notes) to propose t h e following as a working hypothesis:

Church’s Thesis (Church [1936], Turing [1936]) Every eflectively computable function is recursive. T h e Thesis, if true, would have a great relevance as a piece of applied philosophy, since it imposes a precise, mathematical upper b o u n d to t h e vague, intuitive b u t basic notion of algorithm t h a t underlies t h e concept of effective

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computability, an d t h a t has permeated technique and mathematical experience for thousands of years. Post [1944] emphasizes t hat if general recursive functions is t h e formal equivalent of effective calculability, its formalization may play a role in t he history of combinatorial mathematics second only t o t hat of t he formulation of t h e concept of natural number. In applications, t h e Thesis has a n essential use in metamathematics. Limiting t h e extension of t h e concept of algorithm allows for proofs of absolute u n s o lv a b ility : if we prove t h a t a function is not recursive then, by t h e Thesis, it is not computable by any effective means. Thus, t o see t hat a problem is effectively unsolvable, is enough t o faithfully translate it into a function, a n d prove t h a t this is not recursive. Like t h e classical unsolvability proofs, these proofs are of unsolvability by means of given instruments. What is new is t h a t in t h e present case these instruments, in effect, seem t o be t h e only instruments at man’s disposal. (Post [1944]) T h u s undecidability proofs rest on two conceptually different bases: a mathematical proof of recursive unsolvability (independent of the Thesis), and a n

appeal to the Thesis, to deduce from it absolute unsolvability. There is another avoidable use of the Thesis, in Recursion Theory. Giving a n algorithm for a function amounts, by t he Thesis, t o showing t h a t this function is recursive. Although theoretically not important, and in principle always avoidable (if t h e Thesis is true), this use is often quite convenient, since it avoids t h e need for producing a precise recursive definition of a function (which might be cumbersome in details). Strictly speaking, however, this use does not even require a Thesis: it is just a n expression of a general preference, widespread in mathematics, for informal (more intelligible) arguments, whenever their formalization appears t o be straightforward, and not particularly informative. We will d o this (and have already done it) throughout t h e book. T h e meaning of t he above formulation of Church’s Thesis is ambiguous in at least two respects. First of all, t h e statement can be taken as saying t h a t each effectively computable function is extensionally equivalent t o a recursive one or, more strongly, th at every effective rule is intensionally equivalent to, say, some program for a n idealized computer. Following Kreisel [1971]we will distinguish t h e two meanings, and refer t o them, respectively, as Thesis and Superthesis. Second, an d this is t he crux of t h e matter, there are various possible meanings for t h e word ‘effective’, partly depending on one’s philosophy of mathematics. ECxtremal attitudes ar e possible. One (Church [1936]) is t o take recursiveness as a precise definition of t he otherwise vague notion of effective computability: this makes t h e Thesis empty. An opposite one (Kalmar [1959]) is

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t o consider effective computability as an open concept, t h a t can only be successively approximated: this only allows for partial verifications of t h e Thesis, relative to given approximations (although it would allow for a disproval of it). T h e popular attitude is, however, to consider t h e facts t h a t various attempts t o characterize t he notion of effectiveness have all led to t h e sam e class of functions, and th at no counterexample to t he Thesis has ever been found, as conclusive arguments in favor of it, a n d to regard t h e matter as (positively) settled. Kreisel [1972] has stressed t h e fact t h a t equivalence of at t em pt s is not particularly significant (there might be systematic errors), a n d t h a t only t h e intrinsic values of each model can be relevant (one good reason is better t han many bad ones). Church’s Thesis can be analyzed from various points of view, some of which dealt with in th e special issue of t h e Notre Dame Journal of Formal Logic (vol. 28, no. 4, 1987) on t h e subject. T h e task we set for ourselves in this section is t o analyze some meanings of t h e word ‘effective’, a n d t o discuss t h e (lack of) evidence of t h e Thesis for these meanings. Precisely, we consider physical and biological computers. A physical computer, as described here, is a discrete physical system together with a theory for its behavior (according to which t h e values are under experimental control). We restrict our attention to discrete systems because we are considering discrete functions (from natural numbers to natural numbers), although continuous systems can be treated via approximations (see below). T h e fact th at we have a theory (physical laws) to work with, is what makes t h e Thesis in this case less pretentious, therefore less simpleminded, than in t h e original intended meaning (considered afterwards): it allows us t o compare a n abstract model of computability with descriptions of classes of physical devices. Obviously we d o not question here t h e validity of t h e world description in terms of (present day) physical laws: t h e relevance of our discussion will be proportional t o t h e degree of confidence we have in it. Since Turing machines are locally deterministic devices, t o ask whether any physical computer computes only recursive functions actually splits into two questions: it means first t o determine how far a particular model of determinism accounts for all of it, then t o establish the extent of determinism itself. Clearly t h e former is less problematic, and it therefore produces a more satisfactory analysis. For th e biological computer, we d o not have yet a theory, and discussions of human computability are mostly rambling talk. We pursue bot h t h e synthetical (bottom-up) and the analytical (top-down) approaches, by analyzing t h e brain structure and theories of constructive reasoning, but we reach a dead end soon in both cases. Before we plunge into our work, we would like t o warn about what effective computability does not mean: it is not practical (feasible) computability. T h e relationship between these two notions is t he distinction between Recursion

1.8 Church’s Thesis

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Theory a n d Computer Science, i.e. between ideal computers a n d real ones. T h e issue in Computer Science is not Church’s Thesis (whether t h e class of recursive functions is broad enough), b u t its dual (which recursive functions a r e practically computable): from this t h e a t t e m p t to define restricted versions of recursiveness, like polynomial-time computability (considered in C h a p t e r VIII). Actually, from a strict point of view, practical computability is not even interested in asymptotical behavior, a n d it will never use infinitely many values. I n t h i s respect even t h e a t t e m p t to restrict t h e class of recursive functions to other abstract models may be irrelevant.

Historical remarks Post was working, at t h e beginning of t h e Twenties, toward a general formulation of t h e undecidability results h e h a d obtained. He defined t h e notion of canonical systems (p. 143) as a n abstraction of t h e notion of formal s y s t e m , a n d proposed (on t h e basis of reductions h e had of known formal systems to canonical ones) t h e identification of t h e notions of a set of strings effectively generable on o n e side, a n d generable by canonical systems on t h e other. T h i s is equivalent to saying, in modern terminology, t h a t t h e effectively generable sets a r e t h e recursively enumerable ones, a n d it is t h u s indirectly equivalent to Church’s Thesis (for partial functions). However, Post had a Platonist philosophical view, a n d saw his proposal as something t h a t h a d to be proved somehow, by a kind of psychological analysis of t h e mental processes involved in combinatorial niathematical processes. I n particular, h e believed t h a t t h e analysis he had at t h e moment was ‘fundamentally weak’, a n d t h u s t h a t t h e proposal was not completely convincing. All this work (Post [1922]) was left unpublished, a n d so did not influence later developments. At t h e beginning of t h e thirties Church formulated t h e A-calculus, in a foundational a t t e m p t to develop a system of logic from t h e primitive notion of function (Section 6 ) . I t gradually turned o u t (in 1932-33) t h a t there was a natural way to represent integers in A-notation, a n d t h a t a great number of functions were A-definable (ultimately t h a t all t h e recursive ones were, Theorem 1.6.5). I n 1934 Church proposed his Thesis (Church [1936]),as a mathematical definition of t h e informal concept of computability. Meanwhile Godel, dissatisfied with Church’s approach, believed (somewhat following Hilbert [ 19261) t h a t t h e computable functions could all be defined by some general kind of recursion. This again turns o u t to be equivalent to Church’s Thesis, through t h e Fixed-Point Theorem. Godel [1934] even vent u r e d to forniulate t h e notion of Herbrand-Godel computability (Section 2) as

I. Recursiveness and Computability

106

a test, b u t was not at all convinced t h a t this concept really comprises all possible recursion. His proposal was similar to Post’s: to analyze t h e notion of computability, aiming at the isolation of its essential features. T h i s was done (in a way satisfactory to Godel) b y Turing [1936], who unaware of t h e work referred t o above - proposed his model of a n a b s t r a c t computer (Section 4), a n d a n equivalent version of Church’s Thesis. Simultaneously a n d independently, Post [1936] attained a very similar analysis a n d stated t h a t a fundamental discovery in t h e limitations of t h e mathematicizing power of Homo Sapiens has been made. G d e l thought otherwise (see Davis [1965],p. 73): T h e results mentioned . . . do not establish any b o u n d s for t h e power of human reason, b u t rather for t h e potentiality of p u r e formalism in mathematics.

For more information on t h e history of Church’s Thesis see Kleene [1981], [198la], [1987], Davis [1982], Shanker [1987], Webb [1980] a n d t h e original papers in Davis (19651.

Computers and physics T h e notion of a deterministic reality t h a t evolves according to mathematically explicit laws is typical of classical mechanics. Galilei [1623], [1638] introduces t h e modern scientific methodology of experimenting in order to verify t h e results of theoretical reasoning, a n d stresses t h e importance of mathematics (‘the language t h e book of nature is written in’). Newton [1687]achieves a n informal axiomatization of mechanics, for t h e first t i m e unifying large t r a c t s of experience into a coherent, picture. W i t h him t h e mechanization of t h e world picture (see Dijksterhuis [1961] for a n historical account) is accomplished: a system with k degrees of freedom needs only 2k parameters (positions a n d moments) to completely specify every value of physical quantity for t h e system at a given t i m e a n d t h e evolution in t i m e of t h e system s t a t e . If s o m e of t h e parameters a r e unknown, by averaging over t h e m in s o m e way it is still possible to obtain statistical prevision (like in thermodynamics, where the hidden 2k parameters needed to describe a system of k molecules produce a statistical description in terms of pressure a n d temperature alone). Plank’s discovery in 1900 of energy packets ignited a new physics (quantum mechanics, see Feynman, Leighton a n d Sands [1963] for background), with philosophical foundations as distant from those of classical mechanics as they can be. T h e concept itself of reality is at stake: m a t t e r h a s a double

1.8 Church’s Thesis

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appearance] as waves and as particles (Einstein, De Broglie), and its physical quantities cannot in general be simultaneously measured with absolute precision (Heisenberg). A system with Ic degrees of freedom is now described by a wave function \ k ( q l , . .. ,q k ) , which still evolves deterministically in time (Schrodinger), but allows only statistical prevision on the values of physical quantities for the system at a given time (Born). This last fact can be variously interpreted as necessary (change at subatomic level is casual, and can only be accounted for probabilistically), accidental (as in t h e case of thermodynamics, hidden variables might deterministically account for change, a n d give quantum-mechanics states as average), or contingent (the wave function represents not only possibilities, but realities in simultaneously coexisting worlds: a measurement, forcing a possibility into a n actuality, corresponds t o choosing a path in t h e tree of all possible universes, see DeWitt a n d Graham [1973]).

Classical mechanics T h e first aspect t h a t we examine of Church’s Thesis can be phrased as follows: t h e notion of recursiveness (a technical isolation of a restricted class of mechanical processes) captures the essence of mechanism. We can formulate, more precisely:

Thesis M (for ‘mechanical’) (Kreisel [1965])The behavior of any discrete physical system evolving according to local mechanical laws is recursive. This clearly implies our real interest: t h a t any function computable by such a device is recursive as well (each output being obtained by a finite iteration of

a recursive procedure applied t o t h e input). T h e Thesis formulation in terms of behavior of physical systems, in addition t o being more general, has t h e advantage of being directly suitable for analysis (since we d o not need t o know details on how the device computes a function). T h e Turing-Post analysis of Section 4 is certainly not sufficient to prove Thesis M since, being explicitly patterned on human behavior, it sees computations as well-ordered sequences of atomic steps, and thus (at least) it does not account for parallel computations. Arguments in favor of Thesis M fall into three distinct categories] which we analyze separately. a) A general theory of discrete, deterministic devices T h e analysis (Church [1957], Kolmogorov and Uspenskii (19581, Gandy [1980]) starts from the assumptions of atomism and relativity. T h e former reduces t h e structure of matter t o a finite set of basic particles of bounded dimensions] a n d thus justifies t h e theoretical possibility of dismantling a machine down t o

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a set of basic constituents. T h e latter imposes a n upper bound ( t h e speed of light) on t h e propagation speed of causal changes, a n d t h u s justifies t h e theoretical possibility of reducing t h e causal effect produced in a n instant t on

a bounded region of space V , to actions produced by t h e region whose points a r e within distance c . t from some point of V . Of course, t h e assumptions do not take into account systems which a r e continuous, o r which allow unbounded action-at-a-distance (like Newtonian gravitational systems). Gandy’s analysis shows t h a t the behavior is recursive, for a n y device with a fixed bound on the complexity of its possible configurations (in t h e sense t h a t both t h e levels of conceptual build-up from constituents, a n d t h e number of constituents in any structured part of a n y configuration, a r e bounded), and jixed finite, deterministic sets of instructions for local and global action ( t h e former telling how to determine t h e effect of action on structured parts, t h e latter how to assemble t h e local effects). Moreover, t h e analysis is optimal in t h e sense t h a t , when made precise, any relaxing of conditions becomes compatible with any behavior, a n d it thus provides a sufficient a n d necessary description of recursive behavior.

b) Numerical approximations of the local differentiable equations of classical mechanics T h e work in classical mechanics, from Newton to Hamilton, has led to a description of t h e evolution of mechanical systems by local differentiable equations. More precisely, a conservative Hamiltonian system is defined, in local coordinates, by Hamilton’s equations:

where q = ( 4 1 , . . . , q k ) a n d p = ( p l , . . . , p k ) a r e t h e vectors of, respectively, positions a n d momenta of t h e system, k being t h e number of degrees of freedom of t h e system. T h e n t h e evolution in continuous t i m e of t h e system state s (completely describing t h e relevant variables of t h e system) can be expressed by a vector differential equation of t h e form S = f(s). By assuming sufficient smoothness conditions on t h e derivative involved, a n d stepping from continuous to discrete time (in which t h e evolution of t h e system is sampled at regular, sufficiently small, discrete time intervals) we can linearly approximate t h e r a t e of change given by t h e previous equation, as

s(t

+ A t ) N s ( t ) + f ( s ( t ) ). A t .

By taking A t as t h e unit interval of sampling, we get

s(t

+ 1) = s(t) + f ( s ( t ) )

1.8 Church’s Thesis

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a n d t h i s gives, for mechanical (not necessarily discrete) systems, a recursively described system evolution (in t h e form of a simultaneous recursive definition of all t h e relevant variables explicit in s) (Kreisel [1965]). Note t h a t t h e other, equivalent, way t o describe t h e evolution of systems in classical mechanics, namely by global variational principles (like Maupertuis’ principle of least action), does not, seem to b e useful for a similar analysis, because of i t s teleological approach (see Von Neumann [1954]).

c) Discrete models for classical mechanics I n classical mechanics discrete data (coming froni experiments) a r e used to build continuous models, from which discrete data have to be deduced by numerical approximation methods. T h e s t e p from discrete to discrete through a continuous model seems logically unbalanced: as Feynman [1982] p u t s it,

It is really true, somehow, t h a t t h e physical world is representable in a discretized way, a n d . . . we a r e going to have to change t h e laws of physics.

Discrete models, studying the dynamical behavior of systems entirely in terms of (high speed) arithmetic, have been obtained for classical mechanics, including special relativity a n d conservative Ilamiltonian theory (Greenspan [1973], [1980], [1982], L a Budde [1980]). Their dynamical equations a r e difference (opposed to differential) equations, whose solutions a r e discrete functions. T h i s approach still yields various conservation a n d synimetry laws of continuous mechanics, a n d it also has direct applications to non-linear physical behavior. Related t o this, cellular automata have been investigated as a basis

for the representation of partial differential equation models in a direct computer simulation, again avoiding indirect numerical approximation (Vichniac [198,4],Toffoli [1984]). See also (3rd-Smith a n d Stephenson [1975] for a general t r e a t m e n t of computer simulation of continuous systems.

To s u m u p t h e discussion above, it is plausible that the behavior of a discrete physical system, evolving according to the local and causal laws of classical mechanics, can be simulated by a computer, and it is thus, in particular, recursive.

Probabilistic physics We t r y now to formulate Church’s Thesis for abstract machines, in t h e most general way. We will have to account for analog computers, t h a t is any physical system computing some function, by representing numerical data ‘by analogy’ (based on any physical, a n d possibly continuous, quantity, like intensity of a n electrical current, or rotation angles of a watch hand). More

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I. Recursiveness and Computability

precisely, a n analogue computat,ion is a combination of physical processes, behaving (mathematically) in t h e same way as some other process, which is t h e real object of study, b u t which for some reason is more manageable or b e t t e r observable t h a n it (e.g., because of difference in scale). In t h e e x t r e m e case, a n y physical process is a n analog calculation of i t s o w n b e h a ~ i o r . ~We t h u s formulate:

Thesis P (for ‘probabilistic’) (Kreisel [1965]) A n y possible behavior of a discrete physical s y s t e m (according t o present d a y physical theory) i s recursive. Possible behavior means a sequence of states with non-zero probability: we cannot simply talk of behavior according to present d a y physical theory, because this (as opposed t o classical mechanics) is formulated also in ternls of probability. We collect our observations under categories parallel to those used for classical mechanics.

a) A general theory of analog machines Nothing similar in spirit to t h e theory of Gandy [I9801 for discrete deterministic devices has yet been developed. A first step has been undertaken by Shannon (19411, who generalized t h e iiotion of finite a u t o m a t o n into t h a t of general purpose analog computer. This is a device consisting of electronic circuits, a n d a series of black-boxes (hooked u p with lots of instantaneous feedback), of four elementary kinds: constant (producing any desired constant voltage), adder a n d multiplier (producing sum a n d product of t h e inputs), a n d integrator u(s)dv(s) C , where C is t h e (producing, given inputs u a n d v , t h e o u t p u t ‘initial setting’ of t h e integrator). Once t h e connections a n d t h e initial settings a r e made, t h e device is permitted to r u n in real time, a n d a n y voltage t h a t can be read in t h e circuit (as a function of time) is a n o u t p u t . A characterization of t h e behavior of these devices has been obtained (Shan11011 [1941], Pour El [1974], Lipschitz a n d Rubel [1987]): a f u n c t i o n f(x) is t h e output of a general purpose analog computer i f a n d o n l y if it is differentially algebraic, i.e. t h e solution of a n algebraic differential equation

s,’

+

where P is a polynomial (over t h e complex field) in all its variables. Such functions provide a n extremely rich class, including almost all t h e special functions 51n this case, Church’s Thesis amounts to saying that the universe is, or at least can be simulated by, a computer. This is reminiscent of similar tentatives to assimilate nature to the most sophisticated amilable machine, like the mechanical clock in the 17th Century, and the heat engine in the 19th Century, and it might soon appear to be as simplistic.

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in common use (algebraic, trigonometric, Bessel functions), and with strong closure properties (see Rubel [1982], Rubel and Singer [1985]), including the existence of analogues of the universal function (Rubel [1981]). Also, despite t h e fact t h a t some transcendental functions (notably, Euler’s r and Riemann’s 6) a r e not exact solutions of algebraic differential equations, any continuous function can be approximated, with arbitrary preassigned accuracy, by differential algebraic functions. This approach thus describes a wide.variety of physical phenomena, but it is still only a first step toward a general theory of analog computers. Some extensions have been recently proposed, e.g. allowing black-boxes for convolution (which would add memory t o t h e device, since convolutions involve t h e whole past history of their inputs and introduce time delays). Digital simulations with arbitrary precision (which are our real concern, since we talk about functions of integral, not real or complex, variables) should be possible, by replacing t h e black boxes by digital approximations t o them (e.g., integration can be performed by some appropriate numerical integration, say via Simpson’s rule), b u t details have not yet been worked out.

b) Analysis of the formulation of probabilistic physical laws T h e dynamics of classical physical systems with probabilistic behavior may be described by Markov chains, which consist of a finite set of states { q l , . . . ,q n } , together with a n x n stochastic matrix P = ( p i j ) 1 5 , , j j n , whose interpretation is: if t h e system is in a given state qi at a certain instant of time, t h e probability t h a t it be in state qj at t h e next instant (in a discrete time scale) is p i j . A system described by a Markov chain satisfies Thesis P, and actually something more general holds: any sequence of states with non-zero probability, in a stochastic

process with infinitely m a n y discrete states and recursive matrix of transition probabilities, i s recursive (Kreisel [1970a]). T h e r e a o n is simply t h a t such a sequence is a n isolated branch of a finitely branching recursive tree (the tree of possible sequences of states), since there are only finitely many possible sequences of a given non-zero probability. T h e remark just made teaches a more general lesson. Suppose we consider a structurally stable system, i.e. such t h a t slight changes of t h e parameters in t h e equations describing t h e system behavior produce only slight changes in t h e behavior itself. T h e stability of the solutions tends t o require t h a t they be isolated in t h e relevant spaces, and if these spaces are recursively described, t h e solutions a r e recursive as well. Thus it is likely that i f Thesis P fails,

counterexamples have to be looked f o r in unstable systems. It can be noted t h a t it is known t h a t some differential equations in recursive analysis (of t h e kind arising in the description of physical phenomena) have recursive data b u t no recursive solutions (see e.g. Pour El and Richards [1979],

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[1981], [1983]). These results are, however, not directly relevant to Thesis P, since their data a r e mathematically concocted, a n d do not apparently arise from t h e description of physical phenomena (see Kreisel (19821 for a review).

c ) Deterministic models of quantum mechanics We have already noted above t h a t quantum mechanics is not deterministic, as it stands. Hidden-variables theories were postulated to leave o p e n t h e possibility of a deterministic description of subatomic phenomena: their existence would prove quantum mechanics observably inadequate, b u t at t h e s a m e t i m e q u a n t u m theory - although incomplete - could be complemented to obtain a full description of individual systems. Impossibility proofs of t h e existence of hidden-variables theories h a d been proposed, from Von Neumann [1932] on, b u t with unsatisfactory features analyzed in Bell [1966]. A breakthrough was Bell [1964]: h e proved that realism and hidden variables are n o t only philosophically, but also theoretically incompatible with quantum theory. He devised (in t h e style of Einstein, Podolski a n d Rosen [1935]) a simple experiment, a n d computed probabilistic lower b o u n d s to t h e outcome predictions, assuming t h a t well-defined states really exist, prior to their observation. This bound is greater t h a n t h e o n e obtained by q u a n t u m theory considerations. Recently (1981-82) t h e experiment h a s been actually carried o u t , a n d seemingly conclusive evidence provided t h a t the quantistic predictions are correct (see Mermin [1985] for a n elementary description, a n d references). T h i s moves t h e incompatibility of realism with q u a n t u m theory from philosophical a n d theoretical ground t o t h e experimental one, a n d seems to settle t h e matter. At first sight it might seem impossible to simulate Bell’s experiment deterministically, since t h e theoretical outcome predictions would clash with experimental evidence, b u t we should not forget t h a t these predictions are obtained by using a particular kind of inductive inference based, in particular, on classical probability theory. Now t h e s a m e inference theory is used in q u a n t u m mechanics, a n d this flatly produces its incompatibility with determinism. But, as Feynman [1982] points o u t , there could be a problem with probability theory itself, at quantum level: we assume t h a t we can always do a n d repeat a n y experiment t h a t we want, without taking into account t h e constrictions (stressed by q u a n t u m thcory!) imposed by t h e fact t h a t we a r e all p a r t of t h e s a m e universe, a n d t h a t t h e universe does not remain t h e same. O n t h e o t h e r h a n d , we do not know of any version of Bell’s experiment t h a t avoids probabilistic computations. Bwides probability, logic is t h e other tool used in t h e inductive inference of Bell’s theorem, a n d classical logic itself seems to be inadequate to describe phenomena at quantuin level. See e.g. Birkoff a n d Von Neumann [1936], where

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i t is argued t h a t t h e experimental propositions concerning a system in classical mechanics form a Boolean algebra, while (due to t h e fact t h a t only compatible observations commute, a n d incompatible observations cannot be independently performed) t h e y a r e a complemented b u t nondistributive lattice in quantum mechanics.

To sum u p t h e discussion, we have only scarce evidence in favor of Thesis P and, despite the fact that n o outright refutation exists, there is plenty of room t o doubt its validity.

Computers and thought T h e reduction of soul to (atomistic) physics has a n old pre-Socratic tradition, centering around Leucippus a n d Democritus. T h e Socratic revolution, a n d t h e standing success of Plat0 a n d Aristotle, has led to a tradition of organismic physics t h a t has left little space for pure mechanism until more recent times. A notable exception was Lucretius, who devoted two books of his De Rerum Natura to a n atomistic account of t h e mind a n d its functions. Descartes [1637] laid t h e foundation of t h e modern mechanistic world view, by trying to devitalize t h e human organism as much as was logically possible, in particular by including into physics (as he envisaged it) a great deal of what later c a m e to be called psychology, but not t h e mind itself. He saw self-consciousness a n d language sophistication (in particular, t h e ability to see t h e meaning of signs a n d events) as t h e privilege a n d exclusive ability of a n immaterial, unextended mind. Hobbes [1655] provided t h e dissenting note: by relying on t h e apparently effective manipulation, in reasoning, of names as symbols for thoughts, a n d on Pascal’s construction of t h e first calculating machine in 1645, he defended a global mechanism, a n d did not hesitate to obliterate t h e difference between mind a n d matter. In his extreme dedication to mechanism, Hobbes was a rather lonely figure in his day, b u t with t h e advent and t h e success of machines, it was inevitable t h a t mechanism would attract more advocates. L a Mettrie [1748] provided a most notorious a t t e m p t , aimed at wholesale reductionism. Once t h e terms of t h e debate had been set up, endless arguments developed, a n d new life to t h e dispute has been provided by modern advances in t h e areas t h a t supported Hobbes a n d Descartes, respectively. On t h e one hand, much current mathematical practice has been formalized (from Boole t o Bourbaki), a n d thus indeed mechanized (and Godel Completeness Theorem [1930] shows t h a t , for what concerns first-order logic, t h e formalization is complete). M o r e over, t h e quality of computers (from Babbage to t h e Fifth Generation) has improved enormously, a n d machines a r e now capable of quite sophisticated behavior themselves. O n t h e other hand, t h e undecidability a n d incompleteness

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results (Section 11.2) expose the limitations of t h e formalization a n d mechanization programs. Moreover, these results have sometimes been used t o infer the superiority of men over machines, basically with t h e following arguments:

Undecidability. Church’s Theorem 11.2.18 shows t h a t first-order logical reasoning is not mechanically decidable. It might thus appear t h a t man, t h e bearer of this reasoning, is capable of nonmechanical behavior, and thus not a machine. It is easy t o see t h e weak point of this argument: a mechanism is such because of local mechanical behavior, and the Completeness Theorem for Predicate Calculus (Godel [1930]) does indeed show t h a t classical logical reasoning can be formalized, and thus simulated by 1ocally mechanical steps. But a mechanism does not need t o have a global, mechanically predictable behavior, as discussed on p. 151. Undefinability. Tarski’s Theorem (see p. 166) shows that, for a classical formal system, truth is not representable in it. Again, it would seem t h a t man has a notion of truth, and thus that thought has nonmechanical elements. The difficulty of this argument is two-fold: on t h e one hand, it is only global truth that it is not representable, while for each fixed bound of logical complexity there is a representable notion of truth (at t h e next level of complexity); on t h e other hand, not only does man not appear to have a global notion of truth: he even seems unable to handle (by direct intuition) more than four or five alternations of quantifiers, and hence local truth itself, beyond very small levels of complexity. Incompleteness. Godel’s Theorem 11.2.17 tells us that any consistent and sufficiently strong formal system is incomplete, in the sense t h a t it does not prove some numerical sentence which we know is true. It would seem that man, being able to produce, for any machine (formal system), a task that can be solved by him but not by t h e machine, is not himself a machine. The first objection t o this is t h a t the proof of t h e result is effective, i.e. there is a machine that, given the number of a formal system, produces the undecidable sentence (this effectiveness is actually one of t h e crucial features of Godel’s result, since - by showing t h e incompleteness of every sufficiently strong formal system - it points t o inadequacies in t h e concept of formal system itself). Post [1922] has noted that such a n effectiveness does not show up by chance: given an argument intended to prove that man can fool any machine, if this argument can be made sufficiently precise, then it becomes itself mechanizable, a n d it backfires. Another important feature of Godel’s proof is that t h e undecidable sentence is shown to be true only under the hypothesis of t h e consistency of

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the system. Certainly the problem of deciding t h e consistency of given formal systems is not recursively solvable, but there is no hint that man has a decision procedure for such a problem (and hence for t h e truth of the undecidable sentence relative to a given consistent system). If it were so, this would be a direct refutation of mechanism. Finally, although Godel’s Theorem does not by itself refute mechanism, it does so when combined with t h e assumptions that there are no numbertheoretical questions undecidable for the human mind, and that the mind is somehow consistent. But this does not solve the problem, it just moves it t o a different level. Also, although the second assumption seems quite reasonable, t h e first one is certainly more problematic and controversial, even if understood (as it should be) as saying t h a t the human mind has no limitation regarding the problems it poses itself in the limit (given enough time and resources), and taking the words ‘deciding a question’ as meaning ‘settling the problem’, possibly by showing it t o be unsolvable.

See the papers in Anderson [1964], and Hofstadter and Dennett [1981] for some discussions on the relevance of these topics for t h e mind-brain debate, and Popper and Eccles [1977] for a modern philosophical and neurological introduction to the latter.

The brain Our first global approach to Church’s Thesis for human thought is to look at t h e brain, the physical basis of intelligence. We begin by discussing:

Superthesis B (for ‘brain’) (Descartes (16371)The brain is a machine.

It should be stressed that the statement is not t o be taken as reducing t h e brain complexity t o the roughness of present-day machines: a proof of Superthesis B would probably revolutionize the contemporary idea of machine, and precisely in this lies its interest. In particular, Turing (see p. 164) has stressed the significance of the limitation results (Section 11.2) in showing how a purely deterministic model of machine cannot fully account for intelligence. It is, however, instructive t o compare the brain and the most sophisticated available machine, the computer6 (for information on t h e brain see Von Neu‘History teaches us that this should not be taken too literally: Descartes 116641 saw the brain as a complex hydraulic system, permitting the periodic flow of vital spirits from the central reservoir into the muscles; Pearson [1892]described it as a telephone exchange, consisting of fixed wires and mobile switches (a model that proved useful for an understanding of spinal reflex response); Ashby [1952]provided a cybernetic model as a collection of selfcontrolling systems. The computer model might look as simplistic in the not too distant future.

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mann [1958], Arbib [1964], [1972], Eccles [1973], Young [1978], Kandel and Schwartz [1981]). Computers are electromagnetic devices with fixed wiring between more or less linearly connected elements, operating mostly sequentially, a n d at high speed. Brains are dynamical electrochemical organs with extensively branched connections, operating with massive parallel action at a slow speed and low energetic cost, continuously capable of generating new elements, a n d perhaps making new connections. T h e architectural differences a r e great: see e.g. Haddon a n d Lamola (19851 for a survey of t h e technical foreseeable advances regarding chip dimensions. T h e holistic logic employed by t h e brain is simply o u t of reach: we d o not know how it concentrates on essential information and experiences it as structured. We are thus forced t o suspend judgement on the validity of Superthesis B , until enough might be known on these problems. If ever, since it is certainly conceivable (La Mettrie [1748], Von Neumann [1951]) t h a t , due to t h e extreme complexity involved, a linguistic (mechanical) description of t h e cerebral functions might be simply unfeasible or uninforming: t h a t is, t h e system itself could be its own most intelligible description (in t he terminology of p. 151, the brain

could be a random object). On th e positive side there are some results worth mentioning which show t h e mechanical behavior of some simplified neuron nets. As a first approximation t o t h e great complexity of natural neuron systems, McCulloch and Pi t t s [1943] introduce regularity assumptions for artificial neurons: they are infallible all-or-nothing devices with fixed synaptic threshold, firing synchronously at discrete intervals (when t he algebraic sum of t h e adjacent neurons effects reaches th e threshold). T h e behavior of an isolated system of artificial neurons is completely characterized by t h e input conditions, a n d t h e system t hen works as a n abstract machine (actually, this is simply a n equivalent description, and t h e original one, of finite automata, see p. 52). Since t h e control box of a Turing machine can be regarded as a finite automaton, it can thus be seen as a n

abstract brain of the Turing machine. W e thus have two complementary analyses: Turing’s analytical (top-down) approach describes the functioning of the computing device, without further analyzing the way it is actually built, while McCulloch and Pitts’s synthetical (bottom-up) analysis shows how t o obtain the same functioning b y organizing, in a possibly very complex way, simple parts o f described structure. Much work has been done toward a relaxation of t he restrictive assumptions on artificial neuron nets. Von Neumann [1956], a n d Winograd a n d Cowan [1963] consider systems in which t he unreliability of the components does not affect t h e reliability of t h e whole net, by transmitting t h e same information in a highly redundant way, along multiple parallel lines or, respectively, blocks of components. Hebb [1949] and Eccles [1953] permit variable synaptic thresholds: their systems have feedback information, by which they can

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somehow realize whether t h e o u t p u t s a r e conforming to t h e expectations. By trials a n d errors it is possible to gather sufficient information a n d determine thresholds t h a t give t h e expected o u t p u t , t h a t is t h e systems have t h e ability to learn. T h i s represents short-term memory by t h e variable states of t h e system, a n d long-term memory by t h e level of t h e synaptic thresholds. Hopfield [1982], [1984] has considered very general neural networks, with backward coupling (where neurons can a c t indirectly, through other neurons, on themselves), asynchronous firing, graded continuous response (in t h e form of a sigmoid input-output relation, as opposed to a 0,l-valued step function) a n d integrative t i m e delays (due to capacitance). He has shown t h a t (if t h e connect,ions between neurons a r e symmetric) these general networks a r e still computational devices, since a n y set of inputs leads to stable states. T h i s provides a model of content-addressable memory, where a stable state represents memorized information, t h a t can be retrieved by setting t h e right input t h a t would lead to it. Various other models have been proposed, see e.g. Arbib (19731, Bienestock, Fogelman SouliC a n d Wekbuch [1985], Selverston [1985], McClelland a n d Rumelhart [ 19861, a n d recent issues of Bzological Cybernetics. T h e models j u s t discussed a r e all digital (based on neuron nets), a n d have been obviously inspired by t h e structure of digital computers (finite a u t o m a t a , in particular). B u t it is known t h a t p a r t s of t h e central nervous system function analogically, e.g. many neurons never fire, a n d are engaged in different activities (see Rakic [1975], Shepherd [1979], Roberts a n d Bush [1981], Crick a n d Asanuma [1986]). To be closer t o reality, t h e digital model of t h e brain should t h u s be supplemented, a n d substituted by a hybrid one, partly digital a n d partly analog. A natural approach seems to be t h e use of t h e general purpose analogue computer of p. 110 (Rubel (19851): neurons or neuron-circuits t h a t perform t h e functions of t h e black-boxes have been already identified, a n d t h u s at least t h e basic components of t h e general purpose analog computer a r e present in t h e central nervous system. It is only fair to note t h a t (some of) t h e results quoted might be more relevant, to t h e problem of whether machines can think (in t h e operative sense, introduced by Turing [1950], of being able to simulate aspects t h a t we believe to be characteristic of thought) t h a n to our discussion of Superthesis B. I t is not clear whether t h e models proposed above really describe t h e brain’s own s e lutions to problems of unreliability, learning, memorization a n d organization of information. IIowever, they a r e certainly relevant to t h e following ‘Promethean irreverence’:

Thesis A1 (for ‘Artificial Intelligence’) (Wiener [1948], Turing [1950])T h e mental functzons can be simulated by machines.

All work in Artificial Intelligence (pattern recognition, language reproduction, problem solving, theorem proving, game playing, learning a n d under-

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standing) produces inductive evidence for Thesis AI. Note t h a t Thesis A1 is not simply t h e extensional version of Superthesis B. T h e step from t h e latter to t h e former is not automatic: it requires psyche logical materialism, in t h e form (La Mettrie [1748]) of human t h o u g h t being completely determined by t h e brain, with no intervention of a n extraphysical mind. But, stepping down to t h e simulation level introduced by Thesis AI, we a r e not interested - in our discussion of Church’s Thesis - in i t s full version, since our present concern is just mathematical thought.

Constructivism We t h u s isolate our real interest in:

Thesis C (for ‘constructive’) (Kleene [1943], Beth [1947]) Any constructive function is recursive. In view of t h e discussion in t h e last p a r t of t h e previous subsection, even establishing Superthesis B would not automatically establish Thesis C. Conversely, failure o f Thesis C would not disprove Superthesis B. Also, a failure of Thesis C docs rio t, disprove rriaterialism, unless ( t h e extensional version of) Superthesis B holds simultaneously. Actually, establishing b o t h Superthesis B a n d Thesis A1 is probably as far as we can go toward a possible justification of ( t h e above form of) materialism. It is olwiously impossible to disprove t h e existence of mind without using Ockham razor, i.e. beyond showing its unusefulness in explaining thought activities. O n t h e other hand, as Godel has suggested (Wang [1974], p. 326), i t might be possible to disprove mechanism by showing t h a t there is not sufficient s t r u c t u r e (at nerve level) to perform all tasks actually performed by m a n . We t u r n now to a discussion of Thesis C. A first step h a s already been carried o u t by Turing a n d Post, with a n analysis of routine computations (Section 4). T h i s provides, at t h e s a m e time, more a n d less of w h a t we need: it gives a n intensional argument, b u t it concerns only a portion of t h e intended meaning of ‘constructive’. We can however say t h a t in t h e limited ront,ext to which it applies, t h i s analysis is conclusive W e isolate what is proved in t h e following:

Superthesis R (for ‘routine’) (Turing [1936], Post [1936]) Any computation perfonnea‘ b y a n abstract human being working i n a routine way, is isomorphic to a computation performed b y a Tiiring machine. But t h i s is still a far cry from Thesis C. We cannot rely on t h e analysis of mcchanical reasoning given by Turing machines: constructive and mechanical

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are apparently independent concepts. It seems t h a t we understand (by grasping abstract objects) nonmechanical rules, a n d do not understand rules which (although mechanical) a r e too long or too detailed. We t h u s have to lean on analyses of t h e notion of mathematical constructive reasoning. T h e r e a r e many of t h e m , based on different approaches. Following t h e detailed treatments of (and, for more information, referring to) Godel [1958] a n d Kreisel [1965], [1966], we j u s t hint at t h e basic features of t h e most popular, increasingly more comprehensive ones: formalism (Frege, Russell) considers as constructive what can be done on physical objects (symbols), by purely combinatorial (hence mechanical) means (formal rules of derivation for symbols sequences).

finitism (Hilbert) accepts also what can be seen by p u r e intuition, provided only concrete (spatio-temporal) objects a r e used, a n d c l a i m t h a t any thinking process of t h i s kind must be finite (though not necessarily mechanical).

intuitionism (Brouwer) allows for whatever is mentally understandable, possibly using also abstract objects (like higher-type objects, or generalized inductive definitions). Nonconstructive reasoning enters only into platonism, which regards mathematical objects not as thoughts b u t as real objects, t h a t t h e mental process does not create, b u t only discovers. Their properties a r e t h u s perfectly defined as those of physical objects, a n d this justifies use of, for example, t e r t i u m non d a t u r a n d actual infinity. Formalism is directly related to Thesis C . On t h e o n e hand, t h e very formalistic program (of compressing mathematical knowledge into formal systems) rests on t h e belief t h a t some form of t h e Thesis holds ( t h a t t h i s knowledge can be mechanically reproduced); on t h e other hand, since everything computable in formal systems is recursive (by arithmetization, see I.7.7), each success of formalism is partial proof of t h e Thesis validity. Thesis C i s true when constructive is taken in the finitistic meaning as well: indeed, a finitistically defined arithmetical function is certainly (as shown by a n analysis of computations) finitely defined by a system of equations (Section a), a n d hence recursive (by 1.7.12). T h u s the whole problem of Thesis C lies in the intuitionistic meaning of constructive (law-like) function. Since allowing for abstract objects (which a r e not necessarily finitely representable) might make arithmetization of t h e involved mental processes troublesome, we will concentrate our discussion on formal systems capturing aspects of t h e intuitionistic (constructive) reasoning. T h e fact t h a t formal systems usually capture semantical notions of reasoning only extensionally is not important here, since t h e Thesis is precisely extensional.

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T h e technical advantage of formal systems is not only a m a t t e r of convenience (of having a syntactical, concrete version for semantical, a b s t r a c t notions): it could be instrumental in outright proving Thesis C. For this, since we know t h a t only recursive functions a r e representable in formal systems (by arithmetization), it would be enough to find a formal system equivalent to constwctiwe arithmetical intuition (by Kreisel [1972], even a recursive transfinite progression of formal systems would suffice). T h i s is certainly a delicate point: such a n equivalence proof might require unfamiliar principles of evidence, a n d would certainly provide a better insight into t h e notion of constructive validity. O n t h e other hand, failure of Thesis C would show t h e unfeasibility of t h i s reductionist program. Of course, no consistent formal system can be arithmetically complete in t h e classical sense, by 11.2.17 (and t h e s a m e holds for recursive progressions of formal systems, by Feferman a n d Spector [1962]). B u t this is not relevant to t h e reductionist program, since we do not expect constructive intuition to be itself classically complete: t h e problem would be to succeed in deriving w h a t is constructively valid, not to decide (let alone constructively) everything.' T h e realization of this reductionist program does not a p p e a r easy, also in light of a result of Kreisel (19621, [1965], by which Thesis C implies t h a t t h e set of constructively valid formulas of first-order logic is not recursively enumerable. In particular, i f Thesis C holds then there is no formal system capturing constructive logical validity: thus, if a formal system capturing constructive arithmetic validity exists, it cannot be obtained by j u s t extending (by means of arithmetical axioms) a logical system t h a t can be detached from it by recursive means. We t h u s have t h e amusing situation t h a t , in t h e process of searching for a complete formalization of constructive arithmetical reasoning, we might begin by a formulation of t h e purely logical constructive reasoning, a n d discover t h a t we lose t h e war by overwinning a battle: if we a r e completely successful with t h e logical formalization, t h e n Thesis C does not hold, a n d we a r e bound to fail in t h e arithmetical fornialization. Also, a n d t h i s is a situation with no analogue in classical mathematics, constructive validity for first-order logical formulas somehow depends o n what the constructive arithmetical functions are (in particular, on their being or not all recursive). Otherwise said, first-order constructive validity is actually a second-order notion. Short of proving Thesis C by t h e reductionist program, we may consider related questions, technically more manageable b u t , as we will see, more moderately interesting. We isolate two of t h e m . Given a n intuitionistic formal system F,we might see whether in F t h e recursive functions provide uniformization, in t h e sense of 11.1.13. T h i s is 'Note that, as G a e l himself has admitted (see Wang 119741, p. 324), it niight even be possible to find, or have already found, a formal system equivalent to full, not only constructive, mathematical intuition, although of mume, in this case, not provably so.

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expressed in a weak form by t h e following rule:

Church's Rule CR. If FF Vz 3 y R ( z ,y ) then, f o r some recursive function f and all z, FF R(?E,f(z)). By Kreisel [1972], C R is actually equivalent to t h e following:

Constructive 3-Rule. FF 3ycp(y)

* FF cp(?j), f o r some y .

To be s u r e (Kreisel [1972]), there a r e ad hoc intuitionistic systems for which CR fails. B u t t h i s does not automatically disprove Thesis C: it could merely be a s y m p t o m of incompleteness, since Vzcp(z, I(. might )) hold for s o m e recursive

f , b u t we might not be able to prove even its numerical instances in F. O n t h e other h a n d , no intuitionistic system is known to be incowistent with CR, something t h a t would disprove Thesis C . Moreover, for all current intuitionistic systems CR has actually been established, even in t h e stronger form:

t - ~V d y R ( z , y )

t-7 3eVz3z['&( e ,z , z ) A R(z,U(z))]

(see e.g. Kleene [1945], Kreisel a n d Troelstra [1970]). T h i s is however only a very indirect evidence in favor of Thesis C: it merely excludes t h e inconsistency of CR with these systems, a n d it thus shows t h a t they cannot be used to disprove t h e Thesis. We might be tempted (on t h e acceptable argument t h a t constructive validity of a n existential statement should exhibit explicit witnesses) to consider only those systems for which t h e Constructive 3-Rule holds. B u t , since we know t h a t t h e r e must be incompleteness (for any sufficiently strong arithmetical system, see I1.2.17), there is no reason to expect it to show u p necessarily soniewhere else t h a n in numerical instantiations of existential theorems. Only a formal system complete for constructive rawoning would automatically satisfy t h e Constructive 3-Rule (but, then not only C R would hold: Thesis C would indeed be true). Another property at least formally related to Thesis C , is its formal version:

CT1 VJf3eV'z3z['& ( e ,z, 2) A f(z)= U(z)] CT2 V z 3 y R ( r c , y ) 4 3eVz3z['&(e,z,z) A R(z,U(z))]. T h e former tells, via t h e Normal Form Theorem, t h a t every function is recursive a n d is suitable for second-order systems with functional variables. T h e latter is t h e axiom of choice (extracting a function from a V3 form), plus t h e fact t h a t every function is recursive a n d is also suitable for first-order systems. Of course, b o t h forms a r e false in usual classical systems, a n d t h u s CTl and CT2 are not provable in usual intuitionistic systems (in which t h e corresponding classical systems a r e interpretable).

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T h e relevance of t h e two principles t o our discussion is quite feeble. Even if we can disprove one of them in a formal system for constructive mathematics, this would not disprove Thesis C : it would simply mean t h a t it is absurd t h a t all functions can be proved recursive in t h e system (not t h a t some functions a r e not recursive). Kripke has given a formalization of Brouwer’s theory of the creative subject, and has shown t h a t it implies the negation of C T . However, for all current intuitionistic systems (not involving the concept of choice sequence) the consistency with C T has actually been established (see e.g. Kleene [1945], Kreisel and Troelstra [1970]). Once again this is not evidence in favor of Thesis C, not even indirect (as it was for CR): indeed, even a proof of C T would just show that every function we can talk about in the system is recursive and, once again, this would be interesting only for a system complete for constructive reasoning (since these functions would then be all t h e constructive functions). T h e reader will find more information on t h e philosophical analysis and (proofs of) t h e technical results of this subsection in Kreisel [1970], [1972], Troelstra [1973] and McCarty [1987].

To sum up, t h e arguments for Thesis C point out how it could be proved by a formal analysis of constructive reasoning (reductionist program), a n d disproved by showing - for any acceptable constructive arithmetical formal system - t h e inconsistency of closure under Church’s Rule. Both validity a n d failure of Thesis C have interesting consequences for constructive mathematics. Except for these methodological remarks, we have collected only very weak, and

certainly inconclusive, evidence in favor of Thesis C, whose validity must be retained as unproved (which is after all not surprising, since we do not even fully understand it: we still have only a partial grasp of what ‘constructive’ means).

Conclusion To recapitulate our discussion, recursiveness seems to be a model of discrete, deterministic processes general enough t o account for mechanical phenomena, according to classical physics. T h e notion certainly reaches beyond this, e.g. it takes care of probabilistic phenomena described by Markov’s chains, and of a wide variety of structurally stable systems. But we have no positive results, and actually some positive doubts, for what concerns subatomic phenomena governed by quantum mechanics. Turning t o biological computers, only very rough simplifications allow us t o look at the brain as a kind of machine, and we a r e still far from a coniplete theory. T h e analysis of human computations a n d reasoning produces a recursive description only under assumptions of routinnes a n d formal (at most finitistic) manipulation of symbols, respectively. It is a n open problem whether

1.8 Church’s Thesis *

123

constructive reasoning in intuitionistic sense is recursive. Despite t h e weak evidence for some of them, t h e various theses have been proposed not out of empire-builder rashness (with t h e tacit ambition of convincing, short of proving, t h a t recursiveness is somehow a universally permeating concept), b u t rather out of experimenters circumspection (with t h e manifest hope of understanding t h e exact limits of t h e notion). T h e validity of Church’s Thesis (presently proved to some, but certainly not full, extent) is not what would give importance to Recursive Function Theory, although undoubtedly it adds to it (to t h e extent it holds). T h e notion of recursiveness has more than sufficient motivations (reviewed in t h e introduction to this section) to deserve a thorough mathematical study, disregarding its - certainly fascinating connections with mechanism, neurophysiology, and constructivism. But, independently of its practical relevance, work along the line of this section has a n abstract importance. To quote Kreisel [1970]: T h e principal interest is philosophical: not to confine oneself to what is necessary for (current) practice, but t o see what is possible by way of theoretical analysis.

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Chapter I1

Basic Recursion Theory T h i s chapter contains t h e core of Recursion Theory, a n d introduces its basic notions, methods a n d results. We s t a r t , in Section 1, with a n extension of t h e not,ion of recursiveness, by dropping a weak point in t h e various definitions of Chapter I ( t h e request, not effectively verifiable, of totality for a n algorithm). T h i s leads to t h e class of partial recursive functions a n d their set-theoretical counterparts, t h e recursively enumerable sets. T h e elementary properties of these functions a n d sets a r e explored throughout t h e chapter, while a deeper structural analysis will begin in Chapter 111, a n d continue in Volume 11. T w o fundamental tools for nontrivial results are t h e method of diagonalization a n d t h e notion of degree. T h e former, one of t h e innovative inventions of Cantor, is a n extremely helpful technique which h a s become, in various disguises, a pervasive element of Recursion Theory. In section 2 we introduce t h e fundamentals of t h e method, including a codified version of it called t h e Fixed-Point Theorem. T h i s is a powerful a n d somewhat mysterious result which underlies t h e famous undecidability results of t h e Thirties, also treated a n d discussed in Section 2. T h e notion of degree is introduced in Section 3, which is devoted to relative computability as opposed t o t h e absolute computability dealt with so far. We generalize computations t h a t c a n be performed solely by machines, a n d allow t h e machine to stop, from time to time, a n d ask questions. T h e model still describes real computations, b u t t h e machine is not autonomous anymore, a n d may rely on interactions with t h e external world ( t h a t is, also during t h e computation a n d not only, as previously, in t h e input-output activity). T h e distinction between absolute a n d relative computations is t h e one between fully automatic a n d interactive (man-machine) behavior of computers, t h e latter being t h e common practice in sophisticated (not purely computational) projects, e.g in automatic theorem provers, or in Artificial Intelligence tasks. T h e deci-

125

126

11. Basic Recursion Theory

sion to relax t h e autononiy of t h e machine still leaves various possibilities open in terms of t h e amount a n d t h e structure of t h e interaction with t h e outer world. In Section 3 we deal with Turing computability, t h e most general a n d fundamental case, imposing no limitation on t h e help given to t h e machine, except for a n obvious finiteness requirement. Other, more restrictive, notions of relative computability will be introduced in Chapter 111. A fundamental property of partial recursive functions is t h e possibility of enumerating their programs in a n effective way, a n d t h u s of assigning indices to t h e m , according to their place in t h e enumeration. Indices t h u s code descriptions of partial recursive functions, a n d can be used to refer to a function in a n oblique (intensional) way. Section 4 deals with t h e effective operations t h a t can be defined intensionally on t h e (partial) recursive functions (by working o n their indices), a n d their relations with t h e partial recursive functionals, which a r e their exteiisiorial analogues (working directly on functions). Section 5 considers various topics connected with indices. T h e results of this chapter collectively show t h a t the class of partial recursive funct.ions is very comprehensive as a result of i t s striking closure properties. First of all, t h e universal partial function (Theorem 11.1.8) provides a descriptional closure. Second, t h e recursive functions a r e closed under recursive diagonalization, with a two-fold escape from contradiction: for t o t a l recursive functions t h e r e is n o universal function (Theorem 11.2. l ) , hence a n y recursive class of total recursive functions is not exhaustive, a n d diagonalization j u s t produces another recursive function, which is not in t h e given class; for partial recursive functions, diagonalization simply produces particular undefined values (see p. 152). Finally, a n d this accounts for t h e name of t h e class, t h e Recursion Theorems 11.2.10 a n d 11.3.15 ensure closure under recursion of any kind (where ‘recursion’ can be taken to mean, in its greatest generality, t h e definition of a function in t e r m s o f itself a n d of known functions).

11.1

Partial Recursive Functions

We have introduced in Chapter I various independent approaches to t,he notion of effective cornputabilit,y, arid t h e methods of Section 1.7 showed t h e m to be all equivalent. We might t h u s be quite satisfied, b u t t h e r e is still a point t h a t seems a bit o u t of tune: we have been longing for a precise notion of effecLive computable function, a n d all our definitions have a strongly noneffective

element in t h e m , namely t h e infinitary restriction t h a t we consider devices computing only total functions. Having a device potentially computing a function, we did not accept it as an algorithm until we h a d somehow recognized t h a t it p r o d u c e answers for any input: since it is possible to prove (see p. 116) that, t,his cannot be done in general by any recursive means, t h e class of recursive

11.1 Partial Recursive Functions

127

functions seems to depend on something external to it, and it is even conceivable that it depends on the methods of proof allowed for the recognition of the totality of an algorithm. All this might sound quite discouraging, but the final solution to the problem of characterizing effective procedures is at, hand: we only have to set a missing brick, and the construction will be completed. Since it is the verifkation of totality that troubles us, we simply decide to drop it.

The notion of partial function A partial function is simply a function that may be undefined for some (and possibly all) arguments. The set of arguments for which it is defined is called its domain. Of course a partial function is total on its domain, but here we give a privileged status to the set of natural numbers, and consider a function whose domain is properly included in w as only partially defined. The step from total t o partial functions should be appreciated: it was a longstanding philosophical position that there cannot be precise logical laws for propositions about incompletely defined objects, from Aristotle (Metaph@ca, r 7, 1012a, 21-24) to this century. It was probably Brouwer [1919] (see also [1927]) who first corrected this position with his work on choice sequences. We use Greek letters to indicate partial functions, and an extended equality relation,',' meaning that both sides are equal as partial functions (i.e. their respective values are either both undefined, or both defined and with the same value). Also, ~ ( z5-)means that cp is defined (also said: it converges) for the arguments 2, while cp(.') t means the opposite (also said: it diverges). Finally, partial functions can be partially ordered by the inclusion relation naturally defined as: cy

G p e \dz[a(z)1 =+

p ( x ) = a(.)].

Thiis whenever a is defined so is p, and with the same value.

Partial recursive functions We adapt definition 1.1.7 to partial functions:

Definition 11.1.1 (Kleene [1938]) The class of partial recursive functions is the smallest class of functions 1. containing the initial functions 0, S and 2. closed under composition, i. e. the schema that given yl, . . . ,y7,,, $ produces 4 2 )= $ J ( T l ( q , . . . ? Y m ( q ) ,

11. Basic Recursion Theory

128

where the left-hand side is undefined when at least one of the values of y l , . . . ,ymr11) for the given arguments is undefined 3. closed under primitive recursion, i.e. the schema that given @, y produces Cp(.’,O)

V(CY

+(Z) -d.’>Y,Cp(Z,Y))

2:

+ 1) =

4. closed under unrestricted p-recursion, i.e. the schema that given

I) pro-

duces cp(.’)

where

~p(.‘)

= pLY[(V. I Y ) ( I ) ( C

2)

1)A @(ZIY)= 01,

is undefined i f there is no such y.

At first sight, we may think t o define t h e p-recursion schema as: Cp)’(.

r?P Y ( + ( C Y )

= 0).

T h i s would mean t,o look for t h e least, y such t h a t +(Z,y) 2

2:

0 a n d , for every


).

1 =+ I)(.‘, ).

$0

(thus allowing for +(Z, z)T), b u t i t is unacceptable for various reasons. First, on a computational ground: to discover whether a given z has t h e stated property we can only compute +(Z, z ) , a n d since t h e computation gives a n answer only if it converges] this brings us back to t h e original proposal. Second, there is no recursive method to decide whether a partial recursive function converges (see 11.2.7)] a n d t h u s t h e schema just proposed would have t h e s a m e flaw of t h e regularity condition for t h e (restricted) p-recursion. I n particular it would again give rise to a notion which is not self-contained. Finally, even after t h e facts t h e proposal does not work: the partial recursive functions are not closed

under the schema Cp(4

= PY(+(KY) 2: 0)

(Kleene [1952]). This is easy to see, using later results. Let A be a n r.e. nonrecursive set (11.2.3)) a n d define: +(z,y) - 0 0

Then

$J

(?j = O

AZ E

A) V

~= J 1

is partial recursive (II.1.11), b u t if

f(z)= PY(%Y)

= 0)

t h e n f (total) is not partial recursive, otherwise it would be recursive (by I1.1.3), a n d since f(z)= 0 H z E A

11.1 Partial Recursive Functions

129

so would be A . T h e next theorem shows t hat we get an equivalent definition of partial re-

-

cursive function i f we consider the schema p(q

PYR(2, Y)

with R recursive relation (where p(2) is undefined if there is no y such t h a t R(2,y) holds). In terms of later definitions and results, t h e counterexample just given above shows t h a t the partial recursive functions are not closed under the same schema, with R recursively enumerable.

Theorem 11.1.2 Normal Form Theorem for partial recursive functions (Kleene [1938]) There is a primitive recursive function U and (for each n 2 1) primitive recursive predicates 7,, such that for every partial recursive function p of n variables there is a number e (called index o f p) for which the following hold: 1.

~ ( ~ 1 1. ,. x. n )

2-

'~("1,.

1

3 y % ( e , z l , . ..j z n , ~ )

f . 7 X n ) " U ( P Y % ( ~ , ~... I ,> X ~ , Y ) ) -

Proof. T h e proof of Theorem 1.7.3 shows exactly this, by using ( 5 , a ) as t h e index associated t o

p(?) - p Y [ ( v z I Y ) ( ~ ( 2 , z ) l ) A 1 C I ( Z , y ) -01, when a is associated t o $J. Note t hat t he computation tree in t h e case of p-recursion uses indeed only t h e values +(.', z ) for z 5 y. 0

Corollary 11.1.3 The recursive functions are exactly the partial recursive functions which happen to be total. Proof. Obviously, a recursive function is partial recursive and total. Conversely, if p is partial recursive then, for some e ,

= U(pv;T,(e,%. . . zn,Y)). If p is total then Vzl . . .Vx,(p(sl,. . . , z n )I), hence by t h e theorem p(z1,. . . ,%)

1

V z l . . .Vzn3Y%(eIz1,.. . ,zn,Y) a n d 7 n ( e 1zl,. . . ,x,, y ) is regular. Then p is recursive by 1.1.7.

0

By referring t o t h e corollary there can be no confusion when talking of

total recursive functions, meaning recursive functions as in definition I. 1.7, or partial recursive functions which ar e total.

11. Basic Recursion Theory

130

T h e Normal Form Theorem says t h a t every partial recursive function has a n index, b u t t h i s was t r u e for t h e recnrsive functions as well. T h e advantage given by t h e introduction of partial functions is t h a t we can now invert t h e thcoreni, a n d consider every number e as t h e index o f t h e partial recursive function

W @ d e , ~ , ... ,G,Y)), since t h e condition of regularity for t h e application of t h e p-operator h a s been dropped. T h i s will be applied throughout t h e book, a n d we c a n t h e n set up a spccial notation:

Definition 11.1.4 1.

pi

2.

Q:,~

(or (e)")

is the e.-th partial recursive function of n variables:

(or (el,") is the finite approsirnation o f 'pz of level s:

Intuitively, p;,, may be thought of as t h e approximation to p ' : obtained by considering t h e computat,ion of 'pz,a n d cutting it at step s. Note t h a t if 'p:,+(.') 1 t h e n , by t h e properties of t h e coding functions a n d t h e fact t h a t R computation codes everything relevant, inputs a n d o u t p u t must be l e s t h a n s. For simplicity of notations we will drop the indication of the number of variables when this is either n o t important or understood, arid j u s t write qe a n d pe,+in t h a t case. We can now state t h e symmetric version of t h e Normal Form Theorem:

Theorem 11.1.5 Enumeration Theorem (Post [1922],Turing [1936], Kleene [1938])The sequence {'p:}eew is a partial recursive enumeration o f the n-ary partial recursive functions, in the sense that: 1. for each e ,

2. i f

p ';

is a partial recursive function of n variables

+ is a partial recursive function of n variables, then there is e such that

+=

'pz

3. there is a partial recursive function

'p

of n

+ 1 variables such that

11.1 Partial Recursive Functions

131

Proof. Everything follows from t h e Normal Form Theorem for partial recursive functions, a n d t h e definition of 9:. It is enough to let

cp(e,4 =U(PYZ(e,Z,Y)).

0

T h e Enumeration Theorem exposes t h e basw double role of numbers an Recursaon Theory. a p a r t from its intended a n d natural meaning (as a number), a number also has a hidden, second-level meaning as a code of a function. T h i s is t h e basis of t h e self-referential phenomena underlying t h e results of Section 2 (see, in particular, p. 165), a n d it also produces a natural interpretation of X-calculus (see p. 223). W e give now two basic properties related to indices, a n d refer t h e reader to Section 5 for more results on t h e subject.

Proposition 11.1.6 Padding Lemma. Gwen one andex of a partaal recursave functaon, we can effectavely generate anfinitely many other andaces of the same functaon. Proof. Given a n index e for cp as a partial recursive function, we get infinitely many others by attaching to t h e description coded by e any finite number of redundant equations. 0

If we fix a certain number of variables in a partial recursive function $J, we still get a partial recursive function y of t h e remaining variables. Moreover, given a program for $, we can effectively get a program for y. T h e next theorem says t h a t t h i s can be done uniformly in t h e fixed variables. Proposition 11.1.7 Sr-Theorem (Kleene [1938]) Given m,n there is a primitive recursive, one-one function S r ( e, X I , . . . ,x,) such that (Ps,-(e,z,,...,z,)(Yl,.. . 7

~ m )

pe(Xlr.. . , z ~ , Y I. ~ . .

Proof. Suppose we have a description (coded by e ) of a partial recursive function @(XI,. . . ,z,,yl,. .. ,ym). We want from it a description of t h e function defined as

Y(Y1 , . . . ,Ym)

= $(X1

, . . . , X,,

Y11.

..

3

Ym ).

We might think to use t h e description coded by e followed by t h e above equation, b u t t h e n y would be ambiguously defined (depending on t h e values of x i , . . . ,x, which appear in its definition). W h a t we want instead is to define a function for each fixed value of x i , . . . ,x,. But then, instead, we must use t h e constant O-ary functions C,, , . . . , CZn,corresponding to these values. T h u s we have to find an index of t h e function whose description is t h e one coded by e , followed by t h e equation

Y(Yl,...,Ym)= ~ ( C . 2 , , . . . , C Z , l Y l , . . - l Y m ) .

132

11. Basic Recursion Theory

T h i s is now a good description of y,depending uniformly on e a n d t h e values given to X I , . . . , z T L Its. index is t h u s a function S z ( e , z1,. . . , z m ) ,a n d can be made primitive recursive by t h e method of arithmetization used in Section 1.7. It is one-one by its definition a n d t h e properties of t h e coding functions. 0 T h e SF-Theorem (also called t h e Parametrization Theorem, or t h e Iteration Theorem) looks, at first sight, innocuous, a n d simply a p p e a r s to be stating t h a t data can be effectively incorporated into a program. B u t we should not forget t h e fundamental double role of numbers in Recursion Theory: data can code programs themselves, a n d t h u s incorporating t h e m into a program may have t h e effect t h a t t h e program interprets t h e m as subprograms. T h u s the S;r-Theorem actually embodies a notion of subcomputation and a n eflective versaon of composation. In a precise sense, enumeration a n d paranietrization a r e inverse translations, a n d provide t h e technical tools needed t o handle t h e basic duality of numbers: by enumeration a n index can be considered as a n a r g u m e n t , a n d by parametrization a n argument can be considered as a n index. T h i s explains their fundamental role, analyzed in Section 5. I t should be noted t h a t all t h e other approaches of C h a p t e r I could be similarly adapted to t h e treatment of partial functions by dropping t h e totality requirements. T h u s we could consider partial functions computed by Turing machines, a n d say t h a t for given inputs a Turing machine computes a value if it halts in t h e prescribed way, a n d it does not otherwise (i.e. if it does not halt, or it does, b u t not in t h e prescribed way). Flowchart programs, Herbrand-Gijdel computability, A-definability, a n d so on a r e treated similarly. As it was t h e case for total functions, t h e various approaches remain equivalent for partial functions as well, with similar proofs. T h u s the class of partial recursive functions

retains the absoluteness and stability of the class of recursive functions, and it has the extra quality of admitting an intrinsic definition, without reference t o nonconstriictive notions.

Universal Turing machines and computers

*

T h e Enumeration Theorem admits a stronger formulation, d u e to t h e uniformities of t h e definition of 7, w.r.t. 71.

Theorem 11.1.8 Universal Partial Function (Post [1922], Turing [1936], Kleene [1938]) There is a partial recursive function cp(e,z), called universal partial function, which generates all the partial recursive funct i o n s of any number of variables, in the sense that fo r every partial recursive

11.1 Partial Recursive Functions function J I of n variables there is

e

133

such that

Proof. W i t h notations as in t h e proof of 1.7.3,let

a n d recall t h a t , by definition,

T h e n , for every n,

Any Turing machine computing a universal partial function cp is called a universal Turing machine. Since cp is partial recursive, universal Turing machines exist by 1.4.3. T h i s is a n indirect proof, a n d explicit constructions of universal Turing machines a r e in Turing [1936], Wang [1957a] a n d in many textbooks, e.g. H e r n i a (19651, Minsky [1967], Arbib [1969], Hopcroft a n d UIIm a n [1979]. More information on t h e topic is in Davis [1956], [1957], Shannon (19561, Rogers [1967] a n d Priese [1979]. T h e interest of t h e notion is t h a t a universal Tu7.ing machine is a computer in the modern s e w e of the word, and it works as an interpreter, decoding t h e program e given to it as data (in t h e s a m e form as t h e other inputs) a n d simulating it. In other words, a universal Turing machine is not a specialpurpose machine: it is instead programmable in essence, a n d t h u s all-purpose. I n particular, all universal Turing machines a r e equivalent in power, a n d they differ only in speed a n d efficiency. Conversely, any o f the present-day automatic electronic computers (if abstracted from physical malfunctioning) is equivalent to a universal 7hring machine, i f it is given the possibility of having a potentially infinite memory ( t h a t is, of always being able to add more memory units, a n d have access to t h e units already used). In addition to unlimited memory, t h e only necessary properties of a universal Turing machine a r e t h e abilities of performing coding a n d decoding operations (which enable t h e machine to read t h e instructions of a given Turing machine o u t of its index), a n d simulation. Once this level of complexity is reached, t h e machine can perform tasks more complicated t h a n those for which it was directly built (actually, any possible task perforniable by Turing machines): t h e needed complication may be turned over to t h e software, a n d does not need to be built-in. Thus, modulo a universal 7hring machine,

hardware and software are interchangeable.

11. Basic Recursion Theory

134

T h e realization t h a t a machine could be universal-purpose, by being able to simulate other machines through their programs, antedates b o t h Recursion Theory a n d Computer Science: it goes back to Babbage’s [1837] conception of t h e analytical engine. T h i s crucial notion appears natural nowadays, b u t it did not always look so. T h i s was t h e case not only in Babbage’s time, b u t even after Turing’s abstract development, a n d well into t h e process of building real computers: see e.g. Hodges [1983] for a n account of t h e resistance Turing h a d to face in his own computer project, against (in his words) t h e tradition of solving one’s difirulties by means of much equipment rather t h a n by thought which meant a privilege of hardware a n d special-purpose machines over software.

Recursively enumerable sets Having looked at t h e notion of partial recursive function, we t u r n now to its analogue in terms of sets a n d relations. For total recursive functions we h a d no doubts: t h e analogues were just those sets a n d relations whose characteristic functions were recursive. Rut since a characteristic function is always t o t a l , partial recursive characteristic functions would again give t h e recursive sets, by 11.1.3. Natural sets associated to part,ial functions a r e their domains.

Definition 11.1.9 (Post[1922], Kleene [1936]) A n n-ary relation is recursively enumerable (abbreviated r.e.) if it is the domain of a n 11-ary partial recursive function. W e indicate b y W: and W:,,respectively, the domains of cpz and cp:,,.

As we have already done for functions, we will drop the mention of the iiumber of arguments for relations as well, when no confusion arises. Also, we will identify sets and unary relations, a n d thus write x E We for W:(x). I4’rorri t h e definition we have inlmediately:

Theorem 11.1.10 Normal Form Theorem for r.e. relations (Kleene [1936], Rosser [1936], Mostowski [1947]) A n m a r y reZation P is r.e. i f arid only i f there is an 11 + 1-ary recursive relation R such that

i.e. i f and only if there is a number e (called index of P ) such that

P(2)

Wz(Z) ts 3yTn(e,Z,y).

11.1 Partial Recursive Functions

135

Proof. If P is r.e. then P is t h e domain of a recursive function pe, i.e. P is equal t o W e . Then, by 11.1.2,

W(Z)

@

*

%(ql 3Yz(e,5Y).

Conversely, if

P(4

@

3YW,Y)

with R recursive, then P is the domain of the partial recursive function cp().’

N

WW,Y).

0

R.e. relations appear naturally and abundantly in mathematics. Consider e.g. a diophantine equation p(Z,y) = q(.’,y), where p and q a r e polynomials in 5,y with coefficients in the natural numbers. T h e set of non-negative, integral solutions of t h e equation, defined as:

YED

-33[P(.’,

Y) = (I( Y)1 .’,

is r.e., by t h e theorem just proved. Matiyasevitch [1970] has shown t h a t t h e converse also holds, and thus the r.e. sets are exactly the sets of non-negative’ integral solutions of diophantine equations. See Matiyasevitch [1972] or Davis [1973] for a n exposition of this remarkable result, which improves t h e Normal Form Theorem, and also solves Hilbert’s Tenth Problem (Hilbert [1900]). A discussion of its significance is in Davis, Matiyasevitch and Robinson [1976], a n d a n easy proof of a slightly weaker version of it is in Jones and Matiyasevitch [1984]. T h e notion of r.e. set has been defined from t h a t of partial recursive function, b u t the next result shows t h a t the opposite approach is also possible.

Proposition 11.1.11 Graph Theorem. Let cp and f be, respectively, a partial and a total function. Then: 1. cp is partial recursive if and only i f its graph is r.e. 2. f is recursive i f and only i f its graph is recursive.

Proof. Recall t h a t t h e graph G, of cp is t h e set so defined:

G,(.’, z ) e cp().’

N

z.

‘This condition cannot be eliminated: it is known e.g. that there is no diophantine equation whose integral solutions are exactly the prime numbers.

11. Basic Recursion Theory

136

for some recursive R, again by 11.1.10. T h u s

$4.’) l a 323YR(Z, 2 , Y). By coding z a n d y into a single number 1 = (z,y), we h a v e

c p ( 4 = (Wq.’,

(t)l, ( t ) Z ) ) l

Intuitively, t h i s is nothing more t h a n a dovetailed verification of Gv(Z,z ) for every z , until o n e of these verifications succeeds. T h e reason we c a n n o t simply verify t h e 2’s o n e b y one, is t h a t we could get stuck with the first o n e which is not a value of cp(Z),a n d never get to consider t h e remaining possible values. I f f is recursive, so is Gf: CG,

Conversely, let

Cf be

(C.) =

1 iff(.’) = z 0 otherwise.

recursive. Since

a n d t h e hypothesis that, f is total c a n be written as

V232Gf(.’, z ) , we have t h a t f is defined by p-recursion over a regular predicate (1.1.5),a n d it is t h e n rwursive. 0 Exercises 11.1.12 Partial functions with recursive graph. a) Zj cp is a partial junction, then cp has a recursive graph i j and only if there is a mcursive H such that P(Z)= PYR(Z,9 ) . b) There are partial rexursive, nontotal functions with recursive graph. (Hint: let We be a n r.e. set different from w, a n d let cp(z) N ps(z E W e + ) . ) c) There are partial recursive functions with nonrecursive graph. (Hint: let A be a n r.e. nonrecursive set, see 11.2.3, and cp(z) 0 -++z E A . )

137

11.1 Partial Recursive Functions See VIII.2.11 for a refinement of this topic.

T h e close relationship between partial recursive functions and r.e. relations is also indicated by t h e following property, defined by L u s h [1930] in a descriptive set-theoretical context.

Proposition 11.1.13 Uniformization Property (Kleene [1936]) 1. If P is r.e., there is a partial recursive function

3YP(.', 9)

'p

such that

* 4.'1)A P(%4.'))

2. If P is r.e. and regular, there is a recursive function f such that

V2P(.', f ( 2 ) ) . Proof. This is similar t o part 2 of the proof of 11.1.11. Since P is r.e., there is R recursive such that P(Z,y) e 3zR(Z,y, z ) . Then it is enough to let

cpm = (PtR(?, ( t ) l , ( t ) 2 ) ) * . Intuitively, cp chooses the first element y such that P(3,y) has been verified (which, as t h e next exercise shows, is not necessarily the first one for which P holds). If P is regular (1.1.5), then such a y always exists, and cp is then recursive. 0 Exercise 11.1.14 If P is recursive, a unifomizing function is simply p y P ( 5 , y ) . This does not hold in general, for P r.e. (Uspenskii 119571) (Hint: see the remarks after 11.1.1 .) Exercises 11.1.16 Choice functions for r.e. sets. a) There is a partial recursive choice function for the r.e. sets, i.e. a partial recursive function cp such that

We #

0

+- cp(e)l A cp(e) E

we.

y E We.) b) There is no invariant, p ~ r t i a lrecursive choice function for the r.e. sets, i.e. a

(Hint: uniformize P ( e ,y )

choice function cp such that

W, = We #

0

* cp(i)1 A p ( e )1 A ~ ( i=)cp(e).

Thus no partial recursive analogue of Hilbert's &-operatorexists for r.e. sets. (Kleene [ 19521) (Hint: suppose such a function exists. Assume there exists a nonrecursive r.e. set A, see 11.2.3, and let:

YEW,(^) e y = O V ( y = l A e E A ) ~ E W , , (e ~) y=lV(y=OAe€A) Then both pg(e) and p h ( e ) converge and

11. Basic Recursion Theory

138 eE A

* pg(e) = ph(e),

i.e. A would be recursive.) c) There is no recursive choice functionf o r the r . e . sets. (Hint: let f be one such, set W h ( e= ) { f ( e )+ 1 ) and apply the Fixed-Point Theorem 11.2.10.) Having analyzed t h e analogies of r.e. sets versus partial recursive functions, we t u r n now t o t h e differences between r.e. a n d recursive sets. We first characterize both notions in t e r m s of enumeration properties, a n d in so doing we account for t h e name ‘recursively enumerable’ (which suggests ranges, more t h a n domains). T h e next result is very useful a n d will be applied repeatedly.

Theorem 11.1.16 Characterization of the r.e. sets (Kleene [1936])The following are equivalent: 1. A is r.e.

2. A is the range of a partial recursive function cp

3. A = 8 or A is the range of a recursive function f

Proof. We prove t h e result, in a round robin style. b

1*2 If A is r.e. t,hen A = W e ,for some e. Let ~ ( z )

5

cpe(z)

1.

T h e n t h e domain of cpe is equal to t h e range of cp, arid cp is partial recursive, e.g. l ~ e c a u s ecp(z) z 0 . cp,(z).

= +

b

2*3 Let A b e nonenipty, a n d t h e range of a partial recursive function 9,. Choose a E A: we would like t o set

f(x) =

{

z n

if c p e ( z ) L A cpe(z) otherwise.

21 2

As i t stands f is not however recursive, because we cannot decide recursively whether cp(z) (see 11.2.7). To avoid being stuck while waiting for some uridefined value to converge, we dovetail t h e computation of all possible values, a n d p u t t h e m in t h e range of f as soon as t h e y a p p e a r , t h e value a being used for t h e stages i n which no new value appears (to keep f total). T h u s t h e following modified version o f f is recursive:

11.1 Partial Recursive Functions

139

Here 3 is a recursive, onto pairing function (see e.g. p. 27), a n d ontoness is required to have f total. 0

3 3 1 If A = 0 then A is t h e domain of t h e completely undefined function, which is obviously partial recursive. If A is t h e range of a recursive f , we want a partial recursive 'p with domain A , i.e.

'p(z)l

* 3 z ( f ( 4= .I.

T h e n we can j u s t let p(z) N pz( f ( 2 ) = x).

0

Also t h e recursive sets can be characterized in t e r m s of enumerating functions.

Proposition 11.1.17 The following are equivalent: 1. A is recursive

2. A = 0 or A is the range of a nondecreasing, recursive function f Proof. If A is recursive a n d nonempty, let a be its smallest element, a n d

f(0) = a

{ f(n) n+l

f(n+l)

=

ifn+lEA otherwise.

Conversely, if A is finite then it is recursive. If A is infinite a n d t h e range of a nondecreasing recursive function f , to know whether z E A, search for t h e smallest IC such t h a t f(z) > z (which exists because A is infinite). Since f is nondecreasing, t h e n

z~ A e z {f(O), ~ . . . , f (z)}.

0

Exercises 11.1.18 Onsone enumerating functions. a) A n infinite r.e. set is the range of a one-one recursive function. (Kleene [1936], Rosser [1936]) (Hint: define f by primitive recursion, looking at each stage for the next element generated in the set, which has not been previously generated.) 13)

A n infinite recursive set is the range of an increasing recursive function.

(Kleene [1936]) (Hint: define f by primitive recursion, looking at each stage for the next element in the set.)

T h e description of a set consists of t h e infinitely many facts t h a t tell, for any given element, if i t is in t h e set or not. In general, although t h e y always answer yes or no to a question, these facts are just a sequence of accidents, with

140

11. Basic Recursion Theory

no common pattern. T h e recursive sets a r e those for which a p a t t e r n exists, a n d a general procedure to give effective answers can be finitely described. The

recursively enumerable sets present a basic as y n m e t r y between membership, that can be effectively determined b y a finite amount of information, and nonmembership, whose determination may instead rquire a n infinite amount of it ( t h e partial test for menibership of a given element being simply to recursively generate t h e set, a n d wait for t h a t element to appear). T h e r.e. sets a r e t h u s somehow only ‘half-recursive’.* T h e distinction between recursive a n d recursively enumerable c a n t h e n be traced back to t h e irifornial distinction between a decision procedure a n d a generating procedure, envisaged once again by Leibniz [1666], when h e talked of ars iudicandi (checking t h e correctness of a proof) a n d a r s inveniendi (finding a proof). Another reason to see t h e r.e. sets as half-recursive is given by t h e next result, which also characterizes recursiveness in terms of recursive enumerability. I t is sometimes called Post’s Theorem, a n d it will be used repeatedly.

Theorem 11.1.19 (Post [1943], Kleene [1943], Mostowski (19471) A set is recursive i f and only i f both the set and its complement are recursively enumerable. Proof. If A is recursive then both A a n d domain A a n d a r e cp(z)

{ {T

O

@(.)

a r e r.e., since e.g. functions with

if c ~ ( z = ) 1 otherwise if c A ( z ) = O otherwise.

Let now both A and be r.e., arid suppose they a r e nonempty (if o n e is emply t h e other is w ,a n d they a r e already both recursive as wanted). T h e n there a r e recursive functions f a n d g generating them:

T h e two lists a r e disjoint arid exhaustive, a n d to know whether a given z is in A or in 2 is enough to generate them simultaneously, until x appears in o n e of t h e two lists. 2For this reason the r.c. sets are sometimes called semirecursive. We will use this tern] in a different context, see p. 294.

11.1 Partial Recursive Functions More formally, if A and such t h a t

2 are r.e., there are recursive relations

141

R and Q

Proposition 11.1.20 (Post [1944]) Every infinite r.e. set has a n infinite recursive subset. Proof. Let A be infinite, and the range of a recursive function f . Define g recursive a n d increasing as: g(0) = f(0) g(n + 1) = the first element generated in A and greater than g(n) = f(PY(f(Y) > g(n))>. Then t h e range of g is recursive by 11.1.17, and it is a n infinite subset of A by definition. 0 Infinite sets which do not have infinite r.e. (equivalently, by t h e previous proposition, infinite recursive) subsets are called immune, and will be studied in Sections 6 and 111.2.

Proposition 11.1.21 Set-theoretical properties of r.e. sets (Post [1943], Most owski [1947]) 1. With respect to set-theoretical inclusion, the r.e. sets form a distributive

lattice with smallest and greatest element, and with the recursive sets as the only complemented elements. 2. The property of being r.e. is preserved under images and inverse images

via partial recursive functions.

142

11. Basic Recursion Theory

Proof. Sniallest a n d greatest elements are c:learly 8 a n d w. T h e part relative to complementation follows from 11.1.19. Finally, if A a n d B a r e r.e. t h e n so a r e A n B a n d A U B : generate A a n d B simultaneously, p u t in A n B t h e elements appearing in b o t h lists, a n d in A U B those appearing in at least o n e list. Given A r.e. a n d 'p partial recursive, p ( A ) consists of all elements 'p(x),for z E A . To generate 'p(A)is t h u s enough to generate A a n d , simultaneously, to dovetail t h e computations of 'p(z), for t h e various z's which a r e found to be in A . Similarly, ' p - ' ( A ) consists of all z such t h a t 'p(z)l a n d cp(z) E A . 0 Corollary 11.1.22 Set-theoretical properties of recursive sets. 1. With respect to set-theoretical inclusion, the recursive sets f o r m a Boolean algebra.

2. The property of being recursive is preserved under inverse images via recursive functions. Proof. If A is recursive both A a n d 2 a r e r.e., a n d t h e n so a r e f-'(A) a n d a n d so b o t h f - ' ( A ) a n d its complement a r e r.e., a n d f-'(A) is recursive by 11.1.19. 0

f-'(X).B u t f-'(x)= f - ' ( A ) ,

Note t h a t i f A and f are recursive, then f ( A ) is r.e. b y the proposition above, but is not necessarily recursive. Indeed, any noriempty r.e. set A is t h e range of a recursive function f , a n d t h u s t h e iniage of w (which is recursive) via f ,b u t not every r.e. set is recursivc (see 11.2.3). A detailed study of t h e set-theoretical structure of b o t h recursive a n d r.e. sets will be m a d e in Chapter IX. For now we j u s t prove a n additional property, defined by Kuratowski [1936] in a descriptive set-theoretical context.

Proposition 11.1.23 Reduction Property (Rosser [1936], Kleene [1950])The union of two r.e. sets can be reduced to the union of two disjoint r.e. sets. Precisely, given two r.e. sets A and B there are two r.e. sets A' & A a71d B' & B such that

A ' n B ' = 0 and A ' u B ' = A u B . Proof. Given A a n d B r.e., t h e only problem is to decide where to put t h e elements of A n R. We let speed of generation decide: if .z E A n B a n d z is generated by A faster t h a n by B , then z goes into A', otherwise it goes into B'. More formally, if

x E A w 3yR(z,y) EB

*

3yQhy)

11.1 Partial Recursive Functions

143

with R a n d Q recursive, let

Exercise 11.1.24 The reduction property follows from the uniformization property.

R.e. sets as foundation of Recursion Theory

*

Wc have derived t h e notion of recursive enumerability from t h a t of partial recursive function, b u t we have already noted t h a t , by 11.1.11, t h e opposite is also possible: t h e partial recursive functions a r e those with a n r.e. graph. W h a t is needed to avoid circularities, is a n independent characterization of recursive enumerability. T h i s has been provided by Post [1922], (19431 (incidentally, quite before t h e notion of recursiveness had been isolated). His formulation comes from a n analysis of derivations in formal systems, a n d it is t h u s t h e natural conclusion of our journey of Chapter I (see p. 18). T h e underlying idea is t h a t effective mathematical a n d , more generally, linguistical activity can be seen as a way of geiierating words from words (of a given language), according to rules. W i t h coiisiderations similar to those of Section 1.4, one is quickly led to restrict attention to canonical systems consisting of finite alphabets (possibly with distinguished symbols), finite sets of axioms a n d finite sets of finitary rules (called productions), telling how to decompose a word a n d rearrange its p a r t s (by possibly dropping some a n d adding others). Formally, a production h a s t h e form XOpJXI

... xn-1 m x n

-

Y O r n Y l ...

Ym-1

MYm

where i, E (1,. . . , n } , with t h e meaning: if a word c a n be decomposed in t h e way written on t h e left (by somehow filling u p t h e boxes), t h e n it c a n be transformed into t h e word written on t h e right (where t h e boxes on t h e right with a given label a r e supposed to contain exactly w h a t t h e boxes o n t h e left with t h e s a m e label do). Since decompositions of words a r e usually not unique, productions a r e not deterministic rules. I t is possible to show t h a t t h e sets of words produced b y canonical s y s t e m s are exactly ( u n d e r suitable coding) the r.e. ones (one direction comes by arithmetization; t h e other c a n be seen by noting t h a t Turing machines transitions between successive configurations can be written as productions, operating on words expressing t h e configurations). Post also showed t h a t systems with multiple-premise productions a r e reducible to canonical systems, a n d t h a t these a r e in t u r n reducible to normal

144

11. Basic Recursion Theory

systems, with just one axiom, a n d with productions of t h e very specific form

For a proof of this a n d a treatment of t h e whole subject, see Minsky [1967]. T h e approach sketched above is relevant not only to computability b u t also (and even more naturally) to linguistics, a n d it provides a framework for studying structural descriptions of sentences, in formal approximations to natural languages. A system with productions only of t h e form

( t h a t can be written simply as z + y), is called a grammar ( T h u e [1914]). T h e notion is sufficiently general, since t h e simulation of Turing machines referred to above can be naturally carried o u t by means of g r a m m a r productions (instructions a c t only locally o n t h e tape of a Turing machine). Chomsky [1956], [1959] has introduced a hierarchy on t h e types of g r a m m a r productions, which has turned o u t to be strongly connected with machine models. Since we only prove some scattered results a b o u t this in Chapter VIII, we refer to Hopcroft a n d Ullman [1979] for a detailed s t u d y of t h e subject, a n d to Greibach [1981] for a broad overview a n d a n historical account.

A programming language based on r.e. sets

*

Although Post’s productions a r e intended to provide a basis for recursive enumerability, they can be used directly to define partial algorithms for partial functions on strings, by introducing restrictions t h a t make t h e production p r e cess deterministic. A finite sequence of grammar productions (some of which a r e singled o u t as final) computes a partial function on strings ‘p if, given a n y s t r i n g w, t h e following partial algorithm produces t h e string ‘p(w): at each step (starting from w), search for t h e first production in t h e sequence which c a n be applied (i.e. such t h a t t h e premise of t h e production matches a substring of t h e given string), a n d apply t h e production to t h e leftniost possible substring to which it can be applied; then stop t h e process if t h e production is a final one, a n d repeat t h e process otherwise. T h i s approach h a s been introduced by Markov [1951], [1954] a n d it is easily seen (Detlovs [1953], [1958]) to be equivalent to partial recursiveness (one direction follows as sketched above for Turing machines, t h e other by arithmetization). I t is usually referred to EIS Markov algorithms. T h i s suggests t h e possibility of uszng the productzon approach as a basw f o r a prograrninzng language for stnng operations, which has been d o n e with t h e introduction of SNOBOL (String Oriented Symbolic Language) by Farber,

11.2 Diagonalization

145

Griswold and Polonsky [1964] (see Sammett [1969] a n d Wexelblat [1981] for history and references). T h e instructions of this language are labelled, a n d are either assignment statements (giving values t o the variables) or replacement statements (telling t o substitute the leftmost occurrence of a substring in a string by another string), the latter together with conditional j u m p instructions (sending t o other instructions, depending on whether the given substitutions was successfully applied, or could not be applied). T h e language is thus unstructured (see p. 63), and it does not need any primitive operation (pattern matching being sufficient for all purposes: t h e needed operations can be specified each time, as production rules).

11.2

Diagonalization

In this section we introduce one of the basic methods of proof in Recursion Theory: diagonalization. We will give a number of applications, b u t t h e method will be used throughout t h e book either directly or in some codified way (like t h e unsolvability of the Halting Problem, or t h e Fixed-Point Theorem, both proved below).

The essence of diagonalization Given a set S, a function d : S + S which is never t h e identity on S (i.e. d ( a ) # a for every element a of S) and a n infinite matrix of elements of S ao,o

ao,1

a1,o

al,l

a1,2

a2,o

a2,1

a2,2

q

2

... ... ...

we get a transformed diagonal sequence of elements of S d(ao,o) d ( q 1 )

+2,2)

...

which is not equal t o any row of the matrix, because it differs from t h e n-th row on t h e n-th element (by the hypothesis on d ) . That’s all. T h e ingredients of t h e method are two:

1. the use of the diagonal {an,n}nEw This was systematically done by Du Bois Reymond, in his study of orders of infinity (see Hardy [1910] for a neat exposition). 2. the use of the switching function d This crucial part was introduced by Cantor [1874] to prove his celebrated theorem t h a t t h e set of subsets of w is not denumerable.

11. Basic Recursion Theory

146

I n many applications, like t h e results proved or quoted from p. 165 o n , t h e two ingredients t a k e t h e form of self-reference a n d negation.

Recursive undecidability results As a saniple application of the method we prove a constructive analogue of Cantor's Theorem, whose original form can be stated by saying t h a t t h e r e is 110 function on w which enumerates all subsets of w (or, equivalently, t h e set of characteristic functions, i.e. t h e set of t o t a l functions from w to ( 0 , l ) ) . Proposition 11.2.1 Recursive version of Cantor's Theorem (Kleene [1936],Turing [1936])There is n o recursive function which enumerates (at least one index of) each recursive (0,l-valued) function. Proof. Let define

f

be a recursive function such t h a t

s(z) N

' p ~ (is ~ t)o t a l

for every z, a n d

1 - 'pf(z)(").

T h e n 9 is a 0,l-valued function, which is partial rccursive by t h e Enunreration Theorem 11.1.5, a n d total by t h e hypothesis o i l f. Moreover, g is differerlt from 'pf(.) (on tJhe elenient z) for every z, a n d t h u s 110 index of g is in t h e range of

f.

0

T h e proof just given falls urider t h e general framework of tliagonalization, by letting ni,j = c p ~ ( i ) ( j )a, n d d ( n ) = 1 - a. I t also implies t h a t the set

Tot = {x : 'ps is total) is n o t r.e. In particular i t is not recursive, a n d t h u s there is no recursive way to detect whether a number codes a total recursive function or n o t . Note t h a t we expressed o u r results in terms of 0,l-valued functioiis, a n d not of sets. T h i s is because there a r c many different, ways to associatc numbers to recursive sets, a n d different results hold for them (see 11. 226). Exercises 11.2.2 a) A function is called potentially recursive (Church [ 10361) if it ha.. a total recursive extension. There is a partial recursive function which is riot potentially recursive. (Kleene 119381) (Ilint: let p(z) !x 1 - pZ(z).) b) There is a recursive, not primitive recursive set. (Sudan [1927], Ackerrnann 119281) (Hint: consider 0,l-valued functions, and use the function of 1.7.4.)

We now prove one of t h e crucial results o f Rmursion Theory.

147

11.2 Diagonalization

Theorem 11.2.3 Combinatorial core of the undecidability results (Post [1922], Godel [1931], Kleene [1936]) There is an r.e. nonrecursive set. Explicitly, the set defined by X E

K

@

X E

W,

@

cp5(x)l

is r. e. and nonrecursive. Proof. B y t h e Enumeration Theorem 11.1.5, there is a partial recursive function cp such t h a t 4 5 ) cpx(x).

=

T h e n K is r.e., because X E

K@v(x)J.

To show t h a t K is not recursive we give two different proofs, based on t h e two equivalent definitions of K given above (in terms of partial recursive functions a n d of r.e. sets). 0

If Ic were recursive, t h e function

{

=

ifxEK: otherwise

llndefined

would be partial recursive. Then, for some e l cp N cpe a n d

cpe(e)l-

eE

contradicting t h e definition of K . 0

If Ic were recursive, then

K: would be r.e. x E K:=x

a n d so be r.e.

$!

But

w,,

differs on t h e element x from t h e x-th r.e. set, a n d cannot itself 0

T h e argument to show t h a t K is not recursive falls under t h e general framework of diagonalization, by letting

a.

-

1 ifjEWi 0 otherwise,

a n d-d(a) = 1 - a. K: obviously resembles t h e set used in Russell’s paradox (Russell [1903]), is t h e namely t h e set of sets not belonging to themselves (see p. 81). Here

148

11. Basic Recursion Theory

set of numbers not belonging to t h e r.e. sets they code. T h e r e is n o paradox here because Russell’s argument simply shows t h a t such a set is not r.e. itself. We have used t h e word ‘undecidability’ in t h e theorem head, a n d we will use it over a n d over again throughout t h e book, interchangeably w i t h t h e word ‘unsolvability’. In both cases we really mean undecidability a n d unsolvability b y recursive means. T h e reason we do not write t h i s down explicitly is t h a t there a r e good reasons t o suspect t h a t in fact something much stronger is involved here, namely absolute undecidability a n d unsolvability. T h e step from recursive to absolute unsolvability requires a n appeal t o Church’s Thesis (see Section 1.8, a n d p. 103 in particular). We now strengthen t h e theorem just proved. T h e s t a r t i n g point is t h e fact t h a t A is recursive if a n d only if both A a n d 2 a r e r.e. (11.1.19). T h i s suggests t h e possibility of extending t h e theory of r.e. sets to pairs of disjoint r.e. sets. T h e next property is a recursive version of one defined by Lusin, in a descriptive set-theoretical context.

Definition 11.2.4 (Kleene [1950], Trakhtenbrot [1953]) Two disjoint sets A and B are called: 1. recursively separable i f there is a recursive set C such that A

C and

BSC 2. recursively inseparable i f they are not recursively separable. Clearly, A is recursive if a n d only if A a n d 7 a r e recursively separable. T h e existence of a disjoint pair of recursively inseparable r.e. sets is t h u s a stronger result t h a n t h e simple existence of r.e. nonrecursive sets.

Theorem 11.2.5 (Rosser [1936], Kleene [1950], Novikov, nakhtenbrot [1953]) There are two disjoint, recursively inseparable r.e. sets.

Proof. We give two different proofs, which generalize t h e two of T h e o r e m 11.2.3. 0

Define two disjoint r.e. sets as

~

E

z EB

A

cpz(z)czO

e cpz(z) 2-1.

Suppose there is a recursive set C such t h a t A & C a n d B t h e function 1 ifzEC 0 otherwise is recursive. If e is a n index for it, we get a contradiction:

p,(e) p,(e)

eEA 13 e E B

21 0 3 2:

*

e E C cp,(e) N f ( e ) N 1 + e E 77 3 cp,(e) = f ( e ) N 0.

3

c. T h e n

11.2 Diagonalization 0

149

Define two r.e. sets as

xEA xEB

H H

x E W(z), x E W(z)2.

T h e y a r e not necessarily disjoint, b u t A - B a n d B - A a r e recursively for some inseparable. Indeed, suppose t h a t A - R C C a n d B - A & recursive set C. Let C = W, a n d = Wb (since both C a n d a r e r.e.), a n d set x = ( b , a ) . T h e n z @ A n B , because

c, c

x E A H X EW S - X E C E B H x E W, e x E

c.

5

Moreover,

xEA x EB

c

(contradicting A - B

+x E C

(contradicting B - A

3

zE

C)

c),

a n d so C cannot exist. T h e only trouble is t h a t A - B a n d B - A a r e not necessarily r.e., b u t clearly a n y two disjoint r.e. supersets of them will still be recursively inseparable. T h e n it is enough to reduce (11.1.23) A a n d B , to get a pair of disjoint r.e. sets which extend A - B a n d B - A , a n d which a r e t h u s recursively inseparable. 0 Exercises 11.2.6 a) Another proof of the existence of recursively inseparable r.e. sets can be obtained by first getting an enumeration {(A,,, Bn)},,tw of the disjoint pairs of r.e. sets, and then letting

xEA u X E A ~ X E B ts X E B , . (Hint: for the first part, consider a double enumeration of the r.e. sets, and reduce each pair uniformly. Then prove that there cannot be a pair ( A , , & ) such that A C_ B, and B A,.) b) A n y two disjoint w-r.e. sets A and B are recursively separable (Sierpinski [1924], Laventrieff [1925]). (Hint: reduce A and and show that the reduced sets are complementary, and hence recursive, r.e. sets.)

B,

Limitations of mechanisms

*

A slight reformulation of Theorem 11.2.3 is t h e following, which rules o u t t h e existence of a recursive procedure to decide whether a partial recursive function

11. Basic Recursion Theory

150

converges for given arguments. T h e n a m e comes from i t s original formulation, which was in terms of Turing machines, a n d in t h a t setting it shows t h a t t h e r e is no Turing machine t h a t decides whether a universal Turing machine halts or not on given argurrient,s.

Theorem 11.2.7 Unsolvability of the Halting Problem (Turing [1936]) The set defined b y

is r.e. and nonrecursive. Proof. KO is shown to be r.e. as in 11.2.3, by t h e Enumeration Theorem. And K O were recursive so would be K ,because

if

5

E

K

@

(5,z) E

KO.

0

Actually, t h e unsolvability of t h e Halting Problem is j u s t t h e tip of t h e iceberg. T o measure t h e complexity of a problem a b o u t recursive functions] we introduce t h e following notion.

Definition 11.2.8 A set A is the index set of n class A of partial recursive functions i f A = { x :cpz E A } .

If A is the index set of A, we write A = BA. Index sets contain all possible progranis t h a t c o m p u t e functions belonging to a given class, a n d a r e useful in classifying t h e complexity of such classes. In particular, a class of partial recursive functions is called completely recursive if its index set is recursive (Dekker [1953a], Rice [1953]). Completely recursive classes of partial recursive functions correspond to (recursively) solvable problems about t h e m , a n d a r e characterized by t h e next result.

Theorem 11.2.9 Rice’s Theorem (Rice [1953])A class of partial recursive functions A is completely recursive i f and only i f it is trivial, i.e. either empty or containing all partial recursive functions. Proof. If A = 0 thcn its index set is 8, a n d if A contains all partial rccursive functions then its index set is w. Suppose t h e n t h a t A is nontrivial: t h e r e a r e a , b such t h a t pa E A a n d V b !$ A. If t h e coiripletely undefined function is not in A, let f be a recursive function such t h a t

11.2 Diagonalization

151

Then zE

K

@

‘pf(z) E

A

f(.)

E A,

where A is t h e index set of A. T h e n A is not recursive, otherwise so would be K. T h e case of t h e completely undefined function being in A is treated similarly, t h i s t i m e using (Pb, a n d showing t h a t 2 is not recursive. 0 Rice’s Theorem is quite powerful, since it incorporates a number of undecidability results. Its content is t h a t a n y nontrivial property of partial recursivc? j k n c t i o n s is undecidable. Thus, undecidability proofs of given properties a r e reduced to proofs of nontriviality, which a r e usually immediate. E.g., t h e following problems for partial recursive functions a r e undecidable: being t h e completely undefined function, being defined for a given fixed argument, being total, having a given number as value, being onto, being equal to a given partial recursive function, a n d so on. A mechanism is a device with a predictable local behavior, in t h e sense t h a t each move is governed by a mechanical rule. T h e unsolvability results j u s t proved show t h a t t h e behavior of a m e c h a n i s m does n o t need t o be globally predictable, a n d t h a t a purely mechanical analysis of ~nechanismsis b o u n d t o fail. T h i s has consequences for t h e possible description of mechanisms (Von Neumann [1951], [1966]). These a r e of t w o kinds: purely descriptive (telling how t h e device is made, o u t of its constituent parts), a n d operational (telling how it behaves in given situations). T h e t w o descriptions do not need to be at t h e s a m e logical level: t h e formcr is always a n u m b e r ( t h e index of t h e machine), b u t t h e latter is a number only if t h e mechanism h a s a sufficiently simple behavior (describable by a recursive function, a n d t h u s again by a number). T h i s is however not necessary: in general t h e behavior is not recursive, a n d t h e n a f u n c t i o n is needed t o describe it for every possible input. T h u s f o r s u f i c i e n t l y complicated mechanisms, t h e device itself is i t s o w n best (logically simplest) description, a n d it might b e impossible to effectively say substantially morc a b o u t it t h a n how it is made. To measure t h e complexity of (codes of) finite objects, we c a n introduce t h e so called Kolmogorov complexity (Kolmogorov [ 19631, [ 19651, Solomonov [1964], Chaitin [1966]):

whose intuitive meaning is to pick u p t h e smallest description of t h e number z i n t e r m s of programs printing it (on a fixed input, like 0), a n d t h u s to measure t h e quantity of information carried by z. A number z a n d t h e object coded b y it a r e called random (Church [194O]) if z is its shortest description, i.e. if z 5 K ( z ) . Randomness of a n object can be determined by reasons antithetical

11. Basic Recursion Theory

152

in nature, b u t indistinguishable: extreme structural complexity a n d chaos. T h u s a s u f i c i e n t l y complicated m e c h a n i s m is a r a n d o m object, a n d 11.2.7 can be taken to mean t h a t a universal Turing machine is a random object. See p. 261 for a discussion of random numbers a n d their properties. T h e confusion between t h e two meanings of predictability is somewhat widespread a n d harmful. For example, scientific theories describing local mechanical behavior of biological (Darwin [1859]), historical (Marx a n d Engels [1848]) a n d psychological (Freud [1917]) evolution, a r e often rejected o r opposed on social grounds, on t h e false belief t h a t t h e local mechanisms explained by then1 might imply global predictability (i.e. a forecast of t h e final outcome of evolution), something which might be felt as antihumanistic.

Fixed-Point Theorem We t r y now to generalize t h e proof of 11.2.1 a n d , given a recursive function f , we t r y to get a partial recursive function $J which is not in t h e set { ' p f ( z ) } z E w . Of course we do not suppose anymore t h a t t h e 'pf(z)be always t o t a l , since we have already disposed of this case. T h e natural idea is to diagonalize as in 11.2.1, a n d let $J@)

= 1- ' p f ( z ) W

T h e trouble here is t h a t q!~,although partial recursive, is not necessarily different from 'pf(z),since this might be undefined on z, a n d t h e n so would be 1c) too. T h u s t h e n o t i o n of partial recursive f u n c t i o n s e e m s t o h a v e a built-in defense against diagonalization. T h a t it is indeed so is t h e content of t h e next theorem, one of t h e most tricky a n d useful applications of t h e diagonal method.

10 sentiva osannar d i coro in coro a l p u n t o fisso che li tiene alli ubi3 ( D a n k , Paradiso, XXVIII) Theorem 11.2.10 Fixed-Point Theorem (Kleene [1938]) G i v e n a recursive f u n c t i o n f , there i s a n e s u c h that e a n d f ( e ) c o m p u t e t h e s a m e f u n c t i o n , i. e. 'pe = (PI(=) and hence We = W f ( e ) . Proof. We give two different proofs. 1. We s t a r t with a n informal argument. Since f c a n be t h o u g h t of as a program transformation, it would be enough to t a k e a program as follows: 3 I heard 'Hosanna' sung, from choir t o choir, to that fixed point that holds each t o his place.

153

11.2 Diagonalization transform this very program according to t h e result to z.

f, a n d then apply

If e codes this program then, by definition, pe N pf(e).But this program is self-referential, a n d t h u s not well-formed. We have to unravel t h e selfreference, which we will achieve by two successive diagonalizations. Since we a r e searching for a code e of t h e prograni written above, we know t h a t e must also code t h e program: transform t h e program with number e according to f , a n d t h e n apply t h e result to z.

First of all we compute a number coding t h e program j u s t written. T h i s number will depend on e , a n d it will t h u s be h ( e ) , for some recursive function h. If h N pa,this number will be p,(e). W e now know t h a t if such a program exists, it must have a number of t h e form p , ( e ) , for some a a n d e, a n d this number must code t h e program: transform t h e program with number pa(.) according to t h e n apply t h e result to 5.

f,a n d

Now this program depends on two parameters a arid el a n d if we compute a number coding it we a r e going to add a new one, a n d so o n . If we want to avoid a n infinite regress, we have to stop adding parameters. We t h u s t r y to find a program with code number of t h e form p,(e) (first diagonalization), since this depends on just one parameter, a n d it h a s t h e right form. In other words, we consider t h e program: transform t h e program with number p e ( e ) according to f , and t h e n apply t h e rcsult to 5 . Now t h e r e is a recursive function (Pb giving t h e code of this program depending on e , a n d t h e program has thus number pb(e).

We now j u s t have to let, e = b (second diagonalization), since then pb(b) codes t h e program: transform t h e program with number pb(b) according to t h e n apply it to 5 .

f ,a n d

A p a r t from t h e motivation, we can extract from t h e argument j u s t given a formal proof. Let b be a n index of t h e recursive function defined as:

11. Basic Recursion Theory

154

(i.e. of a function giving a rode of t h e p r o g r a m coded by f ( p e ( e ) ) ,as a function o f e ) . Then, for e = b, we have

p'f'h(b)

pf('fh(b))'

T h u s p b ( b ) k a fixcul-point of f .

2. (Owings [1973]) ltcfcrriiig to t h e general framework for diagonalization given at t h e 1,eginiiiiig of t h e section, note t h a t thc diagonal iiiethod c a n also be exprcsscd iii a c*ont,rapositiveform: given a function d 5' + S and a matrix { a , , J } , ,t wJ of elenleiits of $9,if t h e traiisforiiied diagonal sequence is a row of t h e niatrix, t h e n d has a fixtxl-poilit,, i.e. t h e r e is a E s suc.h that, d ( n ) = a . Let d(pe) = ( ~ f ( ~ )we : want, a fixed-point for d , t h a t is ail e s u c h t h a t d(pe) 2i ( p e . Let t h c n S be t h e set of partial recursive functions. Since Y pa for sonic a , t h c valuc; of d a r e of t h e form p'f'n(F!). T h e n consider t h e matrix = PrP,(J) where, if pi(j) diverges, a,,j is t h e coiiipletely undefined function. T h e transformed diagorial sequence is %3

'Pf('fo(0))

Pf('f'pl(1))

' ' '

a n d t h i s is a recursive sequence of partial recursive fuiickions, a n d t h u s a row o f t h e matrix. 'I'hcii a fixed-point for d exists. If we also want to know exactly w h a t t h e f i x e d p o i n t is, just n o t e t h a t it m u s t be t h e elenleiit of the traiisfornied sequence t h a t lies oii the diagonal. So let g be a recursive fiinctioii (which exists by t h t Eiiiiinerat,ion 'I'hcorcm arid t h e S,T--l'heorcm) such t h a t ps(e)

pf(v.(e)).

If b is a n y indcx of y, t h e n g ( b ) = P b ( b ) is a fixed-point, for f , since P S ( b ) fi pf('f'h(b))

2L p f ( g ( b ) ) .

Some remarks iiiight be worthwhile. First, if py(c)= pf('fe(e)) then g(e) is n o t f(p,(e)): indeed ! ~ ( eis) always defined, as an index o f ( ~ f ( ~ < ( ~ while ) ) , f(p,(e)) is undefined when p,(e) is, in which case g ( e ) is an index of t h e completely undefined function. Second, f does not nrm! t o be extensional, i.e. it, does not. need to induce a inap of functions such t h a t

11.2 Diagondization

155

In particular, in t h e second proof of t h e theorem t h e diagonal function d need not be a function o n t h e set { q l p , ( j ) } i , j c w as a set of partial recursive functions, b u t this does not affect t h e proof. Third, the proofs of the Fixed-Point Theorem given above are uniform and constructive: they not only tell t h a t a fixed-point exist, b u t they also explicitly produce it, in a way depending only o n (an index of) t h e given function f . T h u s t h e r e actually exists a recursive function h such t h a t (Pa total

* (Ph(a) =

(Plp,(h(a)).

Finally, the constructions of fied-points in the proofs given above are nothing else than a version of the fied-point operator y i n A-calculus (see 1.6.1), with t h e complications produced by t h e fact t h a t we have to distinguish between numbers as arguments, a n d numbers as codes of functions. Specifically, p,(e) (Pf(lp,(e))

(Pb(e) ps(b)

corresponds to corresponds t o corresponds to corresponds to

xz Y(XX)

Ax. ~ ( x x ) (Az. y(xx))(Xz.~ ( z x ) ) .

Here t h e t w o diagonalizations a r e explicit, in t h e form of t e r m s applied to themselves. Exercises 11.2.11 a) Fixed-Point Theorem with parameters. Zff is a recursive n + 1-anJ function, there is a recursive n-ary function h such that ...,2,))

'Pf(21 ,...,z n , h ( z 1 ,

21 ( P h ( z 1 ,...,z n ) .

Moreover, h may be taken to be one-one. (Hint: for the last part recall, from 11.1.7, that the SF functions can be taken to be one-one.) b) Double Fixed-Point Theorem. If f , g are recursive functions of two variables, there are a and b such that (Pa

(Pf(a,b)

and

(Pb N (Pg(a,b)-

(Muchnik [1958a], Smullyan [1961]). (Hint: first get h such that ( ~ h ( = ci ) (~f(h(=),=), by part a). Then get b such that (pb ci ( p g ( h ( b ) , b ) ,and let a = h(b).) Note that it is in general impossible to find a such that paN ( ~ f ( ~N) ( P ~ ( ~ since ), f and g could be constant functions giving indices of two different functions.

Exercises 11.2.12 Sets of fixed-points. a) No recursive function has only finitely many fixed-points. (Fbgers [1967]) (Hint: if f has only a finite set A of fixed-points, let d~be a partial recursive function different from all those whose index is in A. Then the recursive function g such that

11. Basic Recursion Theory

156

would have no fixed-point.) b) Thew (ire iiontriuial reriirsiw sets which a w sets of fixed-points of sovw rrrursive fimction. (Ilint: let

"(.)

{

undefined 'p=

if 2 E 12 otherwise,

with 3 a recursive set not containing any index for the corripletely undefined function.) c) There (we. nontrivial recursive sets which are not sets of fixed-points of any recirrsive function. Actually, a recursive set is not a set of futed-points if its complement is. (Shore) (Ilint: if R,R are sets o f fixed-points for f , g , a n d

then h has no fixecl-point.) d) There are nonrecursive sets of jixed-points of nzursicie functions, e.g. the set of indices of t h e constant function 0. I n a sense this is a worst-case example, since (with terrriinology to be introduced later, see IV.l.G) it is Il!&cornplete, a n d a set of fixetl-points must be 11:.

We now give a n equiva1f:iit form of t h e Fixed-Point T h e o r e m , wit,h k'leene's origirial proof. T h e reason for the name is purely contingent, n a m e l y the o r d e r of presentation in Kleene's classical Ijook [ 19521. The F i r s t Recursion T h e o r e ~ n will be given in 113.15. Theorem 11.2.13 Second Recursion Theorem (Kleene [1938]) If @ a partial wciirsavc fiinctaon, thprp 2s a n andt'x c' such that

1~

Proof. Fix any recursive function h . Since t h e function $ ( h ( z ) , ~is) p a r t i a l recursive, it ha5 an index (L (depending o n h ) . By t h e S,"-Theoreni,

=

+ ( M z ) > r ) cPs;(a,z)(+

I n particular, this IioIds for h ( z ) = S ; ( z , z ) , for t h e a p p r o p r i a t e a :

By letting

t'

= S': ( a ,a ) we havc

157

11.2 Diagonalisation

Exercise 11.2.14 The Fixed-Point Theorem and the Second Recursion Theorem are in the other the Enumeraequivalent. (Hint: in one direction use the 5’:-Theorem, tion Theorem.)

Although t h e results are equivalent, the proofs of the Fixed-Point and the Second Recursion Theorems require different tools: t h e Sr-Theorem is used in both, but t h e first also uses the Enumeration Theorem. Thus versions of t h e Second Recursion Theorem and t h e Fixed-Point Theorem may respectively hold a n d fail for classes of functions having t h e S c , b u t not t h e enumeration property (such as t h e classes studied in Chapter VIII). T h e Fixed-Point Theorem, in any of its forms, and t h e extensions considered in t h e exercises, assure t h a t it is possible t o define (any finite number of) partial recursive functions in a (simultaneous) self-referential way, by using t h e indices of the functions in their own recursive definitions. Otherwise said, the partial recursive functions are closed under &&-point definitions. But more is true, as the following result shows.

Theorem 11.2.15 (Kleene [1952]) The class of partial recursive functions is the smallest class of functions: 1. containing initial functions and predecessor 2. closed under composition

3. closed under definition b y cases

4. closed. under fied-point definitions. Proof. By definition by cases we mean the schema t h a t produces j(Z)=

{ $1

if t ( Z ) = 0 otherwise

from g , h a n d t . Note t h a t , having composition, closure under definition by cases follows by adding t h e following t o t h e initial functions:

yy

=

{

z

y

ifz=O otherwise.

Let C be the smallest class of functions satisfying t h e stated conditions. Clearly C is contained in t h e class of t h e partial recursive functions, by t h e Fixed-Point Theorem, and because case definition is a primitive recursive operation. To prove t h e converse, we proceed by induction on t h e definition of partial recursive function. Since identities, successor, and composition are given, we only have t o consider:

11. Basic Recursion Theory

158 1. primitive recursion

If

f(z,o) =

g(q f ( ? , y + 1) = h ( % Y , f ( W ) then f can be defined as t h e fixed-point of t h e following q u a t i o n :

'(."?')

=

{ shg!

y - 1, f(Z, y - 1))

ify=O otherwise.

This uses only predecessor, definition by cases, a n d composition

2. p-recursion If

f(2)= py[g(%,?/)= 01, let h be t h e unique fixed-point o f t h e following equations:

h(",g) =

{

0 h(Z,y + 1)

+1

if g ( Z , y ) = 0 otherwise.

T h e n , as in t h e proof of Theorein 1.2.3,

f(Z)= h(Z,O). T h i s uses only 0 , S , definition by cases, a n d composition.

0

I t follows t h a t the Fixed-Point Theorein, together with coinposition and case definition, generates the partial rexursivc: functions, in a n approach with indices. Usually this is clone indirectly, by postulating t h e S ~ ~ - T h e o r camn d the Enumeration Theorein (see Section 6 for a discussion of t h e central role of these two properties in Itwursion Theory), sirice from thcse t h e Fixed-Point T h e w rein follows immediately, as in 11.2.13. T h i s approach has been taken by Kleene [1959], a n d it has proved useful in contexts like recursion on higher t y p e s (see p. 199) or on abstract domains (see p. 203), where no analogue of t h e p o p e r a t o r is available. For further comments on t h e role of t h e Fixed-Point 'l'hcorem, see pp. 182 a n d 184.

Limitations of formalism

*

We a r e now ready t o prove t h e celebrated results of t h e Thirt,ies on t h e liniitations of formalisni. They rest on two main foundations:

11.2 Diagonalization

159

1. a combinatorial argument T h i s is embodied in Theorem 11.2.3, whose proof, as we have already noted, is a positive recasting of t h e diagonalization used by Russell in his paradox, b u t actually goes back to much older paradoxes (see t h e next subsection). 2. a n analysis of formal systems expressiveness T h i s is t h e content of t h e next theorem, which determines exactly what is representable a n d what is not in sufficiently powerful formal systems. T h i s provides t h e link with t h e combinatorial p a r t , by allowing t h e representation of r.e. notirecursive sets in fornial systenls. We now prove t h e missing result, which has interest o n its own: it points o u t another difference between recursiveness a n d recursive enumerability, this t i m e in t e r m s of representability notions. T h e reader interested only in Theorem 11.2.17 should note t h a t , for its proof, only p a r t 1 of t h e next result is needed (and i t h a s already been established, see 1.7.12). P a r t 2 c a n t h u s be skipped, if wanted, b u t it will provide a different proof of 11.2.17.

Theorem 11.2.16 Expressiveness of formal systems (Godel [1931], [1936], Mostowski [1947], =ski, Mostowski and Robinson [1953], Ehrenfeucht and Feferman [1960], Shepherdson [1960]). In any consistent fonnal system extending R: 1. a relation is representable i f and o n l y i f it is recursive

2. a relation is weakly representable i f and only if it is recursively enumerable.

Proof. T h e part relative to recursiveness is immediate, from previous results: t h e remarks following definition 1.3.4 show t h a t , by t h e axioms of R,a relation is representable if a n d only if its characteristic function is representable, a n d t h i s last condition is equivalent to recursivencss (of t h e characteristic function, a n d hence of t h e relation itself), by 1.7.12. Again immediate is t h e fact t h a t if P is weakly representable in a formal system F,t h e n P is r.e. Suppose indeed t h a t P(Z1,.

.

.I%)

-

t F cp(F1,. . . ,%),

for some cp. By arithmetization, t h e predicate

T(zl,.. . ,zn,y)e y codes a proof of

cp(Z1,. . . ,!En) in

is recursive, a n d so P is r.e. by t h e Normal Form Theorem, since P(Zl1.

.

* 3 y T ( Z l ,. . . ,Z n ,Y).

F

11. Basic Recursion Theory

160

T h e only real t h i n g t,o prove is thus the weak represent,ability of I’ r.e., in any consistent formal syst,c:ni ;F ext,cnding R. For simplicity of notatioiis, we rest.rict, ourselves to t,hc (‘asc o f sets. There is it r ~ : u r s i v eIt sudi that f’(z)tj 3 ! / R ( z ,9 ) and, by thc: part, of the theorem already proved, t h c r e I t i u s t bc p which represents It:

is a strong. assuniption: i t woultl follow froill a n ilifiriitary axioni of thc kind !/ = 0 v y = 1 v y = 2 v . . . I3ut, in 12 we only have the fitiitary axioms

froni which it only follows

Actually, we would j u 4

iitrvl

t h r weaker assuitipt i o n

I)tyause then, from Fy- 3 y p ( Z , y ) , w(: would know that,, for

soiiie y, not, Since y st,roligly represetits 12, for that, !/ it. c.ouItl not I x TR(:c,y), otherwise FJ -p(z,!/)would follow, a n d hence R ( z , y ) woultl hold, which is w h a t we want.ed.

t7 -I&,

g).

But, even t,his w m k e r assumpt,ion is a stroiig one, called w-consistency (Giidcl [ 19311, Tarski [ 1933]), a t i d it, does not, follow from s i m p l e consistency. Thus we have only proved the t,heoreiti for w-consistent systeiiis.

11.2 Diagonalization

161

We now modify this naive approach, a n d define a new formula $(x) t h a t still says 3y’p(z, y), b u t which moreover safely bounds y below a numeral, whenever i t is provable (recall t h a t this is exactly where we r a n into troubles above, a n d t h a t bounded quantifiers c a n be handled in R). B u t w h a t exactly do we know when $(z) is provable? Precisely this, a n d t h e n we play a trick ( t h e Rosser trick) a n d use a number coding a proof for it, to b o u n d y: to be sure we s t a y below t h e number of a n y proof, we pick u p only those y’s which do not bound any number coding a proof of $(z). T h i s is t h e intuition behind t h e definition t h a t follows: $(z) w 39 [‘p(z,y) A (Vz

5 y)(z

does not code a proof of $(z))]

If we succeed in defining such a formula (which, as it stays, is selfreferential), then we pay a price by having a more difficult proof for t h e direction that was trivial before, b u t we win in t h e troublesome direction: 1. If P ( z ) holds, t h e n so does R(z,y) for some y, a n d FF ‘p(?E,jj) by s t r o n g representability of R. Suppose $(Z) is not provable: t h e n n o number z codes a proof for it, a n d in particular t h i s is t r u e for z 5 jj, a n d it is provable in 3 because t h e predicate ‘coding a proof’ is recursive, a n d t h u s strongly representable. By t h e axioms o n R, we c a n also show (see e.g. t h e proof of 1.3.3)

(Vz

5 y)(z does not

code a proof of $(C)).

Since we know already t h a t we c a n prove p(?E,ij),we can t h e n prove $(Z), contradiction. T h e n $(%) is provable.

2. If $ ( T ) is provable, there is a number z coding a proof of it a n d (by s t r o n g representability of t h e recursive predicate ‘coding a proof’) we can prove t h i s in 3. But then t h e definition of q!~ implies t h a t t-F (3y < Z)‘p(%,y). T h i s t i m e we can indeed apply t h e axioms of R, a n d get FF cp(?E,jj), for some y. Hence R(z,p)a n d P ( z ) hold.

It only remains to show how to find such a formula $. We use d i a g e nalization as in t h e Fixed-Point Theorem: let { $ n } n E w be a n effective enumeration of t h e formulas of 3 with two free variables, a n d cp1 ( z ,Z, n) be a formula strongly representing the recursive predicate ‘ Z codes a proof of &(z,n)’.Let e be such t h a t $e(z,

(where, recall,

‘p

4

.3

3Y [‘p(Z,Y)

A (Vz

I Y)l(Pl(z, 2, .)I

strongly represents R ) . T h e n

$(?)

* +e(z,e)

11. Basic Recursion Theory

162

satisfies tlie requirements (since then p1( z , x ,e) represents the predicate ‘ z codes a proof of $,(:z, e ) ’ , i.e. ‘ z codes a proof of @(x)’). Following the informal sketch given above, it should now be easy to forrnalize the argument and show that $J weakly represents P . 0 ?‘lie proof of this theorem uses a number of the arguments we have introduced until now: arithmetization (in showing that t h e predicate ‘coding a proof’ is recursive), diagorializatiori in self-referential form, like in the FixedPoint, ‘I’heoreni (in the construction o f @), ant1 tlie so called Rtxsser trick (used, i n a simpler context, i n the proof of the Reduction Property 11.1.23). For similar proofs, see the next subsedion. The difficulty that lies behind the theorem just proved is that weak representahility, unlike representability, ,is not preserved in consistent extensioizs: if xEA FF p(z),and A is not recursive, then A is not, strongly representable. for which -+o(Z) fails. If we take the consistent Thus tlicre must be x E system FU {p(Z)}, then p no longer weakly represents A i n it,.

We now have all the necessary rcsults to approach Godel’s First Theorem in a riioderii versioii. This is one of the jewels of logic in this century, and it should be contcniplatetl with due reverence. Leva dunque, lettore, all’alte ruote ineco la vista4.. . (Dante, Paradiso, X )

Theorem 11.2.17 Limitations of logical systems (Post [1922], Godel [1931], [1934], Rosser [1936], Church [1936], Tarski, Mostowski and Robinson [1953])

I . Every consistent extension F o f R (i.e. a n y corlsisterit set of fonritilas closexi tinder logical coru’egucnct,, and containing all aziorns of R) as tindccidab 1e.

2. If, inoreoiier, F is a f o r m a l system ( i e . the set of its theorems is r.e.), then F i s incomplc1,r. Proof. We give two proofs, exploiting different properties of 0

F.

Let, {@,} be an efrective eiiumeratiori of the formulas in the language of F,with one free variable. If F is dwitlable, t,he diagonal set 11

E

F w FF T+!J,(E)

4Then, reader, lift your eyes with m e t c , see the lofty wheels . . .

11.2 Diagonalization

163

is recursive, a n d then so is F. Every recursive set is representable in F ,by t h e proof of 11.2.16 (which does not use, in this direction] t h e fact t h a t F is a formal system) a n d hence, by consistency of 3,weakly representable. Then there is a n a such that

n E F e I-F For n = a we get a contradiction. If, moreover, 3 is a formal system then, by arithmetization, t h e set of its theorems is a n r.e. set. If F were complete then we would know t h a t either a sentence is a theorem, or its negation is. But then 3 would be decidable: t o know whether a sentence is a theorem, generate t h e theorems until either t h e sentence or its negation appear. 0

This second proof works only for formal systems. Since K is r.e., by t h e proof of 11.2.16 there is a formula cp that weakly represents it: IC

E

K

H l -p ~ (Z).

Then F cannot be recursively decidable] as otherwise sive] contradicting 11.2.3.

K would be recur-

Moreover, K: cannot be represented by cp, otherwise it would be recursive. Then there is at least one IC such that IC

E

K: A

By weak representability, from

IC

not

not E

K

FF -p(z).

we also have

p(Z).

Then 3 is incomplete, since p(Z) and -cp(T) are not provable.

0

T h e two proofs of the theorem are quite different. T h e first requires only t h e weak representability of all recursive sets, which is given by 1.7.12, as well as a simple diagonal argument, given directly in the proof (showing t h a t the set of non-theorems of any consistent system is not weakly representable in i t ) . T h e second requires t h e weak representability of some nonrecursive set, and thus t h e full version of 11.2.16, whose proof uses t h e Fixed-Point Theorem techniques. For a discussion of t h e two methods of undecidability proofs, see p. 352. As far as incompleteness is concerned, the proofs given in 11.2.17 are indirect] a n d d o not explicitly exhibit undecidable sentences, which are neither provable nor disprovable. To obtain this, even under t h e hypothesis of w-consistency, a full use of the self-referential diagonalization must be made

164

11. Basic Recursion Theory

(see t h e next subsection), b u t even t h e examples t h u s obtained have been regarded as somewhat artificial, from a mathematical point of view. G r e a t efforts have been made t o obtain nat,iiral undecidable sentences, for various systems in common use: two extreme a n d classical examples a r c t h e Continuum Hypothesis for Set Theory (Godel [1938], Cohen [1963]), a n d a finite version of Ramsey’s Theorem for Peano Arithmetic (Paris a n d Harrington [1977]). See I-Iarrington, Morley, SEedrov a n d Simpson [1985] for a n account of recent results in t h i s area, d u e mainly to H. Friedman. T h e fatal consequences for formalism embodied in Theorem 11.2.17c a n be expressed as follows: a n y classical f o r m a l s y s t e m is inadequate, being either

inconsistent, or undecidable (and hence also incomplete), or not s u f i c i e n t l y strong (to prove at least t h e elementary arithmetical facts expressed by t h e axioms of R).Turing (see Hodges [1983], p. 361) interprets these facts as saying almost exactly t h a t if a machine is expected to be infallible, it cannot also be intelligent t h u s isolating in t h e rigid, purely deterministic approach to knowledge t h e source of formal systems limitations. A limitation of a different kind (which holds, by G d e l ’ s First Theorem, for every sufficiently strong, consistent formal system) comes from t h e following observation: in a n y undecidable f o r m a l s y s t e m there are infinitely m a n y theo-

r e m s w i t h arbitrarily long proofs, with respect to t h e length of t h e i r s t a t e m e n t s . Indeed, take any recursive function f . If f ( n ) were a b o u n d for t h e length of at least one proof of any theorem of length n, then t h e system would be decidable: to find o u t , for a formula of length 72, whether it is a theorem or n o t , produce all t h e proofs of length at most f ( n ) ,a n d see if one proves it. T h u s t h e r e must be (infinitely many) n for which a theorem of length n has i t s shortest proof longer t h a n f ( n ) . T h e combinatorial p a r t of Godel’s Theorem (11.2.3),together with t h e Matiyasevich rcsult quoted on p . 135, shows t h a t arithmetic is already complicated at low levels of complexity: there i s n o decision m e t h o d f o r one-quantifier f o m i u l a s in the language o f plus and t i m e s . Consider indeed a diophantine representation of K :

y E K: e 3Z[P(Z,y) = q ( Z , y ) ] . If we could decide one-quantifier formulas, then

K

would be recursive.

We conclude our presentation of limitation results by proving a n o t h e r famous one: Church’s Theorem. It concerns first-order Predicate Calculus, a n d shows t h a t t h e dream of Leibniz [1666], of having a calculus ratiocinator t h a t would decide t h e logical truths, is a n unfulfillable d r e a m .

11.2 Diagonalization

165

Theorem 11.2.18 Unsolvability of the Entscheidungsproblem (Church [1936a], Turing [1936]) T h e Predicate Calculus is undecidable. Proof. Let Q be Robinson Arithmetic (p. 23): since Q is a consistent extension of R, it is undecidable. Let $J be the conjunction of its axioms (recall, and this is t h e crucial point, that Q is finitely axiomatized). By the Deduction Theorem of Predicate Calculus, kp- cp holds if and only if ?I) ---$ cp is provable in Predicate Calculus, and thus any decision procedure for this would give one for Q, contradicting 11.2.17. 0 T h e extent of undecidability and decidability of subsystems of the Predicate Calculus has been thoroughly analyzed. See Ackermann [1954], Dreben and Goldfarb [1979] for the former, and Lewis [1979] for the latter.

Self-reference * And God said unto Moses: ‘I a m that I am’. (Exodus, 111, 14) T h e Fixed-Point Theorem can be used directly t o find programs exhibiting self-referential features. E.g., by considering f recursive such that ~ f ( ~ ) ( = x Ce ,) we get a n index e such t h a t cp,(x) = e , which can be interpreted as t h e code of a program printing itself (Lee [1963]). Thatcher [1963] has explicitly written down such a program. A more sophisticated self-referential program, able not only t o print itself, but also t o simulate any given recursive function g , is coded by e such t h a t cp,(x) = ( e , g ( x ) ) ,and can be obtained in a similar way. T h e technique underlying the Fixed-Point Theorem (or its equivalent form, t h e Second Recursion Theorem) is the tool allowing t h e unraveling of selfreferential statements, and their replacement by incontrovertible versions. In particular (being obviously impossible to have a finite phrase with itself as a proper part) self-reference i s never direct: it comes from a controlled confusion of two levels of meaning for integers, which are seen both as numbers and as names for formulas. Then a formula telling some arithmetical fact about an integer may be seen as the translation - by arithmetization - of a m e t a m a t h e matical property. It is easy t o adapt the methods used in 11.2.10 t o build a sentence that, for a given property P weakly representable in a n extension of R, says of itself that it has the property P (Carnap [1934], Godel [1934]). It is enough to consider a n enumeration {$J,},~~of the formulas with one free variable, and let (with t h e notations of p. 145) d($J) = the sentence ‘$Jhas the property P’

11. Basic Recursion Theory

166

a,,j

=

t h e sentence

‘$j

h a s t h e property expressed b y Qi’.

Tlie transformed diagonal seqiicnce is still a row of the niatrix (up to provable equivalence), and this thcrc is a such that d ( $ ) is provably equivalent to q!~, i.c. $ says o f it,self t h a t it has the property P. An old exaniple of paradoxical self-reference go- back to Epirnenides ( 6 t h century B.C.), a C r e t a n w h o said that C r e t a n s a r e always liars. T h i s is known as the liar paradox, a n d it, was quoted (not very synipatlietically) b y P a u l (f
unruly arid vain talkers antl deceiverslS . . . wliose mouths must be s t o p p e d , who siil)vert. whole Iiouses, teaching thitigs which they ought not, for filthy lwre’s sake.

For more elaborated discwsions of the liar paradox, see M a r t i n [1978], [ I 9841. Tlie modern version of the liar paradox is t h e positive r e j u l t (usually referred to as Tarski’s Theorem), that, truth cannot be 7 i I ~ d & ~representable in any c.onsis/ent ezteilsioii of72 (Giitlel [I93 11, l’arski [1936]): otherwise i t s negation l h a i d b y t h e general result o l k i i n e d abovc woiild be weakly r q ~ r ~ e n t a t,oo, we would get a contradictory seritence asserting its own falsehood. I’rovaldity is instcad weakly representable in consistelit extensions of R, antl in t h i s case the geiicrnl rt.siilt, o1)tained abovc gives a sentence asserting i b s o w n unprovability (CLklcl [ I93l]), a n explicit exairiple of a s e n t e n c e not. prova1)Ie in a system which proves only t r u t h s , antl h e n c ~ et r u e . l’kom t h i s we iniitiecliately gct tht: liniitjat,ion results already proved in the prcwious su1)section. Spxifically: 1. tiiid~‘cidabi1Ztyof a n y roilsisfent cztensiori 3 o f R (Church [l936], [193Ga]) S u p p o s e F is clccitlablc: this inpans that ‘being a t,heorem o f F’ is recursive, hence represeiit,abIe (since .F e x t e n d s R) b y some $:

k~ 7/J5 not, k.F

+

k~

+

i-F

$(F) ’$(Z).

(11.1) (112)

(11.3)

11.2 Diagonalization

167

2. a n explicit example of incompleteness, f o r any w-consistent formal system 3 extending R (Godel [1931]) Since 3 is a formal system, ' y codes a proof of $,' is recursive, a n d so representable (since 3 extends R) by some cp:

y codes a proof of y does not code a proof of

-

T h e n t h e formula

11,(x)

+2

+,

*

I-F

3

I-3

cp(Z,?j)

-cp(Z,g).

3Ycp(X,Y)

weakly represents provability in 3 (see t h e proof of 11.2.16, where t h e hypothesis of w-consistency is used to show this). Let now 11,2 assert its own unprovability, i.e.

Then: 0

ll,z is not provable in 3: if it were, both 11,(?t) a n d -11,(Z) would be provable (contradicting consistency), t h e former because 11, weakly represents provability, t h e latter because of 11.4.

0

-$, is not provable in 3: if it were, 11.4 would imply t h a t +(Z) is provable, a n d $, would then be provable (contradicting consistency), because $ weakly represents provability.

Note t h a t we can decide, from t h e outside, t h a t ll,z is true: it is not provable, a n d it asserts its own unprovability. 3. a n explicit example of incompleteness, f o r any consistent formal system F extending R (Rosser [1936]) Let cp be as above, ll,neg -?,b2, and

+(x)

-

litz is provable before 7q2 is 3Y [ c p ( X , Y ) A (Yz I Y ) " p ( 7 W x , 41.

H

+

We cannot assert t h a t weakly represents provability in for a formula t h a t does this). But at least we do have

Suppose indeed t h a t consistency of F,-+2

+,

F

(see 11.2.16

is provable. T h e n FF cp(Z,g), for some y . By is not provable, a n d hence l - ~ " p ( n e g , Z ) , for

11. Basic Recursion Theory

168

every z t.F

$(%I.

5

y.

By the axioiiis of

R,k~

('dz

5 g ) i c p ( m , ~ ) .Thus

1,et now @= assert that it is not provaMe beforc its own negat,ion, i.e. FF

$Jz

++

-?It(%).

(11.6)

l'hen:

is not provable in F,otherwise so would be $(%)and -@(?E), respectively by 11.5 and II.G, contradicting consistency.

$1,

0

is not, provable in F,otherwise so would be $(%) arid -@(T) (contradicting consistency), the former by 11.6, the latter by t h e following reasoning. Note that

T&

-@(:c)

e V'!/[Cp(Z,Y)

4

.)I.

(32 I ?J)P(7%7Z,

If -$I= is provable, let, 2 code a proof of it. Then FF p ( E E j T , Z ) . By the axioms of R,y < 2 V 2 5 y, for any y. I n the first case ?J is a numeral less t h i 2, by the axioms of R,and thus FF -cp(Z,y), because if p(?E,y) holds then $z would be provable, contradicting consistency. 1 1 1 the second case k3- (32 5 g ) c p ( v ,z ) , l~ecause FF p ( n e y ,Z ) . Th(:Il F:y- (yy)[-p(T,y)v (3: I y)'p(7"(1",z),

and hence FF

-'~J(z)

As before, qhz is true becausc it is not provable, i n particular not prova1)le before its negation. It is interesting to note that, the sentence asserting its o w n unprooability is eqwiident to the assc:rt;io7i of rorisistericy o f the s y s t e m : if the system is consisknt, then is riot 1)roval)le (otherwisc both $(Z)atid - @ ( T ) would follow, the first by the properties of ,JI, the second by definition of QZ),atid heticc $, holds; ancl if $, holds then it is not provable, and the systeiii is consistent, (otherwise evcrythitig would be provable). This equivalewe proof is infornial, arid the siniple assumption of represent,ability of provabiIit,y i I i F is r i o t sufficient to reproduce the proof inside t h e syst,em. But undcr some stronger assumptions (scc bclow), it is possible t,o prove it inside F,i.e.

where Con9 is a formal translation of consistency, for exaiiiple t h e assertiori that '0 = 1' is not proval)le. It, then follows, froin tlie u~iprovabilityof $=, that

11.2 Diagonalization

169

the consistency of a consistent formal system is not provable inside the system itself ( G a e l [1931]),a result known as Godel’s Second Theorem, a n d which destroyed Hilbert’s program of justifying abstract mathematics by proving t h e consistency of formal systems of common use by elementary (finitary) means, e.g. in PA, or in any other sufficiently weak formal system in which only finitistically acceptable reasoning could be formalized. T h e assumptions on t h e provability predicate needed for t h e proof of G a e l ’ s Second Theorem ( G a e l [1933], Hilbert and Bernays [1939], Lob [1955], Jeroslow [1973]) are worth examining also for another reason, directly related to t h e subject of self-reference. Since they are outside t h e scope of Recursion Theory, because not purely extensional, we just quote them:

+

*

k3 +)(: This says t h a t if a formula is a theorem of F ,we can prove inside F t h a t it is. It expresses half of t h e condition for weak representability of t h e provability predicate.

1. t-3+,

2. t-3 +(TI

+

+(pro)

where $’pr(z) = +(T). T h e first condition was external t o 3,saying t h a t any single provable formula can be recognized t o be provable by F.This second condition is internal t o F,and says t h a t 3 is aware of the first condition: inside F we know t h a t if a formula is provable, then we can prove this fact.

+(!a

3. k3 A +(imP@,Y)) where @i’impl(z,y) = + +y. This says t h a t F is aware of t h e fact t h a t the provability relation is closed under modus ponens.

+,

+

These conditions are satisfied by usual first-order formal systems extending P A (see Bezboruah and Shepherdson [1976] for a version of Godel’s Second Theorem for Q), and can be loosely stated as: in 3 we can prove that a provable

forrnula is provable, and w e are aware o f this fact and of modus ponens. Going back to self-reference, we have seen t h a t the sentence asserting its own unprovability is t r u e and not provable. We now want to show t h a t , under t h e conditions stated above, the sentence asserting its own provability is true a n d provable. This provides a n example of true self-referential statement which makes no use of negation. T h e general result is t h a t , under t h e conditions stated above, a sentence +(Z) + (asserting t h a t if is provable then it is true, and t h u s expressing a form of soundness) is provable i f and only i f is (Lob’s Theorem, Lob [1955]). T h e only nontrivial direction is to prove is provable then so is Suppose is not provable. t h a t if $(5) + Then F U {-+,} is consistent, and its consistency cannot be proved in it, by G d e l ’ s Second Theorem. 1.e. we cannot prove in it t h a t cannot be proved

+,

+,

+,

+,.

+, +,

+,

11. Basic Recursion Theory

170

i n F,which is expressed by the formula -$(Z).But if .FU{-$,} cannot prove -$(3)then, by the Deduction Theorem and contrapositive, F cannot prove

$@) -4 $z.

-

Since the formula that asserts its own provaldity has the property that $(Z) Q X ,we have i n particular that $ ( T ) -+ $ 5 is provable, ant1 then so is $, it,self, by IJOb’s Theorem. A similar example, relying on the same conditions on the provabilit,y predicate but riot using CWel’s Second Theorem, is t h e sentence asserting of itself that af i t i s provable t h e n i t is t r u e , i.e. I-7

F F $x

- ($(Z)

$,I.

Suppose $, is provable. Then t-F $(Z)by representability of provability, and F J ~$(%) -4 q5 by definition of $,. By niotlus poneris, FT $,. Thus we have provcd that if $, is provable, i.e. if $(Z)holds, then so does $,. This establishes that if $x is provable then it is true. By q X asserts exactly this, and thus it is true and provable. For more informattion on the topics of this subsection, see Siriorynski [1977].

Self-reproduction and cellular automata

*

Vergine madre, figlia del tuo figIio6 (Dante, Paradiso, XXXIII)

On a linguistic level, t o build self-reproducing objects is not difficult. A nontrivial example (quota1 by Hofstadter [1985]) is: Alphabetize and append, copied i n quote, these words: ‘these append, in Alphabetize arid words: quote, copied’ which I)oth lists its parts at tlie word level, and tells how to put them together t,o reconstitute itself. By acting accordiiig to the rules stated i l l it oii the words quoted in it we get the same sentence, but this is somewhat urisat,isfactory as an example of self-rel)roductioti, since it obviously require an external agent that understjarids the rules and performs them. On a mcchanical level, at first sight, it might appear that, a niachiiie can only reproduce less complicated machines, since the building machine must somehow contain a complete tlescription of the built one, together with some additional device to do the actual building. This was a stunibling block for Ilescartes [ 16371, who tliouglit that animals arid human bodics are machines, but had to resort to miracles to explaiii biological reproduction. The crucial point, niisscd by the above discussion is that descriptions of machines are at, a lower logical level than mar:hines themselves, and can be brought down to ‘Virgin niothcr, daughter o f your

SOII

11.2 Diagonalization

171

t h e s a m e logical level of their inputs. T h u s a m a c h i n e c a n reproduce itself, w h e n e v e r it i s able to build m a c h i n e s by following t h e i r descriptions, a n d is fed i t s o w n description. When this minimal complexity is attained, obstacles a r e removed not only for self-reproduction, b u t even for reproduction of more complicated machines. In this subsection we flesh out these observations. T h e ideas of t h e Fixed-Point Theorem can be used to build a self-reproducing mechanical automaton (Von Neumann [1951]). T h e first machine we need is a universal constructor A , with t h e property t h a t , when it is fed t h e description dx of a machine X , it searches in t h e surrounding environment for t h e needed mechanical parts, a n d builds X . In symbols:

Of course universality refers to a fixed class of machines, all particular specimen from a common species, t h a t can be described in a uniform way. T h u s A h a s t h e ability to understand all plans of a given type, a n d realize them. A alone is not yet self-reproducing because, when it is given its own description, it does build a copy of itself, but, t h e result is lacking t h e description t h a t was fed at t h e beginning: (A,dA) A. -+

We t h u s introduce a second machine B , with t h e simple task of reproducing any description given to i t : (B,dx)

-

dx.

To coordinate t h e two machines A a n d B , a third one C is introduced, a n d t h e compound machine A B C will now work in t h e following way. Given a description d x , A builds a copy of X , while B reproduces a copy of dx; t h e n t h e copy of X is fed t h e copy of d x :

+ + (A

+ I3 + C ,d x )

(X dx , ).

4

T h e n , if we call D t h e resulting machine A

+ B + C, we get

T h u s S = ( 0 d, D ) is self-reproducing. T h e s a m e ideas allow not only for self-reproduction, b u t even for production of more complicated machines (a sort of evolutionary process). Indeed, it is enough to insert in D a description dD+F of a machine composed of D a n d some other machine F . T h e n S will produce itself, together with F . Some observations on t h e ideas used in t h e construction j u s t sketched a r e in order. First, note t h a t t h e r e is no circularity involved, since we first obtained

172

11. Basic Recursion Theory

D ,and then we fed it its own description; also, none of t h e three parts A , B , C is self-reproducing by itself, but their combination is. Second, t h e crucial fact t h a t circumvents t h e difficulties pointed out at t h e beginning is t h a t B , while being of fixed size, can copy descriptions of any length: thus t h e description of the constructed machine need not be incorporated in t h e parent machine, b u t only coupled t o it. Third, descriptions (i.e. indirect reproductions) a r e necessary in actual realizations: if we wish t o reproduce a machine X directly, piece by piece, we need t o interfere with it (to know how it is built) in some possibly disrupting way, while t h e reproduction process presupposes a n unchanging original. For similar reasons, descriptions must be directly given, a n d cannot in general be deduced by machine observation. Note t h a t descriptions a r e used at two different levels: one time they are purely duplicated (and so used as raw material), and another time they a r e followed as projects (and so interpreted as instructions). Finally, alterations of some parts of S might produce different effects: a change in D itself might inhibit or perturb t h e whole production process; a change in d ~ + pmight affect t h e constructed machine, directly in t h e copy of D (thus producing a machine t h a t might not be self-reproducing anymore), or possibly only in t h e by-product F . Note t h a t real life reproduces exactly this way: living cells contain universal constructors, basically the same for plants and animals, and only t h e genetical material (the program) is different. More precisely, a suggestive biological reinterpretation of t h e construction of S is t h e following: d x works like a gene (a segment of DNA) that codifies t h e reproduction information; B (a special enzyme - RNA polymerase) has t h e function of duplicating t h e genetic material into a segment of RNA; A (a set of ribosomes) builds proteins by following (a segment of RNA containing) t h e reproduction information; S is a self-reproducing cell. This is thus a n abstract, simplified representation of genetical reproduction. One of t h e simplifications occurs in t h e fact t h a t the gene has only a partial codification of t h e reproduction information, a n d this allows for a partial modification of t h e reproduced object. T h e additional parts possibly produced a r e the analogues of enzymes t h a t t h e gene produces, or whose production it stimulates. T h e effect of alterations on S are reminiscent of mutations: they may be lethal or sterilizing (killing t h e organism or inhibiting reproduction), produce modified (and possibly sterile) successors, or produce fertile successors with changed hereditary strain (generating different by-products). See Watson [1970] for general information, Arbib [1969a] for a discussion of t h e relevance of self-reproducing automata in biological contexts, and Burks and Farmer [1984] for a model of DNA sequences as automata. T h e troublesome part in the discussion above lies in t h e assumption of t h e existence of a universal constructor. We might reason by analogy, and think t h a t a n argument like the one used for universal Turing machines (p. 132) might suffice. However, for Turing machines t h e actions needed for universality a r e of

11.2 Diagonalization

173

q u i t e limited complexity (writing, reading, moving), a n d at t h e s a m e level as those of t h e simulated machines. For actual constructors much more is certainly needed (recognizing, moving, manipulating a n d assembling components). T h u s t h e possibility again arises of a n infinite regress, in t h e sense of needing a n additional level of complexity, t o build machines at a certain level. Von Neumann [1966] was, however, able to solve t h e problem by abstracting some of t h e properties of t h e mechanical model seen above. He noted t h a t constructions can be seen as series of events taking place in a space, a n d h e t h u s considered - by a s o r t of Cartesian representation - a space of cells t h a t c a n be in a certain number of states (intended to represent presence or absence of certain p a r k of t h e mechanical model, so t h a t motion of p a r t s is represented by change of state in cells). Von Neumann t h u s used particular cellular automata, which a r e simply potentially infinite, directed graphs (spaces), whose nodes (cells) a r e finite state machines (see p. 52): t h e global behavior consists of t h e simultaneous a n d coordinated behavior (change of state) of t h e single cells. Von Neumann’s particular automaton consisted of a planar space with 29-state cells of a single type, each connected to t h e four orthogonally adjacent neighbors: w h a t h e d i d was to find a finite quiescent configuration (of a r o u n d 200,000 cells) t h a t , given any other finite quiescent configuration, reproduces it in a different p a r t of t h e space, without erasing itself he thus found a universal constructor for t h e class of quiescent configurations. See some of the essays in Hurks [1970] for more explanations on this model, Codd [1968] a n d Arbib [1969] for improved treatments (with opposite emphasis: t h e former on local simplicity of cells, t h e latter on global simplicity of construction), a n d Langton [1984] for a simple, nontrivial (although not construction universal) cellular model of self-reproduction. T h e simplest automaton admitting self-reproducing configurations we know of is Conway’s Life automaton, so called because modelled on lifelike behavior. It consists of a planar space, with each cell connected to t h e eight adjacent ones. T h e cells have j u s t two states: 0 (death) a n d 1 (life), a n d t h e Life rules a r e t h e following: a dead cell gets born when exactly 3 neighbors a r e alive; a live cell survives if 2 or 3 neighbors a r e alive, a n d it dies otherwise (by overpopulation or starvation). Conway (see Berlekanip, Conway a n d G u y [1982]) has shown t h a t there are self-reproducing Life configurations, a n d t h a t the Life automaton is universal, in t h e sense t h a t any cellular automaton c a n be constructed by taking sufficiently large squares of Life cells a5 its basic cells. These results a r e also interesting because, unlike other cellular a u t o m a t a a d m i t t i n g self-reproductive behaviors, Life’s rules were not introduced with t h e purpose of making self-reproduction possible. Also, Life is a b o u t 8s simple as i t can be (it is known t h a t 2-state cells with a Von Neumann neighborhood do not admit nontrivial self-reproduction), a n d it shows t h a t self-reproduction does not need a complicated universe (since it is logically possible from simple physical

174

11. Basic Recursion Theory

models). For more information on Life, see Poundstone [1985]. A notion somewhat opposite t o self-reproduction is t h e impossibility of being reproduced. A finite pattern in a cellular automaton is called a Garden-ofEden configuration if it cannot be obtained from any previous configuration: it corresponds to a machine t h a t cannot be built, in t h e sense t h a t it cannot arise as a result of any past state of its universe. Obviously, a self-reproducing configuration (being obtainable from itself) cannot contain any Garden-of-Eden configuration: this places limits on t h e possibilities of universal constructors (since it exhibits nonconstructible configurations), as well as o n t h e possible patterns of self-reproducing machines (this being relevant for t h e deterniination of t h e simplest possible conditions for self-reproduction, a n d hence for t h e computation of t h e probability of getting living organisms by chance interaction of nonliving ones). Garden-of-Eden configurations exist o n a cellular automata (satisfying some general conditions) i f and only if the automata is not backwards-deterministic, i.e. if there a r e configurations t h a t c a n be obtained in more t h a n one way (Moore [1962], Myhill [1963]). T h i s condition obviously applies when erasing is possible, erasing being a n irreversible process t h a t loses information. Note t h a t cellular a u t o m a t a a r e quite general devices: given a n y Turing machine, it is possible to build a cellular automaton t h a t simulates t h e computation of t h e machine on any given input ( t h e a u t o m a t o n is a linear tape; t h e state of each cell will reflect t h e situation o f t h e corresponding cell in t h e tape by telling which symbol is written there, a n d whether t h e head is scanning such a cell or not a n d , if so, in which state t h e machine is). In particular, a cellular automata can simulate a universal Turing machine (Smith [1972]). T h e universal constructor we have been quoting above can be m a d e complex, in t h e sense of being simultaneously able to simulate a universal Turing machine ( a n d t h u s being both computation a n d construction universal). For more on cellular a u t o m a t a see Toffoli a n d Margolus [1987], a n d t h e papers in Burks [1970], Farmer, Toffoli a n d Wolfram [1984], Deniongcot, Golks a n d Tchuente [1985],a n d Wolfram [1986]. Actual computer machines niodelled on cellular a u t o m a t a a r e studied in Preston a n d Duff [1984], a n d Toffoli a n d Margolus [ 19871. Problems of reversibility for cellular a u t o m a t a , analogous to those considered on p. 51 for Turing machines, a r e considered in Toffoli [1977], [1981] (where t h e existence of computation a n d construction universal, reversible cellular a u t o m a t a is shown).

11.3

Partial Recursive Functionals

In Chapter I we introduced t h e notion of recursive functions, in various equivalent formulations. In Section 11.1 we t h e n extended t h i s notion to encompass

175

11.3 Partial Recursive Functionals

t h e case of partial functions. Here we present a further, substantial generalization of recursiveness by considering effective procedures t h a t act not only on numbers, b u t o n functions as well. T h a t is, we extend t h e notion of recursiveness from functions to functionals.

Oracle computations and Turing degrees Let us revisit t h e definition of recursiveness: we had a set of initial functions, a n d a set of operations, transforming given functions into new functions. T h e idea was t h a t t h e initial functions were effectively computable, a n d t h e operations transformed effectively computable functions into effectively computable ones. If a function g is added to t h e initial functions, then t h e class obtained is t h e s a m e if g is recursive, b u t it is otherwise more comprehensive.

Definition 11.3.1 (Turing [1939]) If g is a totalfiinction, the class of functions recursive in g is the smallest class of functions 1. containing the initial functions and g

2. closed under composition, primitive recursion and restricted p-recursion.

If A is a set, the class o f functions recursive in A is the class of functions recursive i n C A . A predicate is recursive in g or A i f its characteristic function is. T h e extension of recursiveness relative to a given function corresponds to t h e algebraic procedure of transcendental extension. T h e functions recursive in g a r e not all outright computable, unless g itself is, b u t they a r e still 'computable modulo 9'. A pictorial way to express this state of affairs is to say t h a t t h e y a r e computable with t h e help of a n oracle, a t e r m introduced b y Turing [1939] a n d now s t a n d a r d .

xcri Sij Tor&xcui EL< AsAcpoir< ~ X 8 i j u ~ ~ 6 A procro r p ~ paure6uaa6aL. ?!I&<;

T i TOTE

AeyEI.

ov y h p

6?jTOV $l&ljS&T&i 7 E . 7

0

(Plato, Socrates' Apology) T h e oracle, as t h e word emphasizes, is a n extrarecursive entity, helping t h e computation of a n y function recursive in g in its troublesome spots, when a call to g (which could be effectively answered only if g were recursive) is made. T h e oracle supplies t h e answer to any such call for free. 'And thus once, goue eveu to Delphi, he dared to consult the oracle on this: what does the god say? H e certainly does not lie.

176

11. Basic Recursion Theory

An oracle f may also help tlie computation of f itself, in sonie nontrivial way. For example, some of t h e iriforrriation coded by f niay be r e d u n d a n t , a n d recoverable from t h e rest of it in various ways (see 111.5.9 a n d V.5.15 for s o m e precise foriiiulations) . Definition 11.3.2 Given two fuiictions f and g, we say that:

1. f is Turing reducible t o

( f S T 9) i f f i s recursive in g

2. f is Turing equivalent to g

(f=T

g ) i f f 5 7 1 g and g

ITf.

Note t h a t 57. is a reflexive arid transitive relat,ion,a n d t h u s --T is an equivalence relation. Although trivial, t h i s is a crucial fact for later dcveloptiient, sirice then ET partitions t h e class of t o t a l functioiis ( a n d in p a r t i c u l a r t h e class of characteristic: functions, i.e. t h e class of sets) i n t o equivalence classm, called degrees of unsolvability.

Definition 11.3.3 (Post [1948]) T h e equiualeiice classes of total functions w.r.t. Turing expivalence are called Turing degrees (or T-degrees). (P, 5) is the structure of Turing degrees, with the partial ordering 5 i n d i i r d o n t h e m b?/ I T T w o fiinc:tioris a r e Turing q u i v a l e n t , (in t h e sairie T-degree) w h e n t h e y a r e recursive in each other: t h u s , t h c y help e ~ c hother's c o m p u t a t i o n , and t h e y a r e in the saiiie relalioiiship as t h e rcciirsive functions are airiorig thenisclvcs. T h e coiitiiiuuin is t h u s classified from a recursion-theoretical point o f view. arid t h e s t u d y of such a classification (arid of related ones) is o n e of t h e nii~iris i i l ~ j w t s of our book. Exercises 11.3.4 a) Every T-dcgrrc rontazns a s e t . (Hint: corlsider t h e graph of a given fiinctiori, and code it into a set.) h) 7 % c 7 ~2s u sninllcst Y ' - d q r e r . (Hint: the degree o f recursive scts.)

The notioti o f relative recursiverims is easily g e i i e r a l i z d to p a r t i a l functions: Definition 11.3.5 Zf /3 is a partial fimction, the class of functions partial recursive in p is the srrrallesl class of functions

1. coirtaiiiiiig the initial functions m i d

p

2. closed under composition, priinitive recursion and unrestricted p-recursioii. Relative piirtial recursiveness Ls still called T u r i n g reducibility, write o
aiid

we still

11.3 Partial Recursive E’unctionals

177

By substituting the concept of recursiveness with its version relativized to a given (total) function or set, notions and results dealt with so far generalize. For example, we can define a set as r.e. in A if it is t h e domain of a function partial recursive in A , and prove that

A


B w both A a n d 2 are r.e. in B.

In t h e following we will refer t o theorems in relativized form when needed, but it must be clear t h a t the fact that a result relativizes should not be taken for granted: claiming a relativized form of a result which holds unrelativized requires a proof, even if only a check t h a t everything relativizes in t h e original proof. In Chapter V we will actually see that some results do not relativize

(V.7.13).

The notion of functional A functional is simply a function whose variables range over numbers or over functions of numbers, and whose values are numbers. For simplicity of notations, we will mostly consider the function variables of a functional to range over unary functions, and will leave t o t h e reader t h e care of extending notions and results to the general case. Our convention is not a restrictive one, since any function with many variables can be recursively reduced to a unary function by coding its arguments into a single one. We may suppose t h a t t h e function variables of a functional range only over total functions, or we may allow them t o range over partial functions as well. We use t h e word functional t o refer t o t h e latter case, a n d talk of restricted functional in t h e former. As for functions, a functional can be undefined for some of its arguments. We call total functional a functional which is always defined whenever its function arguments are total. Thus a total functional can be undefined for some of its partial arguments, but a total restricted functional is always defined. Examples of total and nontotal functionals are, respectively, t h e application functional

A d a , 4 ?! a ( 4 , a n d the (unrestricted) p-operation schema

It is precisely t h e existence of nontotal, restricted functionals t h a t makes t h e consideration of nonrestricted functionah natural. We can also talk of (restricted) relations of numbers and functions, by just referring to the characteristic (restricted) functional.

178

11. Basic Recursion Theory

Partial recursive funct ionals We now use the notion of relat,ive partial recursiveness to introduce t h e idea of recursive functional.

Definition 11.3.6 (Kleene [1952], [1969], Sasso [1971]) The Jiinctioiial F(crI,.. . ,n,, :it) i s a partial recursive functional if iL c u i i he obtained Jroin 0 1 , . . . , N, and the initial functions by composition, priinifivc. rectirsion a n d wires tric trA - recim i o 11. A relation oJ Juiickioii (ind nuinbrr variables is recwxive iJ i t s characteristic functional is. The definition just givcri contains more than the mere fact, that the func, ( I ? , ,2) is partial recursive in the 0,’s for each choice of its functioris variablm;. This is certainly inrplied, but. iriore is true: actually A?. I.’(trl,. . . , N, , Z) is uniformly partial recursive i i i the N , ’ s , in the sense that there is a niastm way o f showing the partial recursiveness o f it, in the cri’s, i n which thew appear as paranret.ers. Exercises 11.3.7 a) jf 2s pailzcil wciirsiiie iii a i f and oidy ij lhcir i s (I p(wtiu1 recursiiw functioiial F siicli f.lrci./,/$(.) 2: /..(a, .). 1)) T h e m (im fuiir:lioiis f mid JI such that f is ieculviire iii g , but t h e m is no pc~rlialiecursiwe, t o l d fiiiiclioii(i1 F such that. f (T) --- F ( g , x). (Post [ 1!M4]) (Hint: from 11.3.10 we will gct a notion of index for partial recursive fiiric:tionals. L e t Fe he the functional with index c , and A l)e the set of indices o f partial rc~cursive,total hinctiorials of one funrtiori variable and one niimhx variahlc. I f T

then B CH(S)

=~

S T A, hut

E / j e s E A A F,(c~,s) . o

thr exiskn(:e>of a partial recursive, total functional F siirh that

F ( c A ,S ) Ieacts to a c.oiit.radiction.)

r >

I hc part iwl recursive

which allows for i n the fiinction ones. foriii,

fiin(tioiia1s

arc close:tl i i i i d c r cwnrposit,ion i n a strong oiily i n tlrc: i i i ~ i i i I ~ ( :argiinioiit,s, r I)ut also

siil)st,it,iil,ioii riot,

Proposition 11.3.8 Substitution Property (Kleene [1952]) If F ( N ,z ) (md G(/3,x) are pai~lialr m m i v e Jiirictionab, then so is

Proof. R y iiidiiction on the: tlefinit,ion of G. From a coiiil,iit.;it,ioii~~l point of view, this is cliiit,e clear: whc:n we get, to a call o f /’l i n tlrc coiiipiltat,ion of G,we substitute it by t,ho corrcspoiidiiig call of X2. F(rr,z ) , :ind contiriue the c.oiiil)iit,at,ioii. Thus tho oiily calls to tlie oracle which arc’ not tliscliargcrl arc! those rclat,ivc to or.

0

11.3 Partial Recursive F’unctionals

179

Exercises 11.3.9 a ) Substitution holds for partial mcursive, restricted total functiona k . (Kleene [1955]) b) Substitution fails f o r partial recursive, restricted functionak. (Kleene [ 19631, Kreisel) (Hint: let G ( f , z ) 0, and F ( z , z ) 0 z E Then the function H ( z ) N G ( k . F ( z ,z ) , z) would converge if and only if Xz. F ( z ,z ) is total, i.e. when 2 @ K,and K would be recursive.)

zz

Theorem 11.3.10 Normal Form Theorem for partial recursive functionals (Kleene [1952], [1969], Davis [1958], Sass0 [1971]) There is a primitive recursive function U and (for each m,n 2 1) recursive predicates Tm)TL such that, for every partial recursive functional F of m function variables and n number variables, there is a number e (called index of F ) for which the following hold: 1. F ( ~ I. ,. ,.a r n , z ~. ,. . , z n )1 2. F ( ~3 . I. . , o m

511 .

W L , , ( ~ , .~.,Ix n, ,. G I , . . . ,am, Y)

. . ,zn) U ( ~ 7 m , n ( e51 r .. Yxn 7 .

7

1 .

. . , am, Y)).

Proof. W e cannot use t h e functions cri in computations, since they might not be recursive, a n d we then use finite approximations to them. We refer to t h e proofs of 1.7.3 a n d 11.1.2, a n d just indicate t h e appropriate changes to be made. 1. T h e index of a partial recursive functional can be defined by j u s t a d d i n g clauses for t h e function arguments cri’s, as if they were initial functions. 2. Computations a r e p u t in canonical form as usual. Now some of t h e nodes can be expressions of t h e kind q ( z )= z . 3. Numbers to nodes of t h e computation trees a r e assigned in t h e form

where e is t h e index of t h e functional, and a, is t h e code of a fixed finitc approximation t o a,, e.g. of t h e following function: Liz(.)

2:

{ ezp(a,,p,) undefined

- 1 if ezp(a,,p,)

>0

otherwise.

Numbers to computations a r e assigned in t h e usual way. 4. T h e definition of 7 remains t h e same, except for a local modification of clause B , t o take care of t h e new case corresponding to nodes of t h e kind a,(z) = z , where t h e evaluation is made by means of t h e finite function ii,,in place of 0 , .

11. Basic Recursion Theory

180

6. Finally, since now nodes a r e quadruples, a n d t h e value is t h e last component, u ( y ) = (y)1,4.

In t h e case of restricted functionals, we get a smoother version (where, recall, & is t h e course-of-value of gi, see 1.7.1):

Theorem 11.3.11 Normal Form Theorem for partial recursive restricted functionals (Kleene [1952], Davis [1958]) There is a primitive recurszve fuiiction U and (for each in,n 2 1) primitive recursive predicates I,,,, of only numerical variables such that, for every partial recursive restricted functional F of m function variables and n number variables, there is a number e (called index of F ) for which the following hold: 1.

q a , .. . , g r n , q 1

@

3y7,L,n(eI~,g1(1J), . ,grn(!l),Y) ' '

2. F ( g l i . . . i g r n i Z ) ~ ~ ( ~ ~ y 7 r ,n~ ,i gnl ( ye ) i . . . i g , ~ ( y ) i y ) ) .

Proof. It is enough t o modify, in t h e previous theorern, t h e definition of

Tn,,,

as:

Here 21a now means (Yz I a)(ezP(a,Pz)# 0

eXP(a,PT) = (2)z+1

+ 1).

T h e idea is t h a t we have to t a k e a computation coded by y, consider t h e finite functions used in it (coded by ( ~ ) 1 , 3 , % < y), a n d note t h a t only values u p to y of t h e furiction arguments 9, can be needed in t h e computation. B u t these values a r e all coded in t h e sequence numbers g,(y). T h e definition of is slightly complicated by t h e fact that, t h e g,'s d o not a p p e a r directly in i t , b u t

z,,,,,

181

11.3 Partial Recursive F’unctionals only through t h e numbers j,(y), whose role is taken by z , .

O f course t h e improved normal form does not work for nonrestricted funcfor some tionals, since &(y) could h e undefined for partial N (when a(.) 2 < y), even if all t h e converging values of ct needed for t h e computation a r e relative to argunient,s u p to y. T h e r e is a special case which is going to be particularly useful, a n d we introduce special notations for it, in analogy with t h e notations used for partial recursive functions (11.1.4). I t is t h e case of oracle computations w.r.t. a set A .

Definition 11.3.12 1. 9: (or

(elA) is the e-th function of n variables, partial recursive in A :

(P;(z)2 {e}A(.’) 2. cp&

(or (el:)

-U(~Y7,,,(e,~c’,EA(y),Y))

is the finite approximation of

(P:

of level s:

First Recursion Theorem T h e Normal Form Theorem for partial recursive functionals implies t h a t a cornputation tree of F(cr,z) is finite, a n d t h u s uses only a finite number of values of a . We state explicitly t h e basic properties of oracle computations t h a t c a n be deduced from it,. These a r e key to t h e proof o f t h e First Recursion Theorem.

Corollary 11.3.13 (Kleene [1952],Davis [1958])If F(cr,z) is a partial recursive functional, and F ( N ,x) N y, then: I . compactness: for some finite function u 2. monotonicity: if N

C p,

a, F ( u ,x) 2~ y

then F(P,x) N y.

Proof. By definition of 7,,,,,a value of F is computed by using a finite approximation of N . T h u s compactness is immediate, a n d monotonicity follows from t h e fact t h a t a finite approximation of ct is also a finite approximation of a n y extension of it. 0 Exercise 11.3.14 Compactness and monotonicity are equivalent to the unique w n dition F ( a , z )21’y @ (3u f i n i t e ) ( u C a A F ( u , z )N y).

11. Basic Recursion Theory

182

We can define a function ct by writing down conditions t h a t its values must satisfy. If these conditions involve a itself, then t h e definition takes t h e general form 4 x 1 = F ( a ,21, for some partial functional F (not necessarily recursive). Any function satisfying t h e equation just written is called a fixed-point of F , a n d could be considered as defined by t h e given conditions. But t h e usual intent of a definition is not only to specify all t h e necessary information, b u t also to rule o u t all additional, not explicitly stated information. T h u s we may consider a function as defined by t h e given conditions if it is t h e least fixed-point of F : in this case t h e function does not contain arbitrary information, o n top of w h a t is directly implied by t h e definition. Let 11b consider some exaniplcs: 1. F ( a , z )21 a ( z )

Every partial function is a fixed-point of F , a n d t h e least fixed-point is t h u s t h e completely undefined function. In particular, F is a total functional with total fixed-points, b u t t h e least one is not total.

+

2. F ( c t , z ) N a ( z ) 1, Now there is exactly one fixed-point, t h e completely undefined function. In particular, F is a total functional with no t o t a l fixed-point.

3. Given R ( z , y ) recursive, let

T h e n F is a partial recursive functional, a n d if a is t h e smallest fixedpoint, then ct is partial recursive a n d

a ( z , y )= P

( Z

2 Y/

A

R ( x ,2)).

In particular, a(z,O)N pyR(z,y) (see also p . 158).

Theorem 11.3.15 First Recursion Theorem (Kleene [1952])Every partial recursive functional F ( a , x )admits a least jixed-point, and this is recursive. I n other words, there is a partial recursive function a such that: 1. Y z ( a ( z )E F ( N , x ) )

2. Yz(P(.)

N

* ct

F(P,z))

Proof. Define by induction :

p,

183

11.3 Partial Recursive Functionals N O everywhere undefined

= F ( a n ,).

&+l(.)

’

Intuitively, we first take all t h e values of F t h a t can be computed without any call to a (i.e. by using t h e completely undefined function as oracle). T h e n , at any given stage, we compute those values t h a t use calls to a t h a t c a n be answered because t h e y have already been computed at previous stages. By induction on n, we show t h a t a, crn+1. 0

0

For n = 0 this follows because

a0

is completely undefined.

an+l, let N,+~(z) = F(cr,,x) = y. By monotonicity we have If a , F(a,+l ,x) N y. T h e n a,+2(2) 2: Y, a n d @,+I G a , + 2 .

It t h u s makes sense to consider t h e limit a of t h e an’s: a(.) rz y e 3n(a,(x)

N y).

is partial recursive, because t h e 0,’s a r e partial recursive, uniformly in n. Moreover:

cy

1. a is a fixed-point of F . If ~ ( x1 t)h e n , for some n,

c

a, by monotonicity a(.) N F ( a ,x). If F ( a ,x)1 t h e n , by compactness, only finitely many values of cy a r e used in t h e computation, a n d thus a,, for a big enough n, will suffice. T h e n Since a,

2.

is the smallest fixed-point of F . Suppose F ( P , x ) N p ( x ) , for all x. By induction o n n we have a, a n d t h u s N C_ 0: cy

p because a0 is completely undefined. a, c p then, by monotonicity,

0

cyo

0

If

C_

a n d thus N

G p.

0

p,

184

11. Basic Recursion Theory

T h e proof of t h e First Recursion Theorem gives a computation procedure for the least firced-point of F ( a ,z): take t h e usual computation tree for F , a n d any time a node calling for a computation of a value a(.) is reached, continue t h e computation by substituting F(cy,z ) to a(.). If every branch of t h e tree comes to a n end, then cy has been completely discharged, a n d t h e value is obtained. This shows t h a t there is nothing mysterious about t h e definition of a function in terms of itself. W h a t actually happens is t h a t t h e values are defined in terms of previously obtained ones (in a precise sense, although not always a trivial one, like in t h e ca-je of primitive recursion). Of course, in general, there will be values which can be computed without a n y call to previous values. Otherwise t h e function is really defined circularly, a n d i t is then t h e completely undefined function. As for t h e Fixed-Point Theorem, the First Recursion Theorem, together with composition and case definition, generates the partial recursive functions. Kleene [1978], [198la], [1985] reverses our approach, a n d develops Recursion Theory by taking t h e least fixed-point as a primitive schema. T h i s approach is quite natural, a n d it accounts for t h e name ‘recursive’ for functions whose values a r e somehow defined (by recurrence) using other values of t h e s a m e function. T h e disadvantage of this approach is t h a t it needs partial recursive functionals for t h e definition of partial recursive functions, a n d t h u s it does riot provide a n intrinsic characterization of t h e latter (which is defined simultaneously with t h e former). T h e approach with indices a n d t h e Fixed-Point Theorem (11.2.15) is similar, b u t without this drawback. This brings us to t h e obvious analogies a n d differences between t h e First Recursion Theorem a n d t h e Fixed-Point Theorem 11.2.10 (or, equivalently, t h e Second Recursion Theorem 11.2.13). T h e former produces not only a fixedpoint, b u t t h e least one. T h e latter has a wider range of application, since it does not require any extensionality (in t h e sense t h a t pf(e)is not necessarily a functional on partial recursive functions, see p. 154). Actually, a case can be made t h a t t h e Second Recursion Theorem is more general t h a n t h e First Recursion Theorem:

Proposition 11.3.16 (Rogers [1967]) Let F ( a ,z) be a partial recursive functional, and $J be the partial recursive function N e ,x)

= F(pe.,z).

The fixed-point of ?) produced b y the proof of the Second Recursion Theorem is the least firced-point of F . Proof. First note t h a t $J is indeed partial recursive, by t h e Substitution Property 11.3.8. Let e be t h e fixed-point of $J produced by t h e proof of 11.2.13: then pe(z) 21 F ( ~ e , z ) ,

11.3 Partial Recursive E’unctionals

185

and pe is a fixed-point of F . Recall t h a t cpe(z)21 ~ s ; ( ~ , ~ )for ( za)n appropriate a and, by definition of (see II.1.7), t o compute ‘ps;(a,a)(z) we must first compute $ ( S : ( a , a ) , z ) , i.e. $ ( e , z). In other words, whenever cpe(z)converges, its computation is longer than t h a t of $ ( e , z). To see t h a t cpe is the least fixed-point of F , let p be any fixed-point of F . We show t h a t pe C p, by induction on the length of computations. Suppose p,(z) 1: then F(cp,, z) 1and, by compactness, only a finite subfunction u of pe is used in t h e computation. Choose u minimal: obviously, all the values of u have t o be computed before we get $ ( e , z ) 1~ F(cp,,z). Thus all the computations of values of u have length smaller than the length of computation of cp,(z), and by induction hypothesis we have u C p. Then

Sr

Pe(z) 21 F(perz) 1~ F ( u , z )21 F ( P , z ) P(z), by monotonicity.

0

T h e last result shows t h a t the fied-point operator defined by the proof of II.2.10 (which is the analogue of the fied-point operator y an A-calculus) is actually a least fied-point operator.

Recursive programs

*

Let us look at extended ‘while’ programs (1.5.5), obtained by adding t o the assignment statements the name P (for a n unspecified program). A recursive ‘while’ program is an extended ‘while’ program, preceded by a n instruction of the kind: procedure P. T h e program is interpreted as follows: the extended ‘while’ program defines a program depending on t h e unspecified program P , and the recursive clause added at the beginning tells that P must be the program defined by t h e extended ‘while’ program itself. Thus P is a program with a self-referential flavor. As ‘while’ programs define partial recursive functions, extended ‘while’ programs define partial recursive functionals. T h e First Recursion Theorem tells that there is a ‘while’ program defining a recursive function which is the least firced-point of the extended ‘while’ program. Thus recursive programs are in principle avoidable, in the sense of being equivalent to usual ‘while’ programs (or flowcharts) with no recursive calls. A direct method t o translate recursive programs into flowcharts equivalent t o them is given by Strong [1971]. However, in practice, using recursive programs is a very useful tool, since it often allows us t o write easy and concise programs (with self-referential calh), in situations which might require elaborate nonrecursive programs. Thus

11. Basic Recursion Theory

186

recursive programs a r e widely used (after McCarthy [ 19631) in programming languages allowing definitions (like those of t h e A L G O L family, a n d LISP), as well in languages naturally suited for recursive calls (like P R O L O G ) . A computational approach to recursive programs c a n be deduced from t h e remarks after t h e First Recursion Theorem: it corresponds to substituting every occurrence of P in the computation, with the whole program (complete replacement). A detailed s t u d y of computation procedures more efficient t h a n this (i.e. requiring less substitutions), b u t still sufficient to c o m p u t e t h e least fixed-point of a recursive program, can be found in M a n n a [1974].

Topological digression An instructive way to look at t h e results on partial recursive functionah is by taking a niorc general s t a n d , a n d considering t h e set P of partial (unary) functions as a topological space, following Uspenskii [1955] a n d Nerode [1957] (see Kelley [1955] for a reference on topology). T h i s is easily done, by noting t h a t P can be viewed as a product space S", with S = w U {T}, T being a distinguished element (for t h e undefined value). A natural topology on S is defined by taking as open sets all subsets of w ,a n d t h e space S (thus no nontrivial open set contains T, a n d S is not a Hausdorff space). T h e n P c a n be given t h e product topology. T h i s is called t h e positive information topology, since a countable basis for it consists of t h e finite functions (together with P itself), which contain a finite amount of positive information (specifying t h e values for a finite set of arguments). More precisely, t h e basic open sets a r e

6 = {a:u & a}, where u is a finite function. These sets form a basis because t h e y a r e closed under intersection: Q n 6 = { a :u u v & a } , where u U v is undefined on z if both u ( z )a n d defined a n d different.

Proposition 11.3.17 1. a E X

+

V(Z)

a r e undefined, or b o t h a r e

X 2 P is open i f and only if both of the following hold:

for some finite

'11,

uE X

A

u C a.

Proof. Let X be open. T h e n X is a union of Cis. If Q E X , t h e n a E Q for some Q C_ X . But then u C a , a n d u E X because u E 6.If moreover a C P, t h e n u C p arid p E Q, so p E X .

11.3 Partial Recursive F'unctionals

187

Suppose now that the two conditions hold for X . Then X is t h e union of t h e c's such that u E X (and hence it is open), because: if a E X then, by 1, there is u E X such that u 5 a , so a E 0; and if ct E ii for some u E X , then uCaandaEXby2. An equivalent way of restating the characterization of open sets is: X is open if and only if

aEX

@

(3u finite)(u E X A u C a).

We call X effectively open if the set of finite functions belonging to it is r.e. This requires the identification between finite functions and numbers in any effective way, e.g. let u be the number coding t h e finite function 11 in t h e following way:

In the following, for simplicity of notations, we will not distinguish between a finite function ii and the number u coding it.

Proposition 11.3.18 (Uspenskii [1955], Nerode [1957]) A junction F : P + P is continuous if and only if i t is compact and monotone. Proof. Since the u's are a basis for the topology, F is continuous if and only if F-'(ii) is open, for every finite u. 0

Let F be continuous, and F ( ( Y ) ( zN ) y (note that F ( a )is a function from w t o u).T h e finite function u ( z ) = y (undefined otherwise) defines t h e open set u,so F-'(u) is open, and a is in it. By the characterization of open sets, a E 8 for some .ir F - ' ( u ) : since v E F-'(ii), we have F ( v ) ( z )N y and v 2 a , hence F is compact. If moreover a p, then v 5 p a n d p E 8, so F ( P ) ( z )N y , and F is monotone.

c

c

Let now F be compact and monotone. We want to show that F-'(U) is open, using the characterization of open sets given above.

If a E F-l(fi) then u F ( a ) : u is finite, and for each pair (zi,yi) such that .(xi) 2: y, is F(a)(z,)N y,. By compactness of F , F(ui)(zi) N y, for some finite u, a. Let w be the union of t h e u,'s: then w C a and u C F ( w ) ,so w E F-'(ii) and 1 of 11.3.17 is verified.

c

c

If a E F-'(&)and a C p, F ( a ) E 6 and so u F(a).By monotonicity of F, u F(P). So p E F-'(ii) and 2 of 11.3.17 is verified. Then F-'(fi) is open. 0

c

188

11. Basic Recursion Theory

The behavior of a continuous function on P is then completely determined b y its behavior on finite functions:

U{F(~) : u finite A u

~ ( a=)

a)

(notice t h a t F(u)is not a finite function, in general). T h e function p(u) N F ( u ) for finite u ’ s is called t h e modulus of continuity of F , a n d under suitable coding it can be thought of as a partial function from w t o w.We call a function effectively continuous if its modulus of continuity is partial recursive. We have considered functionals, i.e. functions from P x w to w ,b u t t h e r e is a n obvious correspondence between functionals a n d functions from P to P : if F ( a , z ) is a functional then Xz. F ( a , z ) is such a function, a n d if F is such a function t h e n F ( a ) ( z )is a functional, T h u s t h e result j u s t proved shows t h a t :

Proposition 11.3.19 (Uspenskii [1955], Nerode [1957]) The partial recursive functionals are effectively continuous. O n t h e other hand, t h e converse fails.

Proposition 11.3.20 (Sass0 [1971], [1975]) There are eflectively continuous functionals which are not partial recursive. Proof. T h e idea is simple: define F ( a ) on z by using t w o possible values of a , e.g. if a(2z)N O v a(2z + 1) 21 O F(a)(z) :defined otherwise.

{

T h e n F is effectively continuous, b u t it should not be partial recursive, because a computation tree of F ( a ) ( z )might get stuck on one undefined value of a , even when t h e other is defined. To turn this into a proof, we build a in such a way t h a t F ( N )is not t h e value of any partial recursive functional. Let

Fe(a) = . \ z . U ( P Y q l ( e , z , ~ , Y ) ) b e a n enumeration of t h e partial recursive functionals of one function variable. We want N such t h a t F ( a ) $ F,(a) for every e , a n d we define it by stages. Start with a0 being t h e completely undefined function. Having a,, choose 2 such t h a t both 22 a n d 22 1 a r e greater t h a n all t h e values for which an is defined, as well as those for which a, is a n d must remain undefined because so required by t h e construction at previous stages. Coiisider F,,(a,)(z). T h e r e a r e two cases:

+

Fn(an)(z)l. T h e n only values of a , which have already been defined have been used in t h e computation. Since a, & a , F,(a)(z)1 by rnonotonicity. It is t h e n enough to have F ( a ) ( z ) Ta, n d this only requires leaving a undefined on both 22,22 1 at this a n d later stages.

+

11.3 Partial Recursive Functionals

189

Fn (0,) (z) t . T h e n either it diverges by using only convergent values of a, in t h e computation, a n d t h e n it will remain divergent, or some value of an used in t h e computation is undefined. In t h e latter case, pick u p one: i t is enough t h a t also a be undefined on i t , to force F,(a)(x) as well, since t h e computation is always going to be stuck on t h i s value. We t h e n want F ( a ) ( z )1, a n d this is ensured b u t letting one of a(2z)or a(2z 1) be 0. Since o n e of 2x,2x 1 is free (because both were at t h e beginning of t h i s stage, a n d at most o n e of t h e m is required to remain so by t h e previous work), define a,+l as t h e extension of a , t h a t gives it value 0.

+

+

T h e n a = Untwa, satisfies t h e requirements.

0

A n equivalent way of looking at P is by considering it as a partially ordered set, under C. In t h i s case P is a chain-complete partial ordering, in t h e sense t h a t every linearly ordered chain has a least upper bound (which is j u s t t h e union of t h e chain). We can s t a t e t h e First Recursion Theorem in full generality, by considering a n y chain-complete partial ordering ( D ,L).8 Call U t h e 1.u.b. operation, a n d It h e least element (which exists, being t h e 1.u.b. of t h e e m p t y chain). A function f : D -+ D is:

monotone if it preserves t h e partial ordering continuous if it preserves 1.u.b.k of chains.

A continuous function on a n uppersemilattice D is monotone, since zc_y

* * *

y=zuy

f (Y) = f (I. LJ f(Y) f(.) c f(Y).

T h e general existence theorem for least fixed-points is t h e following:

Theorem 11.3.21 (Knaster [1928], Tarski [1955], Abian and Brown [196l]) If ( D ,C) is a chain-complete partial ordering, and f is a monotone function on it, then f has a least @ed-point. Proof. Iterate f transfinitely, starting with Ia n d taking 1.u.b.k at limit stages (which always exist, since D is chain-complete): zo

=

.a+l

=

xp

=

I

f(4 U,
should not be confused with the partial ordering of sequence ST& partial ordering numbers introduced on p. 89.

11. Basic Recursion Theory

190

T h i s defines a nondecreasing chain in D (because f is monotone), whose length cannot exceed t h e rnaximal length of chains of D . Let ,z, be i t s last element, which must exist by t h e definition at limit stages (otherwise t h e 1.u.b. of t h e otherwise ,z, chain would produce a bigger member). T h e n f(z,,) = z,, would not be t h e last element of t h e chain. And x,, is not only a fixed-point of f , b u t t h e least one: if y is a n y fixedy for each cu (in particular ,z, C y): point of f , by induction we have z,

C y, because Iis t h e least element of D

0

xo

0

if z,

5y

then, by monotonicity a n d t h e fact t h a t y is a fixed-point of f , xCl+l

0

if z, C_ y for all

(Y

= f (xQ)

< p, then z p 5 y

c f (y)

Z Z

y

by definition of 1.u.b.

In particular, continuous functions o n a chain-complete partial ordering have least fixed-points. Continuity being stronger t h a n monotonicity, we might however expect a stronger result. Indeed, i f f is continuous then only w itera-

tions are necessary t o reach the least fied- point:

Exercises 11.3.22 More on Fixed-Points. a) Call f expansionary if z C_ f(z). Every expansionary function on a chain-complete partial ordering has a fixed-point. b) A complete lattice is a partially ordered set in which every subset has 1.u.b. and g.1.b. There is a purely algebraic proof of the existence of least fied-points for monotone functions on a complete lattice. (Knaster 119281, Tarski [ 19551) (Hint: consider the set { x : f ( x ) E z}, which contains every fixed-point of f . Its g.1.b. is the least fixed-point of f .) c) A nonmonotone function on a chain-complete partial ordering need not have a least fied-point. (Hint: on P , consider

F ( a , z )fl'

1 if a(.) - 0 0 otherwise.

Note that F ( a , z ) is defined also when a(.)

is not.)

Before we can apply 11.3.21 to P , we have to show t h a t t h e t w o notions of continuity obtained by viewing it as a topological space a n d as a chain-complete partial ordering coincide.

Proposition 11.3.23 For a functional F on P the following are equivalent:

1. F is compact and monotone

11.3 Partial Recursive Functionals

191

2. F preserves 1.u.b.’~of (countable) chains.

Proof. Let F be compact a n d monotone, a n d { a p } p < p o be a chain of partial functions with 1.u.b. a. W e want

u

F(4=

Fh-4.

B
By monotonicity, from a p 2 a we have F ( a p )

u

F ( a ) ,a n d t h u s

F(ap) G F(a).

B
Now suppose F ( ~ ) ( zP ) y. By compactness, F ( u ) ( z ) N y for s o m e finite function u C a , a n d t h e r e must be p such t h a t u “ 0 . By monotonicity we t h e n have F(crp)(z)= y, a n d hence

F ( @ )G

u

F(@B).

PO0

T h u s F preserves I.u.b.’s of arbitrary chains. Let F now preserve I.u.b.’s of countable chains. I t is automatically m o n e t o n e (see p. 189). For compactness, suppose F ( ~ ) ( zN) y. Since a is t h e 1.u.b. of a chain { Q , } , , ~ ~ of finite functions (e.g. a , can be taken to be t h e restriction of a to t h e arguments from 0 to n ) ,we have

F ( a )=

(JF ( % ) , n Ew

a n d t h u s F(a,)(z) = y for some finite a,,

a.

0

W e then have t h e general fixed-point existence result:

Theorem 11.3.24 (Uspenskii (19551, Nerode (19571) Let F be a functional on P: I . i f F is continuous, F has a least &ed-point 2. i f F is effectively continuous, its least jixed-point i s partial recursive.

We have proved t h e results of this subsection by using only a few particular properties of P , namely:

1. T w o partial functions a r e compatible if they have a common extension

2. A chain of partial functions is a set of compatible elements, containing t h e 1.u.b. of every pair of functions in it.

11. Basic Recursion Theory

192 3. Every chain has a 1.u.b

4. Every function is t h e least upper bound of a chain of finite subfunctions. We can t h e n guess t h a t t h e results j u s t proved would extend to partial orderings with similar properties: we only have to t u r n t h e m into definitions. Consider a partially ordered set (0,E). T w o elements a r e compatible if they have a common extension. A non-empty set of compatible elements is directed i f it contains, together with each pair of elements, a common extension of t h e m . D is a complete partial ordering (c.p.0.) if every directed set h a s a 1.u.b. in D. A function on D is continuous if it preserves I.u.b.’s of directed sets. T o be able to extend property 4 above we need a notion of finite element in U . T h e idea comes from t h e observation t h a t in P t h e finite functions a r e exactly t h o s e functions which, whenever covered by t h e 1.u.b. of a chain of functions, are already covered by some element of t h e chain. We t h e n call a n element of D compact if, whenever it is bounded by t h e 1.u.b. of a directed set A , it is already bounded by a n element of A . T h e compact elements play t h e role of finite elements, a n d we call D algebraic if, for every elenierit 2 , t h e compact elements below it form a directed set with 1.u.b. 2 . Different names used for algebraic c . p . 0 . ’ a~r e complete fo-spaces (Ershov [1972]) a n d domains (Scott [ 19821). IIavirig defined a notion of finite element we can now impose on D t h e Scott topology (Scott [1972]), generated by t h e basic open sets U

= ( 2 : u compact A u

x},

a n d consider t h e associated notion of continuity. O n algebraic c.p.0. ’s the two notions of continuity coincide, and the behavior of a continuous function is completely determined b y its behavior on compact elements. To be able to extend t h e notion of effective continuity, we need t h e possibility of effectively manipulating t h e compact elements. We call D a n effective algebraic c.p.0. if there is a n enumeration (see p. 237) of t h e set of compact elements, which makes t h e relations of partial ordering a n d of compatibility, as well as t h e operation of 1.u.b. for compatible elements, recursive. A continuous function on D is t h e n effectively continuous if its modulus of continuity is partial recursive, w.r.t. t h e given enumeration.

Iteration and fixed-points T h e method of proof used for t h e First Recursion Theorem a n d its refinements is a n old a n d fruitful one. It was first applied by Newton [1669] to o b t a i n zeros of differentiable real funct,ions f , by starting from a n y point xo a n d iterating

11.3 Partial Recursive F'unctionals

193

By t h e geometrical interpretation of derivative this procedure converges to a fixed-point z of F such t h a t f(x) = 0, whenever t h e starting point z o is sufficiently close to z a n d t h e derivative of f is not zero (under these hypotheses, each s t e p doubles t h e number of t h e correct decimal digits of t h e approximation).

It is natural to ask whether t h e method would always produce a zero, starting from a n y point ZO. Cayley [1879] showed t h a t for quadratic functions on t h e complex plane every point not on t h e linc bisecting t h e segment connecting t h e two zeros converges to one of them. Barna [1956] showed t h a t iri general Newton's method works for real polynomials with real zeros, except on a set of measure 0. T h e result fails for complex coefficients (Curry, G a r n e t t , a n d Sullivan [1983]). T h i s p r o m p t s t h e more general question of when t h e iteration of a given function with fixed-points would lead to one of t h e m (in our setting t h a t is always t h e case if f is expansionary, see II.3.22.a). A modern s t u d y case is t h e iteration of t h e quadratic function z2 c on t h e complex numbers, which has at most two fixed-points. For a given c , t h e set of points whose set of iterations is bounded (containing in particular t h e points converging to a fixedpoint, a n d those with a periodic behavior) has a n interesting boundary, called t h e Julia set J , (Julia [1918]), which is either connected or pulverized. T h e shapes of these sets describe, for different c's, a n incredible variety of forms, whose behavior is related to t h e position of c in t h e Mandelbrot set, defined as t h e set of c such t h a t J , is connected (Mandelbrot [1980]). T h i s amazing set contains, on a smaller scale, a reproduction of all Julia sets J,. It also has a sort of universality, since it appears in t h e s t u d y of t h e iteration of a great number of functions (precisely, of any function one of whose iterations behaves like z2 in some portion of t h e plane). O n e aspect of Julia (and, less stringently, of Mandelbrot) sets is their self-resemblance: t h e y contain copies of themselves a n d present, at any scale of observation, t h e s a m e global character. Sets with t h i s property a r e called fractals, a n d a r e useful to describe natural phenomena involving chaotic behavior. See Mandelbrot [1982] a n d Peitgen a n d Richter [1986] for more on this.

+

Another example of a situation in which t h e iteration of a function always produces a fixed-point is t h e one described by Banach Fixed-Point Theorem (Banach [1922]): a contraction function on a complete metric space has a unique fied-point, given b y the limit of the iterations o f f starting o n any point, where f is a contraction if If(.) - f(y)I 5 c . Iz - yl for s o m e fixed constant c such t h a t 0 5 c < 1 1.( - yI being t h e distance between z a n d 9). Indeed, given any x then, by induction,

11. Basic Recursion Theory

194

By t h e triangular inequality,

i<m

T h u s {f(n)(x)}nE,, a n d hence {f(T"l)(x)},E,, converge to a point ZO. Since f is continuous, t h e latter also converges to f(zo),a n d t h u s f(x0) = ZO. Moreover, if x1 is another fixed-point o f f t h e n

a n d , being c

< 1 , it

must be

20 = 5 1

Models of A-calculus (part I)

*

T h e only objects of A-calculus a r e terms, a n d t h u s a model will be a set D over which terms a r e interpreted, in such a way t h a t provably equal t e r m s a r e interpreted by t h e same object. T h e presence of t h e A-operator forces some ternis to be interpreted as functions acting on terms, a n d t h u s some elements of D will have to be interpreted as functions on D , i.e. a n appropriate set [ D 4 D] of functions from D to D will have to be identified with a subspace of D . This rules o u t , by cardinality considerations, t h e naive approach of taking as [ D -+ D ] t h e set of all functions from D to D. Finally, b o t h A-abstraction a n d functional application will have to be interpreted over D , t h e first being a t e r m formation operator, t h e second because of t h e @-conversion rule. See p. 223 a n d Meyer (19821 for a discussion of t h e notion of model of A-calculus. Methods a n d results of t h e last subsection a r e all relevant to t h e present subject. To s t a r t with, P is a model of A-calculus, when [P + P ] is taken to be the set of continuous functionals (this model is connected to t h e graph model of Plotkin [1972] a n d Scott [1975], see below). T h e reason is t h a t continuous functionals a r e completely determined by their behavior on finite functions, a n d thus they can be coded by a single function ( t h e modulus of continuity). T h e only additional piece of information needed to t u r n t h i s into a model of A-calculus is to note t h a t functional application of continuous functionals is continuous, a n d t h u s again representable in P. T h i s also applies to A-abstraction, as well as to t h e operator F i x , t h a t produces t h e least fixedpoint of continuous functionals. By 11.3.16, the interpretation of F i x coincides with the interpretation of the combinator y (Park [1970]). W h a t P does not autoiriatically provide for, is a model of extensionality (77-rule, p. 82, according to which every t e r m is identified with a function): t h e embedding of [P + P ]

11.3 Partial Recursive Functionals

195

into P does not cover P itself (two different moduli of continuity c a n code t h e s a m e continuous function). As it might be expected, there is nothing special a b o u t P : i t s place c a n be taken by a n y other reflexive c.P.o., i.e. any (algebraic) c.p.0. in which t h e set [ D + D ] of continuous functions is embeddable (meaning t h a t t h e r e a r e continuous functions

i : D - + [ D - + D ] and

j:[D--+D)+D

such t h a t i o j is t h e identity on [ D + 01).E.g., in t h e literature P is usually replaced by P(w),t h e set of subsets of w ,with t h e positive information topology generated by t h e finite sets. Here functions a r e represented by their graphs, hence t h e name ‘graph model’ quoted above. To get models of extensionality as well, we need not only to e m b e d [ D + D ] into D , b u t to identify t h e two. T h i s obviously sounds like a fixed-point, a n d we do have a general exist,ence theorem (11.3.21). To be able to apply it, we need to consider t h e function F ( D ) = [ D + D] as monotone over a giant chain-complete partial ordering, whose members a r e (algebraic) c . p . 0 . ’ ~ . We t h u s need a n appropriate partial ordering on them, a n d i t t u r n s o u t t h a t t h e following natural one will work:

D

a D‘

w

there a r e continuous functions

i : D + D ’ and j : D ’ + D such t h a t j ( i ( z ) )= z a n d i ( j ( y ) ) y. T h e intuition behind this definition is t h a t D’ is supposed to contain more information t h a n D , a n d D has to be embedded into it. So i ( z )is t h e element in D’ t h a t corresponds to x in D a n d , going back, we simply recover it (since x a n d i ( x ) have t h e Same information content). B u t a n element y in D’ might not correspond to any element in D , a n d j ( y ) is only t h e closest approximation t o it in D: going back, we might lose some information, a n d t h u s only i ( j ( y ) )& y holds. W i t h t h i s notion of partial ordering for c.p.o.’s, it is easy to show t h a t t h e I.u.b.’s of a chain {D,,}nEwexists: we j u s t have to t a k e t h e c.p.0. D, t h a t s u m s u p exactly t h e information contained in t h e chain. Note t h a t if j , : Dn+l + D,, t h e n j,(x) is a n element in 11, t h a t approximates z. T h u s we only need to consider chains (z,),~~ such t h a t x, E D, a n d jn(z,+l) = x,, as elements of D,. T h e n x, may be thought of as t h e approximation of t h e chain at stage n (think of x as a real number given by i t s decimal expansion: t h e n x, is t h e expansion truncated at t h e n-th digit, a n d t h e projection j , simply cuts o u t t h e last digit of x,+l). First, F is a function on c . p . 0 . ’ ~(since [ D -+ D ] can be t u r n e d into a c.p.0. by pointwise ordering its elements). Second, F commutes with 1.u.b.k of

11. Basic Recursion Theory

196

chains (and i t is t h u s continuous). Moreover, F is expansionary, in t h e sense t h a t D [ D -+ D ] . T h u s not only F admitts least fixed-points, a n d t h e y c a n be reached in at most w iterations of F : they exist above a n y given D . T h u s a n y

(algebraic) c.p.0. D c a n be embedded in a n extensional m o d e l D , of X-calculus (Scott [ 19721). All this machinery can be appropriately formalized in t h e Cartesian closed category of (algebraic) c . p . 0 . ’ ~(where continuous functions, 1.u.b.’~a n d monotonicity of F become, respectively, morphisms, inverse limits a n d functorial covariance, a n d D is a reflexive object if D D D ) . Again, there is nothing special about t h i s category: a n y reflexive object with enough points in a Cartesian

a

closed category gives rise to a model of X-calculus a n d , conversely, a n y m o d e l o f X-calculus i s a reflexive object w i t h enough points in a Cartesian closed category ( t h e condition of having enough points basically meaning t h a t different functions must behave differently on sorne point) (Scott [1980a],Koynians [1982]). T h i s completely characterizes t h e models of X-calculus. For history, exposition, a n d philosophy see Scott [1973], [1975], [1975a], [1976], [1977], [1980], [198Oa]. For technical development a n d other models see Stoy [ 19771, Barendregt [ 19811, Beeson [ 19851, Hindley a n d Seldin [ 19861.

Different notions of recursive functionals We have defined partial recursive functionals, obtained from a uniformization of t h e notion of relative recursiveness 5~ for partial functions. T h e topological approach developed above suggests t h e consideration of t h e broader class of effectively continuous fuiictionals as well. These a r e expressible in t h e form

F ( c Y , z )= z e (3u)(uC N

A

cp(z,u)21 z ) ,

with cp partial recursive a n d u,here a n d in t h e following, (code of a) finite function. By 11.1.11, an equivalent formulation is

F ( c Y , z )2: z e (3u)(uC N A R ( z , u , z ) ) , with R r.e. Of course, such a n expression defines a functional only if the relation R ( t h a t embodies t h e graph of t h e continuity modulus) gives consistent answers. We t h u s have two ways to t u r n this general form into a definition of functional: 1. A partial recursive operator (Myhill [1961a], Rogers [1967]) is a functional F t h a t can be defined as

F(cr,z) N z with R r.e. relation.

@

(3u)(u N A R ( z ,u, z ) ) A (yu’)(y~’)(u’ C N A R(z,u’,z’)

z = z‘),

11.3 Partial Recursive Functionals

197

2. A recursive operator (Davis [1958], Rogers (19671) is a functional F t h a t c a n be defined as

F ( a , z )- z @ ( ( 3 u ) ( u C aARR(z,u,z)) with R consistent r.e. relation, i.e. such t h a t

R ( z ,u,z ) A R ( z ,u‘,z’) A u,u’compatible

*z =

2’.

T h u s t h e difference between partial recursive operators a n d recursive opera t o r s is in t h e consistency condition required on R: only relative to t h e i n p u t function for partial recursive operators, a n d global for recursive operators. It is clear t h a t , from a computational point of view, t h e first notion is not satisfactory, since it requires a consistency check relative to a given partial function. It is not surprising t h a t there is no Enumeration Theorem for partial recursive operators. A b e t t e r way of thinking of partial recursive operators is in ternls of functionals on miiltivalued functions or, bet>terstill, sets, rather t h a n o n partial functions, a n d t h e associated reducibility riotion for sets is called

enumeration reducibility:

A 5, B e for some r.e. relation R, z E A e (3u)(D,C B A R(z,u)). T h e structure of degrees associated with this reducibility (called partial degrees) will be studied in Chapter XIV. An Enumeration Theorem for recursive operators (Rogers [1967])c a n easily be obtained by stepping from a n enumeration {Wz}eeuof all t h e r.e. ternary relations, to a n enumeration {Wf3(e)}eEu of t h e consistent ones, where W j c e ,is obtained from a n enumeration of Wz, by dropping t h e triple (2,u’, z’) whenever a triple (x,u,z ) with u , u’consistent has already been generated. T h e basic difference between recursive operators on the one hand, a n d partial recursive functionals on t h e other, is computational: t h e latter a r e serial, a n d a computation gets stuck if it tries to query t h e oracle for a n undefined value; t h e former a r e parallel, a n d c a n get around undefined values b y dovetailing computations (for general discussions of parallelism, see Elgot, Robinson a n d Rutledge [1967], Shepherdson [1975], Cook [1982]). We t h u s arrive at t h e notion of partial recursive functional by relativizing deterministic approaches to coniputability, like recursiveness (by adding a function to t h e initial ones, as we did in this section), Turing machine computability (by adding a n additional state ishat calls for t h e oracle, Turing [1939]) or flowchart computability (by adding assignment instructions of t h e kind

11. Basic Recursion Theory

198

as in Ianov [1958]). O n t h e other hand, we arrive at t h e notion of (partial) recursive operator by relativizing nondeterministic approaches, like Herbrand-Giidel computability, or representability i n formal systems (by adding a functional letter to t h e constants of t h e language, Kleene [1943]). We t h e n have three notions of relative computability for partial functions: if a = F ( P ) , we say t h a t

a
if F is a partial recursive functional if F is a recursive operator if F is a partial recursive operator.

T h e y a r e respectively called Turing, weak Turing a n d enumeration reducibilities.

Proposition 11.3.25 (Myhill [196la], Sass0 [1971]) 1. N

ST p

a S w b,~ but not conversely

s wp ~* a 5 , p, but not conversely S T ,s w and ~ 5 , cozncide on total functzons.

2. a 3.

Proof. T h e first two implications a r e obvious, a n d 11.3.20 gives a counterexample to t h e converse of t h e first,. A counterexample to t h e converse of t h e second is given by

F ( a , x ) N .z ts ( z = 1 A a(2x)21 1) V ( z = 0 A a ( 2 x + 1) 21 1). Clearly, F is not consistent, in general, b u t it is consistent for t h o s e a’s coding A in ih e following way:

a set

422)

N

a(22+ 1)

21

{ lundefined

ifxEA otherwise

undefined

otherwise.

ifx@A

In this case F ( N ) is t h e characteristic function of t h e set coded by a , arid F ( a ) 5, a . But if F ( a ) <,,T a , then F ( a ) p for a n y (9 extending a , (this uses t h e consistency property, a n d it fails in general for 5,). In particular, this holds e.g. when p is t h e constant function with value 1, a n d t h e n a codes a recursive set (because F ( a ) is t h e characteristic function of t h i s set, a n d it

s,,~

11.3 Partial Recursive F’unctionals is recursive since

/?

is). T h u s , if

ck

199

codes a nonrecursive set, F ( a )

Se a but

F ( 0 ) $wT ff. To show t h a t t h e three reducibilities agree on total functions, suppose t h a t a 5 , g, i.e. a(.) 21 z H (3u)(u E g A R ( z ,u,z ) ) with R r.e. T h e n there is Q recursive such t h a t

4x1=z

* ( 3 ~ ) ( 3 ~ C) (9uA Q(G u,z , ~ ) ) ,

and

( ~ t [ ( t )G l 9 A Q ( x ,( t )1 (~t ) 2 , (t)3)1)2. T h i s is a correct application of t h e p-operator, since u C g is recursive in g (because we know t h a t g is total, a n d we only have t o check t h e values). T h u s ck
a(.)

Since we a r e only dealing with degrees of sets, we will not study t h e degrees of partial functions under LT a n d &,T. We refer to Sasso [1975] for a survey of results (see also Casalegno [1985] for later advances). For degrees under I,, see Chapter XIV.

Higher Types Recursion Theory

*

Let T,,, t h e set of objects of type n, be defined as:

To

=

W

Tn+l = total functions from Ui,,T, -

to w.

We have so far introduced notions of recursiveness for objects of t y p e 1 (functions) a n d 2 (functionals). Recursiveness can be successively extended to objects o f higher types by iterating t h e process t h a t has led to t h e theory of this section, namely by relativizing in a uniform way t h e notion at lower types. T h i s has led (at least for restricted functionals, defined only on total objects) to a highly developed theory, begun by Kleene in two epochal papers ([1959], [1963]). A basic fact which is a source of divergence with t h e classical theory, a n d of technical complications, is t h e failure of compactness: a computation t r e e still has finite branches when a value is defined, h u t it does not need to be finitely branching ( a n d hence finite), since a higher t y p e object might need to know infinitely many values of it,s arguments, already at level two, as for:

0 if (b’z)(f(z) = 0 ) 1 otherwise.

200

11. Basic Recursion Theory

T h i s rules o u t a Normal Form Theorem (since a computation can n o longer be coded by a number), a n d the p-operator loses its central role (as i t is restricted to a search on w , a n d as no natural analogue is available at higher levels). T h e normal form is replaced by t h e Enumeration Theorem, which states t h a t a functional may see one of its number arguments as a n index a n d simulate t h e functional coded by it. T h i s principle is taken as primitive, something which requires a n index approach from t h e very beginning, see p. 158. T h e p-operator is partly replaced by Selection Theorems (Gandy [1967]), which a r e analogues of t h e IJniformization Theorem 11.1.13, a n d provide for choice operators at t h e number level ( t h e choice being now m a d e according to t h e s t a g e of generation in t h e inductive definition of t h e predicate to be uniformized). Another source of trouble is equality for objects of a given t y p e , which is recursive only at t h e integer (type 0) level. T h e theory h a s shown a dichotomy between objects strong enough to compute equality at t h e level of their arguments (normal objects, for which strong regularity results - like t h e Selection Theorems quoted above - hold, a n d which are, from a topological point of view, effectively discontinuous, see Grilliott [1971], Hinman [1973]), a n d those which a r e riot (see below). A treatment of various p a r t s of Recursion Theory for normal, higher types objects may be found in Hinman [1978], a n d Fenstad [1980]. For a n elegant introduction, see Kechris a n d Moschovakis [1977].

A completely different way to extend Recursion Theory to higher types is by taking compactness a n d monotonicity, a n d hence continuity, as a basis. T h i s has been proposed by Davis [1959], Kleene [1959a], a n d Kreisel [1959], a n d h a s led to a theory of countable functionals, quite similar to t h e o n e of t h i s section. T h e basic aspect is t h a t computation trees are still finite, a n d hence at t h e s a m e level of numbers. A functional F ( g ) of t y p e 2 is continuous if a n d only if its values a r e determined by finite pieces of i t s arguments: t h i s can be formally coded by an associate f : w + w which, o n increasingly long segments of g, takes value 0 for a certain time, a n d a constant greater t h a n 0 from a certain point on. T h u s F ( g ) = f ( g ( n ) )- 1, for any sufficiently large n. T h i s notion can easily be generalized to higher types. If F is of t y p e n+ 1, a n d G of t y p e n, a n associate for F is a n y function f as above such t h a t , for any associate g of G,

F ( G ) = f ( g ( n ) )- 1, for any sufficiently large n.

A (recursive) countable functional is a n y functional with a (recursive) associate. Note t h a t countable functionals act on finite pieces not of their arguments (at type > 2), b u t of associates for t h e m . T h u s associates a r e algorithms to compute functionals, t h a t act on algorithms for their arguments.

11.3 Partial Recursive finctionals

20 1

It is clear t h a t a notion of continuity is underlying t h a t of countable functionals. One way t o make this precise (Hyland [1979]) is t o consider filter spaces (see Kelley [1955]), i.e. sets with collections of abstract approximations for each element (each approximation being a filter of sets, and at least t h e following being approximations t o a n element: the principal ultrafilter generated by t h a t element, as well as any filter including a n approximation t o it). A continuous function between filter spaces is a function t h a t respects approximations. Products and sets of continuous maps on filter spaces a r e still filter spaces, a n d they nicely commute. Thus a hierarchy, coinciding exactly with t h a t of countable functionals, can be obtained inductively, by starting with w (organized as a filter space by taking, as only approximation t o n , the principal ultrafilter containing n). Another way t o look topologically at the countable functionals (Ershov [1972]) is via effective algebraic c . p . 0 . ’ (see ~ p. 192). Products of and sets of continuous maps on effective algebraic c . p . 0 . ’ ~are still such, and they nicely commute. Then a class C of partial continuous functionals of higher type can be defined inductively, by letting:

CO C(U,T)

Co-T

= finite sets with t h e canonical enumeration = cu x CT [CU GI. --+

The (effective) extensional, hereditarily total functionals are exactly the (recursive) countable functionals (Ershov 119741). T h e condition on totality is required because the countable functionals are, by definition, restricted functionals.

T h e objects in these spaces are not, in general, extensional or total.

Other approaches to countable functionals can be found in Feferman [1977], Norniann [1981], Longo and Moggi [1984]. For a n exposition of t h e theory, see Ershov [1977], and Normann [1980], [1981]. There a r e relationships between the two extensions of recursiveness to higher types quoted above, surveyed in Gandy and Hyland [1977]. In one direction, a recursive higher type object restricted to countable arguments is countable (Kleene [1959]). In the other direction, there are recursive countable functionals which are not recursive as higher type objects, e.g. the functional t h a t gives a modulus of continuity for any continuous functional of type 2 on compact sets of functions (Tait). However, the countable functionals can be generated by an extension of the schemata generating the higher type recursive objects (Norrriann [ 19811).

202

11. Basic Recursion Theory

Computability on abstract structures

*

Recursion Theory is a notion of computability on t h e s t r u c t u r e of t h e natural numbers. Recursiveness relative to given oracles suggests o n e possible way of extending t h e notion of computability to abstract structures

{A,fl,...,.fn,Rl, ..., Rn) with a given domain A , a n d functions a n d relations on it. Of course, t h e various approaches to relative computability, which a r e equivalent in t h e rase o f t h e structure of natural numbers (see p. 197), generalize in ways which are not necessarily equivalent on abstract structures. We briefly describe t h e most popular ones, a n d leave to Kreisel [1971], Ershov [1981] a n d Shephertlson [ 19851 t h e discussion of other possible approaches, including t h e case of partial structures.

Formalized algorithmic procedures (fap) (Ianov [1958], Ershov (19601, Luckham a n d Park [1964], Paterson [1968], Friedman [1971a], Kfoury [1974]) a r e finite lists of labelled instructions (called program schemata), each one of t h e following kind:

Y := f * ( 4 if R3(Z)go to p , else go to q stop. This is a n extension of (nnstrnctured) flowchart conipat,ability, using t h e functions a n d relations of t h e structure as primitives, a n d with t h e variables ranging over t h e elements of t h e domain. See M a n n a [1974] for an extensive treatment.

Fap with counting (fapc) (Friedman [1971a], Kfoury [1974]) a r e extensions o f fap using also natural nunibers, over which range special variables X , Y,. . . T h e following counting instructions a r e also available: x:=0

x:=x+1 if X = Y go to p , else go to q. This is equivalent to expanding t h e given structure to a s t r u c t u r e

( AU W , f i , . . . , f n , R I ,. . . , & , O , p 4 S ) (with pd a n d S t h e predecessor a n d successor functions for n a t u r a l numbers), a n d considering fap over it.

11.3 Partial Recursive Functionals

203

Recursive schemata (McCarthy [1963], Platek [1966]) are based on fixedpoints, and allow for functions explicitly definable over the structure by case definitions (on t h e relations of the structure), and fixed-point operators. Alternatively, one could simply use recursive schemata

gi(xi) = if Pi(.<) then

...

hl,i(~< else )

hl,z(~i)

gp(x;) = if Pp(x;)then hp,1(xG)else hp,2(x;), where t h e Pi’s are predicates in their variables over t h e structure, a n d t h e h’s are terms in their variables over the structure, possibly using t h e ga’s.

Generalized Turing algorithms (Friedman [1971a]) are programs for extended Turing machines which, on top of performing the usual operations a n d working with their own alphabet, can also handle elements of t h e d e main of the structure on the tape, in t h e following way. First, t h e inputs are placed on t h e tape at the beginning, in adjacent cells t o t h e left of t h e head, the rest of t h e tape being empty. Elements on t h e tape can be moved around, a n d possibly copied, one cell at a time. A new element can be introduced in t h e scanned cell, but only if it is t h e value of one of t h e functions of t h e structure for t h e arguments, placed in consecutive cells t o t h e right of t h e head. Finally, t h e machine can test t h e validity of a predicate of the structure for t h e arguments, placed in consecutive cells t o t h e right of t h e head.

Effective definitional schemata (Friedman [1971a], Gordon [1974]) are r.e. sequences of specifications, each giving a value (in t h e form of a term over the structure) under mutually exclusive conditions, expressed as quantifier free formulas over the structure. They correspond t o effective infinitary case definitions, and generalize the approach t o recursiveness by systems of equations (see p. 39).

Prime computability (Moschovakis [ 19691) extends t h e original definition of (relative) recursiveness reformulated, as on p. 158, with indices a n d enumeration. T h e approach with indices is used because t h e p-operator

obviously works only for well-ordered structures. Indices are here elements of t h e domain of t h e structure, and t o make this approach work we also need a coding mechanism. This is obtained by considering t h e schemata for recursion on the expanded structure

( AU I,fi,.. . , f n , R i , . . . ,R,,O,P~,S,.~,R,L)~ where I is a set of indices containing w,J is a o n e o n e pairing function on I , and R a n d L are its decoding functions.

204

11. Basic Recursion Theory

Search computability (Moschovakis [1969]) adds to prime computability a n unordered search operator, performed by a n oracle t h a t searches through arbitrarily large p a r t s of t h e domain. T h i s requires a n approach with multiple-valued functions because t h e effect of t h e search operator is to take all elements satisfying a given condition, instead of selecting one (since there is, in general, n o canonical choice). P r i m e a n d search computability correspond, respectively, t o deterministic a n d nondeterministic computations on t h e given structure. Despite t h e great variety of approaches, some of t h e equivalences which hold for t h e classical notions retain their validity. E.g., generalized Turing algorithms, effective definitional schemata, a n d prime computability a r e equivalent (Friedman [1971a], Gordon [1974]). We t h u s have only five notions available, namely fap, fapc, recursive schemata, prime a n d search computability. It should be clear t h a t both fapc and recursive schemata extend fap, in incomparable ways, and prime computability extends both. Obviously, search computability is stronger t h a n prime computability. T h c obvious problcm at this point is to choose t h e ‘right’ notion of computability on a n abstract domain, among t h e nonequivalent ones. B y Friedman [1971a] a n d Kfoury [1974], fap’s are equivalent to fapc’s i f the structure has

analogues of the natural numbers, and the counting instructions can be defined. By Moldestad, Stoltenberg-Hansen, a n d Tucker [1980a], fapc ’s are equivalent l o prime computability if term evaluation is fapc-computable (uniformly in the number of variables). T h u s these four approaches a r e equivalent for sufficiently rich structures (like all t h e infinite algebraic structures of common use, e.g. rings a n d fields of characteristic zero). It seems a reasonable assumption to allow t h e use of t h e natural numbers in computations as a n auxiliary tool (for example, as t h e order of a n element in a group, or as t h e characteristic of a n element in a field), a n d to have t e r m evaluation as a computable funct,ion. Under these assumptions, t h e four notions discussed above coincide with prime computability. Search computability is aLso equivalent to a great number of o t h e r notions, see Moschovakis [1969a], Gordon [1970], [1974], a n d Grilliot [1971a]. I n particular, it coincides with extended effective definitional schemata, using existential forniulas in place of quantifier free ones. Also, for countable domains (with computable equality) search computability coincides with t h e very natural notion of V-recursiveness (Laconibe [1964]), so defined. Since t h e s t r u c t u r e is countable, there a r e one-one onto functions between t h e domain a n d w ,each of which allows us to translate functions a n d relations from t h e domain to w. A function or relation over t h e dorriain is ‘d-recursive if its translation is, for all one-one o n t o translating functions.

11.4 Effective Operations

205

I t t h u s seems t h a t p r i m e and search computability are the natural notions of deterministic and nondeterministic computability o n a n abstract d o m a i n .

11.4

Effective Operations

T h e previous section introduced various notions of functionals, with t h e common characteristic t h a t t h e function argurnerits a r e treated extensionally. T h i s is not avoidable in general, b u t for t h e particular case of partial recursive functions we could also t r e a t t h e arguments intensionally, through (codes for) their programs. In t h i s section we s t u d y t h e intensional version of extensional recursive functionals, a n d their mutual relationships.

Effective operations on partial recursive functions Having studied (effective) continuity on P, it is natural to look at t h e s a m e notion on its effective part, namely t h e set PR of partial recursive functions. We will consider functionals o n PR or, equivalently, extensional f u n c t i o n s o n indices, where f is extensional if (Pe 2: (Pel

* (Pf(e)

2 (Pj(el).

O n e approach consists in seeing PR as a subspace of P. T h i s automatically induces a topology, whose (effectively) open sets a r e t h e intersections of (effectively) o p e n sets of P ,a n d whose (effectively) continuous functionals a r e t h e restrictions of (effectively) continuous functionals on P . Another approach exploits t h e fact t h a t t h e members of PR have indices. We call a class of partial recursive functions a completely r.e. class if its index set (see p. 150) is r.e. (Dekker [1953a], Rice (19531). T h e Ershov topology on PR (Ershov [1972]) is t h e topology generated by taking t h e completely r.e. classes of partial recursive functions as basic open sets. T h u s open sets a r e unions of completely r.e. classes, a n d effectively open sets can be defined as r.e. unions of basic open sets (since r.e. unions of r.e. sets a r e still r.e., t h e effectively open sets are j u s t t h e completely r.e. classes). Unraveling t h e definition of continuity shows t h a t a function f induces a continuous functional if, for every e a n d every r.e. index set A such t h a t f (e) E A , there is a n r.e. index set containing e a n d included in j - ' ( A ) . This suggests t h e following notion of effective continuity, obtained by considering extensional recursive functions (since t h e n j - ' ( A ) is itself r.e.).

Definition 11.4.1 An effective operation o n PR is a functional F o n PR induced by a recursive function, i.e. F(cpe) 2: pf(e)f o r s o m e extensional, re-

cursive f u n c t i o n f.

11. Basic Recursion Theory

206

We now wish t o compare t h e two approaches j u s t introduced. To s t a r t with, t h e next result implies t h a t t h e two topologies o n P'R (and h e n c e t h e two notions of continuity) coincide. Recall (see p. 187) t h a t we identify finite functions a n d numbers by explicitly coding t h e graphs of finite functions in some effective way.

Theorem 11.4.2 (Myhill and Shepherdson [1955], McNaughton, Shapiro) A class A of partial recursive f u n c t i o n s is completely r.e. i f a n d only i f there is a n r.e. set A s u c h that peE A H (3u f i n i t e ) ( u E A A

7~

G 9,).

Proof. If there is such a n r.e. set A , t h e n A is completely r.e. To see if peE A, generate simultaneously A a n d t h e graph of pera n d wait until, for s o m e u E A , t h e graph of t h e finite function coded by u (which can be completely decoded from u in finitely many steps) is contained in t h e p a r t already generated of t h e graph of pe. Conversely, let A be nonempty a n d conipletely r.e., a n d OA be its index set. T h e obvious guess for a n r.e. set of finit,e functions generating A is t h e set of its finite functions: they a r e indeed a n r.e set (let g ( u ) be a n index of t h e finite function (coded by) u: t o see if u E A, it is enough to see if g ( u ) E OA, because OA contains all indices of any function in A). W e t h e n have to prove t h e following.

1. A is monotone on partial recursive functions, i.e. if Q E A, D is partial recursive, a n d Q C /3, then p E A. Suppose t h a t (Y a n d ,Ll a r e partial recursive functions such t h a t Q E A, Q p, b u t /3 @ A. Let f be a recursive function such t h a t (Pf(e)(4

N Y -3

41 . = 1J v

[ e E K AD(.)

T h e function pf(.) is well-defined, because eE Indeed, if e E

K then

tj

Q

Y1.

c 0.T h e n

pf(e)E A.

pf(e)= /3, while if e @ K t h e n

= Q.

Since A is completely r.e., K is r.e., contradiction.

c

2. A is compact, i.e. if cy E A then, for some finite function u a , u E A. Suppose t h a t Q is a partial recursive function such t h a t Q is in A, b u t no finite subfunction of it is. Let f be a recursive function such t h a t

11.4 Effective Operations

207

Then

e EE

@

cpf(,) E A.

Indeed, if e E Fc then e E Fc, for all sufficiently big stages x, a n d hence ‘pf(.) is a finite subfunction of a , a n d cannot be in A. Conversely, if e @ K: then e @ Fc,, for any x,a n d cpf(,) 21 a. Since A is completely r.e., E is r.e., contradiction.

0

By t h e definition of a n effectively open set of functions (p. 187), t h e result shows t h a t a completely r.e. class is t h e intersection of PR with a n (effectively) open set of P . Conversely, it is immediate t h a t t h e intersection of P R with a basic open set of P is completely r.e. This shows t h e coincidence of t h e two topologies (and hence of t h e two notions of continuity) on PR introduced above. Moreover, the effectively open sets ( i n both topologies) are exactly the completely r. e. classes. I t only remains to show t h a t the two notions of effective continuity coincide as well.

Theorem 11.4.3 (Myhill and Shepherdson [1955], Uspenskii [1955]) T h e effective operations on PR are exactly the restrictions to P R of effectively continuous functionals on P.

Proof. Let F be a n effectively continuous functional: then t h e function cp(r,x)21 F(cpe,x) is partial recursive because, for some r.e. relation R, F(cp,,z)

21

z

@

(3u finite)(u

cpe A R ( x , u ,z ) ) ,

a n d t h u s t h e graph of cp is r.e. (see 11.1.11). By t h e S z - T h e o r e m , there is f recursive such t h a t P f ( e ) ( x )E ~ ( e , x21) F(Perx), a n d F is a n effective operation. Let now f define a n effective operation F on P R . I t is enough to show t h a t F is compact a n d monotone on P R , i.e. t h a t for some r.e. relation R,

F(cp,, x) 21 z

(321 finite)(u & pe A R ( x ,u , z ) ) .

T h e n we can extend F to all of P , by j u s t letting

F ( a ,x) = z

(3u finite)(u

N A

R ( z , u ,z ) ) .

But since F(cpejx) E

for fixed x a n d z t h e class

q f ( e ) ( x )21 2,

11. Basic Recursion Theory

208

is completely r.e. T h e n all t h e work has already been d o n e in 11.4.2: t h e r e is a n r.e. set A such t h a t

F(pe,x)= z

tj (3u

pe A u E A ) .

finite)(u

T h i s holds, as said, for fixed z, z . But t h e y a p p e a r as parameters in t h e argument, a n d t h e n A depends uniformly on them. In other words, A is really a n r.e. relation R depending o n u, x,z . T h e n , this t i m e for every x a n d Z ,

F(pe,x) N z e (3u finite)(u as wanted.

pe A R ( x , u ,z ) )

0

Note t h a t t h e proof shows t h a t every effective operation is int1uc:ed by a unique continuous functional (which t u r n s out to be recursive), since its values a r e completely determined by t h e behavior on finite functions (which a r e all partial recursive). This, in t u r n , uniquely determines a continuous functional. Exercise 11.4.4 There is a wntinuous functional whose restriction to P'R is not an eflective operation. (Dekker and Myhill [ 19581) (Hint: there are only countably many effective operations on P'R, so it is enough to build uncountably many continuous functionals, with distinct restrictions to P'R. Given a partial function 0, Ict FL3(a) = If donifl"

{

constant function 1 completely undefined

if don1 o n dom otherwise.

#8

# d o m p l , then Fpo and Fp, differ on P'R.)

Effective operations on total recursive functions T h e notion of effective operation can be generalized from PR to a n y class A of partial recursive functions:

Definition 11.4.5 A n effective operation on a set A of partial recursive functions is a functional F mapping A t o A and such that, f o r some partial recursive function I), Ve E A

* I ) ( e )1 A F ( V e )

2 '~q,(e).

Undcr which conditioiis for A does a n analogue of Theorem 11.4.3 hold? We first prove a n important special case, a n d t h e n discuss t h e general situation. In t h e following, R is going to be t h e class of total recursive functions.

Theorem 11.4.6 (Kreisel, Lacombe and Shoenfield [1957], Ceitin [1959]) T h e effective operations o n R are exactly the restrictions of the effectively continuous functionak mapping R to R.

11.4 Effective Operations

209

Proof. As in t h e previous theorem, a n effectively continuous functional mapping R to R induces a n effective operation on R. Let now F be a n effective operation o n R. We approximate t h e result by a series of steps. 1. If f is recursive a n d F ( f ,x) = z , for any preassigned length k t h e r e is a function u of finite support (i.e. 0 almost everywhere) which agrees with f up to k , a n d such t h a t F ( u , x ) = z . T h i s provides a sort of compactness, with functions of finite s u p p o r t taking place of finite functions (which cannot be considered here because t h e y a r e not total). Suppose t h e claim fails: t h e n a recursive function t can be defined, such t h a t pt(e)agrees with f u p to t h e maximum of k a n d (if e E K) t h e least stage in which e is generated in K ,a n d afterwards agrees with a function u of finite support a n d such t h a t F ( u , x ) # z (which exists by t h e hypothesis t h a t t h e claim fails). T h e n , by definition,

and

K is recursive, contradiction.

2. If f is recursive a n d F ( f , z ) = z , there is a fixed length Icf (which can be found effectively) such t h a t , for a n y function u of finite s u p p o r t which agrees with f up to Icf , F ( u ,z) = z . T h i s provides a sort of modulus of continuity: t h e value of F ( f , z ) is determined by t h e initial segment off u p to kf. T h e existence proof is t h e s a m e as given above (dropping Ic), b u t we want to obtain k f effectively. T h e n define pt(e)as above, b u t as a partial function (i.e. look for t h e appropriate u,a n d if this is found, proceed as above). Notice t h a t t h e set

e E C u F(Vt(e),x)

z

c,

is r.e. If a is an index for it, it cannot be a E otherwise by construction vt(,)= f: t h e n a E C = W,, a n d hence a E K . T h u s it must be a E K: let k f be t h e least stage in which a is generated in K. T h e r e cannot exist a function u of finite support which agrees with f u p to k f , a n d such t h a t F ( u , z ) # z , otherwise would be one such, forcing a @ C , against t h e fact t h a t a E K. But F is defined for any recursive function, a n d then if I L a g r e a with f u p to kf , it must be F(u,x) = z . 3. T h e two p a r t s j u s t shown prove t h a t , for f recursive,

F(f,z)= 2 u (3u of finite support) ( u agrees with f u p to k f A F ( u , x )= z ) O n e direction follows from p a r t 1, since if F ( f , z ) = z there a r e functions u of finite support, agreeing with f up to any preassigned length, a n d such t h a t F ( u , z )= 2.

11. Basic Recursion Theory

210

Conversely, let u agree with f u p to k,, a n d F ( u ,x) = z . By p a r t 1 t h e r e is Y of finite support, agreeing with f u p to t h e maximum of kf,k,. By p a r t 2 then F ( v , z ) = F ( f , x ) , because Y agrees with f u p to kf. A n d F ( v , z ) = F ( u , z ) because Y agrees with u u p to k , (since Y agrees with f u p to k,, a n d so does u). 4. F can then be extended to a n effectively continuous functional on all of P , by letting

F ( a , z )N z

(321 of finite s u p p o r t ) ( u agrees with a u p to k , A F ( u , z ) = z ) H

for a n y partial function a .

0

Notc t h a t t h e condition t h a t a n effectively continuous functional m a p R to mapping a n y total function to a total function, not only recursive ones).

R is weaker t h a n totality (which requires

Exercise 11.4.7 There is a n effective operation o n R,which is not the restriction t o R of any total effectively continuous functional. (Rogers [1967]) (Hint: let F ( a ,z) be the first z generated in K such that a(.) = (pl(z),if it exists. F is a n effective operation on R,but if G is a total effectively continuous functional agreeing with F on R,let a ( z ) = ;.(I) 1 ifz E K otherwise.

{

+

Consider any value G ( a , s ) , the finite part of a used in the computation, and a function /3 agreeing with a on that part and on all elements 5 G(a,z), and equal to pa(.) otherwise, where a is the first element generated in K and > G ( a , z ) . Then F ( P ,z) is both equal and different to a, contradiction.)

Effective operations in general * An analysis of t h e proofs of Theorems 11.4.3 a n d 11.4.6 from a constructive point of view has been provided by Beeson [1975], a n d Beeson a n d SEedrov [1984]. In particular, b o t h results a r e derivable in Intuitionistic Arithmetic from Markov’s Principle, b u t without this principle t h e y a r e not derivable even in Intuitionistic Set Theory. Both results describe phenomena which hold in settings much more general t h a n those here considered. For t h e first, given a n effective algebraic c.p.0. X (p. 192) we c a n consider its effective p a r t X,, consisting of t h e elements for which t h e set of compact approximations is r.e. (w.r.t. t h e given enumeration). T h e enumeration of t h e compact elements of X generates a n enumeration of X,, a n d we can t h e n impose on it t h e Ershov topology, with t h e conipletely r.e. sets (w.r.t.

11.4 Effective Operations

21 1

t h e enumeration of X,) as basic sets. Then 11.4.2 and 11.4.3 are respectively generalized as: Ershov’s topology on X, is induced b y Scott’s topology on X , and the morphisms on X , as an enumerated set (ie. functions commuting with

the enumeration, through a recursive function) are exactly the restrictions to X , of effectively continuous functions on X (Ershov [1973]). For t h e second, Ceitin [1959], [1962] and Moschovakis [1964] show t h a t it holds on every effective complete separable metric space, i.e. a space with a recursive metric function on the recursive reals (see p. 213), with a dense r.e. subset, a n d in which one can effectively compute limits of recursive, recursively convergent Cauchy sequences. A more general framework for t h e theorem is t h e notion of effective topological space, see Nogina (19661, [1978]. Spreen a n d Young [1984] give a uniform generalization of both results above by considering countable To-spaces satisfying certain effectivity requirements, which hold for both PR and R. We return now t o the question formulated above, and look for conditions on A ensuring t h a t t h e effective operations on A are exactly t h e restrictions of effectively continuous functionals mapping A t o A. The positive results proved above apply to more general situations, but some conditions on A are necessary. Exercises 11.4.8 The effective operations o n A are exactly the restrictions of effectively continuous functionals mapping A o n A, an the following cases: a) A is completely r.e. (Myhill and Shepherdson 119551) (Hint: see 11.4.3.) b) A is the intersection of a completely r.e. class with R. Such classes are called totally r.e. (Kreisel, Lacombe and Shoenfield [1957], Ceitin [1959]) (Hint: see 11.4.6 ) , functions of finite support which are in A.) and, when defining ( P ~ ( ~ use

Proposition 11.4.9 (Pour El (19601, Myhill) There is an effective operation on a class A, which is not compact (and thus not the restriction of a continuous functional). Proof. Let A consist of the constant functions 0 and 1, and of the recursive functions which take a value different from 0 for an argument smaller than their minimal index. Let f be such that

0 if (Vz 5 e)(cp,(z) 21 0 ) 1 if (32 I e)(cpe(z)I A ~ e ( z$0). ) Then f defines an effective operation F on A which sends the constant function 0 t o itself, and the other functions of A to the constant function 1. Suppose t h a t only finitely many values determine the fact that F ( Q , x )21 0, and let k be bigger than all of them. There is a recursive function which is 0 u p t o k ,

11. Basic Recursion Theory

212

a n d which takes a value different from 0 below its smaller index, e.g.

a(.)

c?

0

ifz#k+l

a ifx=k+l

+

where a is different froni p e ( k l ) , for all e 5 k . T h e n F ( N )is t h e constant function 1, despite t h e fact t h a t a is 0 u p to k. 0

Proposition 11.4.10 (Yates, Young [1968], Helm [1971]) There is a n effective operation on a class A which is effectively continuous on A, and is the restriction to A of a continuous functional, but not of any effectively continuous functional. Proof. Let A be t h e class of (partial) recursive functions (Y such t h a t a(0) is defined, a n d either is not in K , or is promptly generated in i t , at a stage smaller t h a n t h e minimal index of a . Let f be such t h a t

pj(e)(x)

{

0 if pe(0)I A pe(0) E K e 1 if pe(0) 1 A Ve(0) @ Ke.

T h e n f defines a n effective operation F on A, which sends N t o t h e constant function 0 if ~ ( 0is) in K ,a n d to t h e constant function 1 otherwise. And t h e functional F’ which does t h e same on all of P is a n extension of F . Both F and F’ a r e clearly compact (only ( ~ ( 0is) needed to determine t h e value) a n d rnonotone o n their domains, hence continuous. Suppose F has a n extension G which is effectively continuous o n all of P . T h e n K: is r.e., because

xE

ts (311finite)(G(u,O) N 1 A u(0)= x).

Indeed, if there is 7~ as stated a n d x E K ,then n: is generated at s o m e s t a g e s. I t is enough to take a partial recursive function N extending t i , a n d with minimal index greater t h a n s. T h e n G ( N , ~c?) 1 by monotonicity, b u t F ( N , O ) c? 0, contrary to t h e fact t h a t G extends F : thus x E Conversely, if n: E hc, let u(0)= 2. T h e n F(u,O) c? 1 a n d , since G extends F , G(u,O)N 1. 0

c.

T h e t w o examples succeed because t h e y b o t h use not only values of t h e arguments, b u t also algorithms for them. T h e proofs a r e similar b u t q u i t e different, because they fail to extend to all of P for distinct reasons. T h e y provide examples as far a p a r t as they c a n be: t h e first does not e x t e n d for purely topological reasons (not being compact), t h e second for purely computational ones (being extendable to a continuous functional, b u t not to a n effectively continuous one).

11.4 Effective Operations

213

Exercise 11.4.11 Weak effective operations. A weak effective operation on A is defined as in 11.4.5, without the condition that F map A to itself. a ) If A is completely r.e., the weak effective opemtions on A are restrictions of effectively continuous finctionals. (Myhill and Shepherdson [ 19551) (Hint: see 11.4.3.) b) There is a weak effective operation o n R which is not compact. (Friedberg [1958b], Muchnick) (Hint: let (P,(~)(z)be 0 if cpe is either 0 for all arguments u p to e , or it coincides, u p to the first argument in which is not 0, with a function ‘pi that has index i smaller than that argument. Then f induces a weak effective operation F o n R because it is extensional, and F is not compact, as in 11.4.9.)

We have only touched on t h e subject of effective operations. For more information see Grzegorczyck [1955], Friedberg [1958a], Kreisel, Lacombe a n d Shoenfield [1959], Pour El [1960], Ceitin [1962], Lachlan [1964], Young [1968], [1969], Lob [1970], Helm [1971], a n d Freivald [1978].

Recursive analysis T h e real numbers c a n be classically defined in many equivalent ways, either as particular sequences or sets of rational numbers, or axiomatically ( u p to isomorphism) as archimedean complete ordered fields. Since rational numbers c a n be effectively coded b y integers, we have notions of recursive sequences a n d recursive sets of rational numbers, a n d different recursive analogues of t h e real numbers (Bore1 [1912], Turing [1936], Specker [1949]), using e.g. recursive decimal expansions, recursive Cauchy sequences with a recursive modulus of convergence, recursive Dedekind cuts, recursive sequences of nested intervals with length approaching zero. All of these notions t u r n out to be equivalent (Robinson [1951],Myhill [1953]),a n d t h u s a stable notion of a recursive real number is available. The recursive real numbers f o r m a countable subfield R

of the reals, which contains all the algebraic reals, as well as the real zeros of the Bessel functions, and becomes algebraically closed by the adjunction of the imaginary unit (Rice [1954]). Recursive real numbers a r e coded by integers ( t h e indices of their recursive presentations), a n d t h u s a notion of a recursive function of recursive real variable is available, as a n effective operation on t h e codes. Under t h i s definition, t h e field operations o n R a r e recursive (on t h e codes obtained from t h e Cauchy sequences definition), a n d Moschovakis [ 19651 characterizes R up to recursive isomorphism, by a recursive analogue of t h e notion of a n archimedean complete ordered field. T h u s R appears to be a good recursive counterpart to t h e reals. T h e r e a r e t w o different approaches to w h a t may be called recursive analysis, which a r e respectively p a r t s of constructive a n d classical mathematics. T h e first has been followed by t h e ‘Markov school’, which advocates a p h i l o s e

214

11. Basic Recursion Theory

phy (Markov (19541) in which every mathematical object is described by a word of a given alphabet, manipulated by Markov algorithms (p. 144) a n d discussed by means of constructive logic (intuitionistic logic plus Markov's principle). Under these philosophical assumptions, analysis is t h e s t u d y of R a n d of t h e recursive functions on it. T h e second approach remains in t h e realm of classical logic a n d mathematics, a n d it c a n be seen as a s t u d y of t h e extent of constructivity in classical analysis, by a n examination of which results remain valid when constructivized (either by t h e same proofs or by new ones), a n d which results instead fail. Once we live in a classical world, there is no particular reason to stick to recursive functions on R, a n d a notion of recursive functions of real variable has been introduced (Grzegorczyck [ 19551, [ 19571, Lacombe [ 19551, [ 19571) in analogy to t h e effectively continuous functionals, as opposed to effective operations. T h i s provides a new level of analysis, since it isolates computable functions (defined on all t h e reals) a m o n g t h e classical ones. Typical theorems of recursive analysis a r e t h a t every recursive function on R is continuous, although not necessarily uniformly so ( t h e positive part rephrases 11.4.6,t h e negative one follows from t h e failure of t h e recursive analogue of Konig lemma, see V.5.25), a n d t h a t the least upper bound principle fails (Specker [1949]) in t h e sense t h a t there is a bounded, strictly increasing, recursive sequence of rationals which does not converge to a n y recursive real number (simply, t h e sequence of t h e T, = 1/2f(n), with f a recursive o n c o n e function with a nonrecursive range). For a treatment of recursive analysis, see Grzegorczyck [1959], Goodstein [1961], KuSner [1973], Aberth [1980], Pour E l a n d Richards [1983a], Kreitz a n d Weihrauch (19841, a n d Beeson [1985]. Although not especially o n recursive analysis, Bishop [1967] is also relevant. A broader perspective can be obtained by looking at recursive interpretations o f (intuitionistic) set theory, in which analysis c a n t h e n be formalized. T h e effective topos (Hyland (19821) a n d t h e recursive topos (Mulry (19821) generalize, respectively, t h e constructive a n d classical approach to analysis ( t h e first using recursive realizability, a n d t h e second using forcing). Analogies a n d differences a r e investigated in SEedrov [1984], [1987], a n d Rosolini [1986].

CF='=o

11.5

Indices and Enumerations

*

In 11.1.4 a n d 11.1.9 we introduced a particular system of indices for partial recursive functions a n d r.e. sets, a n d later proved s o m e of its properties. In t h i s section we investigate systems of indices in general, b o t h for all partial recursive functions a n d r.e. sets (giving new examples, some equivalent to t h e one we already know, a n d some radically different) a n d for subclasses of t h e m .

11.5 Indices and Enumerations

*

215

Acceptable systems of indices A number of results have been proved so far for recursive functions, a n d some of t h e m mention indices in their statements, or use t h e m in their proofs. T h e question naturally arises: do these results depend on t h e particular formalism chosen t o work with (through which t h e indices were defined), or a r e they instead somehow model-independent? To s t u d y t h e problem, we first note t h a t t h e basic results mentioning indices are: 0

Enumeration Tlieorerri 11.1.5

0

S r - T h e o r e m 11.1.7

0

Padding Lemnia 11.1.6.

Indeed, t h e remaining results mentioning indices, or using t h e m in their proofs, merely refer t o these ( e g t h e Fixed-Point Theorem 11.2.10 follows from Enumeration a n d SF-Theorem alone). We now characterize t h e systems of indices for which these theorems hold.

Definition 11.5.1 (Uspenskii [1956], Rogers [1967]) W e call a system of indices any family of maps $" from w onto the set o f n-ary partial recursive functions (by $2 we will indicate the partial recursive function corresponding to the index e). W e say that:

+

1. $ satisfies enumeration i f for every n there is a number a such that

+ : + l ( e , ~ l r . .. , ~ 7 , )

N +:(zl,.

.. , E n )

2. ?i, satisfies parametrization if for every m,n there is a total recursive

function s such that 711.

z,)(Yl,. . . , Y m )

+8(e,zl ),_.,

?i,jl,m+'l(~l,.

. . ,%,!/I,.

. . ,Ym).

As usual we will drop tlie mention of t h e number of variables, when this is understood. A n example of a system of indices is obviously given by t h e usual cpp, which also satisfies b o t h enumeration a n d parametrization: we call it t h e standard system. A system may resemble t h e standard one, in t h e sense t h a t it is possible to go effectively from t h e former t o t h e latter, arid conversely:

+

Definition 11.5.2 (Rogers (19581) A system of indices is acceptable if, for every n, there are total recursive functions f, and g such that

$2 = (P3(e)

cp:

= +&).

11. Basic Recursion Theory

216

Note t h a t in practice t h e equivalence among different definitions of t h e class of partial recursive functions is usually proved precisely by showing t h a t t h e systems of indices induced by them resemble one another, in t h e sense just given (see t h e proof of 1.7.12). T h e next result shows t h a t they satisfy t h e same basic properties, a n d also gives a n intrinsic characterization of t h e notion of a n acceptable system of indices.

Proposition 11.5.3 (Rogers [1967])A system of indices is acceptable zf and only i f it satisfies both enumeration and parametrization. Proof. Suppose acceptable: 0

0

Given

+:,

+ satisfies both enumeration a n d parametrization.

Then

+ is

by t h e enumeration property this is a recursive function of

n + 1 variables. Since systems of indices must enumerate all t h e partial recursive function, there is a n index a for +cl," with respect to 'p. Then

g is obtained, symmetrically, by using t h e Enumeration Theorem for a n d t h e parametrization property for

+.

'p

Conversely, suppose t h a t $J is acceptable.

To show enumeration, note t h a t

But (p3( (Z) is partial recursive as a function of e, Z (by t h e Enumeration e) Theorem), a n d thus it must admit an index a in t h e system

+.

0

To show parametrization, consider +rtm(Z, 9. By t h e enumeration property just proved, this is a partial recursive function of m n 1 variables (including e ) , and thus there is a n index a for i t with respect to 9:

+ +

Since g allows us to go back to t h e system

+, if

we let

11.5 Indices and Enumerations

*

217

Corollary 11.5.4 E v e r y acceptable s y s t e m of indices satisfies the Fixed-Point Theorem. In other words, given a recursive function f there is an i n d e x e such that

+e

= +'r(.).

Proof. T h e only properties used in t h e proof of t h e Fixed-Point Theorem a r e enumeration a n d parametrization. 0 Exercises 11.5.5 a ) A system satisfying enumemtion alone is not necessarily acceptable. (Friedberg [ 19581) (Hint: a recursive enumeration without repetitions of the partial recursive functions, see 11.5.23, does not satisfy 11.5.6.) b) A system satisfying enumeration and fied-point, i.e. the existence of a recursive function f such that, for $i total, $ @ , ( f ( , ) ) 2: $ f ( , ) , is not necessarily acceptable. (Machtey, Winklmann and Young (19781) (Hint: let (Ph(e) be a recursive enumeration N (Ph(v,(0)) without repetitions of the partial recursive functions, see 11.5.23. Then satisfies enumeration and fixed-point, the latter because $ J ~ depends extensionally on qe, and thus fixed-points transfer from cp to $. Note that any partial recursive function is equal to ( P h ( i ) for a unique i , and then can be equal to only when cp,(O) 2: i : thus it has a n r.e. set of indices w.r.t. 4. But e.g. the completely undefined function cannot have an r.e. set of indices in any acceptable system, a n d thus $ is not acceptable.) c) A system satisfying enurnemtion and composition, i.e. the existence of a recursivc function c such that $=o 41iN $c(e,i), is acceptable. (Machtey, Winklmann and Young [1978]) (Hint: by coding the arguments, parametrization can be reduced to coniposition and its special case for sequence numbers, e.g. if $,(=)(y) = {z, y ) then $ e ( ( z ,Y)) 2: $ e ( $ f ( z ) ( Y ) )

+c(e,f(z))(Y)

And f can be defined by successive compositions of the two functions g(y) and h ( ( z , y ) )= (z I , Y ) . )

=

(0, y )

+

T h e results j u s t proved show t h a t the notion of a universal machine, embodied in the enumeration property, is not suficient for elementary Recursion Theory: it has to be supplemented with the notion of subcomputation, through t h e possibility of either effectively incorporating data into a program (expressed by t h e parametrization property), or effectively concatenating programs (expressed by composition). O n t h e other hand, t h e Fixed-Point Theorem, a n d t h u s t h e possibility of having self-referential programs (so-called recursive programs), does not appear to be as fundamental. Notice t h a t enumeration a n d parametrization imply, among other things, a back a n d forth translation between t h e function spaces

wn x urn+ w a n d

wn 4 (wm + w).

+

Indeed, a partial recursive function of n m variables becomes, by p a r a m e trization, a partial recursive function of n variables, whose values a r e (indices

11. Basic Recursion Theory

218

of) partial recursive functions of m variables. Conversely, a n y partial recursive function of n variables whose values a r e indices of partial recursive functions of m variables becomes, by enumeration a n d composition, a partial recursive function of n m variables. To t h e reader acquainted with Category Theory (Mac Lane [1971]), this is reminiscent of t h e typical property of Cartesian closed categories, i.e. those categories in which t h e sets of morphisms between objects a r e at t h e s a m e level of t h e objects themselves. More precisely, in a Cartesian closed category products A x B a n d exponents B A (which generalize the notions, respectively, of Cartesian product of A a n d B , and of sets of functions from A to B ) nicely commute, in t h e sense t h a t t h e following sets of morphisms a r e isomorphic, in a natural way: H o m ( A x B,C) and H o m ( A , C B ) .

+

For a development of Recursion Theory from a categorical point of view, see Eilenberg a n d Elgot [1970], a n d Di Paola a n d Heller [1987]. We still have to analyze t h e role of t h e Padding Lemma. For explicitly given systems of indices (like t h e ones induced by t h e approaches to partial recursiveness of Chapter I a n d Section 11.1) this property holds trivially, a n d it simply amounts to adding redundant information in t h e given description of t h e function. But having infinitely many indices in one acceptable system does not help in another one, since t h e translations provided by t h e definition of acceptability a r e not necessarily one-one. Nevertheless, t h e result holds in general.

Proposition 11.5.6 Padding Lemma (Rogers [1958])I n any acceptable system, given one index of a partial recursive function, we can effectively generate infinitely many indices of the same function. Proof. I t is enough to show t h a t , given a partial recursive a n d D finite set of indices for it in t h e acceptable system we can effectively find a n index for a which is not in D . Define, by parametrization, a recursive function f such t h a t

+,

"r(e)

{

By t h e Fixed-Point Theorem for cases may occur:

zndefined

ife#D otherwise.

+, there is e such t h a t

+,

1: +f(,).

Now two

e#D Then we must be in t h e first case of t h e definition of f , a n d hence CY +o,(,) N a. T h u s e is a n index of CY which is not in D.

+,

11.5 Indices and Enumerations a

*

219

e E D Now we must be in t h e second case of the definition, and $e is t h e completely undefined function. But e , being in D ,is an index of a,and thus CY is also completely undefined. Then we can play a symmetric game, this time letting g be such that $9(4

=

{

constant 0 if a E D undefined otherwise.

By t h e Fixed-Point Theorem, there is a such t h a t $a $Jg(a). Now it cannot be the case that a E D , otherwise (as above) CY would be t h e constant function 0, while we know it is completely undefined. Then we must be in the second case, i.e. a @ D and is completely undefined. Thus a is a n index of CY which is not in D. 0 Now we have, for acceptable systems of indices, all t h e results proved for t h e standard one, but this does not guarantee that t h e same will remain t r u e for future results. We will now prove two general theorems, showing how much any acceptable system must resemble t h e standard one. They will ensure t h a t acceptable systems are really very much alike. T h e first theorem shows that any acceptable system, viewed as a sequence of partial recursive functions

is nothing else than a recursive permutation of the standard s q u e n c e cpo

'p1 cp2

...

Theorem 11.5.7 (Rogers [1958]) { $ e } , c w is a n acceptable system of indices i f and only i f there is a recursive permutation h such that

Proof. One direction obviously holds: if h exists, its inverse h-' is also a recursive permutation, a n d h and h-' provide t h e needed translations of t h e systems $ a n d cp into each other. Suppose now t h a t $ is acceptable: we define a recursive permutation h t h a t interchanges indices in $ and cp. Since h has to be total, we ensure at even stages t h a t t h e least element not yet in t h e domain gets into it (by defining h on it). Similarly, h has t o be onto, and at odd stages we ensure t h a t t h e least element not yet in the range gets into it (by letting it be the value of h for some argument). We then have to show how to ensure t h a t h is both one-one a n d a function. But h is a function when h-' is one-one, and thus we will have a

220

11. Basic Recursion Theory

symmetric construction t h a t alternates steps to make h t o t a l a n d one-one, to steps to make it onto a n d a function. We j u s t show t h e steps t h a t have to be alternately taken. Suppose h is defined on xo,. . . , x,. If we want to add one element to t h e domain of h (even stages), let e be t h e least number not in ( 2 0 , .. . ,xn}: we have to define h ( e ) in such a way ) . @ is t h a t h(e) # {h(xo),. . . , h(x,)} (to have h one-one), a n d @e = ( ~ h ( ~Since acceptable, given Qe we can effectively find a n index of t h e s a m e function w.r.t. cp. T h e Padding Lemma 11.1.6 ensures t h a t we can effectively generate infinitely many others, a n d t h u s one r a n be found t h a t is not in { h ( x o )., . . , h ( s n ) } .T h e first such one is t h e needed value of h ( e ) . If we want to add one clement to t h e range of h (odd stages), let y be t h e least, number not in {h(xo),. . . , h(x,)}: we have to define e in such a way t h a t e # ( 5 0 , .. . ,sn}(since h h a s to b e a function), a n d Qe = cpu. T h e n we will let h ( e ) = ? j . Since ?) is acceptable, given cpu we c a n effectively find a n index of t h e s a m e function w.r.t. $. T h e Padding Lemma 11.5.6 for @ ensures t h a t we can effectively generate infinitely many others, a n d t h u s o n e can be found t h a t is not in {ZO, . . . ,x,}. T h e first such o n e is t h e needed value for e. 0 . T h e previous result is not completely satisfactory, because i t misses t h e basic duality of Recursion Theory (see p. 131). It translates a number when it is a code of a function, b u t it leaves t h e s a m e number untouched when this behaves as a number (i.e. as an argument or a value). T h e next result shows that, any acceptable system is really the standard one, and merely works with an appropriated reinterpretation of the numbers (with n o distinction m a d e on whether t h e y code programs or not). Thus t h e isomorphism provided here is not a n isomorphism of t h e sequence of function, b u t rather a n isomorphism of t h e underlying structure of t h e natural numbers.

Theorem 11.5.8 (slum) If { @ e } e c u is an acceptable system of indices, then there is a recursive permutation h such that

Proof. T h e r e a r e actually two different things to ensure, namely t w o reinterpretations of t h e numbers, one taking care of t h e case when they a r e seen as arguments a n d values, a n d t h e other when they a r e seen as codes for p r o g r a m . Morcover, the two have to coincide. If we only had to define a recursive permutation h t h a t produces t h e given isomorphism on t h e numbers as codes for a given reinterpretation cr of t h e numbers as values a n d arguments, then we could simply proceed as in t h e previous theorem. Namely, let a-' be any partial recursive function that inverts a , i.e. any function t h a t , given z , dovetails computations of cr until it finds,

11.5 Indices and Enumerations

*

22 1

antl outputs, a number z such t h a t a(.)

E 2.

We have to ensure t h a t

antl this can be restated as both:

If e has to be added to t h e domain of h, we use t h e fact t h a t C Y @ ~ C I - 'is partial recursive, a n d an infinite number of indices w . r . t . cp for it can then effectively be found. If y has to be added to t h e range of h, we use t h e fact that a-'cp,cr is partial recursive, a n d a n infinite number of indices w.r.t. for it c a n t,hen effectively be found. T h u s , given any partial recursive function a , we obtain, uniformly in i t , a recursive permutation ha t h a t satisfies t h e given conditions. If we could s t a r t with a function Q equal to t h e function h, we obtain from i t , t,hen we would have w h a t we want. T h a t this can be done is ensured by t h e Fixed-Point Theorem. Formally, given a: N cpt, let pf(.) N ha be t h e recursive permutation obtained from it:

+

pz(@e(z)) 21 (~'p,(.) Let a be such t h a t pa N

( ~ j ( ~ T h) e. n

( e )( c p Z ( z ) ) .

pa is a recursive permutation such t h a t

pa(@e(Z)) 2~ p p o ( e ) ( p a ( x ) ) ,

as wanted.

0

We t h u s see t h a t acceptable s y s t e m s of indices provide the same structure theory f o r recursive f u n c t i o n s as the standard one we have been using so far, a n d from now on we will j u s t suppose t h a t cp; is any acceptable system. I n part,icular, any of t h e approaches to partial recursiveness discussed in C h a p t e r I a n d Section 11.1 provides a perfectly adequate basis for Recursion Theory. For more o n acceptable systems of indices see Rogers [1958], Lachlan [1964], Schnorr [1975], Hartmanis a n d Baker [1975], Machtey, Winklmann a n d Young [ 19781, a n d Hartmanis [ 19821.

Axiomatic Recursion Theory

*

Thc: central role of enumeration a n d paranietrization, a n d t h e fact t h a t they a r e purely algebraic properties seemingly having nothing to do with computations,

11. Basic Recursion Theory

222

suggest their use in a n axiomatic treatment of t h e part of Recursion Theory developed so far for abstract domains a n d collections of partial functions over them. Wagner [1969] a n d Strong (19681 introduce t h e notion of Basic Recursive Function Theory BRFT as a structure ( D , 3 ,{ ~ p " } ~ ~ , ) , with D a n infinite set, 3 a set of partial functions on D , a n d 'p" a n n 1-ary function of 3 enumerating (over D ) t h e n-ary functions of 3. Moreover, 3 contains t h e identities, t h e constant functions on D , a function

+

f(x,a , b, 4 =

b ifx=a c otherwise

(for case definition), all t h e 'p", a n d parametrization functions. T h e notion of B R F T c a n be variously polished (see e.g. Moschovakis [1971], for an alternative equivalent notion of precomputation theory) . B R F T basically captures the essence of elementary Recursion Theory, t h a t is, t h e p a r t t h a t does not explicitly involve t h e notion of length of computation (like t h e proof of t h e Reduction Property 11.1.23). It t h u s a p p e a r s t h a t t h i s p a r t of Recursion Theory does not use any particular property of t h e set of natural numbers (like being countable, well-ordered, etc.). A particularly interesting special case is t h e one of o-BRFT, where D is w, a n d t h e successor function is in 3. I t is not surprising (see p. 158) t h a t every w - B R F T contains all partial recursive functions: t h i s means t h a t t h e partial recursive functions, together with a n acceptable system of indices, a r e a minimal w-BRFT. T h u s t h e axiomatization extends t h e properties of t h e class of partial recursive functions to bigger classes. T h e result holds in general, a n d provides a connection between t h e axiomatic approach a n d computability on abstract structures (see p. 202): the minimal B R F T o n an abstract domain is the set of prime computable functions over it (Moschovakis [1971]). A strengthening of 11.5.3 shows t h a t any two w - B R F T with the same set 3 of partial functions are inutually traizslatable b y functions in 3,in t h e sense t h a t , a n d ( w , 3 , {$"}nG,), there are, for every n, functions f given ( w ,3,{'p"},~,) a n d g in .F such t h a t

A strengthening of 11.5.8shows t h a t any two such w - B R F T are isomorphic as structures (Friedman [1971]), in t h e sense t h a t there is a one-one, o n t o function i (in 3)such t h a t , for every n,

11.5 Indices and Enumerations *

223

In particular, there is only one w-BRFT with the set of all partial recursive functions as the set of partial functions, up to recursive isomorphism, see 11.5.8. Note t h a t these w-BRFT's a r e exactly t h e ones corresponding to acceptable systems of indices. On t h e other hand, Friedman [1971] shows t h a t there are uncountably many w-BRFT's with the set of total recursive functions as the set of total functions. As we have noted, BRFT captures only elementary Recursion Theory. As a first step toward a n extension t h a t also covers arguments like 11.3.16, Moschovakis [1971] introduces t h e notion of computation theory, basically b y adding t h e primitive notion of length of computation, a n d postulating t h e fact t h a t , ~ )longer ( $ t h a n t h e computation of cp,(.,'f). t h e coniputation of ~ s ~ ( ~ takes T h e n t h e First Recursion Theorem becomes provable in a computation theory, as in 11.3.16. An equivalent formulation of computation theory, based on t h e primitive notion of immediate subcomputation, is given by Fenstad [1974]. For more information on BRFT and computation theories, see Barendregt [1975],Fenstad [1980],Beeson [1985], Byerly [1985]. To consider abstract domains a n d collections of functions over t h e m is taking t h e point of view of Category Theory. Not surprisingly, Recursion Theory can be formulated a n d developed in this setting, see Eilenberg a n d Elgot [1970] for t h e basics, a n d Di Paola a n d Heller [1987] for t h e consequences of enumeration a n d parametrization.

Models of A-calculus (part 11)

*

We have seen (p. 83) t h a t t h e two combinators

S = Azyz.zz(yz)

and

K = Azy.z

allow us to define a version of A-abstraction, and thus to interpret A - t e r m as combinators built u p from S a n d K . Conversely, combinators built u p from S a n d K can be naturally translated into A-terms. This suggests t h e introduction of a first-order theory with equality, called combinatory logic, with two primitive symbols S and K , a n application operation between t e r m s (written as left-associative juxtaposition), and with axioms reflecting t h e behavior of S and K :

Kzy = z a n d

Szyz = zz(yz).

T h e n combinatory logic a n d A-calculus can be interpreted one in t h e other, although (without additional assumptions, such as extensionality) t h e interpretations a r e not inverses. Models of combinatory logic can be easily defined: a combinatory algebra is a structure ( A , , k,s),with . a binary operation, a n d k a n d s distinct elements +

11. Basic Recursion Theory

224

of A satisfying t h e axioms above. An extensional combinatory algebra is a combinatory algebra such t h a t , whenever a . c = b . c for every c in A , a a n d b a r e equal. T h e notions of combinatory logic a n d combinatory algebra c a n be easily extended t,o allow for partial application, with strong equality = substituted by a partial one N (Klop [1982], see Beeson [1985] for details). T h e n the

structure (w, .), with partial application defined as e.z

= cpe(z),

is a partial cornbinatoy algebra, because there a r e numbers k a n d s such t h a t c p k ( z 7 Y ) !x 5

and

(Ps(z,?/,z)

= cp,,(z,(cpy(4).

By 11.5.8, this structure is independent ( u p to recursive isomorphism) of t h e acceptable system { c p e } e E w . Since only enumeration a n d S F - T h e o r e m a r e used to determine s a n d k , any w-BRFT is naturally interpreted as a partial cornbinatory algebra, a n d t h u s provides a model of partial combinatory logic. Models of partial combinatory logic a r e not automatically models of A-calculus. A first problem h t h a t application is t o t a l in t h e latter, a n d t h u s only total combinatory algebras a r e eligible (but t h i s could be solved by defining a version of partial A-calculus). A more serious problem is d u e to t h e fact t h a t t h e translations of A-calculus a n d combinatory logic a r e not inverses. T h e solution here consists of additional requirements t h a t force t h e interpretation of A-terms defined in a total combinatory algebra to more fully reflect t h e structure of A-terms. T h e r e a r e various possibilities. 1. Clearly, a model of A-calculus should identify t e r m s t h a t a r e equal (modulo Q: or @ reductions). A combinatory algebra t h a t preserves equality of A-terms is called a A-algebra. 2. Unfortunately, a A-algebra does not necessarily satisfy t h e requirement t h a t ternls behaving t h e same way (i.e. such t h a t M z = N z for every z) a r e equal as functions (i.e. Ax. M z = Xz. Nz),a kind of weak extensionality t h a t is obviously valid in A-calculus. A A-algebra which is also weakly extensional in this sense is called a A-model, a n d t h i s notion is considered to be thecorrect notion of model (Scott [198Oa], Meyer [1982]).

3. A stronger notion of extensionality requires t e r m s t h a t behave t h e s a m e way to be equal not only as functions, b u t as terms. An extensional combinatory algebra satisfies this, a n d can be interpreted as a A-model (and actually as a model of extensional A-calculus, see p. 82) in a unique way. T h e existence of a n extensional partial combinatory algebra follows from II.5.23.a (Kreisel [1971]).

11.5 Indices and Enumerations

*

225

Not every combinatory algebra can be extended to a A-model a n d , when it c a n , t h e extension is riot necessarily unique. Typical examples of combinatory algebras which a r e A-models in a unique way a r e L), a n d P(w) (p. 194). T h e former, b u t not t h e latter, is also a n extensional combinatory algebra. For a treatment of this topic, see Barendregt [1981], Beeson [1985], Hindley a n d Seldin [1986]. See also Byerly [1982], a n d Fkeyd a n d SEedrov [1987].

Indices for recursive and finite sets Recursive a n d finite sets, being r.e., have r.c. indices which rode ways to gene r a t e t h e m . But t h e y also have special properties t h a t allow for different represent at ions.

Definition 11.5.9 A recursive set A may be given three different types o f indices: 1. characteristic indices, i.e. the indices of its characteristic function. W e write A = Ce if (Pe N C A . 2. complementary indices, i.e. the pairs of indices of the set and its complement, as r.e. sets. W e write A = Re, with e = ( a ,b ) , i f A = W, and 2 = Wb. 3. r.e. indices, i.e. the indices of the set as an r.e. set. Exercise 11.5.10 Characteristic and complementary indices are equivalent, in the sense that it is possible to g o effectively from one to the other, and conversely. (Hint:

see II.1.19.)

Proposition 11.5.11 (Suzuki [1959]) It is possible to go effectively from complementary indices to r. e. indices of recursive sets, but not conversely. Proof. T h e positive assertion is obvious. As a counterexample to t h e converse, suppose t h e r e is cp partial recursive such t h a t

T h c n there is also @ partial recursive such t h a t

Let f be a recursive function such t h a t

w ifeEK: 8 otherwise

11. Basic Recursion Theory

226

T h e n W,,,, is always recursive, so $f ( e ) is total a n d

0

ifeEK

w otherwise. Hence eE

and

E would

w+,(,,is nonempty,

E

be r.e., contradicting 11.2.3.

0

Exercise 11.5.12 There is no partial recursive function cp such that

W , recursiire

+

p(e)

1A

( 3 i _< cp(e))(W, = me).

Thus not only can an r.e. index for the complement of a recursive set not be found effectively, it cannot even be bounded effectively. (Gold [1!467]) (Hint: define Wqe) to contain an element from each nonempty Wi, for Z 5 cp(e), so that its complement cannot be any of the W,, and apply the Fixed-Point Theorem.)

T h e content of t h e results just proved is t h a t t h e characteristic function of a recursive sets a n d a pair of enumerations of it a n d its complement have t h e same information, while a simple enumeration of t h e set is less informative t h a n both. Not,e t h a t proposition 11.2.1 shows t h a t there is no recursive function which enumerates at least one characteristic index for each recursive set, a n d t h a t the

set Char = {x : cpz is total and 0,l-valued }

is not r.e. Different results hold for r.e. indices. By 11.5.19 it is still t r u e t h a t the set Rec = {x : W , is recursive } is not r.e. (a coniputation of its complexity will be given in X.9.12). B u t a recursive function does exist t h a t enumerates at least o n e r.e. index for each recursive set (115.26). Definition 11.5.13 A finiteset A may be given three different types of indices: 1. canonical index, i.e. a number explicitly coding all the elements of the set. W e write A = De i f e = 2zQ . . . 2"-, and A consists of the distinct elements ZO, . . . , x,. By convention, Do = 0.

+

+

2. characteristic indices, i.e. the indices o f its characteristic function. W e write A = C, i f qe 21 c ~ . 3. r.e. indices, i.e. the indices of the set as a n r.e. set.

11.5 Indices and Enumerations

*

227

T h e requirement t h a t t h e xi be distinct ensures t h a t every canonical index e defines a unique set D,. Moreover, t h e relevant initial segment of t h e characteristic function of D, is coded by t h e digits of t h e binary expansion of e , read from right to left (for example, e = 1001101 codes t h e set 0 7 7 = { 0 , 2 , 3 , 6 } ) . As it might be imagined, canonical, characteristic a n d r.e. indices a r e increasingly less effective, a n d give less a n d less information on t h e sets they code:

Proposition 11.5.14 (Rogers [1967]) It is possible to go effectively f r o m canonical indices to characteristic indices, and f r o m characteristic indices t o r.e. indices, but not conversely. Proof. T h e positive directions a r e obvious. Suppose it is possible to go effectively from characteristic indices to canonical ones, i.e. t h a t for some partial recursive function +: p, characteristic function of a finite set A

+

@ ( e )1 A A = D G ( ~ ) .

Define a recursive function 9 such t h a t vg(e) is t h e characteristic function of a nonempty finite set if e E K ,a n d of 0 otherwise, e . g 1 if c = f(x) 0 otherwise

where

f is a one-one enumeration of K. T h e n +g(e) is total, and e E X: tj D$,~(,)= 0 H $g(e) = 0,

a n d K would be recursive. Similarly, suppose t h a t , for some partial recursive function $,

We finite

+ +(e) 1 A p+(.) characteristic function of We

If Wg,,) =

{

0

ifeEK: otherwise

t h e n +g(e) is total, a n d again K would be recursive, since

Exercise 11.5.15 The r.e. indices may be arbitrarily smaller than the canonical ones: if h is recursive, there are 2,y such that W, = D, and y > h ( z ) . Thus less information is nmre easily coded. (Hint: given h, let

11. Basic Recursion Theory

228 and apply the Fixed-Point Theorem.)

As already for t h e r.e. indices of recursive sets, by 11.5.19 the set Fin = { x : W, is finite )

is not r.e. A computation of its complexity will be given in X.9.6. T h e next definition introduces useful terminology, t h a t we will use repeatedly.

Definition 11.5.16 A n array is an r.e. set whose elements code finite sets. It is calld: 1. weak i f its elements are viewed as r.e. indices 2. strong i f its elements are viewed as canonical indices

3. disjoint if its elements code pairwise disjoint, finite sets.

A strong array is nothing more t h a n a weak array together with a recursive function giving t h e cardinality of t h e members of t h e array. Since, if we know how to enumerate a set a n d how many elements t h e r e a r e in i t , t h e n we c a n obtain all its elements: enumerate t h e set, until t h e right number of elements has been generated. We will still call a (weak or strong) array t h e collection of finite sets coded by t h e elements of a given (weak or strong) array. Exercise 11.6.17 T h e m are weak armys which are not strong. (Hint: let A be r.e. and norirecursive, and A, = { z : .z E A A z 5 n}.)

Enumerations of classes of r.e. sets 111 this section we consider classes of r.e. sets, a n d possible ways of enunierating them. Some of t h e results we prove could also bc stated for classes of partial recursive functions, b u t t h e consideration of r.e. sets sometimes allows for snioother formulations a n d proofs.

Definition 11.5.18 (Dekker [1953a],Rice [1953])A class A of r.e. sets is called: 1. completely r.e. i f its index set (i.e. the set containing all indices of each member of A) is r.e.

2. r.e. if there is a recursive function f which enumerates at least one index of each member o f A, i.e. A = {W,(,,}sEw.

11.5 Indices and Enumerations *

229

3. r.e. without repetitions i f there is a recursive function f which enumerates exactly one index of each member o f d , i.e.

A = { W j ( z ) } z ~ uand

(wf(z)= W f ( g ) * 2 = Y).

T h e characterization of completely r.e. classes of r.e. sets is t h e same as t h a t of completely r.e. classes of partial recursive functions (see 11.4.2), a n d we leave t h e routine modification of t h e proof to t h e reader.

Proposition 11.5.19 (Myhill and Shepherdson [1955], McNaughton, Shapiro) A class of r.e. sets A is completely r.e. af and only i f it consists o f the r.e. supersets of the elements of a strong array, i.e. for some r.e. set A : W, E d u (3y)(y E A A D,

C Wz).

We can now show t h a t a number of classes of r.e. sets are not completely r.e. For example: any finite class, any class containing only finite (or only infinite) sets, t h e class of recursive sets (since any finite set admits a n r.e. nonrecursive extension), a n d so on. O n t h e other hand, t h e class of t h e r.e. supersets of finitely many finite sets is completely r.e.

Exercises 11.5.20 a) Not every completely r.e. class is the class o f t h e r.e. supersets of the elements of a recursive strong away. (Rice [1956]) (Hint: let A be the class of r.e. supersets of the strong array {{z} : z E K}.) b) There is a completely r.e. class such that any strong army that generates it must contain superfluous infomaation. (Rice [1956]) (Hint: the only array not containing superfluous information is the core, i.e. the set of rninimal finite sets belonging to the f(z) 1,. . . ,f ( y ) - 1) class. Let A be the class of r.e. supersets of the finite sets (f(z), for f(z)< f(y), with f recursive one-one function enumerating a n r.e. nonrecursive set R containing 0. Then the core of A is not r.e., otherwise B would be recursive.) c) Not every class consisting of the r.e. supersets of the elements of a weak army is completely r.e. (Myhill and Shepherdson [1955]) (Hint: there is an r.e. set A which is hypersimple but not hyperhypersimple, see 111.4.12. Then there is a disjoint weak array B, but no disjoint strong array, whose members intersect 3. Let A be the class of r.e. supersets of the elements of R.)

+

We turn now to r.e. classes. Obviously, t h e class of r.e. sets is r.e. (any number codes a n r.e. set), b u t there seems to be no natural way to extract a n enumeration without repetitions.

Exercise 11.5.21 There is no invariant recursive choice function for indices of r.e. sets, i.e. a recursive function f such that f ( e ) is an index of We,and

Wi = W e

f ( i )= f ( e ) .

11. Basic Recursion Theory

230 (Hint: suppose f exists, and let w

For any a E

ifeEK

K, e E K e fg(e) = fg(a), and K would be recursive.)

Nevertheless, we have t h e following result

Theorem 11.5.22 (F'riedberg [1958]) The class of r.e. sets is r.e. without repetitions. Proof, A natural idea would be to pick up, for each r.e. set, its minimal index, b u t we know froni t h e exercise t h a t this cannot be d o n e recursively. T h e idea of t h e proof is to t r y anyway, with a n indirect approach: we simulate each r.e. set, until we discover t h a t it looks too much like some other r.e. set with a smaller index, in which case we drop t h e finite approximation of t h e former. By doing so, we actually introduce a number of additional finite sets to t h e original enumeration of t h e r.e. sets, b u t this does not interfere with our goal as finite sets a r e r.e. anyway. Consider any acceptable enumeration {W,},cw of t h e r.e. sets, such t h a t Wo = w. We a r e going to define a n r.e. sequence {Sz}2Ew of r.e. sets, in which every r.e. set appears exactly once ( t h e existence of f recursive such t h a t S, = Wf(,, will follow by t h e S ~ - T h e o r e msince , t h e sequence S, is r.e.). Since we enumerate t h e sets SZk, we let S,,3 be t h e p a r t of $7, enumerated by t h e end of stage s. We call x a follower of e if we t r y to make, in t h e e n d , S, = W e . If a n d when we decide, for whatever reason, t h a t we no longer want to pursue S, = We,we release Z, a n d z will never return to be a follower. If, instead, z is never released, i t is a permanent follower. At any stage each e h a s at most one follower, a n d z is called unused at a certain stage if at t h a t stage i t is noi,, a n d has never been, a follower. T h e construction s t a r t s by letting 0 be a follower of 0 (i.e. we t r y to make So = W O )a, n d S,,O = 0. At step n + 1, suppose z is a follower of e (hence, by construction, we have = We,,). W e release it in t h e following situations:

s,,n

1. for some z < e, W,,na n d We,n look t h e s a m e u p to x (i.e. for each z 5 z), t h e reason being t h a t e does not look as a minimal index for W e u p to z. 2. z > 0 a n d , for some y already released, SZ,,, = S,,,, t h e reason being t h a t We might just be t h e finite set S,,,& = S,, a n d e might be a minimal index for it, so if we let z continue to be a follower of e we would get, in the e n d , S , = S,.

11.5 Indices and Enumerations

*

23 1

Note t h a t 0 is never released. If z has just been released, S, will not change anymore after this stage. Since we want t o have all t h e S,'s distinct, let b be a n element larger than UeEwse,n (which is a finite set), and let

s, = Sz,n+l= ( 2 : z < z + b } . Certainly S, is different from all the other sets at this stage, and Sz,n S,. To keep things going we have t o add followers, at least from time t o time, t o elements t h a t d o not have them: e.g. we assign t h e smallest unused element as a follower to e, if e does not have one (either because it never did, or because it lost it) a n d n = (e, t ) (so t h a t e gets infinitely many chances to receive a follower). Finally, if z is a follower of e at this stage, we let Sz,n+l

= We,n+l.

We now prove t h a t the construction works. 1. every r.e. set appears at least once: Ve3z(We= S,). We may suppose t h a t e is a minimal index. Then there is a stage no such that

(Vn 2 no)(Vz 2 no)(Vz < e)(Wi,, and We,%look different u p to z ) . If e has a permanent follower z, then S, = We.And this must be t h e case, otherwise e keeps on getting new followers that sooner or later become released. After stage no the only possibility for release is t h e second one, a n d this means that for each follower z there is a stage n and a released element y such that

We,n= s,,n= s,,n = ( 2 : z < c,) for increasingly big cYk. Hence We = w and, by minimality of e, e = 0. But 0 has a permanent follower, namely 0.

*

2. every r.e. set appears at most once: # y S, # S,. By construction no element is always unused (there are infinitely many r.e. sets, and we always choose as followers the smallest unused elements). So four cases may happen, for different z and y:

both are permanent followers, say S, = We and S, = Wi Since an index has at most one permanent follower, i # ti. Suppose e.g. that i < e. If S, = S, then Wi = W e ,so there is no such t h a t

( V n 2 no)(W+ and We,nlook t h e same up to z). Then z is released at stage n

+ 1, contradiction.

11. Basic Recursion Theory

232 0

b o t h x a n d y a r e released If they a r e released at t h e s a m e stage t h e n S, # S,, because (for t h e appropriate b ) x b # y b. If e.g. 5 is released after y is, then (for t h e appropriate b ) b E S, b u t b # S,.

+

0

+

o n e is released a n d t h e other is permanent Say x is released a n d y is permanent. If S, = S, t h e n S, is finite, so y # 0 (since So = w).Hence, at some s t a g e n, S,,n = S,,n a n d y is released at stage n 1, contradiction. 0

+

T h e method of proof j u s t used (where, as 11.5.21 shows, release of followers is not avoidable) is a weak version of t h e priority method, in which t h e r e a r e requirements, positive (trying t o p u t elements in a set) o r negative (trying to leave t h e m o u t ) , t h a t have to be satisfied, a n d actions to satisfy o n e t y p e might interfere with t h e satisfaction of t h e other type. Here t h e positive requirements t r y to p u t Wein t h e enumeration, while t h e negative requirements t e n d to make t h e SZ's distinct. T h e crucial fact is t h a t t h e action to satisfy a given positive requirement can be interfered with (by releasing a follower) only finitely often, a n d t h u s t h e construction succeeds. T h e priority method will be introduced a n d fully exploit,ed in Chapter X . Exercises 11.5.23 a) The class of partial recursive functions is r.e. without repetitions. (Friedberg [1958]) (Hint: start with an enumeration of the partial recursive functions, given by their graphs, and make the set S, different from all the other Si's as a partial function, instead of simply as a set, when z is released.) b) A n r.e. class A = { A z } l ~ w of disjoint, nonempty r.e. sets is r.e. without repetitions i f and only if it satisfies the effective choice principle, i.e. there is a recursive function f such that f ( z ) E A,, and A, = A, =+ f ( z ) = f(y). (Pour El and Howard

[19641)

Corollary 11.5.24 (Pour El and Putnarn [1965]) Every r.e. class containing all finite sets is r.e. without repetitions. Proof. T h e proof above (lid not use any assiimption a b o u t t h e r.e. sets, except t h e fact t h a t they a r e a n r.e. class containing all finite sets a n d w. Given any r.e. class containing t h e finite sets, augment a n enumeration of it b y p u t t i n g w in t h e first place: t h e proof above will give a n enumeration without repetitions of its elements. If w was not in t h e given class, it is enough to d r o p t h e first element from t h e newly obtained class. 0 Some more criteria for enumeration without repetitions a r e in P o u r El a n d Howard [1964], Lachlan [1965a], [1967], Khntorezkii [1969], a n d Marchenkov [1971]. T h e one j u s t given by t h e corollary provides a number of examples, in view of t h e fact t h a t t h e r.e. classes containing all finite sets a r e completely

11.5 Indices and Enumerations

*

233

characterized by the following result (expressed in terms of a classification of sets introduced in Chapter IV, see IV.1.6).

Proposition 11.5.25 (Yates [1969]) A class of r.e. sets containing all finite sets is r.e. i f and only i f its index set is E!. Proof. If A = {Wf(,)},Euthen

We E A

* *

3z(We = W f ( z ) )

~ ~ V YE(We Y ~ ~ V Y E[ (We Y

Y E Wf(z)) +

YE

wf(z))A (Y E Wf(,)Y E we)] -+

a n d OA is C!. Conversely, suppose A contains all finite sets, and BA is Z:, i.e.

We E A

* 3zV/y3zR(elz,y,z),

with R recursive. Define

t E A,,,,)

* t E We A (VY L t ) ( 3 z ) R ( e , z , y 2). ,

If is finite, then it is in A by hypothesis; if it is infinite, then it is in A because A(,,,) = We E A. Finally, if We E A then Vg3zR(e,z, y , z ) , for some z, and so We = A,,,,). Thus A = {A(e,z)}e,,Eu, and A is r.e. 0

A particularly interesting example of r.e. class is t h e one of recursive sets. This follows from the criterion just given, but it can be easily proved directly: Proposition 11.5.26 (Muchnik [1958a], Suzuki [1959]) The class of recursive sets is r.e. (without repetitions).

Proof. Let Wf(,)be the following r.e. set, obtained uniformly from e: generate W e ,and put a new element appearing in it in Wf(,) only if it is greater t h a n all t h e elements which are already in Wf(,).Each W f ( , )is either finite or enumerated in increasing order, hence recursive by 11.1.17. And each recursive set has a n index coding t h e instructions t o enumerate it in increasing order, a n d so it appears aniong t h e Wf(,)’s.Since every finite set is recursive, from simple enumerability we get enumerability without repetitions, by 11.5.24. 0

W h a t the proposition tells us is that the recursive sets are uniformly r.e., in t h e sense t h a t there is an r.e. relation R ( e ,z) such t h a t , while e ranges over w ,t h e set {z : R ( e , z ) } ranges over t h e recursive sets, and each recursive set is {z : R ( e , z ) } , for some e. Obviously, by diagonalization and closure with respect t o complementation, the recursive sets are not uniformly recursive.

11. Basic Recursion Theory

234

Also, t h e result j u s t proved shows t h a t there is a n r.e. subset of Rec (defined on p. 226), which contains at least one index of each recursive set, although t h e set Rec is not itself r.e. (by 11.5.19). O n t h e other h a n d , n o such property holds for characteristic indices of recursive sets in place of r.e. indices, by 11.2.1. Exercises 11.5.27 a) The classes of the infinite r.e. sets and of the infinite recursive sets are not r.e. (Uspenskii [1955], “571, Dekker and Myhill [1958a]) (Hint: if they were, an infinite recursive set intersecting each element of the given class could be built by enumerating it in increasing order.) b) The class of the coinfinite r.e. sets is not r.e. (Hint: criterion 11.5.25 can be a p plied, but it is difficult to show that the index set of this class is not E; (see X.9.11). A direct diagonalization is easier, although not trivial, and requires the priority method introduced in Chapter X. Elements from complements of r.e. sets, instead of from the sets themselves, are needed. Positive requirements ask for a nonempty intersection with the complement of each set in the class, and negative requirements ask for the infinity of the complement of the set thus built.)

We briefly discuss now classes of finite sets, which a r e at t h e opposite extreme of those taken care of by t h e criterion 11.5.25. Proposition 11.5.28 (Lachlan [1965a],Pour El and Putnam [1965]) There is a class o f finite sets which is r.e. (and it actually admits an enumeration in which every element has at most two indices), b u t is not r.e. without repetitions. Proof. Let

R2=,R2z+l be

generated as follows:

&=: p u t 2x in it, then generate K a n d , if x E K ,add 22 R2=+1:p u t

+ 1.

2x+ 1 in i t , then generate K a n d , if x E K ,add 2s.

Then: if x E

K ,Rzz = R2=+1= {22,2x + 1)

if z

K ,RzZ = (2s) a n d

R2z+1 = (2x

+ 1).

If there were a n enumeration without repetitions of A = { Re}etw t h e n would be r.e., since x E K if a n d only if there a r e two sets in t h e enumeration, o n e with 22 in i t , t h e other with 22 1. 0

+

Exercises 11.5.29 (Pour El and Putnam [1965]) a) For every n 2 1, there is a n r.e. class of finite sets which is r.e with at most n 1 repetitions, but not with at most n repetitions. (Hint: use sets with at most n 1 elements.) b) There is an r.e. class of finite sets which is r.e. with at most finitely many repetitions, but not with at most n repetitions, f o r any N e d n. (Hint: use sets with unbounded cardinality.)

+

+

11.5 Indices and Enumerations *

235

c) Every r.e. class of disjoint finite sets is r.e. with finitely many, possibly unbounded, repetitions. d) There is an r.e. class of finite sets which is not r.e. with at most finitely many repetitions. (Hint: instead of considering, as above, an r.e. nonrecursive set, consider a set E; - II;, see IV.1.13.) e) There is no r.e. class of finite sets such that every enumeration of it repents each element infinitely often. (Hint: if A is r.e. and A E A is finite, then A - { A } is still an r.e. class, since we can simply enumerate the elements of A which contain some element which is not A . ) For more on this topic see Pour El and Putnam [1965],Young [ 19661, and Florence [1967], [1969], [1975).

Exercises 11.6.30 Standard classes of r.e. sets. (Lachlan [1964a]) An r.e. class = We.Thus standard classes are r.e. classes indexed in the same way as the class of the r.e. sets. a) The Fixed-Point Theorem holds f o r standard classes. (Hint: let f be a recursive function such that S, = W,(z). Given Sg(,,) with g recursive, by the Fixed-Point Theorem there is e such that Sg(,)= W,g(e)= W e .Then We E A and We = S,, i.e.

A = {S,},~,of r.e. sets is standard if, whenever We E d,S,

Sg(e)

= See)

b) The clms { @ , A } ,with A nonempty r.e. set, is standard. (Hint: define W,(,) as 0 or A , depending on whether W, is empty or not.) c) If a standard class contains a finite set, it contains a least member. (Hint: let A be a finite set in d, and consider the supersets of A , and the sets intersecting 2: since they are complementary classes, by an analogue of Rice’s Theorem for A their union can cover A only if one of them does already. Since A is in d, either A is the least member of A, or a finite subset of it is in A.) d) A finite class is standard i f and only if it has a least member. (Hint: suppose A = { A l , . . . ,A,,}, and choose finite subsets Bi of Ai such that

Bi C A, Bi C Bj

* *

Ai C Aj

Ai G A j .

If A is standard, as in c) we can show that it has a least member, by considering the supersets of the B’s. And if A has a least member A i o , then we may suppose Bi, = 0, and construct a standard enumeration by letting S, be the union of the A’s corresponding to a maximal strictly increasing sequence of B’s included in W e . ) e) A strong a r m y is a standard class if and only i f there is n o infinite r.e. set which is the union of a n increasing sequence of members of the army. (Hint: one direction , = 0 and Bi = Ai. We may suppose is like part d) above, by letting A = { A , } , E ~A0 that 0 is in d, since by b) above there is a least member, and the class obtained by subtracting it to every element of A is still standard. For the other direction, let C be an infinite r.e. set which is the union of an increasing sequence of members of A. We may suppose C = C,,for a strong array {C,},Ew of members of A. Consider

UzEw

11. Basic Recursion Theory

236

and choose e such that W,(,) = We. Then We is finite by definition o f f , but a t the and hence C . ) same time it should also, inductively, include every C,,, We conclude by giving the relationships among t h e various concepts of enunierability for classes of r.e. sets:

Proposition 11.5.31 (Pour El and Howard [1964])

1. A n y completely r.e. class is r.e. without repetitions, but not conversely. 2. A n y class r.e. without repetitions is r.e., but not conversely.

Proof. Let A be completely r.e.: then A is t h e class of r.e. supersets of a strong array. To get a n enumeration of A without repetitions, start with such a n enumeration {S,},,, for t h e r.e. sets, and put in t h e new enumeration of A only the S,'s for which it is discovered t h a t they extend one element of t h e strong array, in a dovetailed generation of each S , a n d of t h e elements of t h e strong array. Thus a completely r.e. class is r.e. without repetitions. T h e converse does not hold, as t h e example of recursive sets shows (see 11.5.26). A class r.e. without repetitions is obviously r.e., b u t t h e converse does not hold, as 11.5.28 shows. 0

The Theory of Enumerations

*

T h e results of t h e last subsection suggest t h e possibility (proposed by Kolmogorov) of a systematic study (begun by Uspenskii [1955], [1955a], [1956]) of the recursive enumerations of a n r.e. class A of r.e. sets, defined as functions u : w % d such t h a t t h e sets W,(,, a r e uniformly r.e. Given two such enumerations vo and vl, we let v g 5 v1 if there is a total recursive function f such t h a t vo = u1 o f u g = v1 if vg 5 v1 A ul 5 VO. Then is a n equivalence relation (already used in 11.5.3). L"(A)is t h e structure of equivalence classes of the r.e. enumerations of A, partially ordered by the order induced by 5 . T h e Theory of Enumerations studies t h e algebraic structure of C"(d), for any class A. Its emphasis is thus somewhat opposite to t h a t of BRFT (p. 22l), being on subclasses of the class of r.e. sets, instead of on superclasses of it. T h e scattered results of this section can be seen in a new light in this framework. Call u principal if it is in the greatest element of C"(A)(so t h a t vo v , for every v g ) , and minimal if it is in a minimal one (so t h a t , for every vo, if vo 5 u then vo u ) . Then, for the class of all r.e. sets, a n enumeration is principal i f and only if i t is acceptable, and is minimal i f it i s a n enumeration

<

11.5 Indices and Enumerations *

237

without repetitions. T h e enumeration without repetitions given by 11.5.22 is not, t h e only possible one, u p to equivalence: there a r e countably many other, pairwise inequivalent ones (Pour El (19641, Khutorezkii [1969]). O n t h e other hand, t h e r e a r e also countably many, pairwise inequivalent minimal enumerations, which a r e not equivalent to enumerations without repetitions (Ershov [1968b]). Finally, there a r e countably many, pairwise inequivalent enumerations, which do not bound minimal enumerations (Khutorezkii [1969a]). Note also t h a t the fact that the r. e. sets are not uniformly recursive implies that there is n o least enumeration (i.e. such t h a t u 5 Y O ,for every ug). T h e case of finite classes A has been thoroughly examined, a n d it is known t h a t t h e s t r u c t u r e of L o ( d )depends only on t h e set-theoretical s t r u c t u r e of A under inclusion. Moreover, Lo(A) is a distributive uppersemilattice with least a n d greatest element, a n d it is either trivial (a single element), or very rich (having a n ideal isomorphic t o t h e structure of r.e. m-degrees, studied in Chapter X). Since, for Lo(A)a n d Lo(A')to be isomorphic, t h e number of minimal elements (under inclusion) of A a n d A' must be equal, there are countably many isomorphism types, b u t a coniplete classification of t h e m h a s not yet been obtained. T h e case of infinite classes A presents a n even greater variety: not even t h e greatest o r t h e least element exist necessarily in L"(A)(as shown, respectively, by t h e classes of recursive sets, a n d of all t h e r.e. sets ). B u t , even in t h i s case, L"(d)either has only one element (e.g. when A is a disjoint strong array), or it is infinite a n d complicated (not linear a n d not a lattice). A particularly useful notion is t h a t of complete enumeration (Malc'ev [1961], [1963], Lacombe [1965]). This is a n enumeration of A for which t h e r e exists a n element e such t h a t , whenever cp is a partial recursive function, t h e r e is a total recursive function f such t h a t

Clearly, for t h e class of all r.e. sets, t h e s t a n d a r d enumeration is complete (with

e = 0). The classes A possessing a complete enumeration are exactly those containing a smallest set (Malc'ev [1964]). T h e interest of t h e notion comes from t h e fact t h a t it implies versions of (and t h u s it generalizes) t h e FixedPoint Theorem (for every recursive function f there is e such t h a t uf ( e ) = u e ) , of Rice's Theorem (a subset A'of A has a recursive index set .-'(A') if a n d only if it is trivial) a n d of t h e Padding L e m m a (for every element a E A, i t s index set v - ' ( { a } ) is infinite). Lacombe [1960] a n d Malc'ev I19611 have suggested (with motivations from constructive algebra) t h e extension of the notion of enumeration, from classes of r.e. sets to any countable, nonempty set $9.They d r o p effectiveness r e q u i r e ments, a n d consider any function u : w 2 S as a n enumeration of S . T h e

11. Basic Recursion Theory

238

=

equivalence relation still makes sense, a n d L ( S ) is t h e set of equivalence classes of enumerations of S: i t is now a n uppersemilattice, either trivial or uncountable. T h e theory has been here very successful: there are only three possible isomorphism types for C(S), corresponding to cardinalities of S equal to 1 , finite a n d greater t h a n 1, a n d infinite. In t h e first case there is only one element. In t h e second, t h e structure has been characterized (Ershov [1975]) u p to isomorphism, as a strongly universal uppersemilattice with a least element (it is actually isomorphic to t h e structure of m-degrees, which is t h e special case C((0, l } ) , see VI.4.1). T h e last case differs from t h e other two, e.g. because there is no least element. For detailed treatments of t h e Theory of Enumerations see Lacombe [1965], Malc’ev [ 19651, a n d Ershov [ 19771.

11.6

Retraceable and Regressive Sets

*

Chapter I1 has been characterized by t h e search for possible extensions of t h e notion of recursiveness. O u r last a t t e m p t is inspired by t h e property of recursive sets of being effectively enumerable in increasing or decreasing order. Consider a recursive set A = (a0 < al < a2 < . . . }. T h e r e a r e t w o recursive functions f a n d g , such t h a t :

1. f(a,) = a,+], unless a, is t h e maximum element of A, in which case f(a,) = a,

2. g(a,+l) = a, arid g(a0) = ao. Exercise 11.6.1 Property 1 is characteristic of the recursive sets, i.e. if there is a recursive function f as above, then A is recursive. (Hint: use 11.1.17)

I t is natural to see if property 2 is also characteristic of recursive sets, a n d if not, t o s t u d y t h e sets which share this property with t h e m .

Definition 11.6.2 (Tennenbaum, Dekker [1962]) 1. I f { a o , a l , a 2 , . . .} is an enumeration without repetitions of A , and p zs a partial recursive function such that

then A is called regressive via p, and with respect t o the given enumeration.

11.6 Retraceable and Regressive Sets

*

239

2. Under the same conditions, and i f the enumeration is in order of magnitude (principal enumeration), then A is called retraceable. W e stress t h e fact t h a t t h e enumeration of A does not have to be recursive. Exercises 11.6.3 a) If A is retraced b y cp, we can always suppose p(z) 5 x, whenever cp(z) is defined. b) If A is regressed by cp, we may always suppose range cp C domain cp and, whenever cp(z) is defined, cp”(z) N no, for some n.

T h e r e is a surface analogy r.e. - regressive recursive retraceable a n d we now explore t h e extent t o which it holds

Retraceable versus recursive T h e following notion will be helpful in our s t u d y :

Definition 11.6.4 A set A is immune i f it is infinite, but it does not contain injinite r.e. subsets. Note t h a t , by 11.1.20, a n infinite set is immune if a n d only if i t does not contain infinite recursive subsets.

Proposition 11.6.5 (Dekker and Myhill [1958]) Recursive + retraceable recursive or im7nune.

Proof, If A is recursive, t h e n there is a recursive function f which enumerates it in increasing order, i.e.

I t is t,hen enough to define

to have a partial recursive function retracing A . Actually, since f is increasing, we can get cp total recursive, e.g. by letting it be 0 in t h e intervals between a n y two successive values of f. Let now A be retraceable (via ‘p) a n d infinite (if A is finite, it is certainly recursive), a n d suppose A is not immune: then A has a n infinite recursive subset R . R u t t h e n A is recursive: given 5 ,find a n element g ( x ) of B greater t h a n

11. Basic Recursion Theory

240

i t (which exists and can be found recursively, because B is infinite a n d recursive). Now g(x) is certainly in A (because B 5 A ) , a n d we can t h e n repeatedly apply ‘p to i t , to generate t h e elements of A smaller t h a n g(z) in decreasing order, until one is repeated (which means we hit t h e smallest one). And Z, being smaller t h a n g ( s ) , is in A if a n d only if it is generated in this process. 0 We will see, in t h e subsection of existence theorems, t h a t i t is far from being true t h a t every retraceable set is recursive, b u t there is a nontrivial special case in which this holds:

Proposition 11.6.6 (Mansfield) If A and recursive.

7 are both retraceable, then A

is

Proof. T h e proof is in two steps: 1. I f A and

are retraceable. so is the set

A @ X = (22

:z E

A} U { 2 2 + 1 : z E Z}.

+

T h e basic fact here is t h a t exactly one element of each pair (22,22 1) is in A @ 2.Let ‘p a n d ‘p’ retrace A a n d 2,respectively. T h e n A @ 2 is retraced by so defined:

+

+ ( 2 2 + 1)

2:

2 2 - 1 if cp’(z) N z - 1 2 2 - 2 if (~‘(2) N z V ‘p’(z) < z - 1.

@x,

Indeed, if z E A then 22 E A a n d we need to define +(22) as 2 2 - 1 or 2 2 - 2, depending on whether x - 1 is in A or in 2.B u t t h e first case happens when tp(x) cx 2 - 1, a n d t h e second when either a: is t h e smallest element of A (and cp(x) 2: s) or t h e next element of A in descending order is smaller t h a n z - 1 (and ’p(x) < 5 - 1). T h e c a e z E 2,i.e. 2 s 1 E A ex,is symmetric, using ‘p’.

+

2. If A @ 2 is retraceable, A is recursive. We prove t h a t any set B retraceable a n d with exactly o n e element in each pair ( 2 2 , 2 2 l ) , is r.e. (and hence recursive: to know which o n e of 2 s , 2 2 1 is in B , generate B until one of t h e m appears). If retraces B , two cases a r e possible:

+

+

0

+

+

If 2 2 , 2 2 1 a r e both sent by ?1, onto t h e s a m e element c (meaning t h a t some iteration of on 2 2 is c, a n d similarly for some iteration, not necessarily t h e s a m e number of times, of o n 2a: I), t h e n

+

+

+

11.6 Retraceable and Regressive Sets *

241

c E B (because one of 2x,2x+ 1 is in B). And then $ automatically generates every element of B smaller than c. Then, if there a r e infinitely many pairs 2x,2x 1 as such it is enough to look for them, and each of them will generate a n initial segment of B, via $.

+

0

If there are only finitely many such pairs, let b E B be greater t h a n all of their elements (if it does not exist, B is finite and hence recursive). To generate B is now enough to generate all t h e x's such t h a t x > b and x is sent by $ onto b. Indeed, there cannot be one such element t h a t is not in B, since otherwise t h e other element of t h e pair to which z belongs would be sent to b by $ as well, against 0 t h e choice of b.

Note t h a t we proved t h a t A is recursive, but did not produce an algorithm to compute it, since t h e proof is by nonconstructive cases (we only produced two algorithms, one of which will work, b u t we do not know which one).

Exercises 11.8.7 Introreducible sets. The proof of 11.6.5 suggests the following notion: A is introreducible if it is recursive in each of its infinite subsets (Tennenbaum). Not much is known on the results which generalize from retraceable to introreducible sets. In particular it is open whether A is recursive, whenever A and A are introreducible. A stronger and more tractable notion is: A is uniformly ir+ troreducible if there is an index e such that, whenever B is an infinite subset of A, then A = cpf (Jockusch [1968]). a) Every retraceable set is uniformly introreducible. (Hint: see the proof of 11.6.5.) b) Not every unijoonnly introreducible set is retraceable. (Dekker and Myhill [1958]) (Hint: use the intervals (2",2"+l), and put in A exactly one element of each interval, extremes excluded, by letting 2" 1 f(z) be in A if z is, where f is any nonrecursive 0,l-valued function. Then A is retraceable, via the function that sends the interval (2",2"") to z, and nonrecursive, hence immune. Let B code A on the extremes of the intervals, i.e. 2" E B if and only if z E A. Then B is retraceable, since A is. But A U B is not, otherwise an infinite recursive subset of A could be generated, since if z is in A then 2" is in B, and a retracing function would give an element of A greater than z. And A U B is uniformly introreducible, because if C is an infinite subset of A U B then one of C n A and C n B is infinite, and then one of A and B is recoverable, and so is A U B.) c) Not every introreducible set is uni f onl y intmreducible. (Lachlan) (Hint: this uses the priority method, introduced in Chapter X. Build A r.e. such that 3 is as wanted. Given B r.e. and co-retraceable, and such that K
+ +

242

11. Basic Recursion Theory

set of disjoint finite sets intersecting B has members of unbounded cardinality. Apply this to A @ X , which is uniformly introreducible, and to the set of pairs {25,25 l}.)

+

Regressive versus r.e. We follow t h e p a t h set u p by t h e previous subsection.

Proposition 11.6.8 (Dekker and Myhill [1958])

R.e.

3

regressive 3 r.e. or immune.

Proof. If A is r.e. a n d infinite, there is a recursive function f which enumerates it without repetitions: A = { f ( O ) , m , . . ). It is then enough to define

cp(f(0)) = f(0)

cp(f(n + 1)) = f(n)

t o have a partial recursive function regressing A . Note t h a t (unlike in t h e proof of 11.6.5) cp is not immediately extendable to a t o t a l recursive function (see 11.6.11 for a reason). Let now A be regressive (via 'p) a n d infinite, a n d suppose A is not immune: t h e n A has a n infinite recursive subset B. But then A is r.e.: to generate it, repeatedly apply cp to any element of B , until t h e smallest element of A is hit (this can b e recognized, since it is left fixed by cp). 0

It is not t r u e that if both A a n d 2 a r e regressive, t h e n A is recursive: any nonrecursive, co-regressive r.e. set (see 11.6.16) is a counterexample. By symmetry, not even t h e conclusion t h a t A is r.e. holds. The correct generalization of 11.6.6 is t h e following: Proposition 11.6.9 (Appel and McLaughlin [1965]) If A and regressive, then one of A and 2 is r.e.

?I are both

Proof. We t r y to extend t h e proof of 11.6.6. P a r t 2 of it generalizes (using now any enumeration of B ) ,a n d shows t h a t if B is regressive a n d there is a n infinite r.e. set of disjoint pairs intersecting B , then B is r.e. (note t h a t t h e facts t h a t exactly one element of each pair was in B , a n d t h a t t h e pairs covered w ,were used only to deduce t h e stronger conclusion t h a t B was recursive). I t is not however t r u e t h a t if both A a n d 2 a r e regressive, t h e n so is A @ A (see t h e exercises below). We then proceed directly. Let A be regressed by cp, a n d define zE

s, e cp(n+l)(z)21 cp(n)(z)$ ' p ( n - l ) ( z ) .

11.6 Retraceable and Regressive Sets

*

243

T h c Sn's a r e disjoint r.e. sets, a n d each contains exactly one element of A ( t h e n-th in t h e given enumeration). I f there a r e only finitely many Sn's with at least two elements, t h e n A has a n infinite r.e. subset (each S,, with exactly one element contributes to it), a n d is not immune. Being regressive, A is t h e n r.e. If t h e r e are infinitely many Sn's with at least two elements, we c a n get a n infinite r.e. set of disjoint pairs intersecting 2 (each S,, with at least two elements contributes two elements, a n d at most one of t h e m can be in A ) . T h e n A is r.e. by the first p a r t of t h e proof, being regressive. 0 Note t h a t again, as in 11.6.6, we proved t h a t one of A a n d is r.e., b u t we do riot know which one, since t h e proof is by nonconstructive cases. Exercises 11.6.10 (Appel and McLaughlin [1965]) a) If A is regressive, then A is rpcursive. (Hint: use the first part of the proof above.) 1,) If B is regressive and there is an infinite r.e. set of disjoint finite sets of bounded cardinality intersecting €3, then B is r.e. (Hint: consider the greatest number n such that there are infinitely many nituples from the same set, all regressing on the same element.)

We have shown t h a t a n r.e. set is regressive, b u t t h e proof of 11.6.8 does not givc a t o t a l regressive function. T h e reason is t h a t such a function does not always exist.

Proposition 11.6.11 (Appel and McLaughlin [1965]) There are r.e. sets which are not regressed b y any total recursive function. Proof. We use t h e fact t h a t there is a n r.e. set A with immune complement (a siniple set, see III.2.11), a n d we prove t h a t any function regressing such a set cannot have a total recursive extension. Suppose f is such a n extension, a n d let zE f'""'(.) = f'"'(z) # f'"-I)(.).

s,, *

S,, contains exactly one element of A ( t h e n-th in t h e given enumeration). It is enough to prove t h a t there a r e infinitely many Sn's with at least two elements, since then we c a n get a n infinite r.e. set of disjoint pairs intersecting -

A , a n d from i t a n infinite r.e. subset of 2 (against simplicity): either from a certain point on all pairs a r e contained in 2,or there a r e infinitely many pairs intersecting A , a n d t h e n it is enough to generate A to discriminate which ones do, a n d choose t h e element of 2. T h e claim is easy to prove: T h e set

11. Basic Recursion Theory

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(called the splinter o f f at x) is finite for any 2, because either it contains an element of A , and then it stops after t h e first element in t h e enumeration of A is hit, or it is an r.e. subset of 2,and it is finite by simplicity ( Acannot contain infinite r.e. subsets). T h e set S = UnEwS, is then recursive: t o check if x is in it, it is enough t o generate the splinter of f at x, and see whether t h e conditions for membership in S, are satisfied for some n (f being total, and t h e splinter being finite, f must cycle over the elements of t h e splinter, and thus t h e conditions can be checked recursively). 0

Since A C_ S , 3 is finite (being an r.e. subset of 71). But 2 is infinite, and then so is S n 71: thus the Sn's have t o cover a n infinite part of 3. But each S, is finite (it contains only one element of A , and t h e rest of it is a n r.e. subset of Z), and each contains one element of A : then infinitely many Sn's must contain more than one element. 0

Exercises 11.6.12 Splinters. A set A is a splinter if, for some recursive function g and some z, A = {z,g(z),g'2'(z),. . . }.

A splinter is obviously r.e., but the converse fails by III.7.lO.a. a) There are nonrecursive splinters. (Ullian [1960]) (Hint: let A be a n r.e. nonrecursive set enumerated by a recursive function g, and define f recursive as follows. On a sequence number of the form

(g(z),g(O),g(l),... ,g(z),g(z + 1))

+

f gives (g(z 1)). On all other sequence numbers z, f gives z * @(ln(z)- 1)). Then the splinter of f at (g(0)) contains (z) if and only if z E A, and it is nonrecursive.) A different proof will be given in III.7.lO.c, but the present one actually shows that every r.e. degree contains a splinter. b) A n r.e. set is regressed by a total recursive fimction if and only if at is a splinter. (Degtev (19701) (Hint: if A is regressed by g w.r.t. {ao, a l , .. .}, the enumeration can be recovered because A is r.e. Extract a n infinite recursive subset B = {bo < bl < . . .}, with bo = ao and a1 < bl. Define f(z) = g(z) if z E B A z # al,so that f sweeps the intervals between b's, and

f

=

i

if z = a0 = bo g(b1) if 2 = a1 g(b2) g(bn+2) if z = b, A n > 0

so that f visits successive intervals. The converse is similar.) For more information on splinters see 111.7.10 and Myhill [1959], Ullian [1960],

Young [ 19651, [ 19661, [ 19671.

Results proved later (see 111.4.9) will show the existence of

11.6 Retraceable and Regressive Sets

*

245

1. a retraceable set which is not regressed b y total recursive functions 2. a set retraced b y a total recursive function, which is not regressed b y total

many-one recursive functions.

Existence theorems and nondeficiency sets We state our results in strong form, using the notion of degree introduced in 11.3.3.

Proposition 11.6.13 (Dekker and Myhill [1958]) Every T-degree contains a retraceable set. Proof. If A is finite then it is recursive and retraceable. Let A be infinite, and f be t h e enumeration of A in order of magnitude (f is not recursive, in general). Let B be enumerated (in order of magnitude) by the function

Then B is retraceable via the recursive function that chops off the last component of a sequence number of length greater than 1, and leaves unchanged t h e remaining numbers. Clearly B ST A by definition, and A 5~ B because z E A H z is a component of g(z),

since z

5 g(z).

Corollary 11.6.14 There are 2"O retraceable sets. T h e recursive sets are t h e simplest retraceable sets. R.e. nonrecursive sets cannot be retraceable (by 11.6.5 a nonrecursive, retraceable set must be immune), and thus the next level of complexity, and the first nontrivial one, for retraceable sets is being co-r.e. Exercise 11.6.16 Zf A is retraceable and co-r.e., it is retraced by a total recursive function. (Hint: given z, see if p(z)1 or z E 2.)

We now prove that such sets not only exist, but are as abundant as they can be.

Theorem 11.6.16 (Dekker and Myhill [1958]) Every r.e. T-degree contains a retraceable, co-r.e. set.

11. Basic Recursion Theory

246

Proof. If A is recursive t h e n it is itself co-retraceable a n d r.e. Let t h e n A be r.e. nonrecursive, a n d let f be a recursive, one-one enumeration of i t . Let 5

EB

zE

B

*

(3Y > z)(f(y)< f(.)

(YY > X)(f(Y) > f(.)).

T h e elemeiits of B a r e called stages of nondeficiency, or true stages, in t h e enumeration of A given by f , because no new element of A smaller t h a n f(x) is generated by f in t h e future. Hence, for x E B,

B is clearly r.e. Moreover:

AFTB To see if z E A it is enough to find z E

z

E

B snch t h a t f(z)> z , a n d see if

{f(O),. . . ,f(z)).

And x exists because f is o n c o n e a n d B is infinite (given a n element b E B,a greater o n e c a n be obtained by taking first t h e smallest element a E A which is not in {f(O), . . . , f ( b ) } , a n d t h e n t h e stage in which a is generated by f ).

BFTA z E B if a n d only if there is some element in

B is co-retraceable Given x E B,we want to give a n effective procedure t o find t h e greatest element of B smaller t h a n z. Since for y > x is f ( y ) > f(z), i t is enough t o check t h e values o ff for arguments below z. In other words, for z

T h e n it is enough to define g(z) as t h e biggest z

(Y'y)(.

< x,

< z such t h a t

< Y 5 z * f(z) < f(Y))

if t h e r e is one, a n d z otherwise (so t h a t t h e first element of B is left fixed). 0

11.6 Retraceable and Regressive Sets

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247

T h e idea of using nondeficiency stages is a n ingenious one, invented by Dekker [1954].It will show its usefulness time and again, in many different contexts (including infinite injury priority arguments, see Chapter X). As far as retraceable co-r.e. sets are concerned, this idea completely captured the heart of t h e matter:

Proposition 11.6.17 (Yates [1962]) A co-r.e. set A is retraceable if and only ii for some recursive function f, 5

EA

2

E

7i

* *

(YY > .)(f(Y) > fb)) (3> X)(f(Y) I f (z)).

Proof. If a function f as stated exists, the proof of 11.6.16 shows t h a t A is retraceable. If A is finite, it can easily be seen that a function f as stated exists. Let then A be a n infinite, retraceable and co-r.e. set: we want to find f . We have g recursive retracing A , and we may suppose t h a t g is total (11.6.15) a n d g(z) I x (11.6.3). Consider the recursive height function h ( z )= pz [g""(z)

= g(Z)(z)],

and t h e recursive height sets z E H , w h ( z ) = n.

Clearly t h e height sets are disjoint, cover w (since g is total and descending) a n d have exactly one element of A each (since A is infinite). T h e idea is t o define f on H , (ordered by magnitude), by letting it be n until t h e first element of A is hit, and greater than n afterwards. Thus, if E H , n A a n d y < x A y E H,, then y is a deficiency stage, while x becomes a nondeficiency one. Given x E H,, consider the set f = {y : y


E Hn}.

If 5 is empty, z is t h e first element of H,, so let f(z)= n . If P is not empty, enumerate 2 until exactly one element z o f f U {z} has not yet been generated in it. This is possible because is r.e., and H,, has exactly one element in A . If z = z then all smaller elements in H , are in and we can still let f ( z ) = n . If z # x then z has been enumerated in a n d we now know t h a t z has t o be a deficiency stage. Then we want y > z such t h a t f ( y ) I f(z). Note that f(z)= n 1 is not enough, since it might be that H,+1 contains no element y > z. And m > n such t h a t H , has a n element y > 2 is not enough either, since t h e unique element t E H , n A might be smaller than z, a n d setting f (z) = m would make f ( t ) = f(x), while t < X, and t has t o be a nondeficiency stage (since t E A ) . But then let f(z)= m for m > n such t h a t H , contains no element smaller than x. 0

x, x,

+

11. Basic Recursion Theory

248

Exercises 11.6.18 Nondeficiency sets. A set A is a nondeficiency set if, for some

recursive function f,

zE

A

* (VY > Z)(f(Y)

> I(.)).

a) I f f is not finite-one, the nondeficiency set o f f is finite. b) A w-r.e. set is retruceable if and only i f it is the nondeficiency set of afinzte-one function. (Yates [1962]) (Hint: see the proof above.) c) Not every w-r.e. retmcenble set is the nondeficiency set of a one-one b n c tion. (Degtev (19701) (Hint: let A(g) be the deficiency set of g , and A,(g) be its approximation up to n, i.e. 5

E A n ( g ) * z < n A ( % ) ( z< Y

I ~ & A \ ( Y )I g ( z ) ) .

Define f as follows. Given f ( O ) , . . . ,f (n),let 1% = ( e , z) be the first stage in which cp, is total and one-one on (0,. . . ,z}, and An,,(f) Az(cpe),where no = m m { z I n : f ( z ) = e }

Then let / ( n + 1) = e. Otherwise, let f(z)= n. If cpe is one-one and total, then A(cpe) is infinite, and it cannot be A(cp,) = A( f ) , otherwise by construction f takes the value e infinitely often, and then A ( f ) is finite.)

Proposition 11.6.19 (Marchenkov [1976a]) T h e class of r.e. co-retruceable sets is r.e. without repetitions. Proof. Since every finite set is co-retraceable, it is enough (by 11.5.24) to show t h a t t h e class of r.e. co-retraceable sets is r.e. Let

T h e n { A e } e E wis a n r.e. class, a n d each A, is either finite (if peis not total) or co-retraceable (by 11.6.17). 0 T h e class of r.e. co-retraceable sets is not completely r.e. (by 11.4.2), since it is not closed under supersets (see II.6.21.b).

Regressive versus retraceable We now briefly come back to t h e original question of t h e extent of t h e analogy with recursive a n d r.e. sets, arid we show t h a t it fails q u i t e strongly. First we give a positive result, which is t h e analogue of 11.1.20. Proposition 11.6.20 (Dekker [1962]) Every infinite regressive set has a n infinite retraceable subset.

11.6 Retraceable and Regressive Sets

*

249

Proof. Let A be regressed by cp w.r.t. {ao,al , . . . }, a n d bo b,+l

= a0 = t h e first element in t he list of A greater than b,.

Then B = { b o , b l , . . . } is infinite, because A is. T o define t he retracing function for B on a given z, we first iterate p until we stop, which we must if z E A . Then we recreate t h e initial segment of B from bo t o z, by dropping t h e elements t h a t break t h e monotone growing, and take t h e biggest element obtained which is smaller than z. If z E B then we do recreate t h e initial segment of B , a n d we do choose t h e right element. 0 There a r e two properties t h a t we consider essential to claim a nontrivial analogy with recursive a n d r.e. sets, namely:

1. T h e recursive sets ar e closed under complementation.

2. A set which is r.e., together with its complement, is recursive (11.1.19) They both fail here:

1. There is a retraceable set with a nonretruceable complement. Take any retraceable, nonrecursive set: its complement is not retraceable, by 11.6.6.

2. There is u set regressive together with its complement, but not retruceable. Take any set A r.e. and nonrecursive, with a retraceable complement. Then b o th A (being r.e.) and ar e regressive, but if A were retraceable then it would also be recursive, again by 11.6.6. Exercises 11.6.21 (Dekker and Myhill [1958])a) If A and B are retraceable, so is A n B. (Hint: given z, consider the greatest element smaller than it on which 2 is sent by both functions retracing A and B.) Appel [1967] ha7 shown that this fails for regressive sets. b) There are retraceable sets A and B such that A U B is not regressive. (Hint: let A be an infinite recursive set, and B a nonrecursive, retraceable and cer.e. set. If A U B were regressive, it would be r.e. because A is an infinite recursive subset of it, and B would be recursive.)

Despite t h e failure of t h e analogy with r.e. and recursive sets, retraceable a n d regressive sets are interesting on their own, and are useful in some parts of Recursion Theory. See McLaughlin [1982] for a detailed study of them.

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Chapter I11

Post’s Problem and Strong Reducibilities One theme of this chapter is relative computability. In Chapter I1 we introduced t h e most general and fundamental case: Turing reducibility. It will be recalled t h a t no limitation was imposed there on t h e help given t o t h e machine by t h e oracle, except for a n obvious finiteness requirement. Here we take a n opposite stand, and look at various possible limitations. Section 2 deals with t h e most restrictive case of m-reducibility, in which only one question is allowed to t h e machine during a computation, and only at t h e very end of it. Section 3 treats the case of Boolean combinations of atomic questions, called tt-reducibility, while a number of other, less fundamental, reducibilities are dealt with in Sections 4, 7 and 8. Relative computations induce equivalence classes, by identifying functions and sets which have t h e same degree of difficulty of computation. A second theme in t h e chapter is Post’s problem, introduced in Section 1, which asks whether there are only two such classes of r.e. sets. T h e solution, obtained in Section 5, will tell whether the r.e. sets can be distinguished, from a computational point of view, only between recursive and nonrecursive, o r whether instead this rough dichotomy can somehow be essentially refined. T h e strategy for a solution t o the problem is t o analyze the possible structure of r.e. sets (as opposed to giving direct constructions, a n alternative strategy pursued in Chapter X), and t h e tactic is t o solve t h e problem first for m-reducibility, in Section 2, a n d then gradually improve t h e solution for weaker and weaker reducibilities, in Sections 3 a n d 4, until we reach t h e one we are really interested in. T h e original motivation for the study of r.e. sets was t h a t they code (by 25 1

111. Post’s Problem and Strong Reducibilities

252

arithmetization) t h e sets of theorems of formal systems. A third theme of this chapter is the analysis of formal systems, from this abstract point of view. We make the relationship between formal systems a n d r.e. sets precise i n Section 10, where we also revisit some of t h e notions a n d results obtained in the chapter, and discuss their bearing on t h e subject of formal systems.

111.1 Post’s Problem We have so far encountered only two different kinds of r.e. sets, namely t h e recursive sets and K ,and they generate different degree.

Definition 111.1.1 A n r.e. T-degree is a degree containing at least one r.e. set. Two r.e. T-degrees are: 1. the T-degree 0 of the recursive sets 2. the T-degree 0’ of K. Note t h a t , because of Post’sTheorem and the fact t h a t a set a n d its complement are in t h e same degree (being obviously computable one from t h e other), a degree contains only r.e. sets if and only if it contains only recursive sets. This explains why we only require the existence of a n r e . set in a n r.e. degree. Recall t h a t there is a partial order on the degrees, induced by t h e relation ST. It is obvious t h a t 0 is the least degree with respect t o it, a n d t h e next result shows t h a t 0’ is t h e greatest r.e. degree.

Proposition 111.1.2 (Post [1944]) If A is any r.e. set then A 5~ K.

Proof. We prove t h a t there is a recursive function f such t h a t ZE

A

@

f(.)

E K: @ f(Z) E Wf(z),

where the last equivalence holds by definition of be a recursive function such t h a t

w

0

K. By the Sz-Theorem,

let

f

ifsEA otherwise.

Then: 0

* Wf(,) = w * f ( ~ E) Wf(,) + f ( ~E )K J(z)E K * f ( ~ E ) W f ( , )+ Wf,,) # 8 3 z E A.

5

EA

0

Thus, since A S T K: automatically holds for r.e. sets, an r.e. set A is in t h e greatest r.e. degree 0’ if and only if K 51A.

111.1 Post’s Problem

253

Definition 111.1.3 A set A is Turing complete (or T-complete) ij it is r.e. and its degree is O‘, i.e. K: ST A .

In general, given a reducibility S T ,we will call a n r.e. set A r-complete if A , a n d r-incomplete otherwise. As noted above, t h e r.e. sets we know at this point a r e all recursive or T-complete. It is natural t o ask whether there a r e others.

K

Post’s Problem (Post [1944]) Are there r.e. T-degrees different from 0 a n d O’? Equivalently, a r e there r.e. sets which a r e neither recursive nor T-complete? T h e reasons to isolate this natural problem a n d give it a name a r e many. First, despite its technical formulation, t h e problem was motivated by deep methodological questions, related to t h e undecidability results, a n d reviewed in t h e next subsection. Second, t h e solution to t h e problem escaped t h e r e searchers for many years a n d provided, as a by-product, new techniques a n d results, some of t h e m treated in this chapter. Finally, versions of t h e problem arise in different areas of Generalized Recursion Theory, a n d their solution is usually regarded as a proof of maturity for t h e new areas.

Origins of Post’s Problem

*

Post arrived at t h e formulation of his problem after a n exciting intellectual development, which is worth reviewing. I n his dissertation, completed in 1920, h e s t a r t e d by analyzing t h e system of Principia Matheinatica, a n d attacking t h e problem of i t s decidability. He was able to solve a particular case, namely t h e decision problem for propositional calculus, by proving a completeness theorem t h a t showed t h a t t h e theorems were exactly t h e tautologies. He published this in [1921]. In t h e academic year 1920-21, as a postgraduate, Post set down to generalize this decidability result, by attacking t h e general case. Trying to c a p t u r e t h e esscnce of formal systems h e considered, by successive abstractions, a sequence of notions, finally obtaining t h e canonical systems (see p. 143). By showing t h a t t h e system of Principia Mathernatica could be translated in a canonical system, h e convinced himself t h a t h e h a d a siifficiently general notion. Post then turned to t h e decidability problem for canonical systems, hoping t h a t t h e generality of t h e notion would make t h e proof simpler, because independent of details related to particular systems. He soon concentrated on a special problem, called t h e tag, of which he was able to handle some particular casm, but, t h a t turned o u t to be intractably complicated in general (with good reasons: it was undecidable, see Minsky [1961]).

111. Post’s Problem and Strong Reducibilities

254

At this point, t h e unsuccessful a t t e m p t s prompted a revision of t h e plan, a n d Post turned to undecidability. He defined a universal canonical system, which is j u s t a version of t h e set K O of p. 150, a n d showed its diagonal set, i.e. K , to be, in modern terms, r.e. b u t nonrecursive. To be able to deduce from this a general unsolvability result Post needed a version of Church’s Thesis, which h e stated in t h e form: every effectively generable set c a n be generated by a canonical system. From this t h e existence of incomplete formal systems followed easily. All this work, concluded in 1921, a n d anticipating a number of results by Gadel, Church, a n d Turing t h a t would follow much later, was left unpublished (see [1922]), because Post was not convinced of (his version of) Church’s Thesis. H e took it as a working hypothesis t h a t needed verification, in t h e form of psychological analysis of t h e computational process. A step toward such analysis was [1936], in which Post proposed a version of Turing machines, independently of Turing. T h e canonical systems were published only in [1943], a n d t h e form of Godel’s theorem based o n canonical systems only in [1944]. T h e mutual reductions among various notions of canonical systems, as well as particular formal systems like Principia M a t h e m a t i c a , led Post t o t h e concept of m-reducibility between sets. A n d t h e fact t h a t known undecidability proofs, by Post, Church, a n d others, were all obtained by appropriately reducing K to t h e problem in question, prompted t h e problem of whether t h e only undecidable systems were t h e universal ones, in which K could be interpreted. Post [1944] was able to disprove this for m-reducibility, a n d t h e n h e asked t h e s a m e question for t h e general notion of T-reducibility, introduced by Turing [1936]. Despite a good deal of intermediate work, h e could not reach a solution. Sections 2 to 4 a r e a report of Post’s work a n d of modern improvements, a n d Section 5 provides t h e missing brick.

Turing reducibility on r.e. sets Since t h e solution to Post’s Problem will require a detailed s t u d y of t h e struct u r e of t h e r.e. sets, we prove some technical results t h a t will facilitate t h e task.

Proposition 111.1.4 T-reducibility on r.e. sets (Rogers [1967])I f A and B are r.e. sets, t h e n A
2 ts ( 3 u ) ( D , B A R ( z , u ) ) .

Proof. By compactness a n d monotonicity of oracle computations (11.3.13) a n d t h e Normal Form Theorem for restricted functionals (11.3.12) we have, for a given c a n d some r.e. relation Q, (pF(z) N z ts

(3v)(3u)(DU CB

A

D, C B A Q ( z , u , v ,z ) ) ,

111.1 Post's Problem

255

because D, a n d D , together specify a finite subfunction of c g . If A
E A H ( b ) ( z ! u ) (G~ B , AR,

BAQ(Z,U,U,O)).

If moreover B is r.e., t h e expression D, G B is r.e. itself, and thus there is R r.e. such t h a t x E 2 H (3u)(D, C B A R ( x ,u)). Conversely, if this expression for 2 holds for some r.e. relation R, then 2 is r.e. in B . If moreover A is r.e., then it is r.e. in B and, by t h e relativization of Post's Theorem to B , A is recursive in B. 0 T h e next result is a strong generalization of t h e Fixed-Point Theorem, t o any function which is recursive in a n incomplete r.e. set.

Theorem 111.1.5 T-completeness of r.e. sets (Martin [1966],Lachlan [1968],Arslanov [198l])An r.e. set A is T-complete if and only if there is a function f
Proof. If A is T-complete, the set {x : 0 @ W,} is recursive i n A , being co-r.e. By t h e relativized SF-Theorem, there is f 51A such that wf(z) =

{

ifO@W, otherwise

Then Wf(,) # W,, because the two sets differ on 0. Suppose now that A is r.e., and f < T A has no fixed-points. T h e idea to get K ST A , thus showing t h a t A is T-complete, is the following. Consider t h e function ,us(% E K,) if z E K otherwise. Then x E K 2 E K,=.We want t o majorize s, recursively in A , i.e. t o find q!J recursive in A , such t h a t $(x) 2 s,. Then x E K H x E K$(,),and this implies K ST A . Since f
By t h e assumption on f , it cannot be W, = W f ( z ) This . means t h a t , while cp$;(z)

is a n approximation of f(z), it must be a wrong one. Recursively in

256

111. Post's Problem and Strong Reducibilities

A we can first compute ( p t ( z ) ,and find t h e length of t h e least initial segment ~ A ( Y )t h a t gives the right value. Then we can recursively enumerate A t o find a stage s 2 y in which all t h e elements of A used in t h e computation have been generated, i.e. t ~ , ( y )= t ~ ( y ) From . this point on t h e approximations will give t h e right value, i.e. y&(z) = f ( z ) , for any t >_ s. Then it must be s 2 s,. We have only t o do this in general now. By t h e Fixed-Point Theorem with parameters (11.2.11), there is g recursive such t hat

Let $(z) b e a stage in which A has generated all t h e elements needed t o comis recursive pute th e right value of f(g(z)) from then on, as above. T hen in A , an d +(x) 2 s, (if z E K as above, and if z $! K because then s, = 0 ) . 0 Note th a t the condition t hat A be r.e. is essential, since without it t h e result fails in general. A , such that f has Exercise 111.1.6 There is a set A ST K , and a function f no fixed-points, but K & A . (Arslanov [198l]) (Hint: by the proof of 111.2.18, it is enough to find A effectively immune, recursive in K, and such that K $T A . And such a set exists by the Low Basis Theorem V.5.32, applied to the IIy class

{ A : A C SA (Vx)(Dg(%) nA #

0)},

where S is Post's simple set, see 111.2.1 1, and g enumerates a strong array intersecting

-

S.)

For more results on fixed-points, see Arslanov [1981], Jockusch, Lerman, Soare, an d Solovay [1989], KuEera [1986], [1989], a n d Jockusch [1989]. In particular, KuEera [1986] (see X.4.7) has solved Post's problem by showing t h a t any degree below 0' and containing no function without fixed-points bounds an r.e. nonrecursive degree.

111.2

Simple Sets and Many-One Degrees

T h e path we shall follow t o attack Post's Problem is t h e one suggested by Post himself, which is also a natural mathematical practice: confronted with a difficult problem, t r y first with simpler versions of it and, once a solution is found for them, proceed t o more difficult cases, in t he hope of finally reaching t h e solution of the original problem. This will be a long pat h, b u t it will finally pay off in 111.5.20, leaving us with a deep knowledge of t h e structure of t h e r.e. sets.

257

111.2 Simple Sets and Many-One Degrees

Our present approach is structural, since it tries t o solve t h e problem by isolating nonempty properties of r.e. sets t ha t imply nonrecursiveness a n d T-incompleteness. A different approach consists of trying t o build ad hoc solutions by brute force, and will be considered in Chapter

x.

Many-one degrees T h e simplest special case of Turing reducibility, which we have been using already in many of t he previous proofs, is t he following:

Definition 111.2.1 (Post [1944]) A is m-reducible to B (A 5 , B ) i f , for some recursive function f , the following equivalent conditions are satisfied: 1. Vx(z E A

ej

f(z)E B )

2. A = f-'(B)

3. f(A)

B A f(2)

B.

A is m-equivalent t o B (A -, B ) if A I mB and B Sm A . Exercises 111.2.2 a) If A is recursive, then A SrnB for any set B # 0 , w . b) If A I , B, and B is recursive, so is A . c) If A &, B and B is r.e. then so is A . d) If A is r.e. then A Lrn2 zf and only i f A is recursive and A # 0,w. e ) There is a nonrecursive set A such that A SrnA. (Hint: consider K @ f ) If A and B differ finitely, then A =rn B.

Note t h a t 5, is a reflexive and transitive relation, and thus equivalence relation.

G,

z.) is an

Definition 111.2.3 The equivalence classes of sets w.r.t. m-equivalence are called m-degrees, and (D,,,, 5 ) is the structure of m-degrees, with the partial ,. ordering 5 induced on them by I The m-degrees containing r.e. sets are called r.e. m-degrees, and two of thein are: 1. Om, the m-degree of the recursive sets different from 8 and w

2. Oh, the m-degree of Fc. Note t h a t a n r.e. m-degree contains only r.e. sets. T h e m-degrees containing recursive sets ar e three: O m , {u}.T h e last two are incomparable a n d smaller t h a n O m , but every other m-degree is greater t han or equal t o 0,. In t h e following we will always consider nontrivial sets, and thus 0, may be considered as t h e least m-degree. We now show t h a t 0; is t he greatest r.e. m-degree.

{a},

111. Post’s Problem and Strong Reducibilities

258

Proposition 111.2.4 (Post [1944]) A set A is r.e. if and only if A 5 , K . Proof. If A is r.e. then A 5 , K: by t h e proof of 111.1.2. Conversely, if E A f(x) E K then t h e elements of A a r e exactly t h e counterimages of those in t h e intersection of K a n d t h e range o f f , a n d hence A is r e . 0

x

*

Definition 111.2.5 A set A is m-complete if it is r.e. a n d its m-degree is 0; , i. e. K 5, A . Exercises 111.2.6 Both K and the notion of m-completeness are defined w.r.t. the class of the r.e. sets. Let a recursive enumeration {Wh(z)}zEw of the recursive sets be given (see 11.5.26). a) The set x E K * e x E Wh(=), obtained by diagonalization over the recursive sets, is m-complete. (Muchnik (1958al) (Hint: let

{0 w

Wh(g(z))

=

ifzEK otherwise.

Then x E K

H g ( x ) E K*.) b) Zf the recursive sets are uniformly m-reducible to a n r.e. set A, then A is m-complete. (Smullyan 119611) (Hint: if z E Wh(=)++ f ( z ,z) E A, then K * I,,A . )

T h e analogue of Post’s Problem for m-reducibility is: a r e t h e r e r.e. sets which a r e neither recursive, nor m-complete? T h e beginning of our story is t h e following observation.

Proposition 111.2.7 (Post [1944]) , ’ f A is m-complete, then

2

contains an

infinite r.e. subset.

Proof. First consider t h e special case of t h e m.-complete set K. Note t h a t

w,

-IT 3 x E R -

w,.

Suppose indeed t h a t x E W,: then x is in K by definition, a n d in because W , 2 K,contradiction. T h e n it must be x @ W,, a n d thus also x E E. T h e n t h e index of an r.e. subset of is a n element of b u t not of t h e given subset, and this permits us to generate a n infinite r.e. subset of E, by starting with t h e emptyset, a n d getting new elements at each stage. Formally, if f is a recursive function such t h a t :

f(n

f(O)

= a n index of

+ 1)

=

0

a n index of {f(O),

t h e n t h e range o f f is a n infinite r.e. subset of

E.

. . . ,f(n)},

111.2 Simple Sets and Many-One Degrees

259

To extend this to any m-complete set A , let 2 E

Given an r.e. subset -

K:

* g ( 5 ) E A.

B of 7 ,first pull it back (through g) t o an r.e. subset of

K:. Use the property of K proved above, t o get an element in

but not in the pull back of B , then project it via g to an element in 2 but not in B . Formally, let f be a recursive function such that:

f(0) = g(ao) f(n + 1) = g(a,+l)

with a0 index of 8 with an+l index of the r.e. set g-'({f(O)> ' . ., f(41).

Then the range o f f is an infinite r.e. subset of

z. 0

Exercise 111.2.8 A different proof of the result above consists of noting that any infinite r.e. set of indices of 0 is an infinite r.e. subset of R, and can be projected to A because the function that reduces K to A can be taken to be one-one.

Simple sets Since coinfinite recursive sets and m-complete sets have complement with an infinite r.e. subset, a solution to Post's Problem for m-degrees would be given by sets without this property. Recall (p. 141, and 11.6.4) that a n infinite set is immune if it does not contain infinite r.e. (or recursive) subsets.

Definition 111.2.9 (Post 119441) A set is simple if it is r.e. and coimmune, i.e. its complement is infinite and does not contain infinite r.e. subsets. Exercises 111.2.10 a) A coinfinite r.e. set is simple i f and only i j i t has n o coinfinite, recursive superset. (Hint: use 11.1.20.) b) If A and B are simple then A n B is simple, and A U B is simple or cofinite. Thus simple or cofinite sets are a filter in the lattice of the r.e. sets under inclusion. (Dekker [1953]) c) If A is simple and W, is infinite, then An W, is infinite.

It only remains t o show that the notion of simplicity is not empty.

Theorem 111.2.11 (Post [1944]) There exists a simple set. Proof. We give two different constructions 1. Post's simple set S . does The idea is t o build S intersecting each infinite r.e. set, so that not contain any infinite r.e. subset. We cannot recursively enumerate the

111. Post's Problem and Strong Reducibilities

260

infinite r.e. sets (II.5.27.a), so we will intersect S with each r.e. set with enough elements. To prevent t he collapse of t he complement of S , we d o not want t o put too many elements in S, so we make sure t h a t each time a new element goes into S, another one will stay out. Precisely, we put at most half of {0,1,. . . , 2 2 } into S , for each 2. T h e set S is defined as follows: dovetail a n enumeration of all t h e r.e. sets and, for each e , put into S t h e first element greater t h a n 2e enumerated into W e .

S is r.e. by construction. 3 is infinite since a n element of (0,. . . , 2 z } can enter S only if it comes from some We such t h a t 2e < 22: b u t there are a t most z such sets, and each contributes at most one element. T hus 3 has at least z 1 elements, for each z, and it is then infinite. And S is simple, because if We is infinite then it has elements greater t h a n 2e, a n d one of them will be in S.

+

2. A direct construction. We build a simple set A by stages. At stage s we will have A,, a n d we w i l l l e t A , = { a & < a ; < . . . }. I n t h e e n d x = { a o < a l < . . . } , w h e r e a, = lims+oo a:. We want t o satisfy t he following requirements:

Pe : We infinite Ne :

3

Wen A # 0

2 has at least e elements, or lims+m a: < 00.

P stands for positive, because t o satisfy the Pels we have t o d o something positive on A (namely, t o put some element in it). N st ands for negative, because t o satisfy t he Ne's we have to d o something negative on A (namely, to leave some element out of A ) . T h e construction is as follows. We start with A0 = 0 (hence a: = n ) . At stage s 1 we search for t he smallest e 5 s such t hat :

+

We,, n A , = 0 for some

TZ

2 e, a: E We,,.

Note th at both A , a n d We,, are finite, and we only look at e 5 s , so t h e search is effective. If e does not exist, we go to t h e next stage. Otherwise, Pe is th e condition with smallest index which looks unsatisfied, a n d with a chance t o be satisfied. Then we put a: into A , where n is t h e smallest one such th at n 2 e and a: E We,,. This makes all t h e a& with m 2 n move to the next one (since they enumerate t he complemept). In other words, as+' ,= a,s ifm
Since t h e construction is effective, A =

i f m 2 n.

UsEwA, is r.e. Moreover:

111.2 Simple Sets and Many-One Degrees 0

26 1

Z is infinite. It is enough to prove that lim,-,w uk exists. Indeed, u& may move at a certain stage s + 1 only if it happens t h a t uc E W e ,for some e 5 n 5 m. Since each W econtributes at most one element for each e , and there a r e only finitely many e 5 m, uk moves only finitely many times. So all negative requirements are satisfied.

0

A is simple. By induction, suppose that SO is such that SO 2 e , and all Pi’s with i < e have been satisfied at stage SO (i.e. We,,, n A,, # 8 if We is infinite). If We is an infinite subset of 2,there are s 2 SO and n 2 e such t h a t uc = a, E Then one such a, goes into A at stage s+ 1 (because e is the smallest index for which P, looks unsatisfied, and with a chance t o be satisfied), contradiction. 0

T h e second proof given above is a typical priority argument with no injury, with Po > No > PI > Nl > ... as order of priority for t h e satisfaction of the requirements. Indeed, we allow uk t o move only t o satisfy P, for some e 5 n, so e.g. U; can move only t o satisfy PO,ui only t o satisfy PO or P I , and so on. Moreover, we choose t h e smallest

possible positive requirement for satisfaction, and this says t h a t t h e positive requirements are ordered by their indices. There is no injury, because once a positive requirement is satisfied it remains so forever (since we never take elements out of A ) . T h e priority method will be discussed in full generality in Chapter X. T h e two examples of simple sets given above are direct constructions, and thus somehow unnatural. There are however sets which can be naturally defined, and t h a t turn out t o be simple. T h e first uses t h e notion (p. 151) of random number, as a number that is its own shorter description. In terms of the Kolmogorov complexity K , defined (on p. 151) as

a number is random if x

5 K(z).

Proposition 111.2.12 (Kolmogorov (19631,[1965])The set of nonrandom numbers is sample. Proof. Let A = {x : K ( x ) < x}. Then A is r.e., because 2 E

A e (3e < z)(cp,(O)

N

x).

111. Post’s Problem and Strong Reducibilities

262 To show t h a t among

71 is

infinite consider, given any n, t h e converging values

CPO(O),CPI(~),... ,CPn(O).

+ +

1 5 K ( z ) by definition. And if z is t h e least number not among them, z 5 n 1 (since we considered only n 1 possible values, and in t he worst case they are t h e numbers from 0 t o 72). Then z 5 K ( z ) ,and (c is random. Thus there is a random number with complexity at least n+ 1. Since this holds for every n, there are infinitely many random numbers, and 2 is infinite. To show th at there is no infinite r.e. set of random numbers, first note t h a t t h e index e of a n r.e. set We provides a uniform description of t h e elements of t h e set: if We = {zo,z1,. . .}, then z, = t h e n-th element enumerated in W e . By th e S r - Th e o r e m (11.1.7), there is a one-one recursive function h such t h a t ( P ~ ( ~ , , ) ( O ) N 5., If we could prove t hat for some n we have h(e,n) < z, we would know t h a t z, is not random. This is certainly t h e case if n is big enough a n d 2, is greater than t(n)= niax h(e,n),

If z is any number different from them all, then n

+

e
since then h(e,n)is bounded by t ( n )almost everywhere. It is now enough t o note t hat , given W e ,we can uniformly obtain a n r.e. subset W,(,) of it whose n-th element is bigger than t(n),by waiting until such is infinite a n d it a n element is generated in We. If We is infinite, then Wg(e) contains a nonrandom element. Thus there is no infinite r.e. set of random numbers and A is simple. 0 T h e proof just given is a positive use of Berry’s paradox (Russell [1906]), a version of which is: given n, consider ‘the least number t h a t cannot be defined

+

in less than n characters’. This defines, in c 171.1 characters (where In1 is t h e length of n an d c is a constant), a number whose definition needs more t han n characters, and it is paradoxical for all n such t hat c In1 5 n, i.e. for almost every n. This proof contains additional information, exploited on p. 265. A quick, less informative proof is provided by t he Fixed-Point Theorem. Suppose A is a n infinite r.e. set of random numbers, and let h be a recursive function such that ( P ~ ( ~ ) ( O= ) t he smallest z > e generated in A.

+

By th e Fixed-Point Theorem, there is e such t hat ve CI: ( ~ h ( ~By ) . definition cpe(0)is a random number (being in A ) , hence it cannot be bigger t h a n e. But this is exactly how it was defined, contradiction.

The immunity of the set of random numbers is a strong result, implying:

111.2 Simple Sets and Many-One Degrees

263

1. a version of the incompleteness results: in any consistent formal system sound for arithmetic, we can prove that a random number is so only in finitely many cases, since the set of provably random numbers is r.e. (Chaitin [1974]). 2. the undecidability of the halting problem: if we could decide whether cpe(0)J., then we could compute K ( s ) and decide whether a number is random or not. Actually, the halting problem and the Kolmogorov complmity function have the same T-degree: K is obviously recursive in K ,by definition, and the converse holds because the set of nonrandom numbers is T-complete, see p. 265. Exercise 111.2.13 The existence of infinitely many random numbers implies the existence of infinitely many prime numbers. (Chaitin [1979]) (Hint: if there are only n 1 primes, say PO,... ,pn,then for every x, x = pzo ...An. But xi 5 log x , and so x can be described in m n . log x bits of information. For big enough 2, then x is not

+

random.)

The next example of simple sets provides with such sets in every nonzero T-degree, and thus shows that a simple set is not necessarily T-incomplete.

Proposition 111.2.14 (Dekker [1954]) Every nonrecursive r. e. T-degree contains a simple set. Proof. Given A r.e. and nonrecursive, let B be its deficiency set (11.6.16). Then A GT B , and B is a coinfinite and coretraceable r.e. set. So is (by 11.6.5) recursive or immune. Since A is nonrecursive, B is immune, and then B is simple. Exercise 111.2.16 Give a direct proof that the deficiency set o f f recursive and oneone is recursive or immune. (Hint: if B has an infinite recursive subset C, given x find y E C such that f (y) > x. Then x E A x E ( f ( O ) , .. . ,f(y)}. Thus A , and

*

hence B,are recursive.)

Effectively simple sets

*

We have just seen that a simple set can be T-complete, and thus simple sets do not automatically solve Post's Problem. But before we go on with different trials, we want t o make sure that none of the simple sets built above is already T-incomplete. The idea is t o effectivize the notion of simplicity.

Definition 111.2.16 (Smullyan [1964]) A is effectively simple if it is a coinfinite r.e. set, and there is a recursive function g such that

111. Post's Problem and Strong Reducibilities

264

Exercise 111.2.17 We show that other natural attempts to define effective simplicity are either equivalent to the one proposed, or impossible. a) A winfinite r.e. set A is effectively simple i f and only i f there is a partial recursive function cp such that We 2 cp(e) 1 A (We1 5 cp(e). (Sacks [1964]) b) For every sample set A there is a partial recursive function cp such that zf We is infinite then p ( e )4 A p ( e ) E Wen A . c) For no simple set there is a similar total recursive function. (Hint: consider g recursive such that W,(,, = {f (z)}, and apply the Fixed-Point Theorem.)

*

T h u s for simple sets we only know t h a t a n r.e. subset of t h e complement is finite, while for effectively simple sets we also know a recursive bound on its cardinality. T h e interest of t h e notion is t h a t it implies T-completeness.

Proposition 111.2.18 (Martin [1966]) Every effectively simple set is T-complete.

Proof. Let Define f

5~ A

we c 2

/wet _< g(e)-

such t h a t

W,(,, = {the first g ( e ) + 1 elements of 71).

x,

Then f has no fixed-points: if W e = W,,,, then We b u t IWel = g(e) 0 By t h e criterion for T-completeness 111.1.5,A is then 7'-complete.

+ 1.

Exercises 111.2.19 a) There are sample sets which are not effectively simple. (Sacks [1964]) (Hint: a direct proof is difficult, requiring priority and Fixed-Point Theorem. To build A as wanted, we want to destroy

Given z and e, we build an r.e. set W,(,,,, with the property that

Then we need the Fixed-Point Theorem to step from the sets WJ(.,.) to sets of the form W,. What 111.2.18 accomplishes is to separate the use of the Fixed-Point Theorem, to give the T-completeness of effectively simple sets, and of priority, to build a T-incomplete simple set or just, due to 111.2.14, a nonrecursive, T-incomplete r.e. set.) b) There are simple, not effectively simple, and T-complete sets. (Hint: take a simple T-complete set A, a simple not effectively simple set B, and consider A @I B.) c) A sample set A is T-wmplete if and only if there is g IT A such that

We c

=+ [We1 5 g(e).

111.2 Simple Sets and Many-One Degrees

265

(Lachlan [1968]) (Hint: T-completeness is proved as in 111.2.18. If A is T-complete, it is recursive in A to ask if We 2,i.e. if &(z E Wen A ) . If WeE 2 then W e is finite, by simplicity, and recursively in A we can search for an 2 such that (3> z)(y E We) fails.)

We can now show t h a t the simple sets constructed above are all effectively simple, and thus T-complete. We refer t o the proofs of 111.2.11 and 111.2.12, and use t h e same notations as there.

1. Post’s simple set. If We C 3 then IWel 5 2e 1, since otherwise We has more than 2e + 1 elements, so one of them is greater than 2e, and goes into S.

+

2. The set A built in 111.2.11. If We then ]We[5 e, since otherwise there is n 2. e such t h a t a, E W e . By going to a stage s after which t h e Wi’s with i < e do not contribute anymore elements to A , t h e elements of 2 up to a, have settled, and an E We,8,we would then put one element of We into A.

3. The set of nonrandom numbers. If Wecontains at least t ( g ( e ) )elements, then W g ( econtains ) at least g ( e ) elements by its definition. And if Wg(,) contains at least g ( e ) elements, then it contains a nonrandom number (because h ( g ( e ) , n ) 5 t(n)for any n 2 g ( e ) ) . This means that if We contains only nonrandom elements, then it must have at most t ( g ( e ) )elements, i.e. IWel 5 t ( g ( e ) ) . We consider now the simple sets provided by 111.2.14.

Proposition 111.2.20 (Smullyan [1964]) The deficiency set of K: is effectively simple. Proof. Let f be a oneone recursive function with range Ic, and

We want to find g recursive such that

Suppose then We C B. Since we already know t h a t B is simple, We is finite, say (21 < ... < x,}. Consider the behavior o f f on these elements.

111. Post’s Problem and Strong Reducibilities

266

“1

“2

5,-1

2,

Since each xi is a nondeficiency stage, there is no crossover. It is t hen enough t o find effectively a > f(z,):since f is one-one, a 2 IWel. To find a, we use t h e fact t hat

Since K is t h e range o f f , it is enough to choose W , containing all t h e elements of K: below f ( x n ) . This is possible because 2, E B,so

Then

Exercises 111.2.21 Strongly effectively simple sets. (McLaughlin [1965]) A is strongly effectively simple if it is a coinfinite r.e. set, and there is a partial recursive function d such that

We

2 =+ 4 ( e )1 A

(maxWe) < $(el.

An example is given by Post’s simple set. a) may always be supposed t o be total. b) There are effectively simple sets which are not strongly effectively simple. (Martin [1966]) (Hint: this follows from results in 111.4.24, since a strongly effectively simple set is not maximal, and there are maximal effectively simple sets. A direct construction, e.g. in the style of 111.2.19.a, is also possible.) McLaughlin [1973], and Cohen and Jockusch [1975], prove in different ways that the deficiency set of K is not strongly effectively simple.

Exercises 111.2.22 Immune sets. a) There are 2*O sets immune and coimmune, called bi-immune. (Hint: enumerate a set of pairs (zn,yn) such that 2, and yn are different elements of the n-th infinite r.e. set, not yet enumerated in the previous pairs. Any set with exactly one element of each pair is bi-immune.)

111.3 Hypersimple Sets and Tkuth-Table Degrees

267

b) A set A is effectively immune if it is infinite and, for some partial recursive function $, We C A =+ $ ( e l l A IW=II $ ( e ) . $ may always be supposed t o be total. Similarly for strongly effectively immune sets, where instead

We G A =+ $ ( e ) 1 A (maxWe) I $(e). (Hint: let g be a recursive function such that

{0

Wg(e)

we

otherwise. if $ ( e ) L

Then one of $ ( e ) and $ ( g ( e ) ) converges.) c) There are effectively bi-immune sets, i.e. sets effectively immune and effectively coimmune. (Ullian) (Hint: a natural construction, in which each We contributes a t most two elements, one for A and one for 2, gives a bi-immune set A such that if We 5 A or We C 2, then IW,l I 2e 1.) d) If A is strongly effectively immune, then 71 cannot be immune. (McLaughlin [1965]) (Hint: suppose We5 A max(W,) < f(e).

+

*

Let s, be the smallest stage in which x appears in K , if it ever does, and 0 otherwise. Define W t ( z )=

{

p) {sx)

otherwise.

ifx

Then x E K A sz 2 f ( t ( x ) )+ s, E 71. There are infinitely many such s,'s, otherwise for almost every x is x E K ($ 2 E K f ( q , ) ) , and K is recursive. But the condition 2 E K A s , 2 f(t(x)) is r.e., so 71 contains an infinite r.e. subset.) e) A set A is constructively immune if, for some partial recursive p,

W, infinite

+

p ( e ) 1 A p(e) E W, n 71.

If A is constructively immune, then 2 cannot be immune. (Li Xiang [1983]) (Hint: an infinite r.e. subset of 2 can be built, by starting with an r.e. index of w . ) f) The notions of effective and constructive immunity are independent. (Li Xiang [1983]) (Hint: every simple set is constructively coimmune; for the other directions, see c) and e) above.)

111.3 Hypersimple Sets and Truth-Table Degrees We have solved Post's Problem for m-reducibility by constructing a simple set, but have noticed t h a t this does not automatically imply a solution t o Post's Problem for T-reducibility, because there are simple sets which are T-complete

111. Post’s Problem and Strong Reducibilities

268

(actually, all t h e simple sets we built were such). T h e next step is to relax m-reducibility somewhat, a n d solve Post’s Problem for t h e weaker notion. In m-reducibility we only allowed o n e positive query to t h e oracle. T h e r e a r e many possible extensions, depending on t h e number of queries allowed (a fixed bounded number, o r unboundedly many), their n a t u r e (only positive, i.e. asking whether some elements a r e in t h e oracle, or also negative, asking whether some elements a r e not in i t ) , a n d t h e way they a r e combined (conjunctions, disjunctions, o r any possible combination). Some of t h e possible reducibilities, called respectively conjunctive, disjunctive, a n d positive (Jockusch [1966]) a r e t h e following:

A 5, B

H

for some recursive function f ,

zE A AId B

H

H

Df(.)

c B.

for some recursive function f ,

zE A

A SpB

H

D f ( z )n B #

0.

for some recursive function f ,

zE A

* ~ U ( UE Df(.)

A

D,

c B).

T h e consideration of these reducibilities makes good sense for the s t u d y of r.e. sets, d u e to t h e fact t h a t they only use positive information on t h e oracle, which is exactly what we may obtain from r.e. sets (note t h a t A is r.e. i f and only zf A S p K). We will not be too much concerned with t h e m , since t h e picture we get from m-reducibility is finer, b u t we will prove s o m e scattered results here a n d in Chapter X. Since our present concern is t h e solution to Post’s Problem, we might as well consider t h e strongest possible generalization along these lines: to allow for any (finite) number of questions, b o t h positive a n d negative, to t h e oracle. To define t h i s notion precisely, let {rs,},~w be a n effective enumeration of all t h e propositional formulas, built from t h e atomic ones ‘ m E X ’ , for m E w. These are also called truth-table conditions, since they c a n be arranged o,, means t h a t B satkfies on,i.e. t h a t in truth-tables. Given a set B , B t h e propositional formula rs,, becomes t r u e when X in t h e atomic formulas is interpreted as B .

Definition 111.3.1 (Post [1944]) A is tt-reducible to B (A Stt B ) iJ for some recursive function f , 2

EAH B

of(.).

A is tt-equivalent to B ( A E t t B ) if A Stt B and B Stt A . In terms of connectives, t h e various truth-table reducibilities correspond t o truth-table conditions built u p from t h e atomic ones by means of t h e following

111.3 Hypersimple Sets and Truth-Table Degrees

269

connectives:

while Srn correspond to using only atomic formulas. We will see in 111.8.4 that,, on t h e r.e. sets, Srn= {T}. Since 7 ,together with any one of A a n d V, generates all t h e propositional formulas, this would seem to t a k e care of all t h e possible t y p e s of truth-table-like reducibilities. Bulitko [1980] a n d Selivanov [1982] have shown t h a t it is almost so, in t h e sense t h a t only another o n e such rediicibility exists, called linear reducibility, corresponding to t h e logical s u m (i.e. addition modulo 2). Its definit,ion can be p u t as:

A

B

-++

for some recursive function f zEAH

IDf(.)fl BI

=1

(mod 2).

Truth-table degrees O u r first concern is to make t h e difference between truth-table a n d Turing reducibility clear. Recall t h a t a relative computation consists of two different kinds of actions, one purely computational (performed by a machine), a n d t h e other interactive (queries answered by t h e oracle). In Turing computations t h e two p a r t s c a n be strongly interwoven, a n d impossible to unravel: we may come to know t h e questions we need t o ask t h e oracle only during t h e computation itself, a n d t h e r e might even be no recursive bound on their number or size (as a function of t h e input). O n t h e other hand, truth-table computations clearly separate t h e two p a r t s of t h e relative computation, computing a h e a d of t i m e not only t h e elements which need to be queried, b u t also t h e outcome of t h e computation for any possible answer t h e oracle is going to provide for them. Another way t o see t h e difference is in terms of functionals. Recall t h a t A < T B if a n d only if CA N F ( c B ) ,for some partial recursive functional F .

Proposition 111.3.2 (Trakhtenbrot [1955], Nerode [1957]) A S t t B i f and only i f C A N F ( c s ) for some partial recursive, total functional F . Proof. Let 5

EA

* I= Of(.),

a n d define F ( a ,x) as follows. First see if a(.) 1 for every z such t h a t the atomic formula z E X occurs in af(.). If so, consider any set C such t h a t , for a n y z as j u s t said, z E C H a(.) = 1, a n d let

F(a,x)N

1 i f C k Of(.) 0 otherwise.

111. Post's Problem and Strong Reducibilities

270

F is partial recursive a n d total by definition, a n d CA N F(cB). Suppose now t h a t F is a partial recursive, total functional. For each x a n d X , F ( c x , x ) is defined, a n d for every branch of t h e space 2" of sets, t h e information on X needed to compute F ( c x , x) is bounded, by compactness. By Konig's Lemma, there is a bound which works for all branches. Another way to see this is to note t h a t F is continuous a n d total on 2", which is a compact space in t h e positive information topology: then F is uniformly continuous, a n d there is a modulus of continuity t h a t works for every member of 2". In any case, we may thus write down a truth-table t h a t gives F ( c x , z ) for any X , because only values of X up to t h e bound a r e needed, a n d there a r e only finitely many possible combinations. 0 By II.3.7.b we then know t h a t T-reducibility a n d tt-reducibility do not coincide. We now turn to a study of tt-reducibility. Exercises 111.3.3 a) If A is recursive, then A b) If A stt B and B is recursive, so is A . c) A Stt 2.

sitB for any set B .

Note t h a t < t t is a reflexive and transitive relation, a n d thus alence relation.

Ett

is a n equiv-

Definition 111.3.4 The equivalence classes of sets w.r.t. tt-equivalence are called tt-degrees, a n d ( D t t ,5 ) is the structure of tt-degrees, with the partial ordering 5 induced on them by I t t . The tt-degrees containing r.e. sets a r e called r.e. tt-degrees, a n d two of thein are:

1.

Ott,

the tt-degree of the recursive sets

2. O i t , the tt-degree of

K

A n r.e. set A is tt-complete if its tt-degree is Ol,,, i.e. if

K SttA .

Note t h a t an r.e. tt-degree, being closed under complementation, contains only r.e. sets if a n d only if i t contains only recursive sets. Also, O t t a n d O:, are, respectively, t h e least a n d t h e greatest r.e. tt-degrees. T h e analogue of Post's Problem for tt-reducibility is: a r e there r.e. sets which a r e neither recursive, nor tt-complete? We already know t h a t simple sets are not m-complete, a n d we make sure t h a t they a r e not automatically tt-incomplete.

Proposition 111.3.5 (Post [1944]) There is a simple, tt-complete set.

111.3 Hypersimple Sets and 'Ikuth-Table Degrees

271

Proof. Consider Post's simple set S (111.2.11). Any coinfinite r.e. superset of it will still be simple (since any infinite r.e. subset of i t s complement would also be a subset of 3). We t h e n look for a n r.e. set S' such t h a t : 1.

K Stt S ' ,

2. S

i.e.

K is coded into S'

by t r u t h tables

S', a n d S' coinfinite.

be a strong array of disjoint finite sets intersecting 3. I t exists, Let {F,},,, because t h e construction of S ensures t h a t at most z elements of (0,. . . ,22} go into S , so 3 intersects each subset of (0,. . . , 2 z } with at least z 1 elements. It is then enough to let

+

F, = { n : 2" - 1 5 n < 2,'

- 1)

We can t h e n use t h e set F, to code t h e fact t h a t x E K ,by p u t t i n g it into S' if t h i s holds, a n d not otherwise. We also want S' to be a superset of S, a n d we t h u s let S'=SU F,.

u

Z € K

Then:

1. K


2.

is infinite, because there a r e infinitely many elements 2 not in K ,a n d t h u s infinitely many F, not contributing to S * . Each one contains a n element of 3,which cannot go in S' because t h e Fy's a r e disjoint. 0

s*

Note t h a t S* is effectively simple, being a superset of S. T h u s there are effectively simple, tt-complete sets. But t h i s does not mean t h a t every effectively simple set is not only T-complete, b u t actually tt-complete (since hypersimple sets, which a r e not tt-complete, may be effectively simple, see II1.3.15.b). T h e tt-complete, simple set obtained above is a modification of Post's simple set. I t is natural t o wonder whether Post's simple set is already tt-complete itself. T h e surprising answer is t h a t this depends on t h e acceptable system of indices we a r e working with (one half of t h e result is given below, t h e other half in 111.9.2). T h u s Recursion Theory is not completely independent of the acceptable system of indices chosen to work with. T h i s was first shown by Jockusch a n d Soare [1973], who proved t h a t another set constructed by Post [1944] (namely his hypersimple set) could be T-complete or T-incomplete, depending on t h e acceptable system of indices. This cannot be t h e caqe for Post's simple set, which is always T-complete (being effectively simple).

111. Post's Problem and Strong Reducibilities

272

Exercise 111.3.6 Them is a n acceptable system of indices for which Post's sim_ple set is tt-complete. (Lachlan [1975])(Hint: given {W,},,, acceptable, define {W,},t, acceptable, S relative to it, and a strong array { B , } , E , such that

,.

x E

K

H

B, 2 s^. A

To have all the5e. sets in the enumzration, let W p = W,. Note that, by construction, if z > 2e and We = { z } then z E S. If x E K thus let = {2"+' + 2 n + 1)

$i&r+,,

for 0 < n < 2". If x E i? then, let let

A

W2r+,, =

0. For definiteness, let Wo = 0. Finally,

B, = (2"" +2n+ 1 : 0 < n

A

< 2"). )

Hypersimple sets Since simple sets do not solve Post's Problem for tt-reducibility, we look for a stronger notion. T h e idea comes from t h e fact t h a t tt-reducibility uses finite questions a b o u t sets, while m-reducibility uses only one question. W e may t h u s think of replacing, in t h e definition of simple sets, infinite r.e. sets b y disjoint produces a n strong arrays. Of course, a disjoint strong array contained in infinite r.e. subset of it (by choosing one number in each element of t h e array), a n d to rille out disjoint strong arrays contained in 2 is t h u s equivalent to simplicity. We thus relax t h e condition a bit:

Definition 111.3.7 (Post [1944]) A set A is hyperimmune if it is infinite, and there is no disjoint strong array (with members all) intersecting it, i.e. there is no recursive function f such that: z

# Y 3 Df(z) n Df(l/)= 0

Df(2)r-lA #

0.

A set is hypersimple if it is r.e. and co-hyperimmune. We give some conditions equivalent to hyperimmunity.

Proposition 111.3.8 (Medvedev (19551, Rice [1956a], Uspenskii [1957], Kuznekov) For an infinite set A , the following are equivalent: 1.

A is hyperimmune

2. there is no recursive function f such that 2

# Y * Df(2)n Df(l/)= 0

111.3 Hypersimple Sets and lkuth-Table Degrees

273

#0

0

Df(z)" A

0

UzcwDf(2) 2 A.

3. there is no recursive function f such that, for each n, A has at least n elements smaller than f (n)

4.

there is no recursive function f such that, for each n, a, 5 f ( n ) ,where a, is the n-th element of A i n increasing order.

Proof. T h e last two conditions are clearly equivalent. Moreover: 0

0

0

1 =+2 by definition 3 because, if A has at least n elements below f ( n ) , there is a n 2 element of A between n and f ( n 2), a n d thus a disjoint strong array intersecting A a n d covering w can be obtained by iteration:

+

3 1 because if {Dh(z)}zEw is any disjoint strong array intersecting A, then

f(n) = m a ( U D h ( i ) ) i
has at least n elements of A below it.

0

T h e intuitive content of t h e various characterizations of hyperimmunity is t h a t a hyperimmune set has very sparse elements, from a recursion theoretical point of view: for any recursive function f , no matter how fast growing, there a r e infinitely many elements a,, in t h e ordering of A by magnitude, such t h a t

f (n)

an.

Exercises 111.3.9 Domination properties. We say that f dominates cp if, for almost every argument z, f(z) 2 cp(z) whenever cp(z) 1. Let P A be the function enumerating the infinite set A by magnitude (principal enumeration), so that p ~ ( n ) is the 72rth element of A. a) A is hyperimmune i f and only i f p~ is not dominated by any recursive function. b) Call a set A dense immune if P A dominates every total recursive function, and an r.e. set dense simple if its complement is dense immune (Martin [1963]). Then a dense simple set is hypersimple. c) A strongly effectively sample set is not dense simple. (Cohen and Jockusch [ 19751) (Hint: let A be dense, and strongly effectively simple via g:

111. Post’s Problem and Strong Reducibilities

274

x.

Let a, and a: b e the la-th element of 2 a n d If h is a recursive function going t o infinity, there is no such that (Vn 2 n o ) ( % 2 h(n)). We get t h a t A is cofinite if we show ( V n 2 no)(Vs)(an 2 h(z)), since h goes to infinity. This follows by induction if, for n an

2 h ( z )A an+] 2 h ( + ~ 1)

*

an

5 2,

2 h ( +~1).

It is enough to define f and g such that h(z)2 g ( f ( n , z ) ) and ,

W,(n,z)= {a:}, with s minimal such that a:

2 h(s) A

a:+l

2 h(z +

1).

This can be done by the Fixed-Point Theorem. Then a: # a, because A is strongly effectively simple, hence a, 2 a:+l 2 h(s l).) d ) If p A dominates every partial recursive function, then K ST A. (Tennenbaurn [1961]) (Hint: see the proof of 111.1.5.)

+

T h e following result shows t h a t t h e intuition t h a t led to t h e definition of hypersimple sets was correct.

Theorem 111.3.10 (Post [1944]) A hypersimple set is n o t tl-complete. Proof. Let z E K

A a ~ ( ~We ) .prove t h a t A is not hypersimple by finding effectively, given n, a number m 2 n such t h a t -

A n ( 7 1 ,+ ~ ~1,. . . ,7n} # 0.

T h i s will automatically produce, by iteration, a disjoint s t r o n g a r r a y intersecting 7i. Let

zEA*

tj

(-EAAzn

EC

H

-(A*

5

+

C is r.e. Ixcaiise A’ is recursive (being cofinite). So let C = W,: 0

if n E

0

if a E

+

K then A then ’ ( A

of(.) h u t -(A* of(,)) b u t A*

b af(,)), since n E W,

=C

af(,), since a $! W , = C .

S o A af(,)ej - ( A * af(a)): a n d since A arid A* agree below n, they must, disagree above it. But by definition everything above n is in A * , so o n e of t h e elements used in of(,)is not in A . T h e first thought would be to let, m be greater t h a n all t h e elements used i n of(.),but we cannot get this effectively from n, because t h e definition of A* uses A , which is only r.e. But since only elements below n a r e considered, there are only finitely many subsets R, (i < 2”) of (0,. . . , n - I}, a n d o n e of t h e m is

111.3 Hypersimple Sets and Truth-Table Degrees

275

really An{O, . . . ,n-1). So if we consider them all, and let At = B&{z : z 2 n } , we can proceed as before, getting Ci = W a i .Now we cannot anymore assert is not in A , since Af might in general t h a t one of t h e elements used in not be A * , but if we let

m = 1 + max{ elements used in 1<2"

}

'~f(~,)

then m is in particular greater than all elements used in -

A n { n , ..., m } # 0 .

and we still have

0

Exercises 111.3.11 a) A coinfinite r.e. set A is hypersimple if and only if it has no c-complete superset. (Hint: if A B and B is hypersimple, then B is hypersimple or cofinite, hence not ti-complete. For the converse, see the proof of 111.3.5.) b) If A and B are hypersimple sets then A n B is hypersimple, and A U B is hypersimple or cofinite. Thus hypersimple or cofinite sets form a filter in the lattice of the r.e. sets under inclusion. (Dekker [1953])

We have now t o turn t o t h e existence of hypersimple sets. Post's simple set is obviously not hypersimple, but a modification of the second existence proof of simple sets in 111.2.11 will produce a hypersimple one.

Theorem 111.3.12 (Post [1944])There exists a hypersimple set. Proof. We build a hypersimple set A by stages. At stage s we will have A,, a n d we will let A, = {G < a; < - - . } .In the end 2 = {q < a1 < -..},where a, = lim,+- a;. We want t o satisfy the following requirements:

Pe : We infinite disjoint strong array =+ (32 E W e ) ( D z A ) Ne : 2 has at least e elements, or lirn,+-

a:

< 00.

T h e construction is as follows. We start with Ao = 8 (hence a : = n). At + 1 we search for t h e smallest e 5 s such that:

stage s 0

Z E We,,+ D o , n A , # 0

0

for some z E We,,,D, C [a:,00).

Note t h a t both A, and We,,are finite, and we only look at e 5 s,so t h e search is effective. If e does not exist, we go t o t h e next stage. Otherwise, Pe is t h e condition with smallest index which looks unsatisfied and with a chance to be satisfied. Then we put all of D, into A , where z is t h e smallest one such t h a t 2 E We,, and D, C [a:,..). Since t h e construction is effective, A = UsEwA , is r.e. Moreover:

276

111. Post’s Problem and Strong Reducibilities

2 is infinite. It is enough t o prove that lim8+w a: exists. Indeed, a: may move at a certain stage s 1 only if it happens t h a t a: E D,, for some z E We, e 5 n. Since each We contributes at most one finite set for each e , and there are only finitely many e 5 n, a: moves only finitely many times. So all Ne’s are satisfied.

+

0

A is hypersimple. By induction, suppose that SO is such that SO 2 e , all Pi with a < e have been satisfied at stage SO,and no finite set with index in Wi (i < e ) goes into A after stage SO. If We is a disjoint strong array intersecting 2,there are s 2 SO and z E We,,such t h a t D, C_ [a:, co).Then, for one of these z , D, goes into A at stage s f 1 (because e is the smallest index for which Pe looks unsatisfied and with a chance t o be satisfied), contradiction. 0

Also 111.2.14 generalizes, and shows t h a t a hypersimple set is not necessarily T-incomplete. Proposition 111.3.13 (Dekker 119541) Every nonrecursive r. e. T-degree

contains a hypersimple set. Proof. Given A r.e. nonrecursive enumerated by f recursive, let B be its deficiency set (11.6.16):

We already know that B is r.e. and coinfinite, and A ET B . Suppose there is a recursive function g majorizing i.e. g(z) greater than t h e z-th element of B , and in particular of z itself. Then f(g(z)) is also greater than z, because f is increasing on nondeficiency stages, a n d after stage g(z) no element smaller

B,

than f(g(z)) is going to be enumerated. Then

and A would be recursive.

0

Note t h a t , although t h e proof given above produces B Stt A , we only have A ST B , and this is necessary: not every nonrecursive r.e. tt-degree contains a hypersimple set (because a hypersimple set cannot be tt-complete) . Jockusch [1981a] actually shows t h a t not every nonrecursive r. e. tt-degree contains a simple set. Exercises 111.3.14 a) Every r.e. nonmcursive coregressive set is hypersimple. (Dekker and Myhill [1958], Dekker [1962]) (Hint: see 111.5.6 and 111.5.3.) h) There are nonrecursive retruceable sets which are not hyperimmune. (Dekker and Myhill [1958]) (Hint: see II.6.7.b.)

111.3 Hypersimple Sets and Truth-Table Degrees

277

Exercises 111.3.15 A is effectively hyperimmune if it is coinfinite, and there is a recursive function f such that, whenever We is a disjoint strong array intersecting A, IWeI 5 f ( e ) . A set is effectively hypersimple if it is r.e., and its complement is effectively hyperimmune. a) An effectively hypersimple set is effectively simple, and hence T-complete. (Hint: given We,let Wh(e)be the set of canonical indices of the singletons consisting of the elements of We.Then We C 2 =+ IWeI 5 f ( h ( e ) ) . ) h) T h e hypersimple set constructed in 111.3.12 is effectively hypersimple. (Hint: let f ( e ) = e.) Thus there are effectively simple sets which are not tt-complete, and 111.2.18 cannot be improved. c ) There are hypersimple, not effectively hypersimple, T-complete sets. (Arslanov [ 19701) (Hint: see 111.2.19.b.) d) A hypersimple set A is T-complete if and only i f there is f 51-A such that, whenever We is a disjoint strong a r m y intersecting 2, IWel 5 f ( e ) . (Arslanov I19701) (Hint: see III.2.19.c.) e ) A and 2 cannot both be effectively hyperimmune. Note that this is not true for effective simplicity, see III.2.22.c. (Arslanov [ 19691) (Hint: by taking the maximum of the two functions, A and 2 can be supposed effectively immune via the same function. Build a disjoint strong array intersecting both of them. Put in Wg(,,the canonical indices of the singletons {y}, for y 5 f (z) 1, and choose e such that Wg(,)= We. Then We is a disjoint strong array with more than f ( e ) elements, so it intersects both A and 2. Thus (0,. . . ,f ( e ) + 1) is the first set of the wanted array. Continue similarly.) IJor different notions of effective hypersimplicity see Arslanov [1969], [ 19701, 119851, Arslanov and Soloviev [ 19781, and Kanovich [ 19751.

+

The permitting method

*

The proof of the existence of hypersimple sets in each nonrecursive r.e. T-degree used deficiency stages, and w a s thus elegant but a bit artificial. In particular, it does not appear t o be useful to prove different results. But we can isolate from it a useful tool that was used there implicitly, and that has instead a vast range of applications.

Proposition 111.3.16 Permitting method (Dekker [1954], Muchnik [1956],Friedberg [1957],Yates [1965])If A and C are r.e. sets, { A s } s E w and are recursive enumerations of them, and for every x

{c,},,,

x E As+] - As then A

-

(3Y L X)(Y E C,+l - C S ) ,

C.

Proof. To see if x E A , look a t the stages in which some y 5 x is generated in C: recursively in C we may determine if y E C, and if so we just have t o

111. Post’s Problem and Strong Reducibilities

278

generate C until y appears in it. T h e n x E A if a n d only if x is generated at one of these stages. 0 T h e name of t h e method comes from t h e fact t h a t a n element may go into

A only if it is permitted by some element of C (smaller t h a n it). The method is very useful when we have a construction of a n r.e. set with certain properties, a n d we want to build one such set below any given (nonrecursive) r.e. set. Exercise 111.3.17 The permitting method does not produce A < t i C. (Hint: by using the priority method, introduced in Chapter X, we build disjoint r.e. sets A and B , such that A gtt AUB. If C = AUB and C. = A,UB,, z E A.+I -A, + z E C,+I-C.. To satisfy z $! A AUB u,+,=(~), pick up a witness z, $! A, U B,, and wait until (p,(z,) converges. Let A.+I U B,+1 = A, U B, U (2,). To decide where ze should go, see if A,+1 U B,+1 o , + , ~ (If~ so, ~ ) let . z, E B , otherwise let z, E A , )

*

As a n example, we show how t o apply permitting to t h e construction of Post’s simple set. Proposition 111.3.18 (Yates [1965])Every nonrecursive r. e. T-degree contains a simple not hypersimple set. Proof. Given C r.e. nonrecursive, we build A simple, nonhypersiniple a n d such t h a t A ST C, by adding permitting to t h e first proof of 111.2.11: at stage s, for every e 5 s such t h a t We,,n A , = 0,p u t in A t h e smallest x such t h a t

c at stage s , where x is permitted by C at stage s if (3 5 x)(y E C,+, - c,). x > 2e A x E W e , ,A x permitted by

Since A S , where S is Post’s simple set, it is immediate t h a t A is coinfinite a n d not hypersimple. By permitting, A ST C. It remains to show t h a t A is simple, a n d this is not automatic, because less elements go into A t h a n in S . Suppose We is infinite, a n d We C 2.T h e n , by construction,

x > 2e A x E We,,A e 5 s

3

x is not permitted by C at stage s.

We prove t h a t then C must be recursive. Given y, to see if y E C look for x arid s such t h a t x > 2 e A x € We,=A e 5 S A Y s x (which exist, because We is infinite). T h e n y E C H y E C,, because if y E C - C, then, for some t 2 s, y E Ct+l - Ct, a n d x is permitted by y at stage t (since W e , ,C_ W e , , ) . We still have to get, C ST A. Let B = A @ A ‘ , where A* is a hypersimple set such t h a t A* G T C (111.3.13). T h e n C 5~ A* S T B . Moreover, b o t h A

111.3 Hypersimple Sets and Truth-Table Degrees

279

a n d A* a r e reducible t o C , so B
B

Exercises III.3.19 a) C ST A can be ensured directly b y a coding procedure, as follows: at stage s, if y is the element of C enumerated at stage s + 1, put a t (the into A. (Hint: to see if y E C, search recursively in A a stage y-th element of s such that u; = %. Then y E C y E C,. Note also that u: moves only finitely many times for the coding, at most once for each y 5 n.) If this method is used, then we have a different proof of 111.2.14. b) C
*

{a:

where se =

{

,. . .,

if e E C otherwise,

k s ( e E c.) if e E c 0 otherwise.

Let also T , = ps(Vn 5 2g(e))(u: = a,). Since T , is recursive in A, if e E C e+e E C,., then C ST A. So consider C = { e : e E C - Cre}. This is r.e. in A, since C is r.e. and r e is recursive in A. If it is finite, then C 5~ A. So suppose it is infinite. Then there are infinitely many e such that s, > T e , in particular Wg(,) 2,and W,(,)has 2g(e)+ 1 elements, so there is z E Wg(,)h z > 2g(e). Given y , t o test for y E C look, recursively in A , for: A

A

eE

such that 2g(e) > y (C is infinite, and g is one-one)

E Wg(e)A 2 > a stage s such that z E Wg(,),,A g ( e ) 5 s. 5

Then, since z E Wg(,),,A z > 2g(e) A g ( e ) 5 s but Wg(,)5 2,z is never permitted. So no element smaller than it can enter C after that stage. Hence y E C y E C,.) c) The existence of hypersimple sets in any nonmcursive r.e. T-degree can be proved directly, by permitting and coding.

*

To avoid false expectations, we must stress t h e fact t h a t none of the following is true: 1. permitting can be applied t o any construction of r.e. sets

2. when permitting can be applied, it can also be combined with a coding procedure t o give a set with t h e highest possible degree.

As we will see in Chapter X , t h e first fails for maximal sets (since no maximal set can be built below a low degree), t h e second for contiguous degrees or

280

111. Post’s Problem and Strong Reducibilities

7-maximal semirecursive sets (since they exist below any given nonrecursive degree, but not in every degree). It is however true that there seems t o be a sort of maximum degree principle (Jockusch and Soare [1972a]), according t o which natural constructions with ‘weak negative requirements ’ usually give sets with the highest possible degree not explicitly ruled out by the construction. Another fact, related t o the one above and which might be called effectivity principle, is that sets which are constructed in some natural way to satisfy some requirements, tend to satisfy them i n some effective way. For example, the constructions of simple sets (111.2.11) and of hypersimple sets (111.3.12) automatically produced sets which a r e both T-complete, and effectively simple or hypersimple. We will see many other manifestations of the two principles in the following. In particular, Section 6 is a n elaboration of the fact that K is effectively nonrecursive.

111.4 Hyperhypersimple Sets and Q-degrees We have solved Post’s problem for tl-reducibility by constructing a hypersimple set, b u t have noticed t h a t this does not automatically imply a solution to Post’s problem for T-reducibility, because there are T-complete hypersimple sets. Once again the next step is to relax tt-reducibility, and solve Post’s problem for the weaker notion. Since there seems t o be no easy way t o weaken tt-reducibility without falling into T-reducibility, we pursue a complementary tactic, and strengthen T-reducibility instead. Recall (111.1.4) that, for r.e. sets A and B , A
This means, intuitively, that from time t o time we ask the oracle questions about containment of a finite set D,. We thus propose t o ask t h e oracle only questions about singletons. This can be expressed by t h e existence of a n r.e. relation R such that zE

A @ (3u)(u E B A R(z,u)).

By the S,”-Theorem, t h e existence of a binary r.e. relation R is equivalent t o the existence of a recursive function f such that R ( z ,u)@ u E W,(z.. We can then express the previous formulation as

281

111.4 Hyperhypersimple Sets and Q-Degrees

Q-reducibility T h e discussion above leads us t o the following notion.

Definition 111.4.1 (Tennenbaum) A is Q-reducible to B ( A for some recursive function f,


B ) z;f,

Thus Q-reducibility is similar t o c-reducibility (p. 268), only using r.e. sets of questions in place of finite sets. On r.e. sets, which are our real concern, the similarity is even greater, since we can actually use only finite r.e. sets of questions.

Proposition 111.4.2 (Soloviev [1974]) Zf A and B are r.e. sets, then A SQ B i f and only i f there is g recursive such that, for every x, 1.

Wg(,) is finite

2. x E A

Wg(,) C B.

Proof. Given f as in the original definition of SQ, let Wg(,) be t h e r.e. set generated as follows. Generate simultaneously Wf,,),A , and B . At each stage of t h e enumeration, put t h e elements already generated in Wf(,) into Wg(,), unless either x has already been generated in A (this is a permanent block), or some z has already been generated in Wg(,), but not yet in B (this is a temporary block, that could later be removed if z is generated in B ) . If x E A then W,(,) C B , and hence Wg(,)B , because Wg(,) is a subset of Wf(,). Moreover, W,(,) is finite because its enumeration stops no later than x has been generated in A. If x @ A then Wf(,) nB # 8: if z is any element of the intersection, t h e enumeration of Wg(,. stops permanently no later than z has been generated in Wg(,),and thus Wg(,) is finite. Moreover, Wg(,) contains one such z , because x is not in A , and thus Wg(,) B. Notice t h a t if A
111. Post’s Problem and Strong Reducibilities

282

Exercises 111.4.3 @reducibility. We have just noted that 54 is a nonstandard reducibility, and we introduce it mostly for the sake of solution to Post’s problem. Here are some properties of 54 and the associated notion of Q-degree, some of which depend on the definition of the Arithmetical Hierarchy, see IV.1.6. a) 54 is refEexiue and transitive. b) ZfA 54 B and B E then A E H i . c) There is a least Q-degree, containing exactly the II? sets. (Hint: if A E II? and B # w is any set, let b @ B, and

a

*

W,(,) 2 B, and A LQ B.) Then x E A d) A
*

*

As usual we have the notion of completeness: Definition 111.4.4 A set A is Q-complete if it is r.e. and K:
Hyperhypersimple sets The idea t o get r.e. sets which are automatically not Q-complete comes from the fact that the hypersimple sets are not tt-complete, and in particular not c-complete. Since Q-reducibility corresponds t o c-reducibility, with finite sets

111.4 Hyperhypersimple Sets and Q-Degrees

283

given by canonical indices replaced by finite sets given by r.e. indices, we change strong arrays into weak arrays in t h e definition of hypersimple sets.

Definition 111.4.5 (Post [1944]) A is hyperhypersimple i f it is a coinfnite r.e. set, and there is no disjoint weak array (with members all) intersecting A , i.e. i f there is no recursive function f such that

T h i s definition is t h e point where Post got stuck in his epochal p a p e r [1944]. H e left open t h e problems of existence a n d T-completeness of hyperhypersimple sets, both of which will be solved in this section. First we give some characterizations of hyperhypersimple sets. T h e first o n e shows t h a t , in place of r.e. sets of disjoint finite sets given by their r.e. indices, we may only look at r.e. sets of (not necessarily finite) r.e. sets, disjoint o n 3.

Proposition 111.4.6 (Yates [1962]) A is hyperhypersimple i f and only i f it is n coinfinite r.e. set, and there is no recursive function f such that:

.

Proof. Suppose such a n f exists. We first build g recursive such t h a t : 2

0

# Y * Wg(2)n Wg(v)= 0

wg($) nZ # 8.

For this it is enough to generate t h e Wf(,)’ssimultaneously, a n d eliminate repetitions. If at some stage we consider W,(,), a n d a new element of it comes o u t , we p u t it into Wg(,,, unless it has already been p u t in some other Wg(Y) before. T h e n , if z # y, W,,,) n Wg(Y)= 8. And since W,(,) fl3 # 0, we also have Wg(,) # 8, because on 2 t h e W,(%)’Sa r e disjoint. Now from g we build h recursive with t h e additional property that Wh(,) is finite. To define Wh(,), simultaneously generate Wg(,) a n d A , a n d at each stage p u t t h e elements already generated in Wg(,) into Wh(,), unless some z has already been generated in Wh($),b u t not yet in A (this is a temporary block, t h a t could later be removed if z is generated in A ) . Since Wg(,, n7[ # 8, as soon as the first element of t h e intersection is generated in Wh(%) t h e enumeration stops permanently, a n d Wh(,) is finite. B u t since such a n element

nz

111. Post’s Problem and Strong Reducibilities

284

has entered W q z ) its , intersection with

2 is nonempty.

0

T h e second characterization of hyperhypersimple sets parallels t h e definition of simple sets, with r.e. o r recursive sets replaced b y regressive or retraceable ones.

Proposition 111.4.7 (Yates [1962])T h e following are equivalent, f o r r.e.

coinfinite sets: 1. A is hyperhypersimple

2.

2 does

3.

2 does n o t contain infinite regressive sets.

n o t contain infinite retraceable sets

Proof. T h e last two conditions a r e equivalent by 11.6.20, so we c a n consider a n y of them. If contains B retraceable via cp we may suppose t h a t cp(s) 5 s, whenever cp(x)1 (by II.6.3.a). Let

T h e n t h e Wf(,L)’s a r e disjoint. Since t h e n-th element of B in order of magnit u d e is in Wf,,,),Wf(,)fl2 # 8, a n d A is not hyperhypersimple. Let now A be not hyperhypersimple, a n d f be a recursive function such that: 5

# Y 3 Wf(z)n W/(y) = 8

If we could effectively pick u p a n element z, E Wf(,) n 2, we could simply Ict cp send W,(O)over Q, a n d Wf(,+l) over z,. T h e n cp would regress t h e infinite subset { z , } , ~ ~of A. T h i s we cannot do, b u t since t h e effective choice of z, is needed only to have cp partial recursive (because t h e enumeration of t h e regressed set need not be effective), t h e r e is a n easy way out: we j u s t have to do t h e same for each eleriient of each W,(,), not j u s t for z,. E.g., let ‘p’ send W f ( 0 )over Q, a n d Wf((z,y)+l) over t h e s - t h element generated in Wf(y). T h e n cp’ regresses t h e set inductively defined as follows: z,’, = zo, a n d if 2; is t h e s - t h element generated in Wf(y),t h e n z;+, E Wf((s,y)+l)n A. 0 Exercises 111.4.8 Hyperhyperimmune sets. Two definitions are possible. A is strongly hyperhyperimmune if there is no r.e. set of disjoint r.e. sets intersecting it, and it is hyperhyperimmune if there is no disjoint weak array intersecting it. The difference is that in the first cme the r.e. sets are not necessarily finite.

111.4 Hyperhypersimple Sets and Q-Degrees

285

a) A n infinite set is strongly hyperhyperimmune if and only if it does not have infinite retmcxable or regressive sets. (Yates [ 19621) (Hint: see the proof of 111.4.7.) b) A strongly hyperhyperimmune set is hyperhyperimmune, but not conversely. (Hint: build a hyperhyperimmune set in the natural way, with a t least one element from each B, = { ( n , z ): z w } . Then the set has an infinite retraceable subset.) 111.4.6 shows that the converse implication holds for A cer.e. Cooper [1972] shows that it holds for A E A:. c) There are 2N0 hyperhyperimmune, w-hyperhyperimmune sets. (Hint: see II1.2.22.a.)

Exercises 111.4.9 Strongly hypersimple sets. (Young [ 19GSa]) T h e following definitions are patterned on 111.3.8. A coinfinite r.e. set is (finitely) strongly hypersimple if there is no r.e. set of r.e. indices of disjoint (finite) r.e. sets intersecting A, and with union covering it. a) Hyperhypersimple + strongly hypersimple + finitely strongly hypersimple hypersimple, but none of the converse implications hold9. (Young [ 1966a1, Fbbinson [19671)(Hint: by the proof of 111.5.7, a semirecursive hypersimple set is not finitely strongly hypersimple. In Chapter IX we will introduce r-maximal sets, which are strongly hypersimple, and show the existence of r-maximal sets which are not hyperhypersimple (see IX.2.17 and IX.2.20). Finally, we build a finitely strongly hypersimple set which is not strongly hypersimple. Define

*

(z,y)E A’

Hz E

A v

(z

< y).

If we picture w x w as a matrix, then in A’ go full rows if they correspond t o elements of A , and only the part beyond the diagonal otherwise. If A is coinfinite, then the and A’ is not strongly hypersimple. Let A be hyperhypersimple, columns intersect by 111.4.18: then A’ is finitely strorigly hypersimple. Indeed, suppose there is a disjoint weak array f intersecting E , and covering it. We can build a disjoint weak array g intersecting 2,as follows. The idea is to try to put the first component of each element of W,(,) into W g ( = )sirice : W,(.) intersects A*, Wg(.)then intersects A . The trouble is that different W,(.)’Smight have elements with the same first corriponent, and the W g ( e )would ’ ~ then not be disjoint. So we start as we said, and a t stage z 1 we look a t the z-th row: generate simultaneously A and the W , ( e ) ~ ’ , until we find either z E A, or the W,(,)’scovering {(z,O), (2, I), . . . , (z, z)}. One of the two cases must happen, since the second case does if z @ A, and the weak array f covers E . Choose the W,(.) assigned to W,(,), with n minimal, and put z in Wg(,).If some of the remaining W , ( e ) ’was ~ assigned to some other W9(,,,)’s, define can change only new assignments. Note that, by induction, the assignment to Wg(e) finilely many times. E.g., Wg(o)always has Wf(0)assigned and, since this is finite, it can force a change of assignment to W g ( l only ) finitely often.) b) A coinfinite r.e. set A is (finitely) strongly hypersimple zf and only if there is no r.e. set of characteristic indices of disjoint (finite) recursive sets intersecting A. Thus hypersimple, finitely strongly hypersimple, and hyperhypersimple sets are obtained from the same definition, but using respectively canonical, characteristic, and r.e. indices of finite sets. The condition of finiteness may be dropped for r.e.

x,

+

111. Post's Problem and Strong Reducibilities

286

indices, b u t not for characteristic ones. (Martin [ 196Ga1) (Hint: given W,(.) such that

uzcw W,(.) 2 2,let

z E Rg(,.)++z shows u p in W,(.) before that. in A.

Conversely, given I<,(,), it is p x s i l h : to suppose R,(,) priate unions. Let z E

Wg(.)ts ( V y < T ) ( Z @ A',(v)) A ( 2 -

nA infinite, by taking appr(*

2 V (2

> z A z E R,(.))].)

c) A coinfinite r.e. set is (finitrly) stmiigly hypersimple if and only if 2 does not have iiifiiiite subsrts wlmced b y total (and many-one) recursive functioiLs. (Martin [ 1966al) (llirit: use part b) above, and see 111.4.7.) Other parallel characterizations o f strongly hypersimple a n d hyperhypersimple sets, in terms of arrays, are given by Ibbirison [ 1!)G7a].

7'he following result, shows t.liat, t h e intuition t h a t led to t h e dcfiiiition of hyperhyl~ersinrplesets was correct..

Theorem 111.4.10 (Soloviev [1974], Gill and Morris [1974]) A hyperhypersirriplt: set is not Q-complete.

Proof. Let, z E K

W,(,) C_ A , with W,(,) finite. To prove t h a t A is not, hyperhypersimple, siqqmse we already have Wf(o),. . . , W I ( ? ~all ) , finite. We warit Wf(?,+,)finite atid such t h a t :

nx # fl

Wf(?*+l)

W f ( n +n ~ )(Ui
+

We t r y to have f ( u 1) = y(n) for m i i e a E itlrzi is t,o tlefine ail r.c. set B K: such t h a t :I;

E

c

E , so t h a t W,(.) n i l # 0.

The

S + w,(,) n (Uwf(i))= 8. i< ii

T h e n , for n index of U , we have

B=W&C

=3

nEC-B,

a n d t h e two conditions a r e satisficxl. To have R 2 of 9 ,

zE

c we ask, by t h e properties

s + wg(,)n 2 # 0.

P u t t i n g things together, we t r y t,he definition 5

E

W,(.) n A n

cuWf(%)) # fl. i
111.4 Hyperhypersimple Sets and Q-Degrees

287

T h i s gives a sequence of sets which a r e disjoint only on 2,b u t we know from 111.4.6 t h a t this is enough. Since 2 is not r.e., t h e t r u e definition will be: zEB

* (3s 2 4 W g ( z ) , s nA,n (UWf(z),s)# 01. i
u,<, (u,<,

T h e reason t o have s 2 z is t h a t , since Wfct)is finite, for z (and hence s) big enough it will be t r u e t h a t z is in when z is in B . Indeed, t h e r e is a stage SO such t h a t after it all elements of A n Wf(,))(a finite set) have Wf,,),,) is not in A , been generated. For s 2 SO, any element of A,n a n d hence is in A n (Uz<,W f ( , ) ) Thus, . for z 2 SO, i F z is in B t h e n it is in

(u,<,

rlzn (u,<,

W f ( , ) j ,a n d hence in E, because Wg(,) n A # 0. T h u s , even if we do not know whether R C E, at least we know t h a t we miss t h i s only for finitely many numbers. Let now B = Wao:if a0 E B t h e n , by definition of K , Q E E, as wanted. But if a0 E B then a0 E K , t h u s Wg(,,) A , a n d this does not work. I n this case, let W,, = B - {ao}, a n d see if a1 E W a , a n d so on. We thus define a sequence CWJ,a l , . . . After a finite number of steps we must hit a, E W,,, because all t h e a,’s a r e in B , a n d only finitely many of t h e m a r e not in E. T h e n we can just let Wf(n+l)= UW,,,,), a n d note t h a t t h e at’s in K contribute Wg(,,) A , a n d t h u s do not interfere with what occurs on 3. 0 Wg(,)

Exercises 111.4.11 a) A winfinite r.e. set A is hyperhypersimple if and only af it has no Q-complete superset. (Hint: if A B is hyperhypersimple, then B is hyper-

(u,,,

hypersimple or cofinite, hence not Q-complete. I‘or the converse, i f f is a weak array Wfc,,) is Q-complete.) intersecting 2 then B = A U

b) If A and B are hyperhypersimple sets then A n B is hyperhypersimple, and A U B is hyperhypersimple or finite. Thus hyperhypersimple or cofinite sets are a filter in the lattice of r.e. sets under inclusion. (Martin, McLaughlin) (Hint: if A and _ _ B are r.e. and A n B has an infinite retraceable set, so does one of A, B.)

We now want to t u r n to t h e existence of hyperhypersimple sets. We have a great number of hypersimple sets, namely t h e deficiency sets of a n y recursive function enumerating any nonrecursive r.e. set (111.3.13), b u t none of t h e m is hyperhypersimple (111.4.7), because their complement is retraceable (11.6.16). T h i s actually gives:

Proposition 111.4.12 (Yates [1962]) Every r. e. T-degree contains a hypersimple set which is not hyperhypersimple. T h e existence of hyperhypersimple sets seems to be a problematic matter: for t h e first t i m e we a r e in a situation in which the obvious attackfails. Namely,

111. Post’s Problem and Strong Reducibilities

288

if we t r y t o follow t h e line of 111.3.12 t h e n we face t h e trouble t h a t , at some given stage s, we would like to p u t into A t h e elements of a finite set W,, b u t we only have W,,,, a n d there is no way to know whether W, has been generated completely at stage s, or not. Moreover, a n d differently from all t h e cases dealt with so far, t h e construction of a hyperhypersimple set cannot produce a set t h a t satisfies t h e definition effectively. To make this precise, let A be effectively hyperhypersimple as in 111.3.15, i.e. if there is a recursive function f such t h a t IW,l 5 f ( e ) , whenever We is a disjoint weak array intersecting 3. T h e n there is no effectively hyperhypersimple set. Indeed, given a coinfinite r.e. set A , a n d f as j u s t said, we can build W,(,, as a n r.e. set containing t h e indices of f ( e ) 1 finite disjoint r.e. sets intersecting 2 (start with n in t h e n-th set, for n 5 f ( e ) , a n d when t h e last element p u t in a set is generated in A , p u t in the smallest element t h a t is not yet in any of t h e sets). T h e Fixed-Point Theorem produces e such t h a t We = W,(,,, and thus IW,l > f ( e ) , contradiction. These observations show t h a t something new is required for t h e construction of hyperhypersimple sets. As we have said, Post [1944] left t h e problem open, a n d it took a few years to solve it. However, for t h e solution of Post’s problem t h e existence of hyperhypersimple sets is not necessary, a n d t h e reader may t u r n directly to t h e next section, if (s)he wishes.

+

Maximal sets T h e following definition provided t h e starting point for t h e construction of hyperhypersimple sets.

Definition 111.4.13 (Myhill [1956], Rose and Ullian [1963]) A set A is cohesive if i t is infinite, and cannot be split into two infinite parts b y a n r.e. set, i.e. if B is r.e. then either B n A or n A is finite. A set is maximal i f it is r.e. and its complement is cohesive. T h e name maximal comes from t h e fact t h a t maximal sets are the maximal elements in the lattice of r.e. sets under inclusion, modulo finite sets. Indeed, a coinfinite r.e. set is maximal if a n d only if it has only trivial supersets, i.e. if B is r.e. a n d B 2 A then either B is cofinite, or B - A is finite.

Proposition 111.4.14 (F’riedberg [1958]) A maximal set is hyperhypersimple. Proof. Let A be not hyperhypersimple: then there is a disjoint weak array {Wf(,,},,, intersecting 2. We can split 2 by taking ‘half’ of t h e array, e.g. by letting B = UzEwW f ( 2 %B ) : is a n infinite r.e. set, a n d B a n d Bnx a r e

nx

both infinite.

0

111.4 Hyperhypersimple Sets and Q-Degrees

289

Exercise 111.4.15 A marcinla1 set is deilse simple. (Martin [1963], Tennenbaum) (Hint: this follows from the fact, proved in IX.2.25.b, that hyperhypersimple sets are dense simple, but a direct proof is easier. Given an enumeration {(LO < a1 < . . . } of 2, and a recursive function f, suppose there are infinitely many n such that a, < f ( n ) . Let h(n)= max,
W,(,) = {Z : h(n) 5 z < h(n + 1)) have a t least two elements of 2. Build B r.e. by putting in it, for each n, the elements of Wg(n)up t o and including the first not in A. Then B splits 2 into two infinite park.)

Despite their being stronger, t h e requirenients to build maximal sets seem to be easier to deal with t h a n t h e ones for hyperhypersimple sets, since a natural approach to satisfy t h e m works. As a warm up, we first dispose of cohesive sets.

Proposition 111.4.16 (Dekker and Myhill) There exists a cohesive set. Proof. For future reference] we view t h e construction as building a set A with cohesive complement, by stages. At stage s we will have A,, a n d we will let A , = ( a ; < a ; < . . . }. I n t h e e n d A = { a o < a l < . . . } , w h e r e a n = l i n i , E w a ~ , i.e. A = A,. We want to satisfy t h e following requirements:

UsEw

P,: W, n 71 finite or W ,n 71 finite N , : 2 h a s at least e elements, i.e. lim,+m a: < 00. T h e construction is as follows. We s t a r t with A0 = 8 (hence_ a: =-n ) . A t s t a g e e 1 we satisfy P,. Let Be be Wen z if it is infinite, a n d Wen A, otherwise. Note t h a t Be is infinite because A, is, by construction, a n d its elements a r e either all in We, or all in To satisfy P, is thus enough to have almost all t h e elements of Ae+l (and hence of 2)in B e . We t h e n let

+

m.

ae+l n n+l

-

ifnse

a:

= t h e smallest element of B, greater t h a n a;+’

if n 2 e.

Note t h a t t h e construction is highly noneffective, for two reasons: we first ask whether given sets a r e infinite, a n d t h e n we ask membership questions a b o u t B e . 71 is however cohesive: t h e Pe’s a r e satisfied by construction, a n d 71 is infinite because a: may move at a certain stage s 1 only if 71 > s, hence only finitely many times. T h u s lim,E, a: exists, a n d all Ne’s a r e satisfied. 0

+

Exercises 111.4.17 Cohesive sets. a) Every infinite set has a cohesive subset. (Dekker and Myhill) (Hint: given C infinite, let Ao = and proceed as above.)

c

111. Post's Problem and Strong Reducibilities

290

b) There are 2'O cohesive sets. (Hint: any infinite subset of a cohesive set is cohesive.) c) There is no cohesive and co-cohesive set. (Hint: if A is such, and B is a n infinite coinfinite r.e. set, then B n A is_finite, _ by cohesiveness of A . Since B is infinite, B n A is infinite. By cohesiveness of A, B n 2 is finite. So A differs finitely from B or from B for each such B , contradiction.) d ) N o finite union of disjoint cohesive sets can cover w , but an infinite one can. (Hint: see parts a) and c) above.)

We now constructivize t h e proof just given, a n d build a cohesive set with r.e. complement.

Theorem 111.4.18 (Friedberg [1958])There exists a 7naximal set. Proof. We use notations as in 111.4.16. T h e requirements a r e t h e s a m e as there, plus t h e fact t h a t A has to be r.e. T h i s forces us to have a recursive construction, a n d we cannot anymore ask whether given sets a r e infinite. T h i s forbids t h e simple strategy used before, of satisfying o n e positive condition at each stage, a n d forces us to work with approximations. Let us examine PO first: since we cannot ask whether Wo is infinite, a n d at any given stage we only have a finite approximation to it, we w a n t to play two different strategies, one of which will win. Suppose we knew t h a t WOis infinite: t h e n we could force a: on i t , b y waiting long enough. We c a n push this as far as possible a n d , at any given stage s 1, force t h e longest possible initial segment of As+*in Wo. T h i s means t h a t , whenever there a r e elements of Xs in which a r e smaller t h a n elements of 2,in WO,we d r o p t h e m (by putting them into A ) . T h i s produces, at any stage s , a picture as follows:

+

6

If Wo is really going to be infinite, then 3 is going to be included in Wo. Otherwise, except for a finite initial segment, 2 is going to be included in a n d PO is going to be satisfied either way (this does not t a k e care of t h e other requirements, see 111.4.19). We t u r n now to PI. If we knew t h e final outcome of t h e s t r a t e g y for PO, we would not worry: we could play t h e s a m e strategy inside t h e p a r t of 2 on which t h e characteristic function of WOis eventually constant, b e t t i n g on t h e fact t h a t W1 is going to be infinite on i t , a n d having at each s t a g e a n initial segment in W1, a n d t h e rest in Even if we do not know t h e final outcome, nothing forbids us to play t h e same strategy on b o t h approximations, a n d so on. We t h u s t r y t o have 2 look like this:

6,

wl.

291

111.4 Hyperhypersimple Sets and Q-Degrees

This can easily be formalized, by using t he concept of e-state, which codes t h e membership of a given element z into t he first e r.e. sets, at a given stage

s:

a ( e , 5,s)= x{2e-i :i 5 e A z E

w,,~}.

Another way to visualize e-states is by viewing them as binary strings of 0’s and l’s, ordered lexicographically, and with t he i-th digit from t h e left giving t h e value of t h e characteristic function of Wi. T h e following properties of e-states a re immediate: 0

there a r e only finitely many e-states, namely 2e+1

0

t h e e-state of a n element at a given stage can only increase at later stages, i.e. if s 5 t then a ( e , z , s )5 a ( e , z , t ) since, for each i, W,,sC W,,t

0

e-states give absolute priority t o membership in r.e. sets with lower indices, i.e. if a ( e , z , s) < a(e,y, s) then a ( i , z ,s) < a ( i ,y, s), for any i 2 e.

T h e construction of A can now be easily formulated, by trying to let t h e e-th element of 2 have maximal e-state. Precisely, we start with A0 = 8 (hence a: = e). At stage s + l we let, by induction, a:+’ be t h e smallest z E A,greater than a:::, a n d with maximal e-state. Note t ha t , by induction, A, is infinite a n d recursive, so t h e construction is effective, and A is r.e. Moreover: 0

2 is infinite. I t is enough t o prove t hat limsEwa: exists. Suppose, by induction, t h a t limsEwa: exists, for each i < e. Let SO be a stage after which all a8.s have reached their final position. Then, for s > so, a: may move only to reach a higher e-state, and hence only finitely many times.

0

A is maximal. Suppose, by induction, t hat Pi is satisfied, for all i such t h a t

< e.

(Vi)[(Vn 2 nO)(an E Wi)v (Vn L no)(a, E

Knz

Then there is no

m)].

Suppose Wen’ZIa n d are both infinite. There is n 2 n0, e such and a,+l E We. Then there is also so such t h a t , for all t h a t a, E

111. Post's Problem and Strong Reducibilities

292

a n d hence, since n

2 e,

should be t h e smallest element of a n d with niaximal n-state, hence i t cannot

By construction we then have t h a t u;+' -

A , greater t h a n ukPl = a,-l

be a,, contradiction.

0

Exercise 111.4.19 There is an acceptable system of indices { W e } e E w of the r.e. sets such that WOis infinite, but the complement of the maximal set constructed above has n o element in Wo. (Hint: define = w,, and generate elements in Wo only after they have already been generated in A.)

T h e e-state method used in t h e proof above is a kind of priority method, with t h e usual order of priority

Po > N o >PI > N1 > . . . T h e basic difference between t h e constructions of maximal a n d simple (or hypersimple) sets is t h a t the positive requirements P, are infinitary, a n d we cannot hope to satisfy them with a finite action. Actually, since t h e y only allow for finitely many exceptions, each requirement has to be considered cofinitely many times, a n d at any given stage we have to consider many positive requirements all together. But t h e requirements have different priorities, a n d e-states are a device to assign priorities not to single requirements, but t o groups of them. E.g., t h e order of priorities assigned by 2-states is

(Po,P I , Pi?)L (Po,Pl) L (Po,pz) 2 (Po) L (P,,P2) 2 (Pl)2 (P2). We might say t h a t locally this is a finite injury argument, since every element of moves at most finitely many times, b u t globally it is a n infinite injury argument, since a positive requirement may be injured infinitely often, each t i m e for different elements of 2. Exercises 111.4.20 a) Maximal sets are not closed under intersection. (Yates [ 19621) (Hint: take A maximal, and let z E B 2 1 E A.)

+

111.4 Hyperhypersimple Sets and Q-Degrees

293

b) There are hyperhypersimple sets which are not maximal. (Hint: the hyperhypersimple sets are closed under intersection.)

Maximal sets have thinnest possible complement. T h e existence of maximal T-complete sets would show t h a t a notion of thin complement alone is not sufficient t o solve Post’s problem. I n fact, t h e maximal set j u s t built is T-complete, as we now prove, as usual, by showing t h a t it is maximal in a n effective way.

Definition 111.4.21 (Lachlan [1968]) A is effectively maximal i f it is a coin$nite r. e. set, and there is a recursive function g such that, for every e, the sequence of 0’s and 1’s consisting of the values of the characteristic function of We on the elements of 2 in increasing order has at most g ( e ) alternations. Note t h a t having finitely many alternations simply means t h a t one of WJlA a n d W ,n 2 is finite.

Proposition 111.4.22 (Lachlan [1968]) Every effectively maximal set is T-complete. Proof. Let g witness t h a t A is effectively maximal, a n d define f ST A such that

Wf(,) = a finite set with g(e) + I alternations on

T h i s is possible because

2.

is infinite, e.g.

W,(,) = (a2i : i 5 g ( e )

+ 1).

T h e n f has n o fixed-points a n d , by t h e criterion for T-completeness 111.1.5,A is T-complete. 0

Corollary 111.4.23 (Yates [1965]) There exists a T-complete m a x i m a l set. Proof. T h e maximal set built in 111.4.18 is effectively rnaxin~al,since We c a n 0 have at most g ( e ) = 2e+1 alternations on

x.

Exercises 111.4.24

a)

For n o winfinite r.e. set A there is a recursive function g

such that Saniilarly f o r

W, n A infinite

* IW,n;il

-

+ Iw, nZ1 5 g ( e ) .

W , n;i infinite

5 g(e).

Thin these natural candidates for effective maximality fail. (Lachlan [1968]) (Hint: for the second property, put in Wh(,) the first g ( e ) 1 elements of A, by starting with ( 0 , . . . , g ( e ) } and, each time that one element goes into A, adding the first element

+

111. Post’s Problem and Strong Reducibilities

294

not yet in W,,(e), and not yet generated in A. By the Fixed-Point Theorem there is e such that We = Wh(=). Thus n x is infinite, but IW, fl= g ( e ) 1.) b) There is a maximal, eflectively simple set. (Cohen and Jockusch [1975]) (Hint: modify the construction of a maximal set given above by inserting steps to make A simple, as in the second proof of 111.2.11. Note that each a: may move more times than it did for maximality alone, but still only finitely often.) c) A maximal set is not strongly effectively simple. (Cohen and Jockusch [ 19751) (Hint: by 111.4.15 a maximal set is dense, and by 111.3.9 is not strongly effectively simple. See 111.6.21 for a different proof.) d) A strongly effective simple set is not contained in maximal sets. Thus Post’s simple set is not contained in maximal sets. (Cohen and Jockusch [1975]) (Hint: a coinfinite r.e. superset of a strongly effectively simple set is still strongly effectively simple.)

+

111.5

A Solution to Post’s Problem

Post concluded his paper [1944] by saying: we are left completely on t he fence as to whether there exists a recursively enumerable set of positive integers of absolut sly lower degree of unsolvability t han t h e complete set Ic, or whether, indeed, all recursively enumerable sets of positive integers with recursively unsolvable decision problems are absolutely of t h e same degree of unsolvability. On t he other hand, if this question can be answered, th at answer would seem to be not far off, if not in time, then in t h e number of special results t o be gotten on t h e way. This section can be seen as t h e missing conclusion t o Post’s paper, a n d shows t h a t he was indeed right, regarding t he number of special results needed to solve his problem.

Semirecursive sets We know t h a t hyperhypersimple sets are not Q-complete. We are t hen looking for a notion th a t, together with T-completeness, would imply Q-completeness. By coupling it with hyperhypersimplicity we would then have a notion implying T-incompleteness, and Post’s problem would be solved. Since th e difference between Q-reducibility and T-reducibility is t h a t we query t h e oracle on single elements in t h e first case, and on finite sets in t h e second, we need a notion t h a t would reduce a question of inclusion of a finite set t o t h e question of membership of a single element.

111.5 A Solution to Post’s Problem

295

Definition 111.5.1 (Jockusch [196Sa]) A set A is semirecursive i f there is a recursive finction f of two variables such that 1. f(., y ) =

.

Of

f(.,Y)

=Y

2. a:€ A V Y E A * ~ ( x , Y ) E A. Clearly a recursive set is semirecursive, since we can simply decide whether one of x a n d y is in t h e set. Also, the complement of a semirecursive set is semirecursive: i f f witnesses t h e semirecursiveness of A , then t h e function t h a t always chooses, between x a n d y , t h e element not chosen by f , witnesses t h e semirecursiveness of 71. T h e next result is not unexpected, a n d is actually t h e reason why we introduced t h e notion of semirecursiveness.

Proposition 111.5.2 (Marchenkov [1976]) If an r.e. set A is semirecursive and T-complete, then it is also Q-complete. Proof. We only have to show how to reduce finite questions of t h e form D, C 71 to single questions g ( u ) E 2,for some recursive g. Let f be as in t h e definition 111.5.1. Given D, = { x ~ , . . . , x C n } , l e t

Then we obviously have D, in A , then so is g ( u ) . 0

2 * g ( u ) E 71, because

if one element of D, is

Exercises 111.5.3 a) A semirecursive simple set is hypersimple. (Jockusch [1968a]) effectively (Hint: if A is semirecursive so is 2, and thus a finite set intersecting produces a n element of 2.) b) A semimcursive set is not p-complete. (Jockusch [1968a]) (Hint: a sernirecursive pcornplete set would be m-complete, and thus every r.e. set would be semirecursive, contradicting the existence of simple, nonhypersirnple sets.) We a r e obviously interested i n knowing which sets a r e semirecursive. Since t h e definition of semirecursive sets might appear somewhat a d hoc, we first give a n interesting alternative characterization.

Proposition 111.5.4 (Appel, McLaughlin) A set is semirecursive i f and only i f it is a cut of a recursive linear ordering of w.

111. Post's Problem and Strong Reducibilities

296

Proof. Let A be semirecursive via f . We define, b y induction, a recursive linear ordering 4 such t h a t

x 4 y ~y E A Let xo 4 x1 4 extend it to n

. . . 4 x, be

*x E A.

t h e ordering of t h e numbers u p to n. We want t o

+ 1. Three cases a r e possible:

1. f ( n + l , x o ) = n + l If zo E A then n 1 E A , by t h e properties of f , a n d we c a n t h e n let n+l+zo.

+

+

2. the first case fails,and f ( n 1,xCn) = 2, Similarly, if n + 1 E A then x,, E A , a n d we let x, 4 n

+ 1.

3. the first two cases fail Then f ( n 1,xo) = zo a n d f ( n 1 , ~=~n+ ) 1. T h e n t h e r e is i such that f ( n 1,xi) = X i A f ( n l,xi+i) = 7~ 1,

+

+

+

+

+

and thus ~ i +E l

T h e n we can let xi 4 n

A = + n + l E A * X ~E A.

+14

~ + 1 .

Let now A be t h e lower cut of a recursive linear ordering 4 . If

f(x,y) = least element of {x,y } w.r.t.

4

then f ( x , y ) = x or f ( x , y ) = y by definition, a n d if o n e of 2 a n d y is in A t h e n so is f ( x ,y), because A is closed downward w.r.t. 4. T h u s f satisfies t h e conditions of 111.5.1, a n d A is semirecursive. Exercises 111.5.5

a) Every ti-degree contains a semirecursive set. (Jockusch For n. [1968a]) (Hint: let A be infinite and coinfinite, and define r = x n E A 2 any z, let r, = 2-". Then z 4 y T, < r, is recursive. Let B be the lower cut determined by T . A 5~ B because if D, = (A n (0,.. . , n - 1)) U {n} then, by induction, nEA D, E A w r, < r x E B.

xnED,

*

*

*

And B
We turn now t o t h e investigation of which sets, among t h e ones introduced so far for t h e solution of Post's problem, a r e semirecursive.

111.5 A Solution to Post's Problem

297

Proposition 111.5.6 (Jockusch [1968a])Every coregressive r.e. set is semirecursive. Proof. Let A be a n r.e. set, coregressive via cp. Given z a n d y, note t h a t either one of z a n d y is in A , or they a r e both in Ti, a n d thus one of t h e m follows t h e other in t h e given enumeration of 2,a n d it is then sent over it by cp (meaning t h a t some iteration of cp on t h e former produces t h e latter). To define f generate simultaneously A , {cp(n)(z)}nc, a n d {~p(")(y)),~,, a n d let f(XlY)

=

{

if z E A or y E {~p(")(z)),~, if y E A o r z E {~p(")(y)},~,.

z y

Suppose f ( z , y ) E 2:then e.g. f ( z , y ) = y a n d z E {~p(")(y))~~,, so y E z E 2, a n d A is semirecursive. 0

and

T h e result implies, by 11.6.16 a n d 111.3.13, t h a t many hypersimple sets are semirecursive. Actually, every r.e. T-degree contains a semirecursive hypersimple set (in particular there are hypersimple sets which are Q-complete, namely any T-complete semirecursive hypersimple set). But t h e result we were really looking for escapes us.

Proposition 111.5.7 (Martin) No hyperhypersimple set can be semirecursive. Proof. We know (111.4.7) t h a t t h e complement of a hyperhypersimple set cannot contain infinite retraceable sets. Since t h e complement of a set is s e m i r e cursive when t h e set is, i t is enough to show t h a t a n infinite semirecursive set contains a n infinite retraceable set. Let A be infinite a n d semirecursive. We may suppose A immune, otherwise it has infinite recursive, hence retraceable, subsets. Let 4 be a recursive linear ordering of which A is a lower cut: t h e n A is well-ordered by 4 with order t y p e w ,because for any z E A t h e set {y : y 4 z} is a recursive subset of A , hence finite by immunity. Let n 00

if z is t h e n-th element of A w.r.t. 4 if z # A .

Consider t h e elements corresponding to nondeficiency stages of f :

B is a subset of A because, given x E B , there is y E A greater t h a n it (since A is infinite): then y z, a n d z E A because A is closed downward

+

111. Post's Problem and Strong Reducibilities

298

w.r.t. 4 . And B is infinite, as usual for nondeficiency stages (note t h a t f is one-one on A ) . To show that B is retraceable, we cannot appeal directly to t h e fact t h a t so are nondeficiency sets, because f is not recursive, b u t a n argument similar t o t h a t of 11.6.16 can be reproduced directly. Given 5 E B , we want to get a n effective procedure to find t h e greatest element of B smaller t h a n 5. Since for y > z we have y >- z, we only have t o check numbers below z. In other words, for z < z,

Thus it is enough t o define g(z) as the biggest

2

< z such t h a t

if bo 4 z (where bo is the least element of B ) , and z otherwise.

0

Actually, the proof shows that no finitely strongly hypersimple set is semirecursive, because t h e retracing function defined above is total a n d many-one (see 111.4.9). Exercises 111.5.8 Hereditary sets. A set A is hereditary if there is a recursive function f such that z E A A y E A @ f ( z , y ) E A. a) A set A is hereditary af and only i f there is a recursive function g such that D, C A g(z) E A. (Lavrov [1968]) b) For any set A, the set A" = {z : D, A } is hereditary. (Degtev [1972]) c) Semirecursive + hereditary, but not conversely. (Degtev [1972]) (Hint: K is not semirecursive, see III.5.3.b, but is hereditary because it is recursively isomorphic to K", see 111.7.14.) Degtev 119721 has shown that some of the properties of semirecursive sets, e.g. 111.5.7, hold for hereditary sets as well.

*

Exercises 111.5.9 Verbose and terse sets. (Beigel, Gasarch, Gill, and Owings [1993]) We say that we can do ' n for m' on a set A if n questions of membership to A can be answered by a recursive procedure that asks at most rn question to the oracle A. A set A is verbose if, for every n, we can do 2" - 1 for n on it, and it is terse if we cannot d o n for n - 1. a) If we can do 2" for n on A, then A is recursive. Thus the definition of verbose sets is optimal. (Hint: we show the proof for n = 1. Suppose we can do 2 for 1 on A : given two elements z and y, we can compute A ( z ) and A(y) by a recursive procedure that asks only one question to the oracle A. Since the answer to the query can be only 0 or 1, there are two partial recursive functions 'po and (PI, respectively using the answer 0 and 1, one of which gives the correct values for A ( z ) and A(y).

111.5 A Solution to Post's Problem

299

z there is y such that 90 and ~ p 1compute the same value of A(z), then this must be the right value of A ( z ) , and to compute it it is enough to look for such a y. If there is z such that, for every y, cpo and 91 give different answers for A ( z ) , then fix z, and consider the right value A ( z ) . For every y, the right value of A(y) is computed by the one, between 'po and pi, that computes the right value of A(z).) Note that A plays no role as an oracle, and thus A would be recursive even if we could answer 2" membership questions on A by a recursive procedure asking n questions to an oracle B. b) K is verbose. (Hint: we show how to do 3 for 2. To know which of z1,x2,23 are in K, first ask if at least two are. This is an r.e. question, hence K can answer it. If the answer is yes, ask if all three are, otherwise ask if at least one is. The whole procedure requires only two questions, and determines how many of the three elements are in K. To know exactly which ones are, generate K until that many appear.) Since verbose sets are closed are under m-equivalence, every m-complete set is verbose. c) A semirecursive set is verbose. (Hint: let A be the upper cut of a recursive linear ordering, by 111.5.4. To determine which of 2" - 1 elements are in A, first order them according to the linear ordering. To know which are is A and which are not, it is enough to find the first element which is in A, and this can be done by a binary search, thus requiring only n steps and n questions to the oracle A.) d) (R.e.) verbose sets exist in every (r.e.) T-degree. (Hint: by 111.5.5.a and 111.5.6, because verbose sets are closed under complements.)

To compute A recursively, consider two cases. If for every

e) Every n o w e m T-degme contains terse sets. (Hint: given a semirecursive, nonrecursive set A, notice that, given elements 2 1 , . . . ,z, if we know how many are in A, then we can determine exactly which ones: order them according to the ordering w.r.t. which A is an upper cut, and count from the biggest one. Given 2" - 1 elements, less than 2" can be in A, and thus their number can be written, in binary, with at most n digits. Define a set B, by putting ( 2 1 , . . .,X z n - 1 , i) in it if and only if the i-th digit in the binary representation of ( A n {zl,. . .,~ 2 -n1}1 is 1. Clearly A =T B. And 2" - 1 questions on A can be reduced to n questions on B. If we could do n for n - 1 on B then we could do 2" - 1 for n - 1 on A, with oracle B, and hence 2" for n with oracle B @ A, contradicting part a).)

q-hyper hypersimple sets Despite t h e fact t h a t hyperhypersimple sets a r e not semirecursive, we a r e not willing to give u p and waste all t h e work done so far, especially so after having been so close to a solution of Post's problem. T h e idea is to look for a weakening of t h e notion of hyperhypersimplicity t h a t is still incompatible with Q-completeness, b u t is compatible with semirecursiveness. We simply generalize t h e notion of number, a n d substitute it with t h e notion of equivalence class w.r.t,. a n r.e. equivalence relation.

300

111. Post's Problem and Strong Reducibilities

Definition 111.5.10 (Malcev [1965], Ershov [1971]) A n equivalence relation 17 is called positive i f R(x,y) xqy is r.e. Exercise 111.5.11 The class of positive equivalence relations is a sublattice of the complete lattice of all the equivalence relations on w under inclusion. (Ershov 119711) (Hint: the smallest element is the equality relation, the greatest one the trivial equivalence relation. The g.1.b. is the set-theoretical intersection, the 1.u.b. the smallest equivalence relation including the given ones.)

Definition 111.5.12 A set A is q-closed i f it consists of equivalence classes w.r.t.7, i . e . x E A A x r M / * y E A . The q-closure [A],, of a set A is the smallest 7-closed set containing it. Exercises 111.5.13 (Ershov [1971]) a) If A is r.e. then

Z V A ~ * ( Z E A A Y E A v) a : = y is a positive equivalence relation, with (if A has more than one element) A CIS the only nontrivial equivalence class. A set is qA-closed i f and only if it either contains A, or is disjoint from it. b) If 'p is a partial recursive function, then V V Y

* (cp(z) = P(Y)1)v 5 = Y

is positive. A set A is qV-closed if and only if cp-'(cp(A))C A. c) If cp is a partial recursive function, then

is positive. (Hint: recall that, by definition, cp(')(z) N z.) d) q is a positive equivalence relation if and only af, f o r some partial recursive function cp, 77 = 7;. (Hint: let 77 be approximated monotonically by qa. Approximate cp as follows: given psrextend it by defining, when possible, cpa+l(z)N py(zq8+1y).)

Definition 111.5.14 A n 7-closed set A is q-finite or q-infinite, according t o whether it consists of a finite or an infinite number of equivalence classes of 17. By restricting our attention to 7-closed sets, and replacing the notion of finiteness by 7-finiteness, we can relativize to 7 the notions introduced so far.

Definition 111.5.15 (Ershov [1971]) Let 7 be a positive equivalence relation, and A be a n 7-closed and co-7-infinite r.e. set. Then A is: I . q-simple if any q-closed r.e. set contain& in 71 is 7-finite 2. q-hypersimple i f there is no recursive function f such that

111.5 A Solution to Post’s Problem

fY

301

[Dj(Z)IOn P j ( v ) l o = 0 Pf(z)Ia n # 0

5

z

3. q-hyperhypersimple i f there is no recursive function f such that 0

Wj(,) finite

f Y * [Wj(Z,l, n [Wj(,)l, = 0 [Wf(z)I, nx # 0

z

4.

q-maximal q-finite.

iL for every q-closed r.e. set B 2 A, one of B - A and B ZS

T h e implications among the various concepts still hold, but nontriviality conditions a r e not automatically satisfied.

Proposition 111.5.16 A n q-maximal set can be empty. Proof. Let A be any maximal set, and X?]A~

(5 E

A A y E A)

V x

= y.

If B is r.e. a n d ~ A - C I O then S ~ ~either , it contains A or it is contained in 2.In t h e second case it is finite by maximality, hence qA-finite. In t h e first, either B - A is finite, hence B is qA-finite because A consists of just one class, or B is finite, and hence qA-finite. But then t h e empty set is qA-maximal. 0 Exercise 111.5.17 Every nonrezursive q-maximal set A whose equivalence classes on 2 are all finite is simple, although not necessarily hypersimple. (Hint: if B is r.e. and B 2,consider the closure of A U B. It cannot be co-77-finite, otherwise A is recursive. So B is contained in finitely many classes, and thus finite.)

I t is convenient t o picture a positive equivalence relation as a series of boxes, representing t h e equivalence classes. Since the equivalence relations we deal with a r e positive, there is a n effective procedure t h a t builds u p t h e boxes. At some given stage, boxes t h a t were previously separated may be p u t together t o form a bigger box. We may also suppose that, at each stage, all boxes a r e finite. We a r e now ready for t h e final steps of our long journey. Proposition 111.5.18 (Marchenkov [1976]) An 71-hyperhypersimple set is not Q- complete. Proof. T h e proof is like t h e one of 111.4.10, after the following modifications a r e made, for A 71-closed:

111. Post’s Problem and Strong Reducibilities

302 0

0

if z E Ic H Wg(,) 2 A , then Wg(z!is v-closed. If it is not so, tlake its ipclmure: it still works, because A is 7i-closecl itself. A consists of equivalence classes with finitely many elements each. If it, is not so, define a new positive equivalence relation 77’ coinciding with 11 on A , hence with no effect o n ?1-hyperhypersimplicity,as follows. A t stage s 1, a given box of 71 is also a box of 71’ if either it does not intersect A , , or s is t h e first stage in which it does intersect A .

+

Theorem 111.5.19 (Degtev [1973]) There exists an r.e. set A which is n o w rt:ciirsiiie, semirecursive, arid 71-ruaxiina1, for s o i i ~ epositive quivalerice relntIo7i 11.

Proof. We define A , which may thought of as one big box, a n d infinitely iiiany boxes, which a r e going to be t h e ec~iiivalenceclasses of Thus A will aiitoniatically be 71-closed a n d 71-iiifinite. Let {B:}ncw be a n enumeratioii o f tIie boxes of A, at stage s, suc~it h a t if 772 < I L tlieii inax RR < min B:~. We stfart with A” = 0, arid BE = { n } . In t h e construction we alternately take care of t h e positive requirements (for nonraursiveriess a n d semirecursive ness) and of t h e negative ones (for 71-inaxiniality). Thus at stage s + I we t a k e

x.

different actions, according to whether s is even o r odd. 1. s w e n ‘1’0 make A nonrecursive, we make it siniple. T h u s we search for t h e smallest P 5 s such t h a t : 0 0

We,sn A, = 0 for some 11, 2 e ,&;?/

n We,,# 0.

If there is such an e , we p u t all of BA into A , where n is t h e smallest one t h a t n 2 e and , ! I : n W,,, # 0. If there is no S I I C : ~c , we go to t h e

sidi

next stage. ‘l’o ensure sernirecursiveriess we also p u t in A , together with Eli, all t,he following boxes u p to t h e one containing s. T h i s ensures, because o f t h e way t h e boxes a r e grouped together during t h e construction, that, whenever a n clcrnerit 2 goes into A at stage s I , so do all clcnients y such t h a t z 5 y 5 s. ‘1’0 see t h a t this has t h e desired effect, let

+

f h Y )=

(“ Y

niiuc{z,y}

if z E Amax(z,v} i f z @Amaxjz,y} A?/ E otherwise.

Amax{z,y)

We then have z E A V y E A 3 f ( z , y ) E A . T h i s is clear i f f is defiiied i i i any of t h e first two cases. I n t h e last, if one o f n: and ?j goes into A , it

111.5 A Solution to Post's Problem

303

must be at a stage bigger than their maximum (otherwise one of the two other cases would apply). T h e only case of concern is when t h e smallest of 2,y goes in (since f(z,y) is then the other one). But in this case t h e construction also puts the other in A , and thus f ( ~ , y E ) A.

2. s odd We consider the usual e-states

and proceed by induction, by letting BZ+l be the union of all boxes between BZ and Bl, where 2 is the smallest element greater than e - 1 and with maximal e-state.

B:?: 0

BZ+' 1

...

Note t h a t , unlike the case of maximal sets, the construction of q-maximal sets does not force us t o put elements into A , as we already know from 111.5.16,and thus t h e two parts of the construction do not interfere. T h e proof t h a t the construction works is similar to the ones for simple and consists of maximal sets. T h e only slightly different part is the fact t h a t infinitely many classes. But t o show t h a t lims+m B,S exists, it is enough t o note t h a t B,S can change only for two reasons: 0

0

some B: with n < e goes into A . Note that this must happen because, for some index i < n, the box B: or one with smaller index is satisfying t h e simplicity requirement for i. But this can happen only once for each i < e, hence at most finitely often.

B,S reaches a higher e-state, and once again this can happen only finitely often.

We have thus finally reached the end of our journey. We put things t e gether, for completeness and the reader's sake.

111. Post’s Problem and Strong Reducibilities

304

Or quel che t’era dietro t’k davanti: m a perch6 sappi che d i te mi giova, u n corollario voglio che t’ammanti.’ (Dante, Paradzso, VIII)

Corollary 111.5.20 Solution to Post’s problem (Muchnik [1956], Friedberg [1957a]) There emsts an r.e. set whzch zs nezther recurszve nor

7‘-cornplete. Proof. Consider a nonrecursive, semirecursive, 7phyperhypersimple r.e. set: it 0 is not Q-complete by I l l 5.18, a n d hence not T-complete by 111 5 2 T h i s shows in particular t h a t there are more than two r.e. T-degrees, a n d irnmediately raises t h e question of how complicated t h e s t r u c t u r e of r.e. T-degrees is. A complete description of it as a partial ordering is not known, b u t Chapter X will provide a good deal of information a b o u t i t , as well as different methods to solve Post’s problem.

111.6

Creative Sets and Completeness

For a n r.e. set A , to be nonrecursive means t h a t V z ( 2 # W,) may be formalized in different ways: 1. 3y(y E A

ey

E

Ti is not

r.e. T h e fact t h a t

Wz)

2. W z ~ ; T ~ 3 y ( y E X - W , ) 3.

Zc W, + 3y(y E A n w,).

Wt: cannot expect in general to find y recursively in x,a n d we now investigate what happens if we impose this requirement.

Effectively nonrecursive sets We begin by constructivizing t h c first notion, which also appears to be t h e must natural a n d symmetric.

Definition 111.6.1 (Dekker [1955a]) A set A is completely productive if there is a recursive function f such that, for every x,

f(.) E

zef(x) E w,.

‘Now that which stood behind you, stands in front: hut w) that you may know the joy you give m e , I uow would cloak you with a corollary.

111.6 Creative Sets and Completeness

305

A set is effectively nonrecursive i f it is r.e., and its complement is completely productive. K is effectively nonrecursive because, by definition, x E K e x E W,.

Proposition 111.6.2 (Myhill [1955]) A set is effectively nonrecursive i f and only i f it is m-complete. Proof. Let f be a recursive function such t h a t f(s)E A e f(z)E W,. If

w ifxEK 0 otherwise

a n d A is effectively nonrecursive.

0

T h i s shows t h a t any set built by effective diagonalization over t h e r.e. sets must be 7n-complete, a n d t h u s Post's problem cannot be solved b y effectively satisfying the requirements for nonrecursiveness . Exercises 111.6.3 (Rogers [1967]) a) B is effectively nonrecursive in A if, for some recursive function f, f(z) E B ts f(x) E W,". Post's problem cannot be solved b y egectively satisfying the requirements for T-incompleteness, in the sense of building a n r.e. set A such that K is effectively nonrecursive in it. (Hint: if A and B are r.e. and B is effectively nonrecursive in A then A is recursive, because if

{8

w

wgqx) =

ifxEA otherwise

*

f(g(x)) E B.) then z E 2 b) B is effectively not m-reducible to A if, for some recursive function f, pe total

*

[ f ( e )E B

* cp,(f(e)) E 31,

i.e. f ( e ) witnesses the fact that pe is not a n rn-reduction of E to A. Post's problem for m-reducibility cannot be solved b y effectively satisfyzng the requirements for m-incompleteness, in the sense of building an r.e. set A such that K is effectively not

111. Post’s Problem and Strong Reducibilities

306

m-reducible to it. (Hint: as above, by considering as ‘ps(l) the constant function with value x.)

Effective nonrecursiveness gives a nice criterion for m-completeness. With little effort it is possible t o generalize it, and get similar criteria for all the reducitilities so far introdiiced. Exercises 111.6.4 (Friedberg and Rogers [ 19591) Let A

Sr H

H

STbe

for some recursive function f, r E

defined as:

2 H Qr (f(r),R , E )

where Qp is a predicate with the following properties:

Qr(z,W,,Wy) is r.e. as a p r e d i c a t e of x , y , z Q,.(z, X , Y ) is monotone in Y . Note t h a t nionotonicity of Qp for r.e. arguments follows from the first property, a n d t h a t if Qr is also monotone in X then A ST B =+ A 57’ €3. a) All the wdcrcibilities intmduced so far can be so expressed. (Hint: Q l l ( z , X, Y ) holds if is satisfied when priitive a n d negative atorrlic formulas are respectively interpreted over Y and X. &,(z, X,Y ) holds when every finite set with canonical index in 0, intersects Y . For r.c. scts, QT(z,X, Y )holds when 3u(D, Y A u E Wz).) b) An r.e. set A is r-wnipletn i f and only if, for some recursive function g , Qr(g(s)I

A,x)

-Qv(g(x)I A, Wz).

Explicitly, thc interesting criteria are (for some rccursive function 9): &completeness: c-completeness: Q-completeness:

Dg(,)

c2 *

+0

D~(,n ) z Wy(,,n A #

Dg(z)

0 ++

!z w, W -,

w

Wg(,) W , .

Creative sets We now coiihtriictivize t,he second notion of nonreciirsiveness giveii at the beginning.

Definition 111.6.5 (Post [1944], Dekker [1953]) A s e t A z f there 1s a reciirsaue function f such that, f o r e v r r y x,

w, c n

=+

f(.)

E n

A set is creative if i t is r.e. and coprodzictive.

-w,.

2s

productive

307

111.6 Creative Sets and Completeness

As we have already noted, K is creative: if W , G K: t h e n z cannot be in W , (otherwise z E K by definition, and z E K: from W , 5 contradiction). T h e n z @ W,, a n d t h u s also z E E.

c,

T h e t e r m ‘creative’ was introduced by Post [1944] because K (as well as in 11.2.7) embodied t h e essence of t h e incompleteness theorems of t h e Thirties (11.2.17), arid t h e property of productiveness captured t h e main consequence of these results, namely t h a t

K O ,defined

every symbolic logic is incomplete a n d extendible relative to t h e class of propositions constituting KO. T h e conclusion is inescapable t h a t even for such a fixed, well defined body of mathematical propositions, mathematical thinking is, and m u s t remain, essentially creative. T h a t Post got t h e right notion is proved by t h e next result: if F is any consistent extension of R,then every r.e. set is weakly representable in it (see II.2.16),a n d hence m-reducible to t h e set of (codes of) its theorems, which is t h e n m-complet,e.

Theorem 111.6.6 (Myhill [1955]) A set i s creative if and only if it is m-comp lete. Proof. If A is m-complete it is effectively nonrecursive, a n d hence creative. g ( z ) E W,. T h e n Directly, if z E K @ g(z) E A , let z E Wh(,)

*

w, c 2

c c =+

=+ Wh(z)

h ( z )E K: - Wh(l)

* g ( h ( z ) )E 2 - W z ,

a n d f(z)= g ( h ( z ) )is a productive function for 2. Suppose now t h a t A is creative, a n d W z C f(x) E 2 - W,. We want to find a recursive function h such t h a t z E K H h ( z ) E A . Since we want t o use f,we define g such t h a t z E K H f(g(z)) E A .

*

0

0

w,(.) c

If z E E , f gives naturally a n element of 2,if we start from 2. T h e simplest way to ensure this is to let W,(,) = 8 when z E E: t h e n we have f ( g ( z ) ) E 2.

c.

We t r y now t h e converse, i.e. to have f ( g ( z ) ) E 2 =+ z E Iff(&)) E 2 then { f ( g ( z ) ) } 2,a n d by productivity it cannot, be W,(,) = { f ( g ( z ) ) } , otherwise f(g(z)) E 2 - W,,,). We thus define W,(,, = { f ( g ( z ) ) } when z E K : t h e n f ( g ( z ) )E A .

1,et t h e n g be a recursive function such t h a t otherwise.

111. Post's Problem and Strong Reducibilities

308

T h e existence of g is ensiiretl by t h e Fixed-Point Tlicoreiii, e . g . let

W%,(Z) = and choose e such t h a t cpc

{ 8( f ( c p e ( 4 ) )

E (P,(~).

if 2 E K: ot,herwise,

0

Exercises 111.6.7 a) An r.e. set A is crcntiuc i f and only if, f o r som e recursive one-one function f, f(x) E A tj f ( z ) E W,. (Rogers [ 1!)67], Gill a n d hlorris [ 19741) (llint: if A is creative then 1is coriipletely productive and, by 111.7.6, i t h a s a one-one cwrnplrtely productive function f : then A can he represented as stated. Conversely, if 1 is onc'-one then there is a s c t A satisfying the given coritlitions, since mernlwrship o f f ( z ) in A is determined snlely hy W,. Note that, if we h a d f ( z ) = f ( y ) with z # y, then W, and W, could give contradictory answers for rnernbership of f ( z ) in A . ) b) A n r.e. set A is cwcitiiie i f and only i f , for some acceptable system of indices { % z } I E w of /,he r.e. sets, A := K. (Rngers 119581, Lynch [ 19741) (Hint: use 111.7.14 t o gvt a rrcursive isomorphism f between A a n d K , a n d let s E Pbe f(s)E Wfc.,.) c) A circitive set is neither wciirsive nor simple. (Ilint: see 111.2.7.) d) Every iiijnite r.e. sct is the disjoint union of a crmtii~eand a productive set. ( I k k k r r [ 1!)55a]) (flint: if A is the range of a one-one recursive function f, consider t.hr irnagcs o f K ant1 K.) e) 1Si1ery infinite r.e. s e t is the union of a crcdivc and a n infinite recursive set. (Myhill 1195!)]) (Hint: let A 1)e r.e. and infinite, and f l x a recursive one-one function whosc range is an infinite recursive subset B of A . /(K)is creative, and then s o is f(K) U ( A - B ) , and A = f ( K ) U ( A - H ) U B.)

T h c n e x t result, is an analogue of 111.1.5, and provides a criterion for o i i fixed-points. iVe use tlic following generalization ~ ~ ~ - c o i i i p l c t e n based ess of 7ri-rcducibility to functions: f A if a n d only if t h e r e a r e rcxwrsivc: fun(tions f l , f2, a n d g such t h a t

<,,

f (:I;) =

Cleirl~y,H

&1

{ fib)

if g(z) E A f2(z) otherwise.

A if ant1 only if cf+ I,,A ,.

Proposition 111.6.8 (Arslanov) A n r.e. set A i s 7ri-coiriplete i f a i d only ,if them is a f u n c t i o n f A without fmtxi-points.

<,,

Proof. If A is m-complct,c,tlic set { x : 0 E W,} is . ~ r z - r ~ l u c i bto le A , being r.c. Let, g be a recursive fuiict,ioii such t h a t 0 E W, e g(z) E A ,

a n d tlcfine f SnLA as:

n b

if g(z) E A otherwise,

309

111.6 Creative Sets and Completeness

where a a n d b a r e indices of, respectively, 0 and w.T h e n Wj(,) # W,, because t h e two sets differ on 0. Suppose now t h a t A is r.e., a n d f A has no fixed-points. We show t h a t A is m-complete by proving t h a t it is creative. Given W,, let

s,,

cp(z)

=

{

fl

(x)

f2(4

undefined

if g ( z ) shows u p in A before t h a n in W , if g(z) shows u p in W , before t h a n in A otherwise.

Note t h a t if g ( z ) E A U W , then cp(z) 1, a n d if W , C

x

then cp(z) = f ( z ) , if convergent. Consider a recursive function h such t h a t W,,(.) = W,(,), i.e. which is t h e only case of insuch t h a t Wq2) = 0 if cp(z) t: if W , terest, then Wh(,) = Wj(,) if cp(x) converges. Let e be a fixed-point for h: Wh(e) = W e . Since f has no fixed-points p(e) must diverge, a n d t h e n it must be g ( e ) @ A U W,. Since t h e procedure is uniform in x (by using t h e FixedPoint Theorem with parameters 11.2.11.a), we have a recursive function t h a t produces a n element of 2 - W , whenever W , 2,and A is creative. 0

x,

Exercises 111.6.9 Weakenings of creativeness. a) A n r.e. set A is creative if and only i f there is a partial recursive junction cp such that W, empty or singleton A W,

c A * cp(z) 1 A p(z>E 2 - w,.

(Dekker [1955a], Myhill “551) (Hint: see the proof of 111.6.6.) b) For any r.e. nonrecursive set A there is a partial recursive finction cp such that

+

cp(z)lAcp(e)Ez-&.

(Mitchell [1966]) This cannot be interpreted as saying that every nonrecursive r e . set is effectively nonrecursive, because the system of indices {&},Ew for the recursive = Wb let cp(z) be the first element sets is not effective itself. (Hint: if & = then A n Wb # 0.) generated in A n Wb. Since A is nonrecursive, if & = W,, c ) A n r.e. set A is T-complete i f and only if 2 has a productive function recursive in A. (Lachlan [1968]) (Hint: if A is 7’-complete, is recursive in A to ask if W, & 2, i.e. if 3 z ( z E W, n A ) . If W, W, is properly included in 2, since A is not recursive, and a n element in 2 - W, can be found, recursively in A. Conversely, if f 5~ A is a productive function for 2,the function g 5~ A such that

w,,

c x,

has no fixed-points, and A is 7’-complete by 111.1.5.)

Exercises 111.6.10 Productive sets. a) A set A is productive i f and only i f there is a partial mursive function cp such that W,GA

*

p(z)l~p(z)~A-W,.

111. Post’s Problem and Strong Reducibilities

310

(Dekker and Myhill 119581) (Hint: given ‘p and

then one of p(z) and ‘p(h(z))converges.) A. Thus the T-degrees containing b) A set A is productive if and O l d y ifx productive sets are exactly those above 0’. (Dekker arid Myhill [ 19581) (Hint: see the proof of 111.6.6.) c) A set is productive if arid only i f it is completely productive. (Myhill [ 19551) d) There are 2’O productive and co-producticie sets, and the T-degrees containing such sets are exactly those above (Y. (Karp) (Hint: if A is productive, A @ 3 is productive and ceproductive.) is properly contaiiied in Thus even transfinite iterae) The set {z : W, tions of the process starting from 0,and generating new elements of (as in III.2.7), does not exhaust (Dekker 11955al) (Hint: the Fixed-Point Theorem prodiices z such that W, = then z 6 but W, f ) There is a set A pmducti~iew.r.t. j, such that A = {j(z): W, 2 A } . (Dekker 11955a1) (Hint: let A be the intersection of all sets for which f is a productive function.)

s,,,

c.

z}

z.

m:

z,

z

z.)

We still have to take care of the last riotion considered at tlie beginning of this sect,ion.

Definition 111.6.11 (Dekker [1955a]) A set A is contraproductive i f there i s a recursive fiinction f surh that, f o r every Z,

A c W , + f ( ~ E) x n W , . Proposition 111.6.12 (Muchnik [1958a], McLaughlin [1962]) An r.e. set A is crmtive i f and only if is contraproductive. Proof. A creative set is effectively nonrwiirsive, arid tlicri 2 is contraproductive. For the converse, suppose 2 C W , 3 f ( z )E A n W,, arid let

Thus A is m-c.oniplete, and hence creative.

111.6 Creative Sets and Completeness

31 1

Exercises 111.6.13 a) A set is contraproductive if and only if it is productive. b) If A is an r.e. nonrecursiiie set, there is cp partial recursive such that -

A E W , =+ c p ( z ) l A c p ( z ) E A n W , .

(Dekker [1955a])

For more information o n productiveness a n d creativeness see Dekker [1955a], Friedberg a n d Rogers [1959], McLaughlin [1964], Larhlan [1965], Mitchell [19G6], Soloviev [1976], a n d Onianadze [1978].

Quasicreative sets

*

Exercise 111.6.4 considers reducibilities sT generated by relations Q r , a n d generalizes t h e criterion of m-completeness given by effective nonrecursiveness, by coilsidering r.e. sets A such t h a t , for some recursive function f ,

Q r ( f ( ~ ) ~ A ~ x l) Q r ( f ( Z ) , A , w z ) W e might t h e n t r y to extend t h e criterion of m-completeness given by creativeness, by considering r e . sets A such t h a t , for some recursive function f ,

wzC_ 7i

=+

QT(f

(x),A , x i ) A -QT(f (z), A , wz).

Exercise 111.6.14 The criterion fads for c-reducibility. (Omanadze [ 1978a1) (Hint: let A be a strongly effectively simple, not hypersimple, tt-incomplete set. It exists because there is a n acceptable system of indices for which Post’s simple set is tt-incomplete, see 111.9.2, and all the acceptable systems are isomorphic, see 11.5.7. Such a set is thus c-incomplete. Let h be a strong array intersecting 2, and g be such + IW,I < g(z). To get f such that that W,

c

Wz G 2 * D,(.) $7 A A Df(.)C: W,,

T h u s this criterion does not always work, a n d we do not know of a n y othcr formulation as general as 111.6.4 (see Lachlan [1965], Soloviev [1976], Onianadze [1976] for weaker results). T h e r e a r e however two special interesting cases, which we treat in this a n d t h e following subsection, in which t h e proposed formulation succeeds. W e begin with
Definition 111.6.15 (Shoenfield [1957]) A is quasicreative i f it is and, f o r some recursive function f ,

T.e.

111. Post’s Problem and Strong Reducibilities

312

a creative set is quasicreative: if h is a productive function for A , and Df(,) = {h(z)},then

- Obviously,

w,C 3 =+ h ( z ) E 2 - w,* Df(.)

2 7 [ A Df(,)

w,.

Proposition 111.6.16 (Shoenfield [1957]) A set is quasicreative i f and onZy i f it is d-complete. Proof. T h a t a d-complete set is quasicreative follows from t h e stronger criterion for d-completeness given in 111.6.4. Or directly, as in the case of m-completeness, if z E K w Dg(,) n A # 8 let z E Wh(,) H D g ( z )g W,. Then f(z)= g ( h ( z ) )witnesses t h e quasicreativeness of A, since

w,g 7i

=+ Wh(,) g K:=+ h(z) E K:- Wh(,) =+ D g ( h ( s ) ) A Dg(h(s))

$z w,.

c

Conversely, as in 111.6.6, if f witnesses t h e quasicreativeness of A and g is a recursive function (which exists by the Fixed-Point Theorem) such t h a t

then z E

K

H

D f ( g ( z )n) A # 0, and A is d-complete.

0

Proposition 111.6.17 (Shoenfield [1957]) There exists a quasicreative, not creative set. Proof. T h e idea is t o build a quasicreative set A with a simple superset B , such t h a t B - A is r.e. Then A is not creative, otherwise t h e usual procedure (see t h e proof of 111.2.7) would give a n infinite r.e. subset of B,starting with B - A. Fix a strong array {F,},E, intersecting Post’s simple set (see 111.3.5). We proceed as in the construction of a simple tt-complete, with the role of t h e set K = {X : II: E W,} taken by {X : F, 2 W,}. If B=SU

U Fz CWz

F,

111.6 Creative Sets and Completeness

313

then B is simple, if coinfinite. Moreover

F , C B e F,GW,. To have A quasicreative, we can ensure t h e stronger condition

equivalent to

F,nA#0

H

F,GW,

* F,GB.

To build A generate B and, when F, G B is found, put into A t h e element of F, which has been generated last in B. This makes A C B , and both A and B - A r.e. - Since A is quasicreative is not r.e., and so B must be infinite (otherwise A = ( B - A ) UB would be r.e.). 0

A different proof, relying on completeness (i.e. building a d-complete set which is not Q-complete, and thus not m-complete) will be given on p. 342. Exercises 111.6.18 a) A quasicreative set is not simple. (Shoenfield [1957]) (Hint: H D f ( z ) C 2,for some recursive function f. Given distinct elements 21,...,z, of 2,let z E C 6 Df(%) 5 (21,.. . ,zn} . If C = W, then D f ( a )5 2 but {XI,. . . , zn}. This builds an infinite r.e. subset of 3.) Df(.) b) A n r.e. set A is c-complete if and only i f there is a recursive finction f such that A C Wz Df(,) E A A Dj(.) $Z W,.

let z E

*

This provides an analogue of contraproductiveness. (Lachlan [1965]) (Hint: see 111.6.12.) c) An r.e. set A is semicreative if, for some recursive function f,

(Dekker [ 1955al). This provides an analogue of quasicreativeness. Every nonrecursive r.e. 2'-degree contains a semicreative set. (Yates [1965]) (Hint: use permitting, see p. 277. The most natural example of semicreative set is K, but it is not useful for permitting, because each W, contributes at most one element. As a variation consider (z, e ) E A H (z, e) E We,and W j ( , )= {(z, e) : z E w } . A is semicreative because, if We E 2,then ( 2 ,e ) E -A - We,and hence W,(,) C_ 2 and Wf(,) We. Given an r.e. nonrecursive set C, modify the construction of A by adding permitting, in such a way to obtain A @ C semicreative.) d) A semicreative set is not sample.

111. Post's Problem and Strong Reducibilities

314

Subcreative sets We examine now the completeness criterion proposed in t h e last subsection, in t h e case of Q-reducibility. Recall that
QQ(z,X, Y )@ ~

U ( U E

Y A 'u. E W z ) .

T h e criterion thus takes the following form:

Definition 111.6.19 (Blum and Marques [1973])A is subcreative i f it is r.e. and, for some recursive function f ,

WzC2

* Wf(,)$Z A A W f ( z C) W,.

Equivalently, by taking unions with A , we might require t h e existence of a recursive function g such that W , C + A c Wg(,) Obviously, a creative set is subcreative: if h is t h e productive function for A , and Wf(,)= {h(z)}, then

w,.

W, C Z

h(z)E

A - W,

* Wf(,)$Z A A W,(,) C IT',.

Proposition 111.6.20 (Blum and Marques [1973],Gill and Morris [1974])A set is subcreative i f and only if it is Q-complete. Proof. T h a t a Q-complete set is subcreative follows from t h e stronger criterion for Q-completeness given in 111.6.4. Or directly, as in t h e case of m-completeness, let z E K: H Wg(,)2 A , and Z

E Wh(z)

*wg(z)

Wz # 0.

Then f(z)= g(h(z))witnesses t h e quasicreativeness of A , since

Wz 2 2

Wh(z) * h(z)E R - W h ( z ) * W g ( h ( z ) )$Z A A W g ( h ( z ) )C wz-

Suppose now that A is subcreative:

W , C A + W j ( z )$Z A A W f ( z C ) Wz. We want to find a recursive function h such t h a t z E K: @ W q z )C A. Since we want t o use f , we define g such that z E K: W f ( g ( zC_) )A .

111.6 Creative Sets and Completeness

315

If z E E , f gives naturally a set not contained in A , if we only s t a r t from W g ( z C_ ) 2. T h e simplest way t o ensure this is to let Wg(,)= 8 when z E hc: t h e n W f ( g ( Eg) A ). 0

W e t r y now t h e converse, i.e. we want W,(g(z)) A + z E hc. If W f ( g ( zg) )A , there is a n element a E W j ( g ( zn )) By subcreativity C_ A a n d W f ( g ( rC )) it cannot be Wg(,)= { a } , otherwise Wg(,) while a E W f ( g ( z )We ) . t h u s define, when z E K , Wg(,)as a set t h a t chooses a n element of W f ( g ( z ) ) if there is one, a n d is e m p t y otherz E E. wise. T h e n W f ( g ( zn) )2 # 8

z.

*

mg(,),

nx

W e now have to define g . T h e fact t h a t g is self-referential is taken c a r e of by t h e Fixed-Point Theorem, b u t t h e naive approach leads to another problem. If we t r y t h e natural procedure to pick u p a n element from W f ( g ( 2fix, ) ) we siniultaneously generate W f ( g ( za)n)d A . At each stage of t h e enumeration, we p u t t h e elements already generated in W f ( g ( rinto ) ) W g ( z )unless , some element has already been generated in Wg(,,b u t not yet in A . T h i s certainly p u t s a n element of 3 in Wg(,)if there is one in W f ( g ( z )b )u ,t m a y also p u t other elements of A (ones which a r e in A , b u t a r e generated in i t only after having been generated in W f ( g ( z ) T ) )h.e effect is t h a t Wg(,)is not contained in A, a n d t h e discussion above fails. To override t h i s we would like to make sure t h a t , if W f ( g ( z ) ) # 8, one element of t h e intersection is generated in W f ( g ( zas ) )t h e first element. Think of t h e process of generating a n r.e. set, a n d to pass from it to a new r.e. set which consists of t h e elements of t h e given set generated until t h e first stage in which it is found t h a t t h e first element of t h e enumeration is in A . I f t h e given set is infinite, it can be a fixed-point of this process only if t h e first element enumerated in it is in 2. Note t h a t W j ( z )may always be supposed infinite (otherwise consider A U W f ( z ) ) . To define Wg(,),first wait until it is discovered t h a t z E K. T h e n let W f ( g ( z ) ) be a fixed-point of t h e process described above, a n d let Wg(,)consist of t h e first element generated in it (note t h a t t h e definition of g actually requires a double use of t h e Fixed-Point Theorem). Then:

nx

0

E

K: * ITg(,)= 0 C_ 2 * W f ( g ( z ) )A .

eA

Wf(g(2))

z E

K.

Otherwise Wg(,)= { a } , for some a E W f ( g ( zn)2 ) (actually, for t h e first clement generated in W f ( g ( z ) T) )h .e n

c2

Wg(2)

contradiction.

0

c

3 Wf(g(2))

Wg(z),

111. Post’s Problem and Strong Reducibilities

316

Exercises 111.6.21 Strongly effectively simple sets. a) Ewery strongly e$ectively simple set is Q-complete. (Gill and Morris 119741) (Hint: if g is a recursive function such that We& 2 + (rnax W e )< g(e), let W,(+=) consist of all the elements greater than g ( e ). Then f witnesses the subcreativeness of A, because 2 is infinite.) b) A strongly efectively simple set is neither hyperhypersimple, nor contained in maximd sets. (Cohen and Jockusch [1975]) (Hint: by 111.4.10, and the fact that cainfiriite r.e. supersets of strongly effectively hyperhypersimple sets are such.)

Effectively inseparable pairs of r.e. sets T h e fact t h a t a set A is recursive if a n d only if b o t h A a n d 2 a r e r.e. suggests t h e possibilit,y of extending t h e theory of r.e. sets to pairs of disjoint r.e. sets. T h e first step was taken in 11.2.4, with the definition of t h e notion of recursive inseparability as a n analogue of noiirecursiveness. T h e existence of recursively inseparable pairs of r.e. sets was proved in 11.2.5, a n d we now strengthen that, resu 1t,.

Proposition 111.6.22 (Shoenfield [1958])Every nonrecursive r.e. T-degree contains a recursively inseparable pair o f r.e. sets. Proof. Let C be a nonrecursive r.e. set. We modify t h e first proof of 11.2.5, a n d define

C A ‘ p ( z ) z (= ~ )0 before (z)1 E C E C A ( P ( ~ ) ~ (= Z )1 before ( z ) ~ E C‘.

EA

tj

(z)1 E

z E U

tj

(Z)I

z

T h e n A a n d B a r e a disjoint pair of r.e. sets. Moreover:

A
C
A and B are reciirsively inseparable Suppose D is a recursive set, such t h a t A cpe(.)

=

C D and B

1 ifzED 0 otherwise.

B,a n d let,

317

111.6 Creative Sets and Completeness Then

z E C @ z is generated in C before cpe((z,e)) converges. Indeed, if

z E C and cp,((z, e)) converges before z is generated in C,then

But then C is recursive, contradiction.

0

Exercises 111.6.23 a) K is part of a recursively inseparable pair of r.e. sets. (Hint: let A and B be a recursively inseparable pair of r.e. sets. Then A I,,K,and there is a one-one recursive function f such that z E A f(z) E K,namely the function given by 111.1.2, being obtained by the Sr-Theorem. Then K and f ( B ) are recursively inseparable.) b) If B Srn A and B is part of a recursively inseparable pair of r.e. sets, then A is not simple. This generalizes the fact that m-complete sets are not simple. (Hint: i f f reduces B to A, and B and C are recursively inseparable, then D = f (C) is infinite, otherwise f - ' ( D ) is a recursive set separating B and C.) c) If B Stt A and B is part of a recursively inseparable pair of r.e. sets, then B is not hypersimple. This generalizes the fact that tt-complete sets are not hypersimA k ~ j ( ~and ) , B and C are recursively ple. (Denisov [1974]) (Hint: if z E B inseparable, then

*

*

2

EB A

Y E c =+ A

I=Df(z) A

+I=

af(v)).

With notations as in 111.3.10, if 2

E B A Y E C A (A* k

++A'

then A and A' differ on some element used in uf(.) or ~ f ( such that z E B A y E C A (Vi < 2*)(A; k At

*

otherwise we could recursively separate Z ~ Y

~ There ) .

must be x and y

'~f(~)),

B and C, because

(vi < 2")(AZ k uf(.) -e+At t=

~ f ( ~ ) )

is a recursive equivalence relation with only finitely many classes, and we could take the union of the equivalence classes containing elements of B.) d) Not every nonrecursive r.e. tt-degree contains a recursively anseparable pair of r.e. sets. (Hint: from c) above.) Effective nonrecursiveness can be generalized as follows:

111. Post's Problem and Strong Reducibilities

318

Definition 111.6.24 (Kleene [1950],Uspenskii [1953])A a n d B are effectively inseparable i f t h e y are disjoint r.e. sets a n d , f o r s o m e recursive function f ,

T h e recursavely ziLseparable pairs constructed in II.2.5 are both effectively inseparable. For example, if

xEA

tj (p,(z)

=0

and x E B

* cp,(z) = 1,

let f (x,y) be a n index of the partial recursive function which gives z value 1 if the stage i n which z appears in W, is not greater than the stage in which z appears i n W,, and 0 i n the opposite case. Then, if A C W, and I3 W, aiid W,, W, are disjoint, f ( z , y ) E W, U W,, because e.g.

f(Z,Y) E W,

3 cp,(z,y)(f("lW))

= 1 =+ f ( Z , Y ) E B =+ f(z,!/) E W,,

contradiction.

Proposition 111.6.25 If A and U are effectively inseparable, a n y r.e. superset of o n e diSjoint f r o m the other ,iscreative. I n particular, so are A a n d B . Proof. Let, A C a n d B ??, Wg(,l = W, U B and C = W,. If f witnesses the effective inseparability of A and H , then

W,

CC

*

A

C W, A R

W,,,,

3

f(a,y(z:)) E C - W,.

0

Exercises 111.6.26 a) There are recursively iimpumble, not eflectively inseparable puirs of r . e . sets. (Muchnik [l956a], Shoenfield [ 19571, Tennenbaum [ 1961al) (Hint: from 111.6.22 a n d 111.5.20, since effectively inseparable pairs must be T-complete. Or directly, as in the construction of Post's simple set, by building A and H r.e. a n d disjoint, intersecting each infinite r.e. set, a n d with A U B infinite. To achieve the last condition, consider only elements of We greater t h a t 3e.) b) The sets A and B - A in the proof of 111.6.17 a m wcursiuely iiwepamble, b u t not efle~tivelyinseparable. (Shoenfield [ 19571) (Hint: they cannot be effectively C a n d B - A & with C recursive. inseparalde, since A is not creative. Suppose A Sirice A is quasicreative we can build, by iteration, a n r.e. sct D 2 such t h a t D n C is infinite, a n d then B is riot simple, contradiction.)

c, c

111.7 Recursive Isomorphism Types

319

c ) There are two disjoint creative sets which are recwsively iweparable, but not effectively ii~separable. (McLaughlin [ 1962a1) (Hint: use the fact, proved in IX.2.4, that. every infinite nonrecursive r.e. set is the disjoint union of two infinite nonrecursive r.e. sets. Split this way a maximal set, and consider each component as the union of an infinite recursive set and of a creative set, see III.6.7.e. The two creative sets thus obtained are as wanted.) d) There are two disjoint creative sets which are recursively separable. (Hint: let 22 E A e 2 E K, and 22 1 E B e 2 E K. They are creative sets, separated by the even numbers.)

+

Exercises 111.6.27 Conditions equivalent to effective inseparability. Let A and B be disjoint r.e. sets. The following conditions are equivalent to effective inseparability of A and B. a) For some recursive function f ,

A

c W, A B G W, A W, u w,(,) = w + p(2, 9 ) E W, n w,.

This is the dual of the notion of recursive inseparability. (Muchnik [ 1958aj) b) For some recursive function f ,

W, 2

A

W, G B A W,

n W, = 0 * f (z,y ) E A U B - (W, U W v ) .

This is the analogue of creativeness. (Muchnik [l958a], Smullyan [196l]) (Hint: the equivalence is trivial, and it does not use the Fixed-Point Theorem.) c) For some recursive function f , A C W, A C W, f ( 5 ,y ) E ( A n W,) U ( B U W,).

B

*

This is the analogue of contraproductiveness. d) For every pair (W,, W,) of disjoint r.e. sets, there is a recursive function f thal, simultaneously m-reduces t h e m to ( A , B ) , i.e. z E W,

e f ( z ) E A and z E W, e f ( z ) E B.

This the analogue of m-completeness. (Muchnik [1958a], Smullyan [1961]) (Hint: see 111.6.6, and use the Double Fixed-Point Theorem 11.2.11.b.)

Further generalizations of creativeness (e.g. to infinite sequences of pairwisc: disjoint r.e. sets) a r e considered in Cleave [1961],Lachlan [1964], (1964a], [1965b], Malcev (19631, VuEkovich [1967], Carpentier (19681, (19691, (19701, Ershov [1977].

111.7 Recursive Isomorphism Types In this section we introduce two natural generalizations of t h e notion of m-reducibility, obtained by considering special reducing functions. Interestingly, these two reducibilit,ies are different, but, t h e notions of degree induced by them coincide.

111. Post's Problem and Strong Reducibilities

320

Mezoic sets and 1-degrees A natural strengthening of m-reducibility is obtained by asking one-oneness of t h e reducing function. Definition 111.7.1 (Post [1944]) A is 1-reducible t o B (A some one-one recursive function f , x E A H f (x)E B. A is 1-equivalent to B (A ~1 B ) if A < I B and B 51 A .


B ) zf, for

Note t h a t , since t h e proof of 111.1.2 was obtained by t h e S z - T h e o r e m , which automatically provides one-one functions (11.1.7), a set A is r.e. i f and only i f

A 51 K.

Exercises 111.7.2 a) If A 51 B and B is recursive, so is A. b) If A 51B and B is r.e. then so is A. c ) If A is r.e. then A 51A i f and only i f A is infinite, coinfinite and recursive. - d ) If A 51 B then IAl 5 IBJ and 1215 Isl. Thus, if A -1 B, A and B, as well as A and B,must have the same cardinality. e) If A and B are recursive sets,lAi) 5 IBI, and 5 then A 51B. f ) If A and B am infinite and coinfinite recursive sets, then A -1 B.

1x1 Isl,

Note t h a t 51 is a reflexive a n d transitive relation, a n d t h u s alence relation.

is a n equiv-

Definition 111.7.3 The equivalence classes of sets w.r.t. 1-equivalence are 5 ) is the structure of 1-degrees, with the parcalled 1-degrees, and (D1, tial ordering induced on them b y The 1-degrees containing r.e. sets are called r.e. 1-degrees, and two of them are:

<

<,.

1. 01,the 1-degree of the infinite and coinfinite recursive sets 2. Oi, the 1-degree of

K.

A set A is 1-complete i f it is r.e. and its 1-degree is 0'1, i.e. K


A.

Note t h a t a n r.e. 1-degree contains only r.e. sets. T h e 1-degrees containing recursive sets a r e infinitely many: one for each finite o r cofinite cardinality, together with 01,ordered as follows:

32 1

111.7 Recursive Isomorphism Types

where a, is t h e 1-degree of t h e finite sets with n elements, and b, t h e 1-degree of t h e cofinite sets with n elements in t h e complement. Even if we consider only t h e 1-degrees of infinite and coinfinite sets, 01 is not t h e least 1-degree:

Proposition 111.7.4 If a set A has 1-degree above 01, then A is neither immune nor coimmune. I n particular, i f A is r.e. then it is not simple. Proof. If B is a n infinite and coinfinite recursive set, and f is a one-one re(.) € A , then f(B) A and f@) 2, cursive function such t h a t z € B w f a n d both f ( B ) and f(B) a r e infinite r.e. sets. 0

c

c

Simple sets are neither recursive nor 1-complete, a n d thus they solve a version of Post's problem for 1-degrees. But we could also formulate t h e problem as: a r e there (r.e.) 1-degrees strictly between 01 and Oi? Then simple sets a r e of no help any more, since their 1-degrees are all incomparable with 01. To solve this version, we must first understand what the 1-complete sets are.

Theorem 111.7.5 (Myhill [1955]) A set is 1-complete i f and only if it is creative. Proof. A 1-complete set is m-complete, and hence creative (by 111.6.6). Let now A be creative, a n d f be a recursive function such t h a t

w,c7i 3 f(.)

€2-w,.

T h e same proof of 111.6.6 would show t h a t A is 1-complete, if we knew t h a t f can be supposed t o be o n e o n e (recall t h a t t h e reduction function of K t o A is t h e composition o f f and a function g obtained by t h e Fixed-Point Theorem, hence by t h e SF-Theorem, and thus one-one by 11.1.7). We then show how t o build a one-one productive function h for 2,by induction. We can start by letting h(0) = f (0). Suppose h(O),. . . , h(n - 1) have been defined: we want h(n)such t h a t

h(n)@ { h ( O ) , . . .,h(" - 1)) 0

w n c 23

h(n)E7i-Wn.

111. Post’s Problem and Strong Reducibilities

322

Note that, by iteration, if Wtcz, = W, U {f(z)}then, when Wn 2 2,the ~ . 71, elements f ( n ) , f ( t ( n ) ) f, ( t 2 ( n ) ) ., . . are all distinct ant1 i n 2 - w ~Given generate this list, arid sce which of the following happens first. 0

0

C 2,and then h ( n )can be anything, as long as it does not interfere with the requirement that h be one-one. For example, let h ( n ) be the least number different from ti(()), . . . ,h(n - I ) . If we first find a repetition, we know that it cannot be W,

If we first find an element, riot in {h(O),. . . ,h(n - l)}, then we can let h ( a ) be the first such one. Then, as above, h ( n ) E 2 - W, if W, C 7. 0

Exercises 111.7.6 Special productive functions. a) A set is productive if and only if it has n one-one, onto productive function. (Rogers [ 1YG71) (Hint: t h e proof above shows how t o get a one-one productive function f . To get from it an onto one, let g be a recursive one-one enumeration of a n infinite recursive set B of r.e. indices of w , a n d let

h(x) =

{

ifxEB otherwise.)

b) A set is contraproductive if and only i f it has a one-one, onto contraproductive function. (IIorowitz [ 19781) (Hint: symmetric t o p a r t a), using 111.6.12.) c) A set is completely productive i f and only if it has a one-one completely productive function. (Ilorowitz [ 19781) (Hint: if A is completely productive then, by p a r t a), it has a one-one Ixoductive function f . 13y t h e Fixed-Point Theorern there W, n { f ( h ( z ) ) } .T h e n f h , is a is a one-one recursive function h such that Wq,) one-one conipletely productive function for A . ) d) If a set has an onto completely productive function, then its wnipleiiieiit is r.e. (and hence crmtive). (Horowitz [1978]) (Hint: if A is completely productive, there is a recursive function f such t h a t f (z) E 2 ts f ( z ) E W,. If f is onto then, for any y, there is z siich t h a t f ( x ) : y , a n d y is in 2 if a n d only if it is in W z . ) e) The complement of every creative set A has an onto completely productive function. (Horowitz [1978]) (Hint: as in p a r t a), starting with a completely productive function for 2,and using A in place of w.)

Now that we know that the 1-complete sets are exact,ly the creative ones, we have a candidate for the solut,ioii to Post’s problerri for 1-degrees.

Definition 111.7.7 (Dekker [1953])A i s mezoic i f it is an r.e. set which is neither recwsiue, nor creative, nor simple. We only have to show that siic.11sets exist. Actually, they are quite abundaiit.

111.7 Recursive Isomorphism Types

323

Proposition 111.7.8 (Dekker [1953])Every nonrecursive r.e. T-degree contains a mezoic set. Proof. Let A be a simple set in t h e given r.e. nonrecursive T-degree (by III.2.14), a n d (2,~ E )B @ 5 E A . Clearly B is a n r.e. set in t h e same T-degree as A . Moreover B is mezoic, because: 0

0

0

B is not recursive, because so is A . B is not simple, because t h e set { ( a , y ) : y E u } , for any a @ A , is a recursive subset of

z.

B is not creative, because if f were a productive function for z E wz}, then w h ( z ) = {(z,y)

wz

* wh(z)

3 3

a n d A would be creative too.

B, and

f(h(z))E - Wh(z) ( f ( h ( z ) ) h E A - wz,

0

Exercises 111.7.9 A classification of the r.e. sets (Uspenskii [1957]) An r.e. set A is pseudocreative if, for every r.e. subset B of 3, there is a n infinite r.e. subset of disjoint from B. The creative sets are exactly the effectively pseudocreative sets. An r.e. set A is pseudosimple if there is an infinite r.e. subset B of 2,such that A U B is simple. a) The recursive, simple, pseudosimple, and pseudocreative sets are a partition of the c l a s of the r.e. sets. a) Every nonrecursive r.e. T-degree contains a pseudocreative set. (Hint: see the proof of 111.7.8.) b) Every nonrecursive r.e. T-degree contains a pseudosimple set. (Hint: if A is simple, let 22 E B % z E A.) Exercises 111.7.10 Splinters again. a) Every splinter is recursive or pseudocreative, in particular is not simple. (Ullian [1960]) (Hint: let A = { a , f ( a ) ,...} be 2 disjoint from A, A U B is nonrecursive and suppose that, for some r.e. set B simple. Given z, consider {z, f (z), . . .}. If it is finite, then z @ A, otherwise A is finite and hence recursive. If it is infinite, by simplicity of A U B there must be n such that f(")(z)E A or fn)(z) E B. In the second case, z @ A. In the first, f(")(z)= f m ) ( a ) for some nt, and z is in A if and only if m 2 n and z = f ( " - " ' ( a ) . Then A is recursive.) b) Every recursive set is a splinter. (Ullian 119601) c) Every creative set is a splinter. (Myhill [1959],Ullian [1960]) (Hint: let B be creative, and (z,y)E A e z E B. A and B have the same m-degree and hence, by

324

111. Post's Problem and Strong Reducibilities

111.7.5, t h e same 1-degree. By 111.7.13, they are recursively isomorphic. T h u s we can just prove t h a t A is a splinter. Let {an}nEw be a one-one enumeration of R. Picture the pairs of numbers as a double array: we have t o generate the rows corresponding t o pairs with first elements in B, i.e. the elements of t h e kind (an,y). Note t h a t A = A0 U A I , where A,) = {(n,,y) : n < y},and A1 = {(a,,y) : y 5 T L } . Ao is recursive, a n d thus there is a n enumeration {bn}nEw of it in increasing order. I t is then enough t o define f recursive that steps from one element of the following list t o the following: (ao, 0) bo (all O), (01, I ) , b l ( a z , O ) ,( ( ~ 2 , 1 ) , ( 4 2 , 2 ) , b 2 1

Then let f ( ( a n , n ) )= bn, f(L)= (an+lrO), and f ( ( 5 , ~ )=) ( x c l y +1 ) ) otherwise.) Young [1967] has proved t h a t there are pseudocreative sets which are not s p h ters.

Recursive isomorphism types After having strengthened m-reducibility by requiring t h e reducing fiinction to be one-one, we can make a further s t e p a n d ask for o n t o n c j s as well.

Definition 111.7.11 (Post [1944]) A as recursively isomorphic to B ( A B ) aj, for s o m e one-one onto recursave functzon f, x E A # f ( r )E B . The equavalence classes of sels w.r.t. recursave zsoinorphL5in are called recursive isomorphism types. Note t h a t is a n equivalence relation, because t h e class of one-one a n d o n t o recursive functions (called recursive permutations) is closcd under inverses. Exercises 111.7.12 (Rogers 119671) a) The recursive permutatio7is fornl a group, which is not finitely generated. (Hint: if the group were finitely generated, t h e recursive permutations could be recursivdy enumerated, a n d t h e diagonal method would produce a contradiction.) d of the .group b) The gmup of the remrsive permutations is not a 7 l O r ~ subgroup of all permutatio7~sof w . (Ilint let h be a nonrecursive permutation, a n d if x odd

dx) =

z - 1 if x odd z + 1 if 2 even.

Then f-lgf is not a recursive permutation.)

In Set Theory, t h e Caritor-Schroder-I3eriistein Theorem shows t h a t two sets which can be one-one mapped into one another must have t h e s a m e cardinality. T h e next result is a constructive version of it.

111.7 Recursive Isomorphism Types

325

Theorem 111.7.13 (Isomorphism Theorem (Myhill [1955]) 1-degrees a n d isomorphism types coincide, i.e. f o r any pair of sets A and B ,

Proof. Let f a n d g be o n e o n e recursive functions, such t h a t zE

A

H

f(z) E B a n d z E B

* g ( z ) E A.

We want to define a recursive permutation h t h a t interchanges A a n d B , a n d t h u s we will satisfy t h e condition z E A tj h(z) E B . Since h has to be total, from t i m e to t i m e we ensure t h a t t h e least element not yet in t h e domain gets into it (by defining h o n it). Similarly, h has to be onto, a n d from t i m e to t i m e we ensure t h a t t h e least element not yet in t h e range gets into it (by letting it be t h e value of h for some argument). We then have to show how to ensure t h a t h is b o t h one-one a n d a function. B u t h is a function when h-' is one-one, a n d t h u s we will have a symmetric construction, t h a t alternates steps to make h total a n d one-onel to steps t o make it onto a n d a function. We can easily s t a r t by letting h(0) = f(0).Then

0E A

@

f(0)E B

H h(0)

E B.

Now take t h e first y not yet in t h e range (i.e. y # f(0)):we want a n element z such t h a t h(z) = y (towards ontoness), a n d z # 0 (to ensure t h a t h is a function). We obviously t r y g(y): if this is not 0 then we have what we want, a n d can let h(y) = g(y), since

YEB

* S(Y) E A * M Y ) E A

B u t if g(y) = 0 t h e n we have two different elements y a n d f(O), a n d we know t h a t g(f(0)) cannot be 0, since g is one-one. Then we can let h(g(f(0))) = y, since yEB g(y) = 0 E A f(0) E B H g(f(0)) = h(y) E A.

*

*

T h e procedure is perfectly general. At any stage, having defined h on a set of elements ( 5 0 , . . . ,zn}in such a way t h a t

111. Post's Problem and Strong Reducibilities

z

312,

...

111.7 Recursive Isomorphism Types

327

We can then proceed inductively, and extend h t o a new element xn+l, by letting z,+1 be the least element not yet in {zo, . . . ,z,} if n is odd, a n d h(z,+l) be the least element not yet in {h(zo),. . . ,h(z,)} if n is even. The procedure to find h(z,+l) given zn+l must succeed after at most n 1 trials. It is illustrated in Figure 1, and can be described recursively as follows. Given z not yet in t h e domain of h, let z = z, and consider f(z): if it is not yet in t h e range of h, let h(z) = f(z). Otherwise, change z into h - ' ( f ( z ) ) , and try again. The procedure must stop after finitely many steps, as already noted, because the domain of h is finite, z is not in it, and f is one-one. And z E A h ( z ) E B because both f and h have this property, t h e latter by induction hypothesis. The procedure t o find z,+l given h(z,+l) is symmetric, by using g a n d h 0 in place of f and h-'.

+

Corollary 111.7.14 The creative sets are all recursively isomorphic.

Proof. T h e creative sets are exactly t h e 1-complete ones, by 111.7.5, and thus are all 1-equivalent. 0 Note t h a t it also immediately follows that the infinite, coinfinite recursive sets are all recursive isomorphic (Post [1944], Dekker [1953]), although this can easily b e proved directly. Exercise 111.7.15 The effectively inseparable pairs of r.e. sets are all recursively isomorphic, i.e. if ( A , B ) and ( C , D ) are effectively inseparable, there is a recursive permutation that simultaneously exchanges A and C, and B and D. (Muchnik [ 1958a], Smullyan [1961]) (Hint: use II1.6.27.d.) Rogers [1967] introduced Klein's approach in Recursion Theory, a n d stressed the importance of considering properties invariant under recursive permutations. We leave t o t h e reader t h e verification t h a t all concepts introduced so far (with only one exception, see III.7.16.c), as well as those t h a t will be introduced in t h e next chapters, are indeed recursively invariant. Frequently, 111.7.13 is a useful tool in such verifications. Exercises 111.7.16 a) The property of being regressive is invariant under recursive permutations. (Hint: if A = {ao, a l , . . .} is regressed by cp, then the partial recursive function $ such that $(h(z)) 21 h(cp(z))regresses h(A) = { h ( a o ) ,h ( a l ) ,.. .}. Note that the enumeration of the set has changed thus, even if A is retraceable, in general h ( A ) is only regressive.) b) The property of containing an infinite retraceable subset is invariant under recursive permutations. (Dekker [1962]) (Hint: if A contains an infinite retraceable set B, and h is a recursive permutation, then h ( B ) is an infinite regressive subset of h (A ),by part a), and it contains, by 11.6.20, an infinite retraceable subset.)

111. Post’s Problem and Strong Reducibilities

328

c) The property of being retraceable is not invariant under recursive permutations, even for co-r.e. sets. (McLaughlin 119661) (Hint: this requires t h e priority method. We fix the permutation in advance, a n d let h exchange 2z a n d 22 + 1 . We then build A = (a0 < a1 < ...}, a n d cp retracing it. T h e construction has t o b e effective t o give cp partial recursive, a n d this produces A c*r.e. We ensure t h a t h ( A ) is not retraceable. Choose aze and uze+l t o witness the fact t h a t ‘p. does not retrace h ( A ) . Also, let cp(a,+l) = a, a n d cp(ao) = ao, so t h a t cp retraces A. At stage s 1, let cps be the approximation of cp obtained so far. Consider the smallest e 5 s such t h a t the condition ‘cpe does not retrace h(A)’ has not yet been satisfied a n d not = a;, l), a n d y ~ ~ , ~ ( a ; , = + , uZe. ) Then injured afterwards (in particular, cpe,s(h(a;,))= l ~ ( a ; , + ~a)n, d we want t o destroy this. Let z b e t h e smallest odd element which is greater t h a n both az,+l a n d every element in the domain of ( p a , a n d let z = and cp.+l(z) = a;.. T h e reason for doing this is that, if a;e = az,, then p e ( h ( a z e ) )is not in h ( A ) ,arid so cpe does not retrace h ( A ) . ) d) There are r.e. sets coregressiiie, but not coretraceable. (McLaughlin [1966]) (llint: hy the proof of part c), h ( 2 ) =: is such a set.)

+

+

h(A)

Degtev [ 19711 a n d Soare [ 19721 have shown t h a t every norirecursive r.e. T-degree contains a n r.e. set which is coregressive b u t not coretraceable.

Recursive equivalence types and isols

*

Dckker (19551 defines t h e relation of recursive equivalence as:

A

”! B

for s o m e o n e o n e , partial recursive function

(p,

A
cardinals. In Set T h e o r y t h e r e a r e t w o notions o f finiteness: A is finite if it, c a n bc o n c o n e m a p p e d o n a p r o p e r initial segment of w ,a n d is Dedekind-finite if i t c a n n o t be one-one m a p p e d on a p r o p e r subset of itself. T h e t w o notions coincide if choice is assumed, b u t in choiceless Set T h e o r y t h e l a t t e r is weaker, arid sonic infinite sets m a y still be Dedekintl-finite. A set A is recursively equivalent to a proper s u b s e t of itself if a n d only if it is infinite a n d n o t i m m u n e , a n d t h u s the r.e.t. ’s of finite or irnrnune sets (called isols, because such sets are isolated in t h e usual topology, see 11. 186) rnay be seen as a constructive version of Dedekind-finite cardinats (even more appropriately t h e y arise, via Kleene realizabilit,y, from s t a n d a r d cardinal arithmet,ic on s u b s c t s of w ,in Intuitionistic. Set Theory, see McCarty [1986]). T h e set A of isols c a n be given operations of s u m a n d p r o d u c t , induced by

111.7 Recursive Isomorphism Types

329

A @ B and A . B , and a partial order relation 5 defined as 2

5 y e (3z)(2+ z = y).

The structure of isols is quite rich: w is embedded in A (by identifying n with the is01 of finite sets of cardinality n ) ,A has 2N0 elements, and (A, 5 ) embeds every countable partial ordering (as well as many others, see Ellentuck [1973]). Myhill [1958] has introduced a way to extend certain functions from w t o A. First note that each function f : w + w can be written as

where the ti's (Sterling coefficients) are positive or negative integers. If they are all positive or null, then f is called a combinatorial function. Thus every function on w is the difference of two combinatorial ones. A combinatorial function is recursive if and only if t h e sequence {ci}iEwis. Combinatorial functions of many variables are defined similarly. T h e class of combinatorial functions is a rich one, closed under composition, and containing I?,the constant functions, sum, product, factorial and positive-base exponential. A combinatorial set function is a function on P(w) that maps each finite set t o a finite set, and that respects cardinalities, intersections and countable unions (in particular, t h e function is generated by its behavior on t h e finite sets). It is called effective if t h e restriction t o finite sets is recursive (on the characteristic indices). Combinatorial set functions induce functions on w (since they preserve finite cardinalities), and effective combinatorial set functions induce functions on r.e.t.'s, in particular on A. T h e basic connection between the two notions is t h a t t h e (recursive) combinatorial functions are exactly those induced by (effective) combinatorial set functions. Thus for each recursive combinatorial function f on w, there is a function fi\ on I\ extending it. It is immediate to note t h a t for each recursive relation R on w there is a relation RA on A extending i t , since given R there are f,g recursive combinatorial functions such that

and thus it is enough t o define

Note that if a recursive relation R admits an algebraic characterization on w , RA does not necessarily coincide with the interpretation of the characterization

111. Post's Problem and Strong Reducibilities

330

on A. E.g., if Pr is t h e set of prime numbers, PrA is different from t h e set of prime isols, defined as

x prime

2

5x

A b'y'Vz(;y.

z =x

3

y = 1 V z = I).

Nerode [1961], [1962] has used these trarislatioris to show that a substantial part of t h e theory of (finite cardinal) arithmetic c a n be extended, to isols. Consider a first-order language with equality, function constants for all recursive combinatorial functions, a n d relation constants for all recursive relations. T h e n a 1~71111ersal H o r n senteiice (see 11. 39) holds over w if a n d o n l y if its trarislntion holds o w r A. I t imrnediately follows t h a t , e.g., t h e following hold for isols:

T h e result holds in niiich greater generality t h a n stated, b u t sonie restrict,ions a r e needed, because there a r e universal sentences t r u e in w t h a t fail iri A , e.g. the fact t h a t every element is even or odd. Ellentuck [1967] ha3 discovered a notion of universal isol (as a n is01 avoiding RA, for each coinfinite recursive K ,a n d t h u s having a sort of genericity): there a r e Y N o such isols, a n d each of t h e m proirides counterexainples to each universal sentence t n i e in w hut. false in A . I3.g. a universal isol is prime, b u t not a nieniber of P TA . The n o t i o n of isol can be regarded as a n e x t e n s i o n n o t o n l y o f t h e n o t i o n of integer, but also of nonstandard integer. Indeed, Nerode [1966] has proved t h a t aiiy countable nonstantlartl model of Arithiiietic correct for diophantine equations (i.e. such t h a t if an equation has a soliitiori in t h e model, t h e n it has a solution in t h e integers) can be embedded in t h e isols. Conversely, a n y sul>set of t h e isols whose differences generate a n integral domain can be cmbcddcd in a nonstandard model of Arithmetic. T h e theory of ( A , + , . ) is at least as complicated as t h a t of (w,+;) (in particular is not decidable), because t h e set of finite isols is definable over A , by t,he forttiula I i n ( x ) ej ('Vy)(z 5 y v y 5 X) (Dekker and Myhill [196O]). Actually, t h e first-order theory of ( A , 5 ) is rec w s i v c l y isoinorphic t o t h e Second Order A r i t h m e t i c (Nerode a n d Mariaster [1971]), and t h e s a m e holds f o r the theory of ( A , + , . ) (Ellentuck [1973a]). For a 1,re;ttirient of t h e theory of isols (and r.e.t.'s in general) see Dekker a n d Myliill [ 1 !)GO], Dekker [196G], Crosslcy a n d Nerode (19741, McLaughlin [1982]

111.8

Variations of Truth-Table Reducibility

JC

In this srction we play on t h e thcnie of tt-reducibility, by first considering bounds on t h e number of elements used in t h e t r u t h - t a l k s , a n d then by relaxing

111.8 Variations of lkuth-Table Reducibility

*

33 1

t h e condition t h a t we know in advance the effect of t h e answers to t h e queries m a d e to t h e oracle. At t h e end we also introduce a nuniber of other s t r o n g reducibilit ies.

Bounded truth-table degrees T h e variation we consider first is, like c-reducibility or d-reducibility, a strengthening of tt-reducibility, b u t in a different direction. We impose restrictions not on t h e kind of t r u t h tables we allow, b u t rather on their size.

Definition 111.8.1 (Post [1944]) A is btt-reducible to B ( A S b t t B ) if, for some recursive function f and some number m, called the norm of the reduction, 1. z E A

B uses at most ni elements.

2.

If ni is the norm, we also write A &t(m) B. A is btt-equivalent to B (A ?btt B ) i f A Exercises 111.8.2 a) If A is wmrsave, then A h) If A

lbtt

B and B

Sbtt

A.

B for any set B.

Note t h a t S b t t is a reflexive a n d transitive relation, a n d t h u s equivalence relation.

Gbtt

is a n

Definition 111.8.3 The equivalence classes of sets w.r.t. btt-equivalence are called btt-degrees, and ( Z ) b t t , 5 ) is the structure of btt-degrees, with the partial ordering 5 induced on them b y &t. The btt-degrees containing r.e. sets are called r.e. btt-degrees, and two of them are: 1.

Obtt,

the btt-degree of recursive sets

2. Oktt, the btt-degree of K.

A set A is btt-complete if its r.e. and its btt-degree is U b t t , i.e. if K

Sbtt

A.

In general &(m) is not transitive, b u t k. Since a set is always reducible to i t s complement by a bounded truth-table reduction of norm 1, b u t not always b y m-reductions, S m a n d < b t t ( l ) differ in general. B u t t h e y agree on nontrivial r.e. sets.

Proposition 111.8.4 I f A and B are r.e. sets, B # A 5, B .

8 , w and A

&t(l)

B , then

111. Post’s Problem and Strong Reducibilities

332

Proof. Since A

B, there is a recursive function f such that, depending

&(I)

on z,

f(z) E B or z E A @ f(z) $ B. We want a recursive function g such t h a t z E A @ g(z) E B. There are two zEA

@

cases: 0 0

if z E A

* f(z)E B , we just let g(z) = f(z)

f(z)$ B, then exactly one of z E A and f(z)E B happens. if z E A Generate A and B simultaneously, and find out which one. If z E A then we want g(z) E B. If f(z)E B then z $ A, and we want g(z) @ B. Since B # 0,w we can pick u p a E B and b $! B, and let g(z) be a in t h e first case, and b in the second. 0

Exercise 111.8.6 m-ducibility and btt-reducibility differ on the r.e. sets. (Fischer [1963]) (Hint: Let (2, y) E A z E B A y E B, where B is the simple tt-complete set of 111.3.5: then A & t ( 2 ) B. If A &, B, there would be g recursive such that z E BAy E B g(z, y) E B, and B would be creative because, if h is gotten by h ( z ) E B.) iteration of g, z E K: F, B

*

*

*

*

When A &t B , only a finite number m of elements are used in t h e reduction, and hence at most 22m truth-tables may be used. Actually much more is true: a single truth-table is enough (although, in general, with a different norm).

Proposition 111.8.6 (Fischer) ZfB to B by a &ed truth-table.

# 0,w and A S b t t B , then A is reducible

Proof. Consider e.g. the case of norm 1, and let A be reducible to B via of(=). Fix a E B and b $ B. Since of(=) is either y E X or y $ X for some y, let og(z) be the fixed formula (y E X A Z E X ) v (y $ X A Z

ex),

where y is t h e element appearing in of(=), and

-

z = a if of(=) is y E X z = b if of(=) is y $! X.

I=

Then B I= Of(%) B Og(2). T h e general case is similar, although the addition of more than one variable is required, t o be able t o distinguish among many cases. 0 Thus a btt-reduction can actually be seen as t h e assignment, to every z, of a fixed number of elements {b:, . . . ,b}:, such that the answer to t h e question ‘is z in A?’ depends solely on the membership of t h e b4’s in B.

111.8 Variations of Tkuth-Thble Reducibility

*

333

Proposition 111.8.7 A set is btt-reducible to K: i f and only if i t is in the Boolean algebra generated by the r.e. sets. Proof. If a set A is in the Boolean algebra generated by the r.e. sets, it can be obtained from a finite number of r.e. sets A 1 , . . . , A , by the set-theoretical operations of union, intersection and complementation. Express 5 E A as a propositional formula involving the Ai’s, and then substitute each atomic formula z E Ai with fi(x) E K ,where fi is an m-reduction of Ai t o K ,which exists because Ai is r.e. E.g. let A = A1 - (A2 n A3). Then zEA

*

5

and, if

as(,)= f i ( 4 E

E A1 A l ( z E A2 A X E A3)

x A ’(h(4 E x A f 3 ( 4

E

X),

then 2 E A H K: og(,),and A S b t t Fc. Conversely, if A S b t t K then, by the previous proposition, A can be reduced to K by a fixed truth-table, and the procedure just given can be inverted. 0. T h e analogue of Post’s Problem for btt-reducibility is: are there r.e. sets which are neither recursive, nor btt-complete? We already know t h a t simple sets are not m-complete, and we now extend this result to show t h a t they are also not btt-complete.

Theorem 111.8.8 (Post [1944]) A simple set i s not btt-complete. Proof. Let K S b t t A . We try t o define a disjoint strong array {F,},~, intersecting ?I, with every element of fixed cardinality: then A is not simple, because a n infinite r.e. subset of 2 can be obtained by t h e following procedure. Simultaneously generate A and the F,’s: as soon as all but one element of F, have been generated in A , we know that the other is in ?I. T h e only trouble is t h a t there could be only finitely many n for which F, has exactly one element in ?I, and this procedure would not produce an infinite set. But if this happens then (by eliminating the finitely many exceptions) we may suppose that each F, has at least two elements in 2.Again we can proceed as above, if there are infinitely many n for which F, has exactly 2 elements in ?I. In general, let z be t h e least number such that, for infinitely many n, F, n i i has exactly z elements. Then, for some no and all n 2 no, F, n x has at least z elements. Generate simultaneously A and the F,’s, for n 2 no: as soon as all but z elements of some F, have been generated in A , the others must be in 2. If K: S b t t A , then so too is K S b t t A . Let z E K H A of(,),for some recursive function f , with a given reduction {b;, . . . , bk}. Suppose we have already chosen

+

334

111. Post's Problem and Strong Reducibilities

with t h e property t h a t

F, n A # I? a n d A

of(zi).

To define F,,, let C be t h e set consisting of all t h e x for which t h e formula of(z) is deducible, in t h e propositional calculus, from of(zo), . . , ,of(zn-,) a n d the conditions 'zE X',for z E A . By t h e inductlive hypothesis ( t h a t A of(zi)) a n d logical properties, zEC

a n d hence G 0

0

A

tj

K. If C = W,, let x,

o f ( a )because , aE

x E C,

A

= a. From a E

K

-

C w e have:

C

{br, . . . , bk} n A # 8. Otherwise, being of(a) t r u e in A , a n d using only t h e elements by,. . . , bk which a r e a l l in A , of(,)would be dediicible f r o m t h e conditions 'zE X ' for z E A . T h e n a E C, contradiction.

However, we c a n n o t prove t h a t t h e F,'s a r e disjoint o n A. B u t t h i s is unnecessary: it is enough to show t h a t , given F,, t h e r e a r e only finitely m a n y sets of o u r sequence, with t h e s a m e intersection o n 2.Indeed, consider {b:, . . . , bk}: since only membership in A or 2 m a t t e r s , t h e r e are only 2* possibilities, If t w o given conditions have s a m e elemcnts o n 2,t h e y a r e not only equivalent: t h e y a r e also deducible o n e from t h e o t h e r froni t h e conditions ' z E X ' for z E A . T h u s in our sequence, b y definition of C , at triost 2" conditions m a y have the s a m e intersection o n 2. T h e n t h e procedure described at t h e beginning still produces a n infinite subset of A. 0

Corollary 111.8.9 There a r e tt-complete, btt-incomplete sets. Proof. By 111.3.5 t h e r e is a siiriple tt-corrrplcle set: by simplicity i t c a n n o t be btt-complete.

0

Exercises 111.8.10 a) A pseudosimple set is not btt-complete. (Shoenfield [ 19571) b ) If B < b t t A and I3 is part of a recursiiiely inseparable pair of r.e. sets, then A is not simple. (Kobzev [1973]) (Hint: by induction o n ni, prove t h a t if A is simple a n d {F,},G~ is a strong array of nL-tuples intersecting 2, there is a finite set D 2 such that, for all R, F, n D # 0. Then let B < b t t A via {b;",. . . , b:}, a n d consider the z E B , and { b y , . . . ,b z } C A =+ z @ B. If e.g. t h e first two cases {bq, , . . ,bk} C A {b?, . . . ,bk} n 2 # 0. holds, and B a n d C are recursively inseparable, then z E C Let D 2 be finite, a n d such t h a t z E C =+ {by,. . . ,b z } n D # 0. Moreover, let xER (07,. . . , n D # @.Then R is recursive, C C R, a n d B n R a n d C are recursively inseparable. Split R into nt recursive parts

*

*

*

bz}

z E RI * x € R A b y E D

x E R2

*X

E R-R1 A b ; E D

...

111.8 Variations of Truth-Table Reducibility

*

335

For at least one i, B n& and C n R, are recursively inseparable, and B n & i b t t A with norm m - 1. S o A is not simple by induction hypothesis. For m = 1 recall that < b t t ( l ) coincides with Srnon the r.e. sets, and see 111.6.23.)

Theorem 111.8.11 (Kobzev [1974], Lachlan [1975]) A btt-complete set .is d-complete. Proof. Let K

Sbtt

A: for some recursive J

and, for some n, of(,)uses exactly n elements. We want t o get

for some recursive h. The obvious approach is t o consider

G, = {elements used i n of(,)}. We can certainly modify the reduction in such a way t o suppose that, whenever

x is enumerated in K ,then some element of G, is enumerated in A at t h e same stage. This gives in particular 2

E

K + G, n A # 0,

but we do not have the opposite: some element of G, could go in A even if X

#K.

We thus consider the set G, dynamically: at every stage s we see the set G x n A,+1, and can consider the remaining elements of G,. We thus have n r.e. sets F,, and Fi consists of elements zk which, at a certain stage s 1, are put in Fi because IC,: nA,+,I = i. We try t o consider zk in place of x and, as above, if X E K ~ ~ then ~ E~ EK K + ~ G , ~ ~ A , + ~ I > ~ .

+

If, for j > i and at stage s + 1, we put in Fj only elements z not yet in Ks+l,and such that IG, n As+ll = j,and we avoid putting in K elements Z: which are also in Fj (to avoid a situation in which Z; goes in K when y @ K), then for t h e greatest z such that Fi is infinite, and for almost every z, we also have

IG,: n A1 > i

+ z E K.

Indeed, only finitely many zk’s can be in Fj for some j > i, and thus only for finitely many z’s not in K it will be IG,, n A1 > i. There are however two problems.

111. Post's Problem and Strong Reducibilities

:Mi

1. We want z E K 3 2: E K , but we do i i o t . have any control over K ,and t h u s we (:annot directly force 2: into K. What we can do, is t o build a n r.e. set, D , a i i d usc a recursive function y such t h a t z E 1) tj g(z) E K : and g ( : a i i be used i n the constructiori of D itself, by tlic Fixed-Point Thmreiii. Thus putting z in D will force g(z) into K. We will thus consider C:g(r:), in place of G,, .

The const,riir:t,ion is finally fatiefollowirig: at, stage s + 1, if n: E K,+l -K, and 2L.E F,,s (i.e. the z-th element of Fi has already bc:cri gcncrated), p u t 22. in D , unless zk E Fir for some j > i. 0

if, for some i and some z 5 s ,

t,hen clioose i iiiaxiinal, n: niininial, and put z i n Ft. 2 . Let, now i be the greatest, siicli that lf; is illfinite. Let

where s

+ 1 is the stagc i n which 2:

is gciicrated in Fi. We would like t,o

show :c E

K e D h ( s )n A #

a,

but here conies the sec:oiitl proldem: by cwistruction, n A # 0 only for those z which are generated in K after 2: is generated i n F,. So consider these clernents:

'l'heii K - K * is recursive, enoiigli to prove :I,.

so

K'

is crwtivc, arid t o have A ti-complete is

E K * tj D,+) n A #

Cn.

If z E K * tlieii D h ( z )n A # fl by constriictiori, since a n eleiiicnt of Cg(t:) is generated i n A a t the saiiie stage i n which g ( z i ) is generated in K , hence aft,c:r t,lic stage zi is generated in b;, The opposilc holds for almost, every z, 1xcx:aiisez is t h e greatest such that 1;1 is infinitc, a r i d thus, except for finitely many z, we have l l , L ( xn)A # 0 only if :I: E K * ,by t h e first part of the coiist,riict,ioii. 0 Since btt-conipletdesets arc d-coniplete and hence quasicrmtivr:, and quasicwative sets art: i i o t , siniplc (III.6.I8), we have a different proof o f 111.8.8.

111.8 Variations of Truth-Table Reducibility

*

337

Exercises 111.8.12 a) A semirecursive set is not btt-complete. (Jockusch [ 1968a1) (Hint: a semirecursive set is not pcomplete, by III.5.3.b.) b) A set is btt-complete af and only af, for some recursive function f and some n, l D f ( z ) l5 n, and W , C_ 2 =+ Df(,) 2 A D j ( z ) W,. (Kobzev 119741) (Hint: a btt-complete set is bounded d-complete, and thus the bounded version of the quasicreativeness criterion for d-completeness applies.)

Weak truth-table degrees Recall t h a t truth-table reducibility differs from Turing reducibility in t h a t it is possible to foresee a h c a d of time, in a computation, b o t h t h e elements on which t h e oracle is going to be queried, a n d t h e outcome of all possible answers. A natural intermediate reducibility is obtained by retaining t h e first characteristic, while relaxing t h e second.

Definition 111.8.13 (Friedberg and Rogers [1959])A is wtt-reducible t o B (A I w t t B ) i f c~ cx pf for some number e and some recursive function f, and the calculation of p,(x) requires only queries to the oracle B on elements less than f(x). A is wtt-equivalent t o B (A G W t tB ) if A Swtt B and B Swtt A . T h e main difference with truth-table reducibility is in t h e fact t h a t weak truth-table reductions may diverge. If p$ is a weak truth-table reduction of A to B , t h e n pf needs to be total only for X = B , b u t not for oracles X different from B . Actually, by 111.3.2, this must be t h e case if t h e reduction is not a truth-table one.

Exercise 111.8.14 A set A is wtt-reducible to K if and only i f it is tt-reducible to it. (Hint: if cp: is a wtt-reduction to K with bound f , given z we may consider all the sets X C_ (0,. . . ,f(z)}. Each of them is recursive, and so we can ask, recursively in K,if cpf (z) converges. This makes it possible to build the appropriate truth-table.) Note t h a t Swttis a reflexive a n d transitive relation, a n d t h u s equivalence relation.

E w t tis

an

Definition 111.8.15 The equivalence classes of sets w.r.t. wtt-equivalence are called wtt-degrees, and ( V w t t ,5 ) is the structure of wtt-degrees, with the partial ordering 5 induced on them b y I w t t . The wtt-degrees containing r.e. sets are called r.e. wtt-degrees, and two of them are: 1. O w t t , the wtt-degree of recursive sets

2. O L t t , the wtt-degree of K.

111. Post's Problem and Strong Reducibilities

338

.4 set A

LS

wtt-complete af zt

LS

r.e. and itsuitl-degree as O L t t , a.e. afK

swlL A.

Proposition 111.8.16 (Friedberg and Rogers [1959]) A hyprrsLmple set L ) not uitt -complete. Proof. We refer to the proof of 111.3.10, which goes through practically changed, by letting 2 E e p,A'(2)= 0,

1111-

c

wliere c h zz p,", with 1,oiind f. If C = W, then p,"'( a ) and 9," ( a ) are 110th convergeiit a n d different, so t~iereniust Iie an element of A between the given n and f ( a ) . The rest p r o c d s as before. 0 .

A simple but useful observation is that permitting preserves wtt-redwibiM y . This allows us to extend inaiiy results froni T-degrees to uitt-degrees. G.g. cwry n o ~ ~ r e c u r s r. i ~e.~ ewtt-degree contains a simple set (111.3. 18), although not always a hypersimple one (since a hypersimple sct, is not, wlt-cwmpl .loc:kusch [1981a] shows that no1 every nonrecursive r.e. tt-degree contains a simple s e l . Another useful fact to iiotice is that the coinpleteness criterioii for T/'-reducil)ility (111.1.5) extends to wtt-reducibility ELS well, with the obvious tlefinitiori of uitt-reducil~ilityfor func:tiotis, and a similar proof Proposition 111.8.17 (Arslanov [1981]) A n r.e. set A is ukl-completc if and oiil~jif there is a fiinction f S w t t A without jikd-points. Rwall that, an application of the original criterion showed that every e l k 1.ivc:ly simple set is T-coiiiplete (111.2.18). Hcre the same results holds, for all the effectively simple sets not ruled o u t by the previous result.

Proposition 111.8.18 (Kanovich [1970 , [197Oa], Arslanov [198l]) 1 5 ~ t'r ?J effectively .stmple, not hypersimple set i wtt-complete. Proof. IA

WrC;;l*lW, I>c41ief


hllcli

tliat

Wr(., = {t,Iie first, !/(el

+ 1 c:Ienieiii,s o f A}.

'I'hcn f has n o fixed-poiiks, a5 in 111.2.18. Wc: show that f A . Since A is not, hypersimple, there is a strong array h intersecting 3.Thus the first g ( c ) + 1 clcinciits o f a r e below t,hc rriaxiriiiiiri of U z < g (D e h) ( z ) and , this provida the n c c d ~ rccursive l bound to the qucstions f litis to answer. 0

n

111.8 Variations of Tkuth-Thble Reducibility

*

339

In particular, Post's simple set is wtt-complete (Ladner). As we have already noted, t h e tt-completeness of Post's simple set depends on t h e acceptable system of indices for t h e r.e. sets (111.3.6, 111.9.2).

Exercise 111.8.19 Not every strongly effectively simple set is wtt-complete. (Hint: let A be any coinfinite r.e. set. If it is not hypersimple itself, it has a hypersimple superset, namely A U UItBD,(,), where f is a disjoint strong array intersecting A, and B any hypersimple set. And if A is strongly effectively simple, so is any coinfinite r.e. superset of it.) Exercises 111.8.20 a) A n r.e. set A is wtt-complete if and only i f W, E 3

* Dg(,) AUW,

for some recursive function g. (Kanovich [1969], [197Oa]) (Hint: if such a g exists, Then f A , since it uses the oracle only for elements let W,(,) = Dg(,) in Dg(2). Suppose W,(,) = W,: then W, C 3, so Dg(,) A U W,, contradicting W, -= Dg(,) n x . Then f has no fixed-points, and A is wtt-complete. Conversely, let f CY (p! with bound h be without fixed-points. Given 2,let

nz,

{

1 if z shows up first in A 0 if z shows up first in W,.

Since pi is recursive, there is z such that Wrpz,(,) = W, (by the Fixed-Point Theorem). Since W,(,) # W,, ' p r ' ( z )# f (2). If W, 2, then 'pi is correct on A U W,, hence there must be an element of A U W, below h(z). Then let Dg(%) be the set (0,. . ,h(41.) b) For every nonrecursive r.e. set A there is f recursive such that f

W, & 2 * W,(.) finite

A

W,(,)

Z ! A u W,.

(Soloviev [1976]) (Hint: put in W,(,) the smallest element that does not appear to be in A U W,.) c ) The class of wtt-complete sets is properly included in the class of T-complete, not hypersimple sets. (Soloviev [1976]) (Hint: let A be T-complete and hypersimple, and B be an infinite recursive subset of it. Consider A - B: it is T-complete because A is, and is not simple. It is also not wtt-complete, since A - B A, and A is not w tt-complet e.)

Other notions of reducibility

*

T h e reader certainly feels t h a t we have introduced enough reducibilities, b u t many more have been considered. We just review some of them. A natural way t o weaken reducibilities is by letting t h e finiteness condition in t h e sets defining them become an r.e. condition, like we did e.g. for c-reducibility, obtaining Q-reducibility. In a similar way, d-reducibility is weakened into

111. Post’s Problem and Strong Reducibilities

340

s-reducibility, defined as follows: A 5, B if a n d only if, for some recursive function f, R: E A W f ( 5n) B # 8. T h e n t h e r.e. sets fall in j u s t two s-degrees, one consisting of all t h e nonempty r.e. sets, a n d t h e other consisting of 8 alone. Similarly, positive reducibility is weakened in enumeration reducibility, introduced in Section 11.3 (p. 197). Recall t h a t A 5, B if a n d only if, for some recursive function f, R: E

A ts ( ~ u ) ( DC, B A u

E

Wj,,.).

T h e r e is just one r.e. e-degree, which is also t h e least e-degree. As we have already seen, this rediicibility is particularly suitable for t h e s t u d y of partial function (through their graphs), arid we will deal with it in C h a p t e r XIV. A number of other reducibilities can be introduced b y limitling t h e size of t h e sets used in defining known reducibilities, like we did for btt-reducibility. T h u s we can define, i n a natural way, riotions of bounded conjunctive, disjunctive, positive, weak truth-table a n d Q-reducibility. Also T-reducibility rail be restricted in a similar way, once we recall (111.1.4) t h a t , on r.e sets, A LT B i f a n d only if there is a recursive function f such that,

T h e n A is bounded Turing reducible to B if, moreover, there is a fixed number n bounding t h e size of W f c 5 , . T h c r e a r e otlher ways of imposing bounds on known reducibilities. E.g. wtt-reclucibility is obtained by restricting t,he number of elements queried in a coniputation. A restriction on t h e number of queries can be imposed too. Jockusch [1972a] calls A bounded search reducible to B ( A
Jockiisch [1972a] proves t h a t
111.9 The World of Complete Sets *

34 1

A different way t o weaken reducibilities is by considering partial function in place of total ones, thus obtaining partial reducibilities. E.g. Ershov [1977] calls a set A partially m-reducible t o a set B ( A Spm B) if, for some partial recursive function cp,

This reducibility is particularly important for t h e study of the A: sets. For results on some of t h e reducibilities quoted above, see Degtev [1979], [1981], [1982], [1983], Soloviev [1976a], Omanadze [1976a], [1980], Zakharov (19841, [1986].

111.9 The World of Complete Sets * We summarize here the work done in this chapter with respect t o completeness properties. We have already proved most of the positive results, and only some counterexamples are missing. Some of them are constructed by using the priority method, and should therefore wait until Chapter X. Since however this is t h e right place for them, we sketch their proofs here anyway, for the reader already acquainted with t h e method.

Relationships among completeness notions Our goal is t o show that the following implications hold, and no other one does:

Q

btt

d

All the implications are trivially true, except for the fact, proved in 111.8.11, that a btt-complete set is d-complete. We thus have only t o provide counterexamples t o the missing implications. They come from different sources, including Post [1944], Lachlan [1965], Young [1965], Jockusch [1968a], Gill and Morris [1974], Odifreddi [1981]. We first prove t h a t no other arrow holds.

111. Post's Problem and Strong Reducibilities

342

1. There i s a Q-complete, not wtt-complete set A hypersiniplc set is not zutt-complete, by 111.8.16, and we proved on 11. 297 that there is a Q-complete, hypersimple set. 2. There i s a btt-complete, not Q-complete set We build two r.e. sets A and B such that K: S b t t A , and R $Q A . The first condition cnsiires t h a t A is htt-complete, t h e second t h a t it, is not Q-complct,e. To get K: &,A we let

X E K

Q

+

{2~,2~+l}nA#0.

If at stage s 1 we see z E K , but { 2 2 , 2 2 + 1) nA , = 0, we put i n A the first element between 2 : ~and 22 1 which is not, restrained. If both are resitrainerl, we put in A the one restrained by the condition with lower priority.

+

To get B $Q A , we want, to spoil every reduction IC E

U

WpcCz,

A.

Pick u p a witness a,, and wait until cp,(a,) converges. If i t does not, then (p, was not a &-reduction. Otherwise, wait for a later stage s 1 for which W,c(ac) A , , i.e. Wpe(n,) n& # 0. If it never comes, then W,e(,e,C A , and a, never gets into B , so cp, does not Q-reduce U t o A . Otherwise, choose a n elenient IC, of W,c(,c)which is riot, yet i n A , restrain it, from entering A , and put a, into B . If the condit,ion is never irijured, i.e. X , does not go into A for satisfact,ion of a requirement of the first type (to make A btt-complete), then W+,c(ae)A , while a, E B , and again p, does not reduce R to A . Otherwise, a new attenipt will have to be made, to satisfy the rcx~nirer~ieiit.

+

3 . There is n c-complete, not d-coinplete set Post's example 111.3.5 of a siniple set which is tt-complete, actually produces a c-complete set. R u t no simple set ran be &complete, by 111.6.18.

4. There is a btt-comp l'he set built in pa

not c-complete set above is btt-coniplet,e, and not Q-complete. It cannot be c-coniplete, since c-reducibility implies Q-reducibility.

We now prove that, no arrow can be reversecl. 5. Th.ere i s n T-complete, n,ot Q-complete set, By 111.4.23, there is a T-coniplete niaxiriial set. Being hyperhypersiniple, it cannot be Q-coriipI(

111.9 The World of Complete Sets

*

343

6. There is a Q-complete, not c-complete set We noted on p. 297 that a hypersimple set can be Q-complete, but by 111.8.16such a set cannot be wtt-complete, and in particular it cannot be c-complete. 7. There is a c-complete, not m-complete set There is a simple tt-complete set, by 111.3.5, but a simple set is not m-complete, by 111.2.7.

8. There is a btt-complete, hence p-complete, set which is not c-complete, hence not m-complete The set built in part 2 is p-complete but not Q-complete, and hence not c-complete. 9. There is a d-complete, not btt-complete set This is a modification of the proof of part 2. We build two r.e. sets A and B such that K I d A , and B f - b t t A . Thus A is d-complete, but is not btt-complete. To get

K

I d

A we let

+

where I0 = { 0 } , 11 = {1,2}, . . . , and 1, has z 1 elements. Of course we have t o use tables with unbounded number of elements, otherwise we would have K: < b t t A , and there would be no hope of satisfying the remaining condition. To spoil btt-reductions with norm n we have t o take care of n elements, and almost every I , has more than n elements. This makes t h e argument work. To get B

&t

A , we want t o spoil every reduction of bounded norm

Pick up a witness ae, and wait until cp,(a,) converges. If it does not, then cp, was not a btt-reduction. Otherwise, wait for a later stage s 1 for which A , eqe(,). If it never comes, A fails but a, never gets into B , so cpe does not btt-reduce B t o A. Otherwise, restrain from entering A all t h e elements used in and not yet in A , and put a, into B . If t h e condition is never injured, 1.e. no element used in and not in A, goes into A for satisfaction of a requirement of the first type (to make A d-complete), then A fails, while a, E B , and again pe does not reduce B to A . Otherwise, a new attempt will have t o be made, t o satisfy t h e requirement.

+

344

111. Post’s Problem and Strong Reducibilities

10. There is a p-complete, not d-complete set T h e simple tt-complete set of 111.3.5 is actually c-complete, hence p-complete, but is not d-complete by 111.6.18, being simple. 11. There is a tt-complete, not p-complete set A semirecursive set is not p-complete, by III.5.3.b, but may be tt-complete, by 111.5.5.a.

12. There is a T-complete, not wtt-complete set A hypersimple set is not wtt-complete by 111.8.16,but can be T-complete, by 111.3.13. As the reader will have noticed, only one result is missing. Its proof is more complicated than the ones given so far.

Theorem 111.9.1 (Lachlan [1975]) There is a set which is wtt-complete, but not tt- complete. Proof. We want t o build A wtt-complete, but not tt-complete. We will reduce K to A , with a recursive bound. We will thus define a recursive function f (such that f(0) = 0), and a sequence of boxes

I , = ( 2 : f(x) 5

2

< f(.

+ 1)).

To get K: &,tt A is enough t o ensure that, whenever x E K,+l - K,, some element of I , n2, enters A,+1. Then, t o know if x E K ,we look for s so big that all elements of I , n A have been generated in A , , and then see if x E K,. To get A not tt-complete, we define B r.e. such that B g t t A . T h e strategy for this is the usual one: t o spoil t h e e-th reduction, choose a witness a,, wait for pe,,(ae) t o converge, and then put a, in B if and only if ‘T+,,(~,) fails on A . Since requirements of the first kind have t o be satisfied immediately (i.e. they have highest priority), they might interfere with the satisfaction of requirements of t h e second kind. To avoid this, we define a positive equivalence relation 7 which breaks down the boxes into pieces, i.e. such t h a t , at any stage, its equivalence classes are subintervals of the boxes. Also, A, is 7-closed, and consists of initial segments of t h e boxes. In other words, at stage s a typical box I , looks like this:

111.9 The World of Complete Sets *

345

A n Z,

T h e construction is as follows. Suppose t h a t K is enumerated in such a way t h a t , for infinitely many stages, nothing is enumerated in it. At stage s 1 we d o t h e following:

+

0

0

x,,

if z E Ks+l - K,, take the smallest element of Z, n a n d put its equivalence class into A . We will define f in such a way t h a t I, nii, is always nonempty, so t h a t this is always possible.

if no element is generated in such t h a t t h e requirement

K at stage s + 1, look for t h e smallest e 5 s

Re : ' ( B I t t A via p,) has not yet been satisfied and not injured afterwards, and pe,,(a:) converges (where a: is t h e current witness for Re). Consider t h e smallest m 2 e such t h a t all elements used in D , + , ~ ( are ~ ~ in ) Ui<, Ii. We act in such a way to ensure that the truth value of A D ~ ~ depends, , ~ ( ~ after ~ ) stage s 1, only on A n (UiCeI i ) , i.e. on t h e first e boxes. This requires action on I,, . . . ,I,. We show how t o act on I,: t h e same action will be taken on I r n p 1 ,... ,I,. Consider I, at stage s and let no,.. . , ng be t h e largest elements of t h e equivalence classes of I,.

+

Anz,

Since, by construction, we put equivalence classes into A , t h e final value of A n I, will be one of { z : f (m)5 z 5 ni}, for some i 5 q. T h e final value of Zi) is one of 2 f ( m )many (since there are f (m)elements in t h e first m boxes). For each nil there are thus 22f'"' possibilities: it follows that, if q 2 22f'm', then at least two ni's determine t h e same truthvalue of o v e , s ( a zfor ) , any possible choice of A below f ( m ) . In general,

+

if q 2 2T'2f'm)then there are r 1 of t h e ni's which determine t h e same truth-value of for any possible choice of A below f ( m ) .

We will define f in such a way t h a t r is big enough t o ensure t h a t , for any t , ,Z fl& # 8. Thus let r 5 q be the maximum number such t h a t

346

111. Post’s Problem and Strong Reducibilities

there exists a subsequence p o , . . . ,p r of n o , . . . , 71q which determines t h e same truth-value of ope for any possible choice of A below f ( m ) . Let, I , n As+l = (2 : f(711) I I P O } , and, for i < r , let { z : p , < z 5 p,+1} be t h e new equivalence classes of on I,. Since in t h e future

I,

71

n A = { z : f(771) i z 5 p i }

for some i 5 T (by construction), t h e truth-value of only on t h e first, m boxes.

depends now

W e proceed similarly on I m P l.,.., I e by descending induction, determining I , n A,+1 for e 5 i 5 m,a n d in t h e e n d we have

Note t h a t t h e truth-value of A uve,s(a:) deperids only on t h e first e boxes. P u t a: into B if a n d only if A,+, ove,s(az) fails. T h e requirement R, is now satisfied, a n d can be injured only if something changes on t h e first e boxes. Of course we could have gone all t h e way down to 10,to fix t h e truth-value of up=s(az) once a n d forever. T h e trouble with this is t h a t t h e size of t h e boxes

is finite, a n d we a r e not able to avoid a collapse, if every condition is free t o interfere with every box. By lei,i,irig Re interfere with I, only for i 2 e, we ensure t h a t only finitely many conditions interfere with a given box I,, a n d we a r e t h u s able to show t h a t I , n & is always nonernpty (something which is needed t,o satisfy t h e first kind of requirements). It only remains t o determirie f. Note t h a t t h e internal situation of I , can be changed only if either z gets into K ,or some Re with P 5 z is satisfied. Now Re can be satisfied at two different stages only if it gets injured between t h e m , a n d to injure Re we must t a k e action on Ut<eI,. T h i s is possible only if s o m e i < e get,s into K (which c a n happen only e times), or some R, with i < e is satisfied. By induction, it follows t h a t Re can be satisfied less t h a n 2e+1 times, 2e+* = Yt2 times. I t is t h u s enough to let: hence I , can change at most Ce<= -

f(0) = 0 f(z+ I ) = 22=+*2’(”).

0

Exercise 111.9.2 Them is a n acceptable system of indices for which Post’s simple set is tt-incomplete. (Lachlan [1975]) (Hint: apply the method of 111.3.6 to the above

111.9 The World of Complete Sets

*

347

proof.)

We briefly discuss now bounded truth-table reducibilities (p. 340). The interesting fact is that the notions of bd, bp, btt and bwtt-completeness coincide (Lachlan [1975], Kobzev [1974], [1977]), the first three because a btt-complete set is d-complete and p-complete (III.8.11),and the last two with a proof similar t o 111.9.1. Moreover, from the next theorem it follows that bc and m-completeness coincide, and thus there are only two interesting notions of completeness for bounded truth-table reducibilities.

Theorem 111.9.3 (Lachlan [1966]) If A . B is m-complete, so is a t least one of A and B . Proof. We want t o build an r.e. set D such that one of the following holds: 1. K < , D < , A 2. K

<,

D 5, B.

Note that, no matter how we define D , it will be an r.e. set. Since A . B is m-complete, there will be a recursive function h such that

X E D e h(z)EA.B, and thus there will be two recursive functions f and g such that

x E D e f(x) E A A g(x) E B. By the Fixed-Point Theorem we can use D itself in its own definition, and hence we may suppose f and g given beforehand. Consider the set

D* = (2 : ( 3 t ) (@~Dt A&) For z E D* we have z E D two cases: 0

6 f(z) E

E Bt}.

A , since g ( z ) E B already. There are then

If D* is infinite, let {Q, z1,. . . } be a recursive enumeration of it. Then z, E D e f(z,) E A , and we can easily ensure condition 1 by letting

xEKez,ED. 0

If D* is finite we have z E D H g(z) E B , with at most finitely many exceptions. Indeed, if z E D then g(z) E B ; and there are only finitely many z such that g(z) E B A z 6 D , since they must be in D'. Then we can easily ensure condition 2 by letting D and iC differ only finitely.

111. Post's Problem and Strong Reducibilities

348

The corist,ruction of D is the following. At stage s an enunieration (20,.. . , z I } of the element,s of

+ 1 we have K,,D,, and

D,* = ( 2 : (35 S ) ( Z !$ Dt A ~ ( z E ) Bt}. Then: 0

0

if x E Ks+l,let z , E Ds+l (if z, exists, i.e. if enough elements have already been generated i n D * )

if z E Ks+l and x # L):, let x E Ds+l (this kills z,in the sense that it ensures x # D ' , since IC goes into D ) .

If L)' is infinite then x E K H z, E D: if x E K then z , E D by the first part of the construction; and if z , E D then it must be z E K ,since the second part of the construction puts in D only elements of F , while z, E D * . If D* is finite, then the second part of the construction applies, except for finitely many cases, and thus, for almost every z , z E K tj x E D. 0 Corollary 111.9.4 A bc-complete set is 7n-complete.

Proof. A set A is bc-complete if there is a recursive function f such that

where the size of Df(,) is bounded by a fixed riiini1,er 12. By possibly adding element,s of A to it, we can suppose that D f ( , ) always has exactly 71 elements. But then there are n recursive functions fi such that

But then A . . . A is m-complete and thus, by repeatedly applying the theorem, so is A .

0

Structural properties and completeness We now sum tnarize the connections between structural properties on one side, and completeness properties on the other. We first look at properties implying completeness.

1. A set is creative if and only if it is 1-complete (or m-complete) (111.6.6, 111.7.5).

2. A set is quasicreative if and only if it is d-complete (111.6.16).

111.10 Formal Systems and R.E. Sets

*

349

3. A set i s subcreative i f and only i f it is Q-complete (111.6.20).

4. An effectively simple set is T-complete (111.2.18), but i t m a y be Q-incomplete (III.4.24.b, 111.4.10) or wtt-incomplete (111.8.19). 5 . A strongly effectively simple set i s Q-complete (III.6.21.a), but i t m a y be wtt-incomplete (111.8.19).

We then look a t properties implying incompleteness. 6. A simple set is not d-complete, and hence not btt-complete (111.6.18.a), but i t m a y be c-complete (111.3.5). 7. A hypersimple set is not wtt-complete (111.8.16),but it m a y be Q-complete (p. 297). 8. A hyperhypersimple set is not Q-complete (111.4.10), but i t m a y be T-complete (111.4.23).

9. A semirecursive set is not p-complete, and hence not btt-complete (111.5.3.b), but it m a y be tt-complete (III.5.5.a) or Q-complete (p. 297). We finally look a t naturally defined sets with completeness properties.

10. K is 1-complete and m-complete. 11. Post’s simple set (111.2.11) is wtt-complete (being effectively simple and not hypersimple, 111.8.18) and not m-complete (being simple), and can be tt-complete (actually c-complete, 111.3.6) or not (111.9.2), depending on the acceptable system of indices. 12. T h e deficiency set of (being hypersimple).

111.10

K is T-complete (111.3.13) but not wtt-complete

Formal Systems and R.E. Sets *

As we have seen (p. 253), the theory of r.e. sets arose from problems connected with the study of formal systems, and Post’s problem was motivated by methodological considerations. But we have pushed the study of r.e. sets quite far, and it is natural t o enquire whether we have lost touch with ‘reality’. We will see here that, far from being so, many of the notions so far introduced have bearing on our original study of formal systems, and provide appropriate tools t o describe natural phenomena.

350

111. Post's Problem and Strong Reducibilities

Formal systems and r.e. sets

*

An abstract approach t o formal systems, which isolates their basic properties, is t he following. Fix a formal countable language, a n d identify (by a r i t h m e tization) th e sets of well-formed formulas and of sentences (formulas with no free variable) with two recursive sets F a n d S , with S C F .

Definition 111.10.1 A formal system F in the given language is a pair (T,R) of r.e. sets contained in S and interpreted, respectively, as the sets o f (codes of) theorems and of refutable formulas. .F is said to be: 1. consistent i f T n R = 0

2. complete i f T U R = S 3. decidable i f T and R are recursive

4.

undecidable i f T is not recursive.

By Post's Theorem, a consistent and complete formal system is decidable (Janiczak [1950]): given t h e code n E S of a sentence, enumerate T and R simultaneously, until n appears in one (and exactly one) of them. Usually T is generated by isolating a n r.e. subset A z of S (set of axioms), a n d w a r y functions r : S" + S (called rules of deduction). T hen T is t h e closure of A z under t h e rules. R can be generated in t h e same way, a n d independently of T, or there may be a function n : S --* S (called negation) such th a t R = n(T).

Exercise 111.10.2 Every formal system can be generated by means of finitely many recursive rules of deduction, fmm a finite set of a x i o m . (Kleene [1952]) (Hint: since T is r.e., for some recursive R is z E T e 3yR(z,y). As a system of axioms take the equations of a system which Herbrand-Gael computes the characteristic function f of R. As rules, take R1 and R2, together with the rule that produces z from

f(c,y) = i.)

While t h e abstract notion of formal system is certainly broad enough to include all t h e common examples, the formal systems associated to r.e. sets (like in 111.10.2)may look unnatural and ad hoc. In mathematics one usually uses first-order formal systems, whose axioms include all axioms of firstorder logic with equality, t h e set of theorems is generated from t h e axioms by t h e first-order logical rules, and t h e refutable formulas are t h e negations of theorems. For first-order formal systems, inconsistency implies that every sentence is a theorem, bccause if 'p and "p ar e both provable then so is ' c p V (i.e. 'p -++) for any sentence +, and then follows by modus ponens. T h e next result shows t h a t t he r.e. sets d o describe (at least u p to T-equivalence) formal systems of common use.

+

+

111.10 Formal Systems and R.E. Sets

*

35 1

Proposition 111.10.3 (Feferman [1957])E v e r y r. e. T-degree contains a first-order fo m a 1 s y s t e m . Proof. Let A r.e. be given. The idea is to consider a decidable theory, and code A into it. Consider the language of field theory (containing =,O,l,+,x). First note t h a t : The theory o f algebraically closed fields of any given characteristic is complete and decidable. If neither cp nor -cp are provable, both are consistent with the theory, and by the Completeness Theorem of first-order logic there are two algebraically closed fields of the given characteristic and same uncountable cardinality, satisfying one 'p, and the other -9.But this is against Steinitz Theorem, according to which any two such fields must be isomorphic, and in particular must satisfy the same formulas. Thus t,he theory is a complete and consistent formal system, and it is then decidable.

-

We use field characteristics to code the given r.e. set A : let TAbe the t,heory of algebraically closed fields whose characteristic is not p,,, for any n E A (i.e. add to the axionis of algebraically closed fields the statements 1 1 # 0,

+ +

for

71

E A ) . Then:

AITTA Indeed n E A

H krA

1

pn times

+ . . + 1 # 0, for the following reasons.

pn times

If n E A then 1 if n

#A

+ . . . + 1 # 0 is obviously provable, being an axiom. And

pn times

their 1

+ . . . + 1 # 0 cannot be provable, because in this case TA

pn times

is consistent with the theory of algebraically closed fields of characteristic Pn T A

ST A

By definition TA is r.e. in A . It is then enough to show also that its complement is r.e. in A . Since a formula cp is provable in T A if and only if it is true in some algebraically closed field of characteristic 0 or p , , for some n $ A , by completeness of the theories of algebraically closed fields of given characteristic we have t hat cp is not provable in TA if and only it,s negation is true in some field of characteristic either 0 or p , , for some n # A . Then, by decidability of the same theories, is r.e. i n A . 0 More results on the connections between degrees and formal systems are in Feferman [1957], Shoenfield [1958], and Ilanf [1965]. I n particular, e v e r y

352

111. Post’s Problem and Strong Reducibilities

r.e. tt-degree contazns a Jinztely axioinatazable first-order formal system. On t h e other haiid, not every r.e. m-degree contazns a first-order formal system, because closure under conjunction implies t h a t if A is t h e set of t h e o r e m of a first-order formal system, then A . A A , a n d this fails in general for r.e. sets (see 111.8.5).

s,,%

Undecidability To express in full generality t h e methods to prove undecidability, suppose t h a t t h e language contains constants Ti, for every n. Fix a n effective enumeration {$,},,, of t h e formulas of t h e language with o n e free variable, a n d suppose t h a t there is a recursive substitution function s, such t h a t s ( e , n ) is tlie code of $,(Ti) as a sentence. T h e notion of representability of predicates (13.4) c a n be easily generalized to this abstract setting: a set A is weakly representable in .?=if, for some t, z E A ts s ( c , x ) E T. Note t h a t every set weakly representable i n a fornial system iiiust be r.e. T h e direct methods to prove t h e uridecidability of a forrrial systeni a r e suniniarizecl in t h e following:

Theorem 111.10.4 Direct undecidability proofs (Tarski, Mostowski and Robinson [1953], Bernays [1957], Putnam [1957], Smullyan [1958], Vaught) A formal system F is undecidable i f one of the following holds: 1. Every recursive set zs weakly representable in F.

2. Some nonrecursive r.e. set is vimkly representable in 3. Proof. If 3 k decidable, t h e diagonal set E 7”

is recursive, a n d then so is there is e such t h a t

T*. I f 72

E

s ( n , n )E T every recursive set is representable in

F,

ts s ( e , n ) E T .

71 = e we get a contradiction, and this proves t h e first part. T h e sccontl method is trivial. if .F is decidable, every set weakly r e p r e sentable in it m u s t be recursive. 0

For

T h e two methods a r e silhstantially different in principle: tlie undecidalility is forced in o n e case b y t h e quantity of sets represented, despite t h e fact t h a t all

111.10 Formal Systems and R.E. Sets

*

353

of t h e m may be computable ( t h e underlying reason being t h a t there is n o recursive enumeration of all t h e recursive sets), in t h e other by t h e (noncomputable) quality of some of them. I t is not obvious however t h a t they a r e distinct, since i t could be t h a t once all t h e recursive sets a r e weakly representable, also some nonrecursive r.e. set is. Actually, for t h e examples of 11.2.17 much more is true: every r.e. set is weakly representable. In particular, all these examples weakly represent K ,a n d t h u s t h e sets of their theorems a r e all T-complete. T h u s the

positive solution to Post’s Problem implies that not every undecidable formal system can weakly represent every r.e. set, b u t this does not solve our question yet. T h e answer has been obtained by Shoenfield [1961], with a formal system in which the weakly representable sets are exactly the recursive ones. T h e proof introduces a difficult extension of t h e priority method used for t h e solution of Post’s Problem, a n d will be given in X.3.10. T h e r e is also a n indirect method to prove undecidability, a n d t h i s was Post’s direct concern. Formal systems can be interpreted one into another. T h e simplest way is having a recursive function f t h a t translates forniulas a n d preserves theorems: given 3 a n d 3’ with sets of theorems F a n d F’, 2

E F

e f (x)E F’.

But t h i s is only one possible way, a n d we can say in general t h a t 3 is interpretable in 3’ if F
positive solution to Post’s Problem implies that there are undecidable formal systems that cannot be proved undecidable b y interpreting R in them.

Essential undecidability Since a consistent formal system 3 is described in t e r m s of a pair of disjoint r.e. sets ( T ,R ) contained in a recursive set S , a n d respectively coding t h e theorems a n d t h e refutable formulas, we may first of all expect t h a t t h e theory of pairs of r.e. sets might be relevant.

Definition 111.10.6 (Tarski [1949]) A consistent formal system F = ( T ,R ) is essentially undecidable if T and R are recursively inseparable.

111. Post’s Problem and Strong Reducibilities

354

If we say t h a t F‘ = (TI,R’) is a n extension of 3 = ( T ,R ) (in t h e s a m e language) when T‘,R’ C S , T C T‘ a n d R C R’, t h e n essential undecidability means t h a t not only the system itself, but also every consistent extension of it in the same language is undecidable. T h e notion of representability of sets can be generalized to pairs. Recall t h a t s is t h e substitution function, a n d s ( e , n ) is t h e code of t h e formula + , ( R ) , for a given enumeration {$,},Ew of t h e formulas of t h e language with o n e free variable.

Definition 111.10.7 (Kleene [1952], Putnam and Smullyan [1960]) Given a formal system F = ( T , R ) and two disjoint sets A and B , we say that A and B are separable in 3 i f there is e such that zEA

+ s(e,z)

E

T and z E B

5 s(e,z) E

R.

As in t h e case of simple undecidability, we have two methods to prove essential undecidability. Theorem 111.10.8 Direct proofs of essential undecidability (Kleene [1952], Tarski, Mostowski and Robinson [1953], Smullyan [1958], Putnam and Smullyan [1960]) A consistent formal system F is essentially undecidable i f one of the following holds: 1. Every pair of disjoint recursive sets is separable in F. 2. Some recursively inseparable pair is separable in

F.

Proof. Note t h a t everything separable in F remains separable in a n y consistent extension of it. And if every pair of disjoint recursive sets is separable in a consistent system (T’,R’), every recursive set is weakly representable in it (given A recursive, let e be t h e niimber of t h e formula separating A a n d 2: then

SEA zE

3

2+

s(e,z)ET’ s ( e , z ) E R’ 5 s ( e , z ) @ T’

s ( e , z ) E TI). T h e n every consistent by consistency, a n d hence z E A extension of F is undecidable, by 111.10.4 (because every recursive set is weakly representable in i t ) , a n d 3 is essentially undecidable. T h e second p a r t holds because F ,if not essentially undecidable, has a decidable consistent extension (T’,R’),in which t h e given recursively inseparable pair ( A ,B ) is still separable: zE

A

+ s ( e , z) E T’

and z E B

3

s(e, z) E R’.

But t h e n t h e recursive set {z : s ( e , z ) E TI} would s e p a r a t e A a n d B.

0

111.10 Formal Systems and R.E. Sets

*

355

Exercises 111.10.9 A set A is representable in F = (T,R ) if, for some e, z

EA

+ s(e,z) E T

and z E 2

* s(e,z) E R,

i.e. if A and 2 are separable in F. a) Every recursive set is representable in

F i f and only i f every pair of disjoint recursive sets is separable in it. b) If some nonrecursive set is representable in F,then some recursively insepamble pair of sets is separable in it, but not conversely. (Hint: in R only recursive sets are representable, by 11.2.16, but every pair of disjoint r.e. sets is separable, see below.)

As already for t h e undecidability proofs, t h e two methods for essential undecidability a r e substantially different a n d distinct. We now see t h a t they both apply, as usual, to consistent extensions of R. Theorem 111.10.10 (Rosser [1936],Putnam and Smullyan [1960])I n a n y consistent formal system 3 extending R,every disjoint pair of r.e. sets is separable. Proof. If A a n d B a r e disjoint r.e. sets, there exist recursive relations R a n d Q such t h a t zE

A

R

3yR(z,y) a n d z E B

R

3gQ(z,y).

We already know t h a t all t h e recursive relations a r e representable in .F (11.2.16), a n d then there a r e formulas $1 and $2 such t h a t

qz,Y) l-F Q(z,Y) 3 I-F

$11

(TIV ) a n d

l R ( 5 ,Y)

and

~Q(X,Y)

$2(%V)

3

I- l?h(%V)

* l- 7$2@,?7).

T h e n t h e formula

cp(4

~Y[$&,Y)

A (vz


separates A a n d B : z E A 3 l - cp(Z) ~ If z E A then, for some y, R(z,y) holds, a n d hence I-F

$l(?f,jj). Since

A a n d B a r e disjoint, z @ B: hence, for every z , ~Q(x,z) a n d so I-F 7$2(T,Z). By t h e axioms of R we can treat bounded quantifiers, a n d hence we also have I-F (b'z

xEB

< ?7)7$2(Z,z).

I-F -cp(T) Note t h a t -cp(x) H b'y[+~l(z,y) V (32 for some z , Q(Z,Z) holds, and hence I-F

Then I-F cp(T).

< y)$z(z,z)]. If z

E B then, $2(T,Z). Since A a n d B a r e

356

111. Post’s Problem and Strong Reducibilities disjoint, 5 @ A and so, for every y, -R(z, z ) holds, a n d l - ~~@ ~ ( T , jBy j). t h e axioms of R we know t h a t , for every y a n d a fixed Z , y 5 Z V 7Z < y. T h u s we only have to consider finitely many cases because, again by t h e axioms of R, y 5 Z means t h a t y is actually t h e numeral corresponding to some y 5 Z . If y 5 Z then l - ~-+l(Z,jj).If Z < y t h e n , from +z(Z,Z), we have (32 < y)+2(Z, 2)). T h u s l - ~i(p(5). 0

Corollary 111.10.11 If F is a consistent formal system extending R then 3 is essentially undecidable, and the sets of (codes o f ) theorems and of refutable formulas of F are effectively inseparable (and in particular creative). Proof. Both criteria for essential undecidability a r e applicable, since every pair of disjoint r.e. sets is separable in 3, in particular every pair of disjoint recursive sets, a n d any recursively inseparable pair. Moreover, separability of A a n d B provides a simultaneous m-reduction of them to t h e sets of theorems a n d of refutable formulas. These a r e t h e n effectively inseparable, because there is a pair A a n d B of effectively inseparable r.e. sets. 0 Smullyan [1961] calls F = (T,R) a Godel theory if every r.e. set is weakly representable in it, a n d a Rosser theory if every pair of disjoint r.e. sets is separable in it. These two notions simply express m-completeness, respectively for r.e. sets a n d pairs o f disjoint r.e. sets. T h u s i f F is a Godel theory then T is creative, and i f F is a Rosser theory then T and R are effectively inseparable. T h e fact t h a t two disjoint creative sets may be recursively separable (III.6.26.d) can be seen as saying t h a t a Godel theory i s not necessarily essentially undecidable. O n t h e other hand, t h e existence of recursively inseparable pairs in any nonrecursive r.e. T-degree (111.6.22) shows t h a t a n essentially undecidable

formal system is not necessarily a Rosser theory. Pour El [1968] calls effectively extensible a theory

F = ( T ,R ) for which there is a n effective procedure t h a t produces, for any consistent extension 3’of it, a formula which is neither provable nor refutable in F’. T h i s notion is clearly equivalent to t h e effective inseparability of T a n d R (see II1.6.27.b), a n d stresses t h e effective content of t h e proof of Gijdel’s Theorem, which effectively exhibits undecidable sentences for any sufficiently strong consistent formal system. T h e last results and comments show t h a t t h e notions of creativeness a n d effective inseparability a r e useful tools for t h e description of common theories, for what concerns t h e description of undecidability a n d related phenomena. T h e limits of this usefulness a r e pointed o u t by 111.7.13 a n d 111.7.15 (and their extensions considered in Pour El a n d Kripke [1967], Pour El [1968]): f r o m a recursion-theoretic point of view, all effectively inseparable (and, in particular, a great number of common) formal systems are isomorphic, a n d t h u s mere

111.10 Formal Systems and R.E. Sets

*

357

variations of one another. To get a finer analysis t he recursive, purely extensional approach (only considering the sets of theorems) has t o give in t o a proof-theoretical, intensional one (accounting for t he way t he theorems are obtained). As Kreisel [1971] puts it, Proof Theory begins where Recursion Theory

ends.

Independent axiomatizability Since consistent theories have disjoint sets of theorems and refutable formulas, if we call a formal system nontrivial when it has at least infinitely many theorems a n d infinitely many refutable formulas, then (the sets associated to) consistent nontrivial formal systems are not simple. This does not mean t hat t h e notion of simplicity, as well as its strengthenings, has no relevance for t h e description of formal systems, as we now see. Until now we have looked at decidability questions, and thus recursive enumerability was seen as opposed t o recursiveness. This depended on t h e fact t h a t we considered formal systems only, t hat is r.e. sets of formulas. But in mathematics theories need not be r.e. (and t h e results on t h e limitations of formal systems show t hat there is no other way t o override t he incompleteness phenomena, while preserving expressive power).

Definition 111.10.12 A set of formulas in a given language is called a firstorder theory i f it is closed under logical consequence. A theory is: 1. axiomatizable i f it is the closure under logical Consequence of an r.e. set of formulas { c Y ~ }called ~ ~ the ~ , axioms 2. independently axiomatizable i f moreover, for every e, the axiom a, is not a logical consequence of the remaining axioms 3. finitely axiomatizable i f it has a finite set of axioms. Note t h a t the axiomatizable theories are just the first-order formal systems, because t h e closure of a n r.e. set of formulas under recursive rules (like t h e ones of any standard complete formalization of t h e Predicate Calculus) is still a n r.e. set of formulas Moreover, axiomatizable theories always have a recursive set of axioms (Craig [1953]), because any formula is implied by t h e conjunction of itself with other formulas, and thus we can substitute t h e n-th axiom with t h e conjunction of t h e first n ones, obtaining a set enumerated in increasing order, a n d hence recursive. Finally, a finitely axiomatizable theory is independently

axiomatizable. Proposition 111.10.13 (Kreisel [1957], Pour El [1968a]) The following are equivalent, for an axiomatizable first-order theory F:

111. Post’s Problem and Strong Reducibilities

358

1. 3 is independently miomatizable 2. for any axiomatization {a,},Ew of 3, the set nEA

@ a0

A-.. Aa,

k

is not hypersimple 3. for some axiomatization {cY~},.~ o f 3 , the set

nE A

($

cro A s . . A a,

k

is not hypersimple.

Proof. Note th at t he conditions are trivially equivalent when 3 is finitely axiomatizable (because in this case t he set A is finite, a n d certainly not hypersimple), an d then we can suppose t hat 3 is not finitely axiomatizable. We first prove t h a t if 3 is independently axiomatizable, and {a,},cw is any axiomatization of it, then t h e set TZ

E A @ a0 A

... A LY,

I= C Y , + ~

{Pn}nEw

is not hypersimple. Fix a n independent axiomatization of 3. It is enough t o show th at , given any n, we can effectively find a n m > n such t h a t {n,. . . ,m - 1) # 8 (since this permits a n inductive generation of a strong array intersecting Given ao, . . . , a, we can find first of all a number p such t h a t they ar e deducible from PO,.. . , &. Then we can find a number m such t h a t &+I is deducible from ao, . . . ,a,. Now one of n, . . . , m - 1 is not in A otherwise, inductively, all of a,+l,. . . , a , would be deducible from (1~0,. . . ,a,, and hence from p0,. . . ,&. But then so would be &,+I, contradicting t h e fact t h a t th e /?’s are a n independent axiomatization. We now show th at if, for any axiomatization of 3,t he set

nx

x).

n E A@aoA

... Acr,

+antl

is not hypersimple, then 3 is independently axiomatizable. We d o this in two steps:

there is an effective procedure that produces, given any theorem y of 3, another theorem yl that is not deducible from it By hypothesis, we have a strong array { D f ( z ) } z E w t h a t intersects 2. Given any theorem y of 3,first find a n m such t h a t y is deducible from 0 0 , . . . , a m and r then an z such t h a t Df(.) contains only numbers greater than m. T h e conjunction yl of t h e a,+l’s such t h a t n is in D f ( = )is then a theorem of 3 (because a conjunction of axioms) t h a t cannot be

111.10 Formal Systems and R.E. Sets

*

359

deduced from y. Indeed, one number n in Df( is not in A , i.e. an+l ? cannot be deduced from t he a’s with smaller indices, and in particular is not deducible from y. But a,+1 is a conjunct of yfrand thus y’is not deducible from y either. there is a31 independent axiomatization of F Let { a n } n E w be a n enumeration of t h e theorems of F. We want t o generate this set with a set of independent axioms {Pn}nEw.T h e first part of th e proof produces, given any formula y, a formula y‘ not deducible from it. T h e opposite is not necessarily true, since y f could be t o o strong, an d imply y. To have independent formulas, we have t o relax y’a little, a n d this can be done by considering yy V y’,i.e. y + y’. T hi s is not deducible from y as before, since otherwise (by modus ponens) so would be y’.And if y is not valid, it is not deducible from y + y’, otherwise from +y (which is stronger than -y V y’) we would get y,a n d then y would be valid. More generally, t h e sequence

y

y + y’ y’4 y”

..

is independent. We still have t o make sure t h a t all t he an’s are going to be deducible from the axioms, and we simply add them one by one as conclusions, so t h a t an can be obtained from t he first n 1 axioms. Th u s our set of independent axioms for 7 is a set of formulas inductively defined as follows. First we let

+

po = ah. This makes sure t hat we start from a formula t hat is not valid. Then, if Pn = 6n ynr +

Exercise 111.10.14 For any hypersimple set A there is a consistent first-order formal of it, such that system F extending R , and a n miomatiration 7~

EA

* a0 A ... A an

a,+ 1

In particular, F is not independently axiomatizable. (Kreisel [1957])(Hint: let (xo be the conjunction of the finitely many axioms of Q, see p. 23, and of the formula Vx(cp(z) + P ( x ) ) , where P is a new predicate, and ‘p weakly represents A in R. Moreover, let on+] be P(5i). Then a,+] can be deduced from a0 only when n E A . )

Pour El [1968a] shows t h a t Q, and more generally every theory with effectively inseparable sets of theorems and of refutable formulas, has a consistent

360

111. Post’s Problem and Strong Reducibilities

extension which is not independently axiomatizable, a n d with t h e same language as t h e original theory. For more recursion-theoretical results a b o u t fornial systems see p. 510, as well as Smullyan [1961], Martin a n d Pour El [1970], a n d Downey [1987].

Chapter IV

Hierarchies and Weak Reducibilities T h e t h e m e of this chapter is definability in given languages, a n d a classification of sets a n d relations according to their best definition. Of course definability is not a n absolute notion, a n d it depends on t h e given language. Here we will consider three natural ones, t h e first two for Arithmetic, t h e last o n e for Set Theory. T h e first two will differ in t h a t we will allow only number quantifications in o n e case, b u t also function (or set) quantifications in t h e other, a n d will define, respectively, t h e arithmetical a n d analytical sets. T h e t h i r d approach will lead us to t h e constructible sets. Definability is a linguistical, more t h a n computational, notion. However, we already know t h a t it is possible to characterize t h e recursive sets in a purely linguistical way, see 1.3.6. T h i s suggests t h e possibility of considering definability as a n abstract version of computability, and we will see in later chapters t h a t this program is indeed feasible, for some of t h e definability notions t h a t will be introduced in this chapter.

We only scratch t h e surface of t h e subject in here, a n d refer to Volumes I1 a n d I11 for a detailed s t u d y of t h e arithmetical a n d t h e analytical sets, to which t h e two volumes a r e respectively dedicated. B u t we will prove a number of interesting results already in this chapter, providing some nontrivial characterizations of a number of classes. In particular, we deal with: t h e limit sets, t h a t can be obtained as limits of recursive functions; t h e hyperarithmetical sets, t h a t can be effectively computed modulo number-theoretical quantification; a n d t h e Xi sets, t h a t c a n be defined over t h e constructible universe in a particularly simple way.

36 1

362

IV. Hierarchies and Weak Reducibilities

A: = analytical I I I I I I

I

n: liyperarithmetical I

I I I I

I I

A: = Ah = arithnictical I I I I

I I I

Figure IV. 1 : The Aritlirnetical and Analytical Hierarchies

IV.l The Arithmetical Hierarchy

IV.l

363

The Arithmetical Hierarchy

We start by considering, as t h e simplest framework for t h e definition of sets of natural numbers, First-Order Arithmetic. We first introduce t h e notion of arithmetical definability, a n d then classify sets definable in Arithmetic by looking at their best possible definitions. We will t h u s obtain classes closely related, b o t h in computational content a n d in structure theory, to the classes of recursive a n d recursively enumerable sets.

The definition of truth

* Pilate said unto him, W h a t is t r u t h ? And when he had said this, he went o u t . (John, Gospel, XVIII, 38)

T h e idea for a definition of t r u t h comes from Aristotle (Metaphysica, r 7, 1011b, 25-27): it is false to say t h a t t h e being is not or t h a t t h e non-being is; it is t r u e to say t h a t t h e being is a n d t h a t t h e non-being is not. T h u s e.g. ‘it is raining’ is t r u e if a n d only if it is raining, where t h e quoted phrase is t h e one whose t r u t h we a r e trying to establish, a n d i t is considered as a purely syntactical object, while t h e unquoted phrase is taken semantically, for what it means. And t h e quoted phrase is t r u e if a n d only if it reflects what happens in t h e world, as expressed by t h e unquoted phrase. For natural languages this explains t h e meaning of t r u t h , b u t it is not particularly manageable. T h e advantage of formal languages is t h a t they a r e built by induction, a n d thus we can actually apply t h e ideas just introduced t o produce a n inductive definition of t r u t h . T h e original definition will be applied directly only to atomic formulas, while for compound formulas we will rely on induction, a n d will force only t h e interpretation of t h e new symbols introduced. T h i s was first done by Tarski [1936].

Truth in First-Order Arithmetic T h e r e a r e two natural first-order languages t h a t come to mind, for t h e purpose of classifying arithmetical definitions. They a r e in some sense extreme examples, allowing respectively for a minimal and a maximal set of nonlogical primitive functions a n d predicates. We consider both of them, a n d prove t h a t their definitional power is t h e same.

364

IV. Hierarchies and Weak Reducibilities

Definition IV.1.1 Definition of truth in Arithmetic (Tarski [1936]) Let C be the first-order language with equality, augmented with constants ?I for each number n, and binary function symbols and x, and let A be the intended structure for C, i.e. the natural numbers, with the usual s u m and product. cp) is inductively Given a closed formula cp of C , cp is true in A (A defined as follows:

+

+

A n n-ary relation P is definable in First-Order Arithmetic (briefly, is arithmetical) if, for some formula cp with n free variables,

As we have already noted, this definition explicitly defines the meaning of a n y syntactical first-order formula over Arithmetic by first reducing t h e nieaning of compound statements to t h e meaning of simpler ones, a n d t h e n by forcing t h e meaning of t h e function constants + a n d x t o agree with t h e usual standard meaning of sum a n d product. After this, being arithmetical is then defined in t h e same way as being representable (1.3.4),b u t with t h e notion of provability in a formal system replaced by t h e notion of t r u t h in First-Order Arithmetic. T h i s procedure of defining t,ruth for formulas of a given language over some structure is quite general, a n d in t h e future we will simply indicate t h e changes needed to extend t h e above definition of t r u t h to different languages a n d structures. Definition IV.1.2 (Kleene [1943],Mostowski [1947])Let L' be the firstorder language with equality, augmented with constants 7 i for each number n, and a relation symbol c p for ~ each recursive relation R, and let A* be the intended structure for C * , i.e. the natural numbers, with all the recursive relations. Given a closed formula cp of C*,A* cp is defined inductively as above, starting from

+

IV.1 The Arithmetical Hierarchy

365

A n n-ary relation P is in the Arithmetical Hierarchy if, for some formula cp of C' with n free variables,

P ( z ~ ,.. ,zn)H A*

+

~p(Z1,.. . ,?En).

I3riefly stated, the Arithmetical Hierarchy consists of the relations definable in First-Order Arithmetic, with the recursive relations as parameters. This seems natural from our point of view, since we want t o classify sets and relations according t o their noneffectiveness, as expressed by t h e complexity of their definition, a n d thus t h e recursive relations may be given for free. T h e next result shows t h a t we are classifying t h e same sets as before.

Theorem IV.1.3 (Godel [1931]) T h e Arithmetical Hierarchy contains exactly the arithmetical relations. Proof. One direction c o m e from t he fact t hat t h e relations represented by t h e atomic formulas of t h e language C are recursive, being built up from plus a n d times only. T h e other direction follows from t he fact, proved in 1.3.6, t h a t t h e recursive relations are representable in R,an d hence (since t h e axioms of R are tr u e in d)are arithmetical. 0

The Arithmetical Hierarchy Before we can classify relations according t o their definition in C*,we need to be able t o p u t these definitions in some kind of normal form. This is easily accomplished in a standard way, by manipulation of quantifiers.

Proposition IV.1.4 T h e following transformations of quantifiers are permissible (up to logical equivalence): 1. permutation of quantifiers of the same type 2. contraction of quantifiers of the same type

9. permutation of two quantifiers, one of which bounded

4. substitution of a bounded quantifier with a n unbounded one of the same type. Proof. P ar t 1 is obvious. P a r t 2 can be accomplished by codifying t h e various quantified variables into a single one. E.g., given a formula VZl . . .'JZ,R(ZI,.

. . ,Zn),

this is equivalent t o t h e formula ' J 4 ( 4 1 , .

where z = ( 2 1 , . . . ,xn).

. .,

b ) T L ) ,

IV. Hierarchies and Weak Reducibilities

366

Pa r t 3 is obvious when t h e two quantifiers are of t h e same type. For t h e remaining cases, consider e.g. (Vz

I a)(3y)R(z1y).

If we let ys be a number such t hat R(z,ys),for z 5 a , then y = (yo,. . . ,y,) witnesses th e tr u th of

(3Y)(V's I a)%

(Y)Z+d,

a n d thus t h e former formula implies t h e latter, while t h e converse implication holds trivially. T h e other case is treated similarly. P a r t 4 is standard:

(Vz

5 a)R(z)

H

(3z 5 a ) R ( z ) H

Vz(z I a + R(z))

3z(z I a A R(z)). 0

Proposition IV.1.5 Prenex Normal Form (Kuratowski and 'Igrski [1931]). A n y relation in the Arithmetical Hierarchy is equivalent to one with a list of alternated quantifiers in the prejk, and a recursive matrix. Proof. T h e previous transformations allow to contract quantifiers of t h e same type, without changing t h e recursiveness of t he matrix. T hus we only have t o show how to push quantifiers in front. This is accomplished by t h e following well-known transformations, read from left t o right, and which again work up t o logical equivalence. First of all note t hat bound variables can be renamed, according to t h e rules: 3 4 4 Yza(z)

* *

34Y) VW(Y),

where y is any variable t h a t does not occur free in a. T hus we can always suppose t h a t , in th e following rules, z does not occur free in p.

7(3z)a

*

-(Vz)a

e (3z)m

(Vz)-a

(3za)A @

H

3z(aA@)

(b'za)Ap

($

Vz(aA/?)

(3za)v g e 3z((Yvp) (Vxa)vp Vz(avp)

*

IV.l The Arithmetical Hierarchy (3zcr) ( k x )

--p --p

-p-(3sa)

p-

tj

@

@

367

vz(a+p) 3z(a--p) 34a-P)

(Vza) e. Vz(a+-p)

0

It should be noted that the rules t o bring a formula into prenex normal form can be applied in any order. In particular, prenex normal forms are not unique, and may have different numbers of quantifiers. However, since there are only finitely many possible manipulations, the prenex normal form with smallest number of quantifiers can always be found, starting from a given formula. Note t h a t we are not claiming that there is a n algorithm t h a t gives t h e best prenex normal form to which t h e formula is equivalent, since better prenex normal forms might be produced by equivalent formulas. T h e Prenex Normal Form suggests that we only have t o count t h e number of alternations of quantifiers in the prefix, t o measure t h e distance from recursiveness. And of course there are, for each such number, two possibilities, according to whether the first quantifier is existential or universal. It just remains t o give everything a name.

Definition IV.1.6 The Arithmetical Hierarchy (Kleene [1943],Mostowski [1947]) 1.

X : is the class of relations definable over A* by a formula of 13; in prenex form v i t h recursive matrix, and n quantifier alternations in the p r e f i , the outer quantifier being mktential.

2.

II: is defined similarly, with the outer quantifier being universal.

3.

A: is Z:na, forms.

i.e. the class of relations definable in both the n-quantifier

4. A: is the class of the arithmetical relations. By extension, we will call a formula E: or @, if it is in prenex normal form, with n quantifier alternations in t h e prefix, the outer one being, respectively, existential or universal. Note also that, by contraction of quantifiers, n quantifier alternations are equivalent to n alternated quantifiers.

The levels of the Arithmetical Hierarchy T h e first levels of the hierarchy are inhabited by old friends. First of all, t h e level 0 obviously consists of the recursive relations (because no quantifier is involved). More interestingly,

IV. Hierarchies and Weak Reducibilities

368

A:

= recursive

E:

= recursively enumerable.

This obviously follows from 11.1.10 and 11.1.19. If we wished to, we could define a similar hierarchy for formulas of C. If we count quantifier alternations, and ask the matrix t o be quantifier free (i.e. a Boolean combination of diophantine equations), we still get t h e r.e. sets at the first existential level, by Matiyasevitch result (see p. 135), a n d thus this hierarchy coincides, from t h e first level on, with t h e Arithmetical Hierarchy. By using C' we simply avoid the proof of this representation theorem for t h e r.e. sets on one side, and have more freedom in t h e computations on t h e other. If we a r e willing t o compromise a little, we may allow t h e matrix to contain bounded quantifiers. Then the proof of IV.1.3 would suffice t o show t h a t t h e r.e. sets a r e at the first existential level of this hierarchy (because of t h e form the formulas t h a t represent recursive relations in R have). T h e A: relations of this hierarchy (namely t h e sets definable with plus a n d times, by using connectives and bounded quantifiers) form the interesting class of rudimentary predicates (Sniullyan [1961]),t o which we will return in Chapter VIII. Exercises IV.1.7 The Bounded Arithmetical Hierarchy (Davis 119581, Harrow 119781) Let A: be the class of relations definable over C by using only connectives and bounded quantifiers. We can stratify it by counting the number of bounded quantifier alternations, getting classes E:,,, and @,n in the natural way. Because of the collapse in d) below, A;,,, is defined as the class of sets A such that both A and 3 are in the n-th level of the hierarchy. and E;,,, are closed under conjunction and disjunction. a) b) For n 2 1, II;,,, is closed under universal quantification bounded by a polynomial, and E&, is closed under exktential quantification bounded by a polynomial. (Hint: by induction, since e.g.

(32I P ( q + 9(q)P(z) (32i P ( q . S ( 4 ) P ( 2 )

* *

+

(3a I P(Z))(3bI 9 ( 3 ) P ( a b ) (3aI P ( q ) ( 3 b < 9 ( q ) q P ( q . b + a).)

c) The matrix can always be reduced to a diophantine equation. (Hint: diophantine equations p ( 2 ) = 0, where p is a polynomial with integral coefficients, are closed under Boolean combinat ions. Precisely,

P ( q =SF) P(3C3fO

* *

P(q=0 P(q
p(q=ovq(q=o

*

p ( q=OAg(q =O

@

p'(z)++'(5)=

*

(32 I q ( q ) ( P ( . 3

P(3c3 < Q ( Z )

p(Z).q(z)=O

0

+ 2 + 1 = 9(Z)).

By part b), quantification bounded by a polynomial is allowed.)

I V . l The Arithmetical Hierarchy d) @,2,+, = Hierarchy wduces to

and

C;,zn+2

@I$

c

369

= C:,2n+l,

c

.G,l

a , 2

and thus the Bounded Arithmetical

G

G , 3

.. .

(Hint: the bounded universal quantification of a diophantine equation is still diophantine. Indeed, consider

R ( z ,.3

* (VY I Z ) @ ( Y , .3 = 0) * ( C P 2 ( Y 1.3) = 0. v
This is a polynomial in y and 2, while we want it to be a polynomial in z and 2. Since p2 is a polynomial, it can be written as qn(2). yn. Thus we have -

n
v
cv<,

It only remains to note that, by induction, yn is indeed a polynomial in z, of degree n+ 1, and with rational coefficients, that can be turned into integral coefficients by taking common denominators.) e) The negation of a set in a given level belongs to the next level, and thus

f ) If for any n two of the previous classes coincide, the hierarchy collapses. (Hint: if two classes coincide, then the common class is closed under both bounded quantifications, and thus is the whole hierarchy.) It is not known whether the Rounded Arithmetical Hierarchy collapses or not.

T h e properties of t h e first level of the Arithmetical Hierarchy a r e inherited at higher ones, by very similar proofs, which we just sketch.

Proposition IV.1.8 Closure properties (Kleene [1943], Mostowski [19471)

1. R i s E: i f and o n l y i f 7 R is rr", R is II", if and only i f T R is E: 2. A: is closed under negations

3. E:, I: and A: are closed under conjunction, disjunction and bounded quantification

4. for n 2 1, X:

is closed under existential quantification, and under universal quantification

5. the universal quantification of a E: relation is quantification of a Pnrelation is E:+~.

I: is closed

Pnfl, and the existential

IV. Hierarchies and Weak Reducibilities

370

Proof. Everything easily follows from logical operations and quantifier manipulations. E.g., let 3xVyR and 3xVyQ be E! formulas. Then, if v , w, z , and w do not occur free in Q or R, (3xVyR ) A (3zVyQ)

and this is a C! formula.

e (3vVw R) A (3zVtQ) e 3v3zVwVt ( R A Q ) ,

0

Theorem IV.1.9 Enumeration Theorem (Kleene [1943], Mostowski [1947]) For each n,m 2 1 there is an m + 1-ary C: relation that enumerates the in-ary C: relations. Similarly for rr",. Proof. This is simply a consequence of the Normal Form Theorem for r.e. relations, 11.1.10, according to which there is an m 1-ary Cy relation enumerating the m-ary Cy relations. Then its negation enumerates the rrr-ary f l relations. Inductively, let R ( e ,x , q be an in 2-ary C: relation enumerating the m 1-ary C: relations. Then V x R ( e ,z, is an rn 1-ary relation enumerating the m-ary fln+l relations, and its negation enumerates the m,-ary C:, relations.

+

+

+

+

Definition IV.1.10 A set A is E:-complete i f it is C:, and every C: set is m-reducible to it. II:-complete sets are defined similarly. The concept of Cz-completeness is the analogue, at level n, of the concept of m-completeness for r.e. sets. By induction we thus have:

Proposition I V . l . l l For each n 2 1, C:-complete and a - c o m p l e t e sets exist.

I n Chapter X we will see that many natural index sets (p. 11.2.8) are plete a t some level of the Arithmetical Hierarchy. Exercises IV.1.12 Partial truth definitions. Let meration of the closed formulas of C. a ) For each n 2 1 , the set

eETn

@

{$e}eew

COIII-

be a n effective eriu-

4Je i s C I : A d b 4 J e

is C:-wmplete. Similarly for rI;. (Hint: first note t h a t t o check, of a given formula, whether it is CE is a recursive procedure. To show t h a t Tn is CE proceed indiictively, using the definition of truth in A: note t h a t if there are no quantifiers then the formulas a r e recursive, a n d their t r u t h can be effectively determined. To show completeness, use the existence of CE-complete sets.)

I V . l The Arithmetical Hierarchy

371

b) The set

(e,n)E T - e E T , , is not arithmetical. (Tarski [1936]) (Hint: if it were, it would be C: for some n. This would contradict the Arithmetical Hierarchy Theorem given below.)

Theorem IV.1.13 Arithmetical Hierarchy Theorem (Kleene [1943], Mostowski [1947]) T h e Arithmetical Hierarchy does not collapse. More precisely, for a n y n 2 1 the following hold: 1. E: - p",# 2.

3.

8,

c E:

and hence A:

fl",- E: # 8, and E$ uII: c A:+1.

c@

hence A:

Proof. T h e first two parts a r e equivalent, by taking negations. T h e proof of t h e first one is similar to t h e one of 11.2.3: if R ( e , x) enumerates t h e unary E$ relations, t h e n P(.)

is E:. But i t cannot be would be e such t h a t

e,

@

R(x,z)

otherwise its negation would be C:, a n d there

R ( e , s ) e -P(x)

@

~R(x,z).

For x = e a contradiction would follow. For t h e last part, note t h a t we can always a d d dummy quantifiers in front or at t h e end of t h e prefix, a n d thus E: U @ A:+1. To show t h a t t h e inclusion is strict, let P E E: - I: then -P E p", - E:. If

s ( x ,Z ) then Q E A:+l.

-3[ P ( x )A

B u t Q is not in

,:I

z = 01 V [-P(x) A

z = 11

otherwise so would be

P ( z ) @ Q ( x ,01, a n d similarly Q is not in Ez.

0

T h e results just proved justify t h e picture of t h e Arithmetical Hierarchy given in Figure 1 (p. 362), where upward connections mean strict inclusion, a n d n o other inclusion holds. We have defined t h e Arithmetical Hierarchy by iterating quantifiers a n d proved, by direct arguments, properties of t h e various levels similar to those of t h e first level. T h e next result gives a different, purely recursion-theoretical, definition of t h e Arithmetical Hierarchy, and explains t h e similarities of t h e various levels. It also suggests t h e possibility of different proofs for t h e results proved above, by relativization of t h e results of t h e first level.

IV. Hierarchies and Weak Reducibilities

372

Theorem IV.1.14 Post’s Theorem (Post [1948],Kleene) A relation is: 1. A:+] if and only if it is recursive in a E: 2. E:+]

or a

I:

relation

i f and only i f it is recursively enumerable in a C: or a I:

relation.

Proof. First note t h a t , for what concerns relative recursive computations, C: a n d I: a r e interchangeable, because a n oracle a n d its complement a r e equivalent. We can t h u s use any of t h e m . W e first prove p a r t 2. Suppose P is C:+]: then, for some R in

q,,

P(.’)

* 3Y R ( E Y)

By t h e relativized version of 11.1.10, P is t h e n r.e. in R , a n d hence r.e. in a @, relation. Conversely, let P be r.e. in a relation &. T h e n P is r.e. in a set A , obtained by coding Q:

re

(XI

, . . ,)z,

qL

EA

*

&(XI,.

. . ,z,).

By t h e relativizcd version of 11.1.10, together with compactness a n d monotonicity (11.3.13), there is a n r.e. relation R such t h a t

P(.’)

* 3u3v(D,

A A D,

2 A R ( Z ,u , ~ ) ) .

Note t h a t

D , C_ A e (Vz E D , ) ( z E A ) . Since D, is finite, t h e quantifier is bounded: b u t A is in I:, whole expression D , C A . Similarly,

a n d t h e n so is t h e

D, C 2 e (Vz E D,)(x $2A ) is E:, because t h e negation of A is used. T h u s b o t h expressions a r e E:+,, a n d so is R, being r.e. ( a n d hence Ev). But E:+l is closed under existential quantification, a n d t h u s P is in it,. P a r t 1 follows from p a r t 2 a n d t h e relativized version of 11.1.19. If P is recursive in a C: relation, then both P a n d 7 P a r e r.e. in it, a n d hence C:+,. But then P is both Xi+, a n d a n d hence A:+, . T h e converse is similar: suppose t h a t P a n d 7 P a r e each r.e. in some (not necessarily t h e s a m e one) E!L relation. T h e n they a r e r.e. in t h e complete C: set, a n d hence P is recursive in t h e complete E: set. 0

q+l,

Exercises IV.1.15 a) For any n 2 1, the union of two Cz sets can be Educed to the union of two disjoint C z sets. (Hint: see 11.1.23.) b) For any n 2 1, there are two disjoint E: sets that cannot be separated by a A: set. (Hint: see 11.2.5.) c) A n y two CE-complete sets are recursively isomorphic. (Hint: use 111.7.13.)

373

I V . l The Arithmetical Hierarchy

A; sets As a special case of Post's Theorem, we get: Proposition IV.1.16 A set is A; i f and only if it is recursive in K. Proof. K: is C!-cornplete, and hence any Cy or @ relations is recursive in it. Thus, being recursive in some Ey or relation is equivalent to being recursive inK. 0 This allows us t o prove a n interesting characterization, useful in constructions.

Proposition IV.1.17 The Limit Lemma (Shoenfield (19591) A is A! if and only i f its characteristic function is the limit of a recursive bnction g , i.e.

Proof. If g is given, then 2

EA

W

W

2 s A g(z,t)= 1) 3stlt(t 2 s + g(2,t) = l ) ,

'ds3t(t

and A is A!. Conversely, if A is A; then A

ST K. Let e be such t h a t

9(x13) Then

CA

is t h e limit of g.

=.CP:

C A CY

,cp:

and

(.).

0

This characterization of A! suggests a possible hierarchy for it, obtained by bounding t h e number of times t h e approximating function g may change. T h e r.e. sets can be obtained by approximations t h a t change at most once (let g ( x , s ) be 0 until it is discovered t h a t z is in the set, and 1 afterwards). If we consider t h e sets whose approximation may change only n times, we have t h e Boolean Hierarchy, which stratifies t h e set belonging to t h e smallest Boolean algebra generated by t h e r.e. sets (or, by 111.8.7, t h e sets btt-reducible t o K ) (Addison [1965], Gold [1965], Putnam [1965], Ershov [1968]). Obviously, A! is not exhausted by t h e Boolean Hierarchy (since there are sets recursive in K and not btt-reducible t o it), but this can be achieved by a n appropriate transfinite extension of t h e hierarchy (Ershov [1968a], [1970]). Exercises lV.1.18 The Boolean Hierarchy (Ershov 119681). For n 2 1, let C,' be the class of sets with a recursive approximation g which changes at most n times,

IV. Hierarchies and Weak Reducibilities

374

n;'

and such that g(z,O) = 0. is the class of complements of sets in Xi', and A,' = C,' nn;'. A set is called 71rr.e. if it is EL1, and weakly n-r.e. if it is A;:-' a) C;' is the class of r.e. sets. b) For n >_ 1, C;' is the class of sets of the f o r m (A1 - Az) U (As - A4) ... , with

Ai r.e. c) For each n 2 1, there are E,'-complete

sets. (Hint: let

(z,y) E A . B * z G A A ~ E B and ( z , y ) E A + B e z ~ A v y € B ,

and consider the sequence of sets

K

K C K

( K . K ) + K : *..)

d) For each n 2 1, E,' - I;' # 0, and I,' - C,' # 0. (Hint: use the sets given in part c.) e) For each 7~ 2 2, there is a A;' -complete set. (Hint: if A and B are, respectively, E;'-complete and II;'-complete, then A @ B is A,:,-complete.) c A;:.' (Hint: use the set given in part e.) f) For any n, C,' UII,' g) For n 2 1, a set is A* :; zf and only if it has a recursive appmximation that Then both A and changes at most n times. (Hint: suppose A is both C,' and I;'. its complement can be approximated by recursive functions g and h that change at most n+ 1 times, and with value 0 at stage 0. For any z, we then know that one of g(z, 0) and h(z,0) is wrong. Look for the first s such that g(z, s) # h ( z ,s), and let g(z, S) be the first value of a new function f approximating A. Let f change only when g does, and g and h differ. Then f changes at most n times.)

Note that there is nothing special about the second level: the characterization of A! obviously extends, by relativization, t o all levels.

Proposition IV.1.19 Shoenfield [1959]) For n 2 1, A is A!+l if and only if there is a n n + 1-ary recursive function g such that

Relativizations

*

We can relativize all the work done so far t o a given oracle X, defining the classes E:x, @" and A:x , and obtaining similar results, simply by substituting 'recursive' with 'recursive in X ' . In particular, we have

ASTB

* A E A:'".

We might thus think that a number of other reducibilities can be defined, by looking at higher levels. The next result shows that this is not the case.

IV.2 The Analytical Hierarchy

375

Proposition IV.1.20 T h e relation

A iao BwAE is transitive only f o r n = 1. Proof. T h e transitivity for n = 1 follows from the fact that in this case l a y is simply Turing reducibility. For any n > 1, we show a counterexample. Let A E E! -R, which exists by the Hierarchy Theorem. Then there is B E pn-l such that z E A w 3yB(z,y). By definition A is in E;lB, and hence in A:B (since n > 1). Since B is in A! by its choice, if 1, is that quantifiers simply sum up: if B has n quantifiers, and we stick n more in front of it, one might collapse (if the leftmost quantifier of t h e prefix of B is of the same type of the rightmost added in front of it), but the others are going t o remain. This ceases to be a problem, if we do not care anymore for a fixed number of quantifiers.

Definition IV.1.21 A is arithmetical in B ( A Sa B ) i f i t is in the Arithmetical Hierarchy relativized t o B. A is arithmetically equivalent to B ( A = a B ) if A l a B and B l a A. Exercises IV.1.22 a) I f A is arithmetical, then A B and B is arithmetical, so is A. b) If A


B for any set B.

Note that 5, is a reflexive and transitive relation, and thus alence relation.

-a

is a n equiv-

Definition IV.1.23 The equivalence classes of sets w.r.t. arithmetical equivalence are called *degrees, and (Wa, 5 ) is the structure of a-degrees, with the partial ordering 5 induced o n them by ia. The structure of the a-degrees will be studied in Chapter XIII.

IV.2 The Analytical Hierarchy The Arithmetical Hierarchy certainly includes only a small portion of all possible sets, being countable (each set has a definition in e*,and there are only

IV. Hierarchies and Weak Reducibilities countably many definitions). Moreover, by Tarski’s Theorem (see p. 166, or IV.1.12.b), there are natural examples of nonarithmetical sets (namely, t h e set of codes of sentences of C t r ue in Arithmetic). We thus feel t h e need of enlarging th e Arithmetical Hierarchy. One natural way t o proceed is by extending t h e language and going t o t h e second order, i.e. by considering quantifiers not only over numbers, but also over functions.

Truth in Second-Order Arithmetic As already for First-Order Arithmetic, there are at least two possible languages of interest. Definition IV.2.1 Definition of truth in Second-Order Arithmetic (Tarski [1936]). Let C 2 be the second-order language with first-order equality, augmented with number constants 7i for each number n, and function constants f for each total function f , among them and x. Let also A 2 be the intended structure for L 2 , i.e. the set of natural numbers and total functions over them. Given a closed formula cp of 132, A 2 cp is defined inductively as usual. A relation P of n number variables and m jknction variables is definable in Second-Order Arithmetic (from parameters) if, for some closed formula cp of with n free number variables and m free function variables,

+

If cp contains no function constant except plus and times, then P is simply analytical. Notice th at ‘analytical’ is defined without t he use of parameters (i.e. function constants other t han and x). Since t h e use of parameters is like t h e use of oracles (see Section 11.3), t he analytical relations are those t h a t can be defined in Second-Order Arithmetic] without any help. T h e reason for t he name ‘analytical’ comes from t h e fact t h a t Second-Order Arithmetic allows for the formalization of elementary Analysis, by coding real numbers as functions of integers.

+

Definition IV.2.2 (Lusin [1925], Sierpinski [1925], Kleene [1955]) Let

C; be the second-order language with first-order equality, augmented with number constants ii for each number n, function constants 7 for each total function ( P R for each restricted recursive relation of numbers and total functions. Let also d a be the intended structure for C;, i.e. the set of natural numbers and total functions over them, with all the restricted recursive relations of numbers and total functions. Given a closed formula cp of La,A; cp is defined inductively as usual.

f , and relation symbols

IV.2 The Analytical Hierarchy

377

A relation P of n number variables and m function variables is in the P r e jective Hierarchy i f , f o r some closed fonnula 'p of Lc,t with n free number variables and m free function variables, P(Zl,.

. . ,ZTL, fl, .. . , fm) @ A3

+

Cp(:1,.

.. ,%l

-

-

f 1 , . . ,f,). *

If

'p contains no function constant except plus and times, then P is in the Analytical Hierarchy.

I t follows from IV. 1.3 relativized t h a t the Analytical Hierarchy contains exactly the analytical relations.

The Analytical Hierarchy Function quantifiers can be manipulated in a way similar to number quantifiers.

Proposition IV.2.3 The following transformations of function quantifiers are permissible (up t o logical equivalence): 1. permutation of quantifiers of the same type

2. contraction of quantifiers of the same type 3. permutation of a number quantifier and a function quantifier

4.

substitution of a number quantifier with a function quantifier of the same type.

Proof. P a r t 1 is obvious. P a r t s 2 needs only a way to code a n d decode finitely many functions, e.g.

P a r t 3 needs a way to code a n d decode infinitely many functions, e.g.

fn(z) if x = ( n , z ) otherwise

IV. Hierarchies and Weak Reducibilities

378 Having this, a n d given e.g.

PX)

(3f)R(f,).

7

let f n be such t h a t R ( f n , n )holds. Then t he given formula is equivalent to

( W ( V X c ) R ( ( f ) z , .). P a r t 4 is easily obtained, noting t hat

3aR(z)@ 3 f R ( f ( O ) )and V x R ( x )@ VfR(f(0)).

0

Proposition IV.2.4 Prenex Normal Form (Kuratowski and Tarski [1931]). A n y relation in the Analytical Hierarchy is equivalent to one with a list of alternated. function quantifiers in the p r e f i , and a n arithmetical matrix. Proof. Similar to IV.1.5.

0

Proposition IV.2.5 (Kleene [1955]) The matrix of a relation in the Analytical Hierarchy can always be reduced to a single number quantifier, opposite in type to the rightmost function quantifier of the p r e f i , followed b y a recursive predicate. Proof. Having a formula in prenex normal form, with a prefix consisting of alternated function quantifiers, and a n arithmetical matrix, there a r e t w o cases. If there are at least two function quantifiers, then all t h e number quantifiers can be eliminated, by first moving near t o t h e function quantifier of t h e same type, then rising t o function quantifiers, and being contracted; afterwards, one dummy number quantifier can be reintroduced. If there is only one function quantifier, then this works only for t h e number quantifiers of its same type, while t h e other remain, a n d can be contracted to a single one. More precisely, proceed as follows. 0

0

0

Reduce the matrix in prenex normal form, with a prefix of alternated number quantifiers, and a recursive matrix. Take t h e leftmost number quantifier, and confront it with t h e rightmost function quantifier of t he prefix. If they are of t h e same type, raise t h e number quantifier t o a function quantifier, and then contract. If they a r e not of t h e same type, permute them. See if there is a function quantifier t o t h e left. If so, it must be of t h e same type as t h e number quantifier, which can thus be eliminated as above. If not, proceed with th e next number quantifier in t h e matrix, inductively.

IV.2 The Analytical Hierarchy 0

379

At t h e end, either all number quantifiers have been eliminated, or a sequence of them, of t h e same type, lies to t h e left of t h e (necessarily unique) function quantifier. Contract them, a n d permute t h e resulting number quantifier back to t h e matrix. 0

Exercises IV.2.6 Set quantifiers. a) The same analytical classes are obtained i f we use set quantifiers in place of function quantifiers, and matrices with at most two number quantifiers. (Kleene [ 19551) (Hint: sets and their characteristic functions can be identified, and we can thus work with the latter. In one direction, to say that a function is a characteristic function only involves a number quantifier, e.g.

* gf[Vz(f(z)5 1)

(3A)R(cA)

A

R(f)l.

In the other direction, functions can be substituted by their graphs, and saying that f is the characteristic function of the graph of a total function is an arithmetical condition: f has values bounded by 1; whenever it is 1 for 2, then z = (z, y ) , i.e. it codes a relation; whenever it is 1 for (z, y ) and (z, y ' ) then y = y', i.e. the relation is a function; and for every z there is y such that f ( ( z , y ) ) = 1, i.e. the function is total. Note that this last condition introduces two number quantifiers.) b) Set quantifiers and matrices with only one number quantifier are not SUBcient to generate the whole Analytical Hierarchy. (Kreisel) (Hint: the predicate t l A l z R ( ? ~ ( z )with ) , R recursive, is r.e. Indeed, the set of sets is finitely branching. Thus, by Konig's Lemma, if for every A there is z such that R(?A(z))holds, there must be a n 2 that works for every A. This means that we only have to look for z such that ( t l A ) R ( ? ~ ( zholds. )) For a fixed z this is a recursive condition, because there are only finitely many possible sequence numbers of length z. Thus the whole condition is r.e.) Exactly like we did for t h e Arithmetical Hierarchy, we can now count t h e number of alternations of function quantifiers, a n d stratify t h e Analytical Hierarchy.

Definition IV.2.7 The Analytical Hierarchy (Lusin [1925], Sierpinski [1925], Kleene [1955]) 1.

i s the class of relations definable over A; by a formula of C;l (without parameters) in prenex f o r m with arithmetical matrix, and n quantifier alternations in the prefix, the outer quantifier being existential.

2.

IIi i s defined similarly,

3.

with the outer quantifier being universal.

A; i s EinrIk, i.e. the class of relations definable in both the n-quantifier forms.

4. A:

i s the class of the analytical relations.

380

IV. Hierarchies and Weak Reducibilities

By extension, we will call a formula without parameters EA or HL if it is in prenex normal form with n function quantifier alternations in t h e prefix, t h e first of which, respectively, existential or universal. Note also t h a t , by contraction of quantifiers, n function quantifier alternat,ions a r e equivalent to n alternated function quantifiers.

The levels of the Analytical Hierarchy A good p a r t of t h e theory of t h e Arithmetical Hierarchy goes through for t h e Analytical Hierarchy as well, with similar proofs, t h a t we do not repeat.

Proposition IV.2.8 Closure properties (Kleene [1955]) 1. R i s EL i f and only i f 7 R is 11:1 R is IIA if and only i f 7 R is CA

2. A: is closed under negations 3. CA, II; and A; are closed under conjunction, disjunction and number quantification

4. for n 2 1, E; is closed under existential function quantification, and KI; is closed under universal function quantification

5. the universal function quantification of a E: relation is HA,,, existential function quantification of a n; relation is E,’,,, .

and the

Proof. Everything easily follows from logical operations a n d quantifier nianipulations. E.g., closure under number quantification follows by moving t h e number quantifier to t h e matrix, by successive permutations. 0 Theorem IV.2.9 Enumeration Theorem (Kleene [1955]) For each n and m 2 1, and any 1, there is a CA relation with m + 1 number variables and 1 function variables that enumerates the .Xk relations with m number variables and 1 function variables. Similarly for II;. Proof. Siniilar t o IV.1.9, using this t i m e t h e Normal Form Theorem for r.e. relations of numbers a n d total functions, t h a t follows from 11.3.11. Note t h a t , because of IV.2.5, t h e hierarchy is inductively built u p by s t a r t i n g from t h e r.e. 0 predicates, by negations a n d function quantifications. Corollary IV.2.10 Normal Form Theorem for IIi sets (Lusin [1917], Suslin (19171, Kleene [1955]) A is H: i f and only if, for some recursive predicate R, 3: E A V’f3yR(z7

*

m).

IV.2 The Analytical Hierarchy

38 1

Definition IV.2.11 A set A is Xi-complete if it is Eg, and every E i set is m-reducible to it. IIi-complete sets are defined similarly. Proposition IV.2.12 For each n 2 1, Ch-complete and IIh-complete sets exist. Theorem IV.2.13 Analytical Hierarchy Theorem (Lebesgue [1905], Lusin [1925], Sierpinski [1925], Kleene [1955]) The Analytical Hierarchy does not collapse. More precisely, for any n 2 1 the following hold: 1. Eh - IIk 2.

# 8,

and hence Ah

IIi - EA # 8, and hence A;

c Ei c IIi

We thus get a picture for the Analytical Hierarchy similar t o t h e one obtained for t h e Arithmetical Hierarchy, see Figure 1 (p. 362). Despite t h e similarities between t h e Arithmetical and Analytical Hierarchies, there a r e differences. T h e most striking one seems to be t h e fact t h a t there is no recursion-theoretical generation of the Analytical Hierarchy from below. Thus function quantifiers, unlike number quantifiers (that correspond t o relative recursive enumerability) seem t o have no computational content. See p. 395 for a precise statement of this fact.

II: sets T h e Normal Form Theorem for II: sets suggests some natural representations for them, in set-theoretical terms. To state them precisely, we first introduce some terminology.

Definition IV.2.14 A set of sequence numbers T is a tree i f it is closed under subsequences. (8) is the root, or vertex, of the tree. Elements of the tree are called nodes. A branch of T is a maximal linearly ordered subset of T . The terminal node of a branch is a leaf. A tree is well-founded if it has no infinite branch. Th u s a tree is simply a subset of t h e tree of all sequence numbers ordered by extension, such t hat whenever a node is in TI so are all nodes between it and t h e vertex (8). Moreover, an infinite branch of a tree can be thought of as a function f whose course-ofvalues determines the branch (recall, see p. 89, th a t th e courseof-value o f f is the function f , where f(y) is t he sequence number coding t h e first y values of f).

IV. Hierarchies and Weak Reducibilities

382

Figure IV.2: A tree Well-foundedness of a partial ordering usually refers to lack of infinite descending sequences. Here a branch can be thought of as a descending sequence (if we picture t h e tree as growing downward, as in Figure 2): t h u s t h e terminology is consistent, a n d a tree is well-founded as a t r e e if a n d only if it is so as a partial ordering. W h a t makes well-founded trees interesting is t h a t it is possible to proceed by induction on t h e m , from t h e leaves to t h e vertex. T h i s procedure of backward induction is called bar-recursion. T h e next result is mostly a rephrasing of t h e Normal Form Theorem IV.2.10.

Theorem IV.2.15 First Representation Theorem for II: sets (Lusin and Sierpinski [1923], Kleene [1955]) A set A is IIi i f and only i f there is a recursive sequence {T,},,, of recursive trees, such that x E A ts T, is well-founded.

Proof. If A is IIi t h e n , for sonie recursive R, z

E A tsVf3yR(x,f(y)).

It is now enough to define u E T,

@

S e q ( u ) A (Vv c u ) ~ R ( z , v ) ,

where u a n d u range over sequence numbers a n d (see p. 89) c means proper initial subsequence. Clearly T, is a recursive tree, uniformly in x . A leaf corresponds t o a first place in which R holds. An infinite branch of T, corresponds t o a function f such t h a t b ' p R ( x , f ( y ) ) , a n d it exists if a n d only if z @ A. T h u s x E A if a n d only if T, is well-founded. Conversely, let {T,},cwbe a recursive sequence of recursive trecs, a n d suppose z E A ts T, is well-founded.

383

IV.2 The Analytical Hierarchy T h c n A is

II;, since

by t h e interpretation of infinite branches of trees as funct,ions. Note t h a t t h e relation f ( y ) $ T, is recursive, as a relation of f , x, a n d y , because T, is uniformly recursive in 5 0

Corollary IV.2.16 The set 7 of charactenstic andices of well-founded recursaw trees as ni-complete.

n: because ‘being t h e index of t h e characteristic function of a tree’ is a n arithmetical condition, a n d ‘having no infinite branch’ is n:: Proof. 7 is

eE7

*

YI

( v x ) ( 3 ~5 ) [ 1~A p e ( . ) cx A (vx)[pe(x:) 1 4 seq(z)] A

(v.)(v~)[z L Y A ( P ~ ( Y )21 1

4

pe(x) fi 11 A

(vj)(jy)[pe(.f(y)) 2: 01.

7 is II!-cornplete by t h e proof of t h e First Representation Theorem. More precisely, if 5

E

A

vf3YR(z,I(!/))

for some recursive R then there is, by t h e SAn-Theorem, a recursive function g such t h a t 1 Seq(u) A (YZJc v ) ~ R ( z , v ) P g ( z , ( 4 21 0 otherwise

{

7,, because T h e n A I, zE A

@

g(x) E 7.

0

Not,ice t h a t it is possible to assign to each well-founded t r e e T a n ordinal, as follows (by bar-recursion):

Definition IV.2.17 If T as a well-founded tree, the ordinal of a node u on T as defined as:

The ordinal of a well-founded tree T is defined as:

IV. Hierarchies and Weak Reducibilities

384

T h u s we assign 1 to t h e terminal nodes of each branch of a tree, a n d t h e n we climb t h e t r e e by assigning to each nonterminal node t h e least ordinal not yet assigned to any of its successors. A nonempty tree has t h e ordinal of its vertex (8). Of course t h e whole procedure works because t h e tree is well-founded, a n d thus t h e inductive procedure of climbing t h e tree from t h e terminal nodes makes sense. T h e interesting fact is t h a t this is really a n equivalent way to look at countable ordinals.

Proposition IV.2.18 An ordinal a is countable i f a n d o n l y i f there is a wellfounded tree T such that a = o r d ( T ) . Proof. Note t h a t a n ordinal is assigned to a node of T only if all the smaller ordinals have already been assigned to sonie of its successors. Since a t r e e has only countably many nodes, being made of sequence numbers, i t s ordinal can have only countably many predecessors, a n d it is t h u s countable. Let now a be a countable ordinal: we define a well-founded t r e e T such t h a t a = o r d ( T ) , by induction on t h e countable ordinals. 0 ff=o.

Let T = 8. 0

0

a=p+1 Let To be a well-founded tree such t h a t p = ord(T0). We only have to build a tree T t h a t consists of a subtree isomorphic to TO,a n d o n e vertex on t o p of it. Let T = ((8)) U ((0) * u : u E T o } . ff = supncw an

Lct Tn be well-founded trees such t h a t an = o r d ( T n ) . W e only have to build a tree T t h a t consists of s u b t r e e isomorphic to t h e Tn's, a n d one vertex on top of t h e m . Let

T ={

(a)} U u{ (n)* 'u. : 'u. E T n } . ntw

Since t h e notion of well-founded tree is easily constructivized, by considering recursive well-founded trees, we have a natural constructive analogue of t h e notion of ordinal: a recursive ordinal is a n ordinal Q such t h a t Q = o r d ( T ) , for some recursive well-founded tree T . Since t h e node of a t r e e is assigned a n ordinal only if all smaller ordinals have already been assigned to s o m e of i t s successors, a n d t h e subtree consisting of t h e extensions of a node of a recursive tree is isomorphic to a recursive tree, t h e recursive ordinals form a n initial segment of t h e ordinals. T h e first nonrecursive ordinal is indicated by w i k ,

IV.2 The Analytical Hierarchy

385

because it is t h e constructive analogue of t h e first uncountable ordinal w1, a n d it was introduced by Church a n d Kleene [1937]. Clearly wfk is countable, because there a r e only countably many recursive trees, a n d thus only countably many recursive ordinals. T h e theory of recursive ordinals will be developed in Volume 111, a n d will be useful for a finer analysis of t h e hyperarithmetical (p. 391) a n d II; sets. Another way to introduce countable ordinals is b y looking at countable well-orderings (equivalently, at well-orderings of L J ) , a n d t h u s we may g u e s t h a t t h e r e is a result similar to IV.2.15, using well-orderings instead of wellfounded trees.

Definition IV.2.19 A (recursive) partial ordering 5 is a (recursive) binavy relation on a (recursive) set A of numbers which i s reflexive, anlisynmetric and transitive, i.e. such that, for every x , y, z E A,

I. x 5 x 2. x 5 y A y 5 x

+x =y

3. x 5 y A y 5 z + x d z .

A linear partial ordering is a partial ordering which is total on A, i.e. such that, for every x, y E A ,

4. x 5 y v y 5 x . A well-ordering is a linear partial ordering with no infinite descending sequence of elements.

As usual, we will write x

+ y for x 5 y A x # y.

Theorem IV.2.20 Second Representation Theorem for II: sets (Lusin and Sierpinski [1923], Kleene [1955], Spector (19551) A set A is II; if and only if there is a recursive sequence { + z } z E w of recursive linear orderings, such that x E A w +z is a well-ordering. Proof. By IV.2.15, it is enough to show t h a t it is possible to go back a n d forth, in a n effective way, from recursive trees to recursive linear orderings of w,preserving well-foundedness a n d well-ordering. Let 4 be a recursive linear ordering of w:we want a recursive t r e e T4 such that

+

is a well-ordering

@

T+ is well-founded.

I t is enough to define

(XI, ..., x n ) w x 2 , + . . . + x 1 .

386

IV. Hierarchies and Weak Reducibilities

T h u s branches of T4 code maximal descending sequences of elements w.r.t. 4 , a n d t h e r e a r e infinite branches in T4 if a n d only if t h e r e a r e infinite descending sequences in 4. Let T be a recursive tree: we want a recursive linear ordering 4~ such t h a t

T is well-founded ts 4~ is a well-ordering.

A natural way t o linearly order a tree is t h e lexicographical one: given two sequence numbers, we look at t h e branches on which they lie, a n d give t h e precedence to t h e one on t h e leftmost branch when t h e branches differ, a n d to t h e one on t h e higher level if they a r e on t h e same branch. B u t for our purposes this is overkilling: t h e resulting order is always a well-ordering (since each node has only finitely many predecessors at t h e s a m e level, on different branches, a n d only finitely many predecessors at lower levels, on t h e s a m e branch). T h e reason why this ordering is not sensitive to infinite branches of t h e tree is because it follows each branch in increasing order: we t h e n have only to modify i t , a n d reverse t h e order on single branches. T h u s we will still order lexicographically sequence numbers lying on different branches, b u t reverse t h e Icxicographical order on single branches (and t h u s t h e only way to produce infinite descending sequences will be to have infinite branches). T h e wanted linear ordering is thus, for z , y E T ,

T h i s is sometimes called t h e Kleene-Brouwer ordering (Lusin a n d Sierpinski [1923], Brouwer [1924]) associated to T . 0 Exercises IV.2.21 a) The set W of characteristic indices of recursive well-oderings of w is IIi-complete. (Lusin and Sierpinski [1923], Spector [1955]) (Hint: the proof above shows that 7 SrnW , if +T is extended to a total ordering, e.g. by putting all niimbers not in T before everything, with their usual ordering. Thus it is enough to show that W is H:.) b) 7 and W are recursively isomorphic. (Hint: this holds in general, for II:-complete sets.)

Since well-orderings of w a r e another way of looking at countable ordinals, we might think of defining t h e recursive ordinals by looking at recursive wellorderings, instead t h a n recnrsive well-founded trees. Happily, by t h e proof j u s t given, the class of ordinals defined by recursive well-founded trees o n one side and recursive well-orderings on the other coincide, t h u s showing t h a t t h e notion of recursive ordinal is natural a n d stable. We have j u s t scratched t h e surface of t h e theory of IIi sets, which will be developed in detail in Volunie 111. We j u s t hint here at t h e fact t h a t ni sets

IV.2 The Analytical Hierarchy

387

admit a natural interpretation in terms of infinitary computations. Consider a recursive finitely branching tree: by Konig's Lemma, t h e process of determining whether it is well-founded or not is r.e. (see IV.2.6.b). T hus t h e process of determining whether a recursive tree is well-founded can be seen as a n infinitary analogue of a n r.e. question. This can be made precise, and it t urns out that, t h e IIi sets are exactly t h e r.e. sets relative to t h e ordinal wEk (see p. 385).

A: sets T h e level 0 of t h e Analytical Hierarchy corresponds t o t h e arithmetical sets, b e cause we allow for arithmetical matrices. To describe A: we take a constructive stand, an d will show how t o obtain any member of it from below. Recall t h a t we constructed t he smallest Boolean algebra generated by t h e r.e. sets by closing them under complements a n d finite unions, and by so doing we did not even exhaust A!. T h e idea here is t o consider not only finite unions, but infinite ones as well. Of course we have t o be careful, since t h e singletons are r.e. sets, and if we allow arbitrary countable unions then we get any possible set: this shows t hat infinite unions are t oo powerful. We consider only constructive unions, i.e. unions of r.e. families of sets, and a great deal of t h e power is maintained, since we obtain this way all t h e A: sets.

Theorem IV.2.22 Suslin-Kleene Theorem (Suslin [1917], Kleene [1955a]) A: is the smallest class of sets 1. containing every singleton set { n }

2. closed under complements 3. closed under effective unions.

Proof. To be able t o talk about effective unions, we are going to assign indices t o sets in t h e class, a n d operate on them by using recursive functions. We thus call a class of sets C = {Cz}zGca n SK-class (for Suslin-Kleene) if there are recursive functions f l , fi and 'p such t h a t

0

if pn is total then p ( n ) 1 and Cp(n)=

UzEwCpn(,)

T h e theorem thus amounts t o proving t h a t A: is t h e smallest SK-class. We do this in two steps, first explicitly defining t he smallest SK-class, and then showing t h a t it coincides with A:.

388

IV. Hierarchies and Weak Reducibilities

1. Definition of the smallest SK-class 6 We first define, by induction, the system of indices we are going to use, as the smallest set G such that 0

(0,n)E G

0

if n E G then (1,n) E G

0

if cpn is total and Vx(cpn(x)E G) then (2,n)E C.

These indices obviously correspond t o the three clauses of t h e definition of a SK-class. We can then define the class 6 = { G z } z Ein~t h e natural way: G(0,n)= 0

{n)

-

G(1,n) = Gn G(2,n)= U z E w G V n ( Z ) .

By definition, 6 is a SK-class. It is also t h e smallest such one, intuitively because we put in it only what is strictly required by t h e definition of SK-class, and formally because if C = {C,},cw is a SK-class, and f i , f 2 , cp are the recursive functions associated t o it, then

xEG

L

i.e. each element of

* G,

6 is in C, where g

= Cg(,),

is any recursive function such t h a t

if x = (0, n) (n) f2(9(72)) i f x = (1,n) if x = ( 2 , n) and cp,(.) cp(e)

J1

g(x) =

= g(cpn(Z))

otherwise.

Note t h a t such a function exists by the Fixed-Point Theorem. 2 . 6 is contained in A: We have t o show that if x E G then G, E A:. It is enough t o prove that G, E for every x E G: then, since = G(I,,), C, E IIi and G, E C i , hence G, E A:.

c,

T h e reason why G, E n: is that 6 is the smallest class of sets satisfying an arithmetical condition, which means that every class satisfying t h e same condition must contain it: this can be expressed by a universal quantifier over classes of sets, and an arithmetical matrix, and it is ‘almost’ n;. We only have to talk about sets, rather than classes of sets, a n d this can easily be accomplished by considering a n indexed class of sets as a binary

IV.2 The Analytical Hierarchy

389

relation, which can then be coded as a set. Precisely, we a r e going to show t h a t 6 is uniformly Hi, i.e. that there is a II: set A such t h a t

z E G, H ( z , x ) E A. We will write a n arithmetical predicate P ( z ,x,X ) , meaning t h a t X is a set coding a class satisfying the conditions of t h e definition of SK-class, a n d then we will say t h a t A is t h e smallest such set, as follows:

( z , x )E A Then A will be

@

(‘dX)(P(z,x,X) --+ ( z , z ) E X ) .

II:. It just remains t o define P :

3. A: is contained in 6 Suppose A is A:: there are two recursive relations R a n d Q such t h a t

xEA

*

w!”f,S^(Y))

E

@

~~YQ(~,O(Y)).

5

To show t h a t A E G , it is enough to find a recursive function h such t h a t

Gh(s)=

{

ifxEA otherwise.

Then A = UrEwGh(,), i.e. A is a n effective union of members of hence is in 6.

6, a n d

Since A is Hi, consider t h e representation of A by uniformly recursive trees T, given by t h e First Representation Theorem IV.2.15. If z E A then T, is well-founded, and thus each branch is finite. We might label t h e leaves with t h e set {x}, and climb our way to t h e top of t h e tree, by giving t h e same label {x} to any node whose successors all have t h a t label. Then, if x E A, the vertex (corresponding t o t h e empty sequence number (0)) also has that label. This is easily defined in recursive terms, by first giving t h e function

{

$n’Ew

if R(z,u)A (‘du C u ) i R ( x u, ) cpo(x,u * ( n ) ) otherwise,

IV. Hierarchies and Weak Reducibilities

390

a n d then considering cpo(z,(8)). function $o(x,U ) such that orem:

Note that there is a partial recursive

c ~ ~= ~(O ~ (Z,U , )by ~ , th )e Fixed-Point The-

{ ( 2 ,x)e ) (1,

'@O(z'u) 21

if R(x,u)A (VW C u)-R(x, ?J) if cp,(n) -$o(x,u* ( n ) )otherwise. ,

This almost produces what we want, a n d there is only one problem: 'po(x,(8)) is undefined when x E 2,because in this case some branch of the tree has no leaf, a n d thus we cannot climb our way t o t h e top, since we never define cpo on that path. However, if x E 2 then we can play a similar strategy on t h e other tree, corresponding to Q. We can thus get a function 'p1 such t h a t ' p l ( x ,(8)) gives 8 if x E a n d is undefined otherwise. Moreover, we can find a partial recursive function $1 such t h a t c + l ( z , u ) = 'pl(",'u.).

x,

z,

Since x is either in A or in one a n d only one of cpo(x,(8)) a n d cpl(x,(8)) is defined, and we can thus let

Then, as wanted, Gh(z) =

{ $"'

ifxEA otherwise.

0

Corollary IV.2.23 T h e arithmetical sets are properly included an A:. Proof. We prove, by induction on n,t h a t every EE set is A:. Each r.e. set can be obtained as effective union of recursive sets. Precisely, if 5

EA

* (3y)R(Z,Y)

then A is t h e union, over y E w, of the sets A, = {x : R(z,y)}. By relativization, each E;, can be obtained as t h e effective union of n",sets. By induction hypothesis, every E: set is A:, and by closure under complements so is every set. Thus every Xi+, set is A:. Let now B, be a C:-complete set, for every n 2 1. T h e set

C, = { (x,n) : z E B,}

39 1

IV.2 The Analytical Hierarchy is still E:,

and hence A:. Then the set

is also A:. If it were arithmetical, it would be E! for some n. But then e.g.

z E Bn+1

H

( z , n +1) E C,

and the E:+l-complete set B, would be E:,

contradiction.

0

The characterization just given suggests the possibility of building a hierarchy for A:, by looking at the stages in which a given relation is generated, in the inductive definition of the class. This will be done in Volume 111, and will define the Hyperarithmetical Hierarchy (Davis [1950], Mostowski 119511, Kleene [1955a]), which can be seen as a transfinite extension (of length w?, see p. 385) of the Arithmetical Hierarchy (by the proof of the corollary). A suggestive reformulation of the Suslin-Kleene Theorem is that the A: sets are exactly the sets computable with the help of a n oracle that embodies number quantification (disguised in the operation of union). Otherwise stated, t h e A: sets are the sets computable modulo a finite number of quantifications over w. This can be made precise with the extension of the notion of relative recursiveness t o higher types (see p. 199). Then the A: sets are the sets recursive in the type-2 object 0 if 3 z ( f ( x )= 0) E(f) = 1 otherwise

{

(Kleene [1959]). Before too great expectations arise, we immediately show that no result similar to the Suslan-Kleene Theorem m a y possibly hold f o r higher levels of the Analytical Hierarchy, in a precise sense. Thus IV.2.22 is an exceptional characterization, working only because one function quantifier is needed t o express the notion of ‘smallest set satisfying a given condition’: A: has no power t o swallow it up, but all other higher Ak’s do.

Proposition N.2.24 (Addison and Kleene [1957])There i s no n > 0 such that Ah+l i s the smallest indexed class of sets uniformly satisfying a Ah co nditio n. Proof. Let P ( z ,x , X ) be a A: formula meaning that X codes a class of sets indexed by numbers, as follows: z E G,

* ( z , z )E X .

IV. Hierarchies and Weak Reducibilities

392

Then the smallest class of sets 6 defined by P is strictly contained in Indeed, let A be the set coding it: (z,z) E

Then A is

A

H

( t I X ) ( P ( z , z , X+ ) (z,z) E X ) .

and hence, since n > 0,

T h e set B defined as

X E Be ( . , . ) @ A is thus too, but it cannot be in t h e class coded by A , otherwise there would be an e such that

xEB

@

(z,e)E A,

and for x = e we would get a contradiction.

0

Descriptive Set Theory * T h e Arithmetical and Analytical Hierarchies resemble very much, both in notions and results, classical hierarchies that were studied at t h e beginning of t h e century in Descriptive Set Theory, which started as a way of generating from below interesting and graspable classes of functions a n d sets (on T e a l s , not on natural numbers), as opposed t o t h e definition of the continuum as a whole, by the power axiom. Part of the motivation was t o prove t h e Continuum Hypothesis for larger and larger classes of sets, with t h e hope of finally getting t o a complete solution of the problem. T h e classical development has its first landmarks in Borel [1898], who introduces Borel sets (as t h e smallest class containing t h e open sets, a n d closed under complements and countable unions), Baire [1899], who introduces Baire sets (via Baire functions, defined as the smallest class containing t h e continuous functions, and closed under limits), and Lebesgue [1905],in which the classes of Borel and Bake sets are proved to coincide. Lebesgue also introduces the analytic sets (not t o be confused with t h e analytical sets dealt with in this section) as projections of Borel sets, and falsely claims t h a t a n analytic set is Borel. This is corrected by Suslin [1917], where it is proved t h a t a set is Borel if and only if it is both analytic and coanalytic. T h e Projective Hierarchy, obtained by iterating projections and complements, is defined and studied in Lusin [1925] and Sierpinski (1925). T h e classical period ends in the late Thirties, when a stumbling block is reached, with t h e impossibility of extending the theory beyond the second level of t h e Projective Hierarchy. This will be explained later, after the introduction of the methods t o prove t h e independence of the Continuum Hypothesis (Godel [1940], Cohen [1963]): the theory of higher projective classes is mostly independent of ZFC, and thus it cannot be pursued without additional set-theoretical hypothesis.

IV.2 The Analytical Hierarchy

393

A second period of development for the subject comes when the recursion theorists attempt t o classify sets of natural numbers, from their own point of view. Kleene [1943] and Mostowski [1947] introduce the Arithmetical Hierarchy, and while the former works without awareness of the classical work, the latter explicitly develops the hierarchy as an analogue of the Projective Hierarchy. After t h e introduction by Kleene [1955] of the Analytical Hierarchy, the analogies begin t o clarify. Addison [1954], [1959] not only makes them precise, but also sees that there is more: classical Descriptive Set Theory can be obtained b y relativization of the theory of recursion-theoretical hierarchies, using the following translations (in which the left-hand side is the effective version of the right-hand one): recursive function r.e. set hyperarithmetical set set analytical set

continuous function open set Bore1 set analytic set projective set.

In particular, the recursion-theoretical versions of classical results are stronger, and imply their classical counterparts. This explains the double assignment of credit t o results, in this section. After the classical and t h e effective periods, the subject has entered its modern era with the introduction of new set-theoretical axioms (which, as we have quoted, are necessary t o go beyond the first two levels of the Analytical Hierarchy). The first axiom t o imply results about analytical sets was t h e Axiom of Constructibility (IV.4.2): Gijdel [1940] and Addison [1959a] showed t h a t a coherent theory for all levels of the Analytical Hierarchy can be obtained from it. Although provably consistent with Z F C , the Axiom of Constructibility is however taken more as a useful technical tool than as a real additional axiom of Set Theory. The existence of measurable cardinals is taken more seriously, and it does provide for additional results about analytical sets (Solovay [1969], Martin and Solovay [1969]), but its influence does not seem t o extend much beyond the fourth level of the hierarchy. The most successful axiom t o date for the development of Descriptive Set Theory is the Axiom of Projective Determinacy, a restricted version of the full Axiom of Determinacy (V.1.15) saying that all projective games are determined. This can be seen as an axiom about the existence of very large cardinals, and it implies an extremely coherent picture of the Analytical Hierarchy (the first results in this direction have been obtained by Blackwell [1967], Addison and Moschovakis [1968] and Martin [1968]). We will come back t o this subject in the last chapter of our book. Classical references for Descriptive Set Theory are Hausdorff [1917], Lusin [193Oa],Sierpinski [1950],Kuratowski and Mostowski [1968]. The definitive test

IV. Hierarchies and Weak Reducibilities

394

about the subject, treating classical, effective and modern theory, is Moschovakis [1980].

Relativizations

*

As for the Arithmetical Hierarchy, the work done for t h e Analytical Hierarchy can also be relativized t o a given oracle X , defining t h e classes C > x ,f i x a n d A;x, and obtaining similar results. This time a number of new reducibilities do arise. Proposition IV.2.25 (Shoenfield [1962]) The relation

is transitive, for any n. Proof. For n = 0 this is obvious, since then 0 , A E A$B, and B E A$c. There are P E 1B R E E;-l such that z E A e 3f P ( z ,f ,B ) a V f R ( z f, , B ) .

To show t h a t A E C$c, note t h a t zEA

3 D [ D = B A 3 f P ( z ,f,D ) ] .

T h e expression B = D can be rewritten as

V x [ ( zE B

+

x E D ) A (z E D

+z

E B)].

By using t h e IIkC form for B in t h e first conjunct, a n d t h e E$c form in t h e second one we get, by manipulation of quantifiers, a E$c form for B = D which, substituted above, produces a CkC form for A. Similarly, noting that Z E A@VD[D=B +VfR(z,f,D)],

we get a II$c form for A, which is then A$C. T h e relation
0

is obviously reflexive and thus it induces, as usual, an

Definition IV.2.26 For n 2 1, the equivalence classes of sets w.r.t. the equivalence relation A E A ; B e A l ~ Bi A B < a t , A

are called A:-degrees, and ( D : , 5 ) is the structure of A:-degrees, with the partial ordering 5 induced on them b y
IV.2 The Analytical Hierarchy

395

The A:-degrees are also called hyperdegrees, and will be studied in Volume 111, together with the other A;-degrees. For an anticipation, see p. 441. The nice closure properties that produce the transitivity of
A:. Proposition IV.2.27 (Addison and Kleene [1957], Shoenfield [1962]) is the class of sets A: in some Ek set. There i s no n such that Proof. By the result proved above, the class of sets A: in some E; set is contained in A;+l. We have t o show that the inclusion is always proper. For n = 0 this is clear, since there are A: sets which are not arithmetic. Suppose then n 2. 1: we prove that there is A E A;+l such that, for any set B E EL, A is not A: in it. Let C be a Ck-complete set: since any other EA set is recursive in it, it is enough t o get A not A; in C. Since we want t o limit its complexity, the obvious first choice for A is a Eic-complete set, which is certainly not A; in C. Moreover, by the transitivity of l ) , no such characterization is possible anymore.

Post’s Theorem in the Analytical Hierarchy * We have seen that, in a precise sense, Post’s Theorem IV.1.14 has no analogue in the Analytical Hierarchy. But we should not forget that, in stepping up from the Arithmetical t o the Analytical Hierarchy, the universe has changed quite radically: functions are now treated as objects, like we did with numbers all along, and we quantify over them. But there has been no corresponding increase in t h e power of computations: we have retained the notion of computability that we found suitable for numbers. In particular, the p-operator is available as a search operator over numbers, but nothing of the kind exists over functions. The negative results obtained above could thus be interpreted as signs of the

IV. Hierarchies and Weak Reducibilities

396

inadequacy of t h e notion of recursiveness as a notion of computability over t h e continuum. Notions of computability over abstract domains have been proposed, see p. 202, a n d they all basically reduce t o either prime or search computability (p. 203), which are abstract analogues of deterministic and nondeterministic computability. We have made a great use of searches over numbers in t h e proofs of th e results of Section 1: computationally, t h e existential quantifiers are handled by searching for elements t h a t satisfy t h e matrix. In particular, Post’s Theorem is a relativization of 11.1.19, whose proof amounted to a parallel search. This suggests t h a t , in an abstract setting, it is search computability t h a t we need. Moschovakis [1969] has indeed proved t h a t , once t h e notion of recursiveness is replaced by t hat of search computability, a n d a Projective Hierarchy is defined in analogy t o t he Arithmetical Hierarchy (i.e. by allowing search computable matrices, and quantifications over t h e domain), t hen the theory of the Arithmetical Hierarchy, including Post’s Theorem, generalizes to

abstract domains. I t only remains t o apply t h e general theory, and translate t h e notion of search computability over t h e continuum in t he familiar terms of t he Analytical if and only if it is search Hierarchy. I t turns out t h a t a relation is computable in IIk,,, and this provides for a n analogue of Post’s Theorem for t h e Analytical Hierarchy (Hinman and Moschovakis [1971]). T h e idea of th e proof for n = 0 (the other cases being similar by relativization) is th e following. T h e main computation concerns t h e complexity of domains of functions search computable in II:, which are t h e analogues of (relativized) r.e. relations. One direction requires a n arithmetization of t h e computation tree similar t o t hat of 1.7.3. Since t h e relevant objects can be coded only as functions (instead than as numbers), a search computable function converges if and only if there is a function coding a computation tree for it, a nd this introduces a n outer existential function quantifier. Moreover, since computation trees are not finite objects anymore, they must well-founded, and t o express this we need a universal function quantifier. T hus t h e whole relation is Ci. If the function is search computable in some oracle t h e complexity of t h e computation tree is accordingly increased, but II: oracles only require a universal quantifier, which is present anyway because of well-foundedness, and in this case t h e total complexity is still Xi. Conversely, let A be Xi: then t h e function t h a t searches for a witness of t h e II: matrix is search computable in t h e characteristic function of t he latter, and converges if and only if t h e witness exists. Thus A is the domain of a function search computable in IIi. By a refinement of t h e argument sketched above, Hinman a n d Moschovakis [1971] show th at a relation is A; if and only if it is search computable in the

equality predicate for functions. These results point out t hat t he proper analogue of recursiveness on t h e

IV.3 The Set-Theoretical Hierarchy

397

continuum seems t o be A;, rather than A:. See also p. 442 on this.

IV.3

The Set-Theoretical Hierarchy

Arithmetic is a natural environment for the study of sets and functions of natural numbers, since these can arguably be seen as primitive objects. But modern mathematics is usually done (formally or informally) in the framework of Set Theory, in which t h e natural numbers can be defined. In this section we look at definability in the language of (formalized) Set Theory, with a n eye t o t h e applications to our subject. Since this is not a book on Set Theory, we refer t o Godel [1944], [1964], and Kreisel and Krivine [1966] for general discussion, F’raenkel, Bar-Hillel, and Levy [1958] for foundations and history, Monk [1969] and Levy [1979] for the fundamentals of t h e subject, and Cohen [1966], Jech [1978], and Kunen [1980] for more advanced topics. However, this section and t h e next are mostly selfcontained, and require only the working knowledge of elementary Set Theory used until now. We will also recall the needed concepts and results.

Truth in Set Theory A primordial notion of set (F’rege [1893]) corresponds to an extensional percept i o n of properties. It can be formulated on two simple principles: extensionality, which ensures that a set is completely determined by its elements, and full comprehension, according to which one can consider the set of all t h e objects for which a given property holds, abstracting from the intensional way t h e given property is expressed. Unless one admits partial properties, t h a t may possibly be undefined for a given argument, this naive point of view is shaken by Russell’s paradox (p. 81), since t h e set corresponding t o the property of not being a member of itself is contradictory. One possible remedy to the situation is to switch t o a different notion of set, corresponding t o a n arbitrary collection of elements taken from a given collection, like when we consider a set of natural numbers. This implies a cumulative conception of t h e universe, in which sets are obtained by stages, from simple to more complicated ones, and it suggests a number of properties of the membership relation E. They fall into five different categories: 1. extensionality Membership being t h e only concept at hand, a set is completely determined by its elements, and thus two sets with t h e same elements are indistinguishable. As a particular instance of a more general principle going back to Leibniz (identitas indiscernibilium),according to which

IV. Hierarchies and Weak Reducibilities

398

two objects t h a t cannot be distinguished a r e equal, we get t h e Axiom of

Extensionality : (Vz)(z E

2

c-)

z E y)

2

= y.

2. comprehension Full comprehension is simply false for t h e present notion of set ( G a e l [1958]),since now a set is a collection not of arbitrary elements, b u t only of elements already belonging t o a given collection. In this setting it takes t h e form of t h e Axiom of Separation,according t o which one can consider the subset of a given set consisting of all t h e objects in it for which a given property holds. 3. set construction Since t h e Axiom of Separation only isolates subsets of given sets, without additional a x i o m we could not produce sets bigger t h a n t h e given ones. On t h e other hand t h e paradoxes show t h a t there a r e collections t h a t a r e simply too big t o be sets, and thus some restrain is needed. We compromise by adopting the doctrine of limitation of size (Cantor [1899], Russell [1906]), according t o which a construction principle should produce only sets which are not too large, compared t o sets on which they build. Typical examples of natural construction principles a r e t h e following: Pairing, Union, Power Set (taking t h e set P(a) of all subsets of a set a ) , a n d Replacement (taking t h e image of a set under a function). To avoid t h e consideration of functions replacement can be substituted, in presence of t h e other axioms, by Collection (taking a set b such t h a t (Vx E a)(3y E b)cp(z,y), provided (Vx E a)(3y)cp(z,y) holds). 4. set existence T h e axioms given until now do not ensure, by themselves, t h e existence of any set: separation only produces subsets of given sets, while t h e construction principles have to build on something. We t h u s need t o postulate the existence of at least a set. Even a single set, together with t h e previous axioms, makes t h e subject nontrivial, as we will see in this section. But t o make possible t h e definition of the usual mathematical objects, like the real numbers, we actually need a n infinite set (say LJ), whose existence is ensured by t h e Axiom of Infinity. T h e existence of a single infinite set, combined with t h e construction principles, produces large sets. But t h e existence of very large sets (e.g. of sets closed under replacement a n d power set) does not follow from t h e axioms, and it must be postulated independently, if needed. See Drake

399

IV.3 The Set-Theoretical Hierarchy

[I9741 for a treatment of such axioms t ha t , like t h e Axiom of Infinity, usually postulate t h e existence of large cardinals. 5 . foundation If t h e membership relation is well-founded, not only sets a r e obtained from simpler constituents: there ar e atomic constituents (urelements), which all sets are ultimately built of. Then t h e process of generation of sets can be seen as a well-ordered sequence of stages, through which t h e cumulative universe comes into being. This implies a form of t h e vicious circle principle, according t o which a totality cannot contain members t h a t presuppose it (PoincarC [1906], Russell [1908]). In particular, there cannot be sets x such t h a t x E x, otherwise they would give rise t o a n infinitely descending chain . . . E x E x. T h e well-foundedness of E is stated by t he Axiom of Foundation (Mirimanoff [IgI7])

(3Y)(Y E ).

+

(3Y" E x

A PzE

Y)(Z @ 511,

which says t h a t every nonempty set has a minimal element w.r.t. E (note t h e analogy with t h e Least Number Principle, p. 21). T h e usefulness of foundation is t hat it allows for recursion on E , also called €-induction:

(V'z)[(VY E

Z)cp(Y)

+

cp(z)l

(Vz)cp(.).

Suppose indeed t h a t (3x)-(p(s). We have t o find a nonempty set t o which apply t he Axiom of Foundation. Fix z such t h a t -cp(x) holds, a n d consider 2 = {x} U t he downward closure of x w.r.t. E, which is a set (see IV.3.12). By separation, consider t h e set consisting of t h e elements of 2 t h a t do not satisfy cp. By foundation, it has a minimal element z w.r.t. E: by minimality, no element of 2 belonging t o z satisfies cp. B u t every element of z is in x, by its downward closure, a n d thus every element of z satisfies cp. Then, by hypothesis, also z satisfies it, contradiction. T h e Zermelo-E'raenkel system Z F (Zermelo [1908], Fraenkel [1922]) is t h e first-order theory with equality, t h e relation symbol E as t h e only nonlogical symbol, an d th e axioms just stated. Z F C is obtained by adding t o it t h e Axiom of Choice or, equivalently, t he Well-Ordering Principle (every set can be well-ordered). We will mostly be concerned with t h e Generalized KripkePlatek system G K P (Kripke [1964], Platek [1966]), which is obtained from

IV. Hierarchies and Weak Reducibilities

400

Z F by dropping Infinity a n d Power Set (see also p. 421), a n d with Z F - , which instead only drops t h e Power Set Axiom. For convenience, t h e language of Set Theory is enriched of a t e r m formation symbol { }, defined according to t h e following rule: whenever we can prove t h a t

Wz(z EY

-

cph)),

then we can denote such a y (which, by extensionality, is necessarily unique) by

{x : cp(4). T h e existence of distinct urelements is not necessary for a development of

Set Theory, since a single object is enough to s t a r t t h e whole process, a n d it is provided by t h e empty set (which has no elements). By extensionality, t h e empty set is t h e only urelement. T h e existence of other urelements is sometimes useful, a n d in these cases o n e can state t h e Axiom of Extensionality only for nonempty sets.

Definition IV.3.1 T h e Cumulative Hierarchy is described, in terms of ordinals, as follows:

vo

= 0

Va+l P(VJ Vp = U,,pV, v = U,V,.

(/3 limit)

In Z F we can prove, by €-induction, t h a t Vz-Ja(z E V,). Indeed, suppose (Vy E x)(3a)(x E V,). By collection, (3a)(Vy E x)(x E V a ) ,a n d t h u s z 5 V,, i.e. z E V,+l. T h e notion of extension of a property, which axiomatic Set Theory meant to tame, can now be dealt with precisely. For any property cp, {x : ( ~ ( 5 ) ) is a class. A class can be either a proper class, or a set. Sets a r e exactly those classes which a r e members of (or, equivalently, a r e contained in) some V,, a n d a r e thus not too big. Proper classes a r e exactly those classes which a r e equinumerous to V . V can be seen as t h e intuitive universe of Set Theory, a n d Z F as its axiomatization. To gain a better understanding of t h e m we will consider different structures, t h a t can be seen as ‘models’ of (parts of) Z F , in t h e sense t h a t they satisfy some of t h e axioms of Z F . To make this more precise we need a notion of t r u t h for formulas of t h e language of Set Theory in given structures, which we can define as usual (see I V . l . l ) .

Definition IV.3.2 Definition of truth in Set Theory (*ski [1936]) Consider A = ( A , € ) ,where A is a class, and E is a binary relation on zt. Let

IV.3 The Set-Theoretical Hierarchy

401

LA be the first-order language with equality and E, augmented with constants ii f o r any a in A . Given a closed formula cp of LA, 'p is true in A (A cp) is inductively defined as usual, starting with:

* *

A b Z E z-

dbzi=b

acb a=b.

A n n-ary relation P is definable over A if, for some formula cp with n free variables and xl,. . . , xn E A, P(z1,.. . ,xn) a A

+

-

~ ( T I , .. ,. ~ n ) .

Intuitively, cp is true in (A,€) if the formula obtained from cp by replacing E , a n d restricting all quantifiers t o A, is true (i.e. it holds in (V,E)).

E with

Definition IV.3.3 A structure A is a model of a theory T with axioms in the language of Set Theory, i f all the axioms of T are true in A. This notion is standard in logic, and so is the result that a first-order theory is consistent if and only if it has a model whose universe is a set (Godel [1930]). By Godel's Second Theorem (p. l69), we cannot then prove in Z F t h a t there is a model of Z F with a set as universe. We will however find models whose universe is a class.

Standard structures Among t h e various possible structures for Set Theory we isolate a particularly interesting class. T h e idea is that we would like our structures t o satisfy some minimal conditions, namely extensionality and well-foundedness. T h e first property is completely captured by t h e Axiom of Extensionality, which is satisfied by A if

( A ,E)

+

[(Vz)(z E z

c-)

z E y)

-+

z = y].

By t h e definition of truth this means that, whenever z,y E A, (Vz E A)(z E x

++

z E y)

3

z = y,

t h a t is

znA=ynA

~ = y .

A condition implying extensionality is thus transitivity of E (i.e. if z E x a n d z E A then z E A), which amounts t o saying that if x E A then z C A, a n d hence z n A = z.

IV. Hierarchies and Weak Reducibilities

402

For what concerns well-foundedness, there is some difficulty. A relation E on A is well-founded (on A ) if every nonempty subset of A has a minimal element with respect t o

E.

This can be written as:

If we think of simply requesting (A, E ) t o satisfy this, t hen we do not ensure wellfoundedness of E . T h e reason is t hat t he intention is to consider every possible subset x of A , but either we leave the schema as such (by substituting x with any formula cp), and then we only talk about t h e countably many definable subsets of A , or we turn t h e schema into a second-order axiom, by universally quantifying x,a n d then we only talk about t h e members of A (because of t h e interpretation of quantifiers over a structure). In either case, a great deal of subsets of A are not considered, by Cantor's Theorem. One reason t o insist on well-foundedness of E is t h a t it almost justifies (in presence of separation) transfinite induction on E , i.e.

P Y E A)[(Y'zE Y>cp(.)

+

4Y)l

* P Y E A)cp(Y).

T h e only thing th at does not go through in t h e proof of recursion on E (p. 399) is t h a t we cannot prove t h a t x is a set. This is taken care by t h e additional assumption t h a t th e predecessors of any element of A w.r.t. E form a set (in which case E is called left-narrow). One obvious way t o have a well-founded membership relation is t o let it be t h e usual E . We are thus led t o t h e following notion.

Definition IV.3.4 A structure (A,€) is standard if A is transitive, and the membership relation restricted to A .

E

is

Since E is intended as E, a standard structure is completely determined by its universe A , and we will usually refer only t o it. Also, by definition of ordinal (set of smaller ordinals) a n d transitivity, t he ordinals of a standard structure are closed downward. If A is a transitive set it thus makes sense t o talk of t h e ordinal of A, meaning t he smallest ordinal not in it or, equivalently, t he set of ordinals in it. T h e next result shows t hat standard structures really capture t h e essence of extensionality and well-foundedness, a n d it can be seen as a generalization of t h e fact th at a well-ordered set is isomorphic, in a unique way, t o a n ordinal (which is obtained as a special case of i t , when E is a total ordering on t h e set A).

Theorem IV.3.5 The Collapsing Lemma (Giidel [1939], Mostowski [1949]) If (A, E ) is an extensional, left-narrow., and well-founded structure, i.e.

IV.3 The Set-Theoretical Hierarchy

403

1. for every z,y E A , i f (Vz E A ) ( z E z

H

z

2. for every z E A , the class {y : y E A A y

E

E

y ) then z = y

x} is a set

3. every nonempty subset o f A has a minimal element w.r.t.

E

then ( A , & )is isomorphic, in a unique way, to a standard structure. Proof. Suppose ( A ,E ) is extensional a n d well-founded. T h e idea of t h e proof is q u i t e simple. We picture t h e internal structure of A w.r.t. E as a tree, which is well-founded because so is E . We t h e n relabel t h e tree, by assigning 0 t o t h e leaves, and t h e set of i t s predecessors (which, by left-narrowness, is really a set) to any nonterminal node. T h u s we obtain a set which is transitive by construction (because we never skip a level, a n d proceed from t h e leaves in a n orderly fashion), a n d is isomorphic to ( A ,E ) because their internal structures a r e represented by t h e s a m e tree. Formally, we define t h e collapsing function f as

f(.) a n d show t h a t

=

:Y E

A

A

Y E 211

f is t h e required isomorphism between ( A , & )a n d ( f ( A ) ,E).

1. f is a function on A By induction on E it is easy t,o prove t h a t , for each z E A, f(z)exist a n d is unique. Existence follows by collection a n d separation (or replacement), applied to t h e set of predecessors (w.r.t. E ) of z in A , which is a set by left-narrowness. Uniqueness follows by t h e Axiom of Extensionality, since a set is completely determined by its elements. For more details, see t h e proof of IV.3.10.

2. f ( A ) is transitive Let z E u E f ( A ) : by definition of f , there exists z E A such t h a t u = f (z) = {f(y) : y E A A y E z}. Since z E u, t h e r e is y E A such t h a t z = f ( y ) , i.e. 2 E f ( A ) .

*

3. i f y E A then y E z f ( y ) E f(x) Since f(z)= {f(y) : y E A A y E z}, from y E A A y f(y) E fk). 4. f is o n c o n e We show, by induction on t h e well-founded relation if z,y E A then f(z)= f ( y )

3

E,

E

z i t follows t h a t

that

z = y.

By induction hypothesis, (Vs E A)(Vt E A ) ( sE z A t

E

y A f(s) = f ( t )

s = t).

IV. Hierarchies and Weak Reducibilities

404

By extensionality, we will have z = y (since z,y E A ) if (Vs E A ) ( s E z ++ s E y).

Suppose s E A A s E z. Then, by part 3, f(s) E f(z).If f(z) = f ( y ) then f(s) E f(y) = {f(t) : t E A A t E y}. T h u s f ( s ) = f ( t ) , for some t E A A t E y. By t h e induction hypothesis s = t , a n d hence s E y. T h e converse in similar. 5. if y E A then f ( y ) E f(z) 3 y E z Since f(z)= {f(t) : t E A A t E z}, if f ( y ) E f(z)then f ( y ) = f ( t ) for some t E A A t E z. From t E A A y E A we have y = t by p a r t 4, a n d from t E z then y E z.

6.

f is unique Suppose there a r e two such isomorphisms: they induce a n isomorphism of two transitive classes. We prove, by induction on E, t h a t if g : B + C is a n isomorphism a n d B , C are transitive, then g is t h e identity. Suppose z E B , and g(y) = y for every y E z n B : we show t h a t g(z) = Z.

c

x g(x) If y E z then y E B by transitivity, a n d thus y E z n B . By induction hypothesis, g(y) = y. Since g is a n isomorphism, g(y) E g(z) because y E z, a n d thus y E g ( z ) .

g(z) G 2 If z E g(z) then z E C by transitivity a n d , since g is onto, there is y E B such t h a t g(y) = z . Then g(y) E g ( s ) a n d , since g is a n isomorphism, y E z. From y E z n B a n d t h e inductive hypothesis, 1~ = g(y) = z . T h u s 2 E 5 . Since g is t h e identity, there can be only one isomorphism of ( A , & )with a transitive class.

0

Corollary IV.3.6 An extensional class A can be collapsed t o a n isomorphic transit.ive class in a unique way, with the transitive subclasses of A left unchanged.

Proof. T h e Collapsing Lemma can be applied to ( A , E ) , since extensionality holds by hypothesis, a n d E is automatically left-narrow a n d well-founded. T h u s there is a unique isomorphism with a transitive class. Since now E is E, t h e definition of f is simply

IV.3 The Set-Theoretical Hierarchy

405

A is transitive. We want t o show, by induction on E, that Suppose z f (x) = 2 for x E z , so that f does not change z. But if x E z then x z by transitivity, thus z C_ A and z n A = x. Then

f ( 4 = {f (Y) : Y E 4. By induction hypothesis, f(y) = y if y x (since then y E z, by transitivity), and thus f(x)={y:yEx}=x. 0 Note that, in particular, the transitive collapse of a n extensional class does not change t h e ordinals of t h e class. For any Set Theory with the Axiom of Extensionality, the Collapsing Lemma shows t h a t there are only two kinds of models for it: the ones which are not well-founded and, up t o isomorphism, the standard ones. We will thus identify t h e latter with the transitive models.

The Set-Theoretical Hierarchy Manipulation of quantifiers similar t o those already seen for number and function quantifiers in Arithmetic are possible also in our present context. Here we call bounded quantifier a quantifier of the type w x E a) or (32E a).

Proposition IV.3.7 The following transformations of quantifiers are permissible (up to provable equivalence in GKP): 1. permutation of quantifiers of the same type

2. contraction of quantifiers of the same type 3. permutation of two quantifiers, one of which is bounded

4.

substitution of a bounded quantifier with an unbounded one of the same typeProof. Part 1 is obvious. Part 2 uses t h e Pairing Axiom: 3z3y(P(x,y)

.3

jz(3a: E 4(3Y E

-4. = {wlA Cp(Z,Y)ll

where z is a variable not occurring in cp. Part 3 uses the Axiom of Collection: (YE E 4(3y)(P(x,y)

43

( 3 a W x E 4(3Y E a)cp(z,y),

where a is a variable not occurring in cp. Part 4 is just the definition of bounded quantifiers:

(Vx E a)cp(z) (3x E a)cp(z)

H .3

Vx(x E a -+ (~(5)) 3z(x E a A cp(x)). 0

406

IV. Hierarchies and Weak Reducibilities

Proposition IV.3.8 Prenex Normal Form (Kuratowski and Tarski [1931]). Any relation definable in the language of Set Theory is equivalent in G K P to one with a list of alternated quantifiers in the prefi, and a matrix without unbounded quantifiers. Proof. T h e previous transformations allow us t o contract quantifiers of the same type, without introducing unbounded quantifiers in the matrix. And quantifiers can be pushed in front by t h e usual transformations (see the proof of IV.1.5). 0 In the style of the Arithmetical and Analytical Hierarchies, we can now introduce a hierarchy for formulas of Set Theory.

Definition IV.3.9 The Set-Theoretical Hierarchy (Levy [1965]) 1.

En is the class of relations definable over the language of Set Theory by a formula in prenex normal form with no unbounded quantifiers in the matrix and n quantifier alternations in the p r e f i , the outer quantifier being existential.

2.

n,

is defined similarly, with the outer quantifier being universal.

3. A, is C,nrI,, forms.

i.e. the class of relations definable in both the n-quantifier

4. Xz, IIz, A;

are the classes of relations definable by formulas provably equivalent, in the theory T , t o (respectively) C,,II,, A, formulas.

5. Xf, llf, Af are the classes of relations definable over the structure A b y (respectively) C,,rI,, A, formulas (also called C,, rIn, A,-definable over A). By extension, we will call a formula C, or II, if it is in prenex normal form with n quantifier alternations in the prefix, the outer one being, respectively, existential or universal. We will also say that a function is in a given class if its graph is.

AFKp functions The level 0 of the Levy’s Hierarchy, which consists of t h e relations definable by formulas without unbounded quantifiers, is quite interesting: inclusion, set equality, ordered pair and relative projections, function, domain, range, graph, w, and ordinal, ordinal successor, limit, n, natural number (i.e. x E w), x x = w are all AFKP notions, as their natural definitions show. E.g., a n ordinal is a transitive set which is totally ordered by the E relation (which induces the order relation on the ordinals); x is a natural number if and only if it is either

IV.3 The Set-Theoretical Hierarchy

407

0 (i.e. 8) or a successor ordinal (i.e. of t h e form x U {x}, with x ordinal), together with all its predecessors; x is w if it is a limit ordinal (i.e. neither 0 nor a successor) such t h a t all its elements a r e natural numbers. However, t h e existence of w as a constant (is equivalent to, a n d hence) requires t h e Axiom of Infinity, a n d is t h u s not provable in G K P alone. T h u s we c a n use individual natural numbers, b u t not their collection. Of course, w is A i F - . The level 1 contains various other interesting notions. For example, wellfouridedness is AFKP:a relation on a set x is well-founded if a n d only if every nonempty subset of x has a minimal element w.r.t. it (this provides the nfKP form), a n d if a n d only if t h e r e is a function with domain x a n d range contained in t h e ordinals which is order preserving (this provides t h e C F K P form). Note t h a t t h e last assertion exploits t h e possibility of defining a function by recursion on a well-founded relation (see IV.2.17 for details) To prove a strong a n d useful closure property of A f K P ,we consider t h e analogue of primitive recursion for set functions. Recall t h a t a nonzero natural number is inductively generated by its predecessor. A set is instead inductively generated by i t s elements. In both cases, primitive recursion gives t h e value of a function on a given element when its values a r e given for t h e elements t h a t inductively generate it.

Theorem IV.3.10 Primitive recursion on E (Von Neumann [1923], [1928], Karp [1967]) The class AfKP is closed under primitive recursion o n E. F'onnally, let g be a total A f K p function, and

&, Y) = {f(?, 2) : 2 E Y}. Then the function

f(?,Y) = g ( % ? / , i ( W ) is total and AfKP. Proof. We ornit t h e parameters 2, which a r e kept constant in t h e proof. We consider t h e functions t h a t satisfy t h e recursion equation on their domain: if their domains a r e closed downward w.r.t. E (i.e. transitive), they c a n be seen as approximations to f . Formally, let

P ( h ) e h is a function A D o m h is closed downward w.r.t. E A (YY E

0 0 7 1 1 h)[h(Y)= S(Y,

h(Y)l,

where D o m h is t h e domain of h. By t h e hypothesis o n g, a n d t h e fact t h a t t h e remaining notions used in t h e definition a r e AFKP,P is AFKp. If we let

f(Y)

= 2 . 3 ( 3 h ) [ P ( hA) h(Y) = 21,

t h e n f is C F K p . W e now show t h a t f is t h e required function.

IV. Hierarchies and Weak Reducibilities

408

1. f is unique By induction on E. Suppose P(h1)A P(h2),a n d let hl a n d hz be defined o n y . Then

MY)).

hl(Y) = S ( Y , h l ( Y ) ) a n d h2(Y) = S(Y,

If z E y then, by closure downward of t h e domains of hl a n d h 2 w.r.t. E , z is in them, a n d t h u s hl and h2 a r e both defined o n z . By induction hypothesis they agree on z E y: t h u s i 1 ( y ) = k 2 ( y ) , a n d h l ( y ) = hz(y).

2. f is total

By induction on E we show t h a t f is defined o n y. T h i s means finding a function h defined on y a n d satisfying P . By induction hypothesis, (Vz E y)(3h2)[P(h,) A z E D o m h,].

By collection and separation, there is a set A containing t h e functions satisfying P , a n d with some z E y in their domains. T h e union of members of A is t h u s a function defined on all z E y, a n d we c a n let h be this function extended to y by:

M Y ) = d Y ,h(Y)). Note t h a t i ( y ) is determined by t h e (union of) functions in A . I t remains to be shown t h a t h satisfies P , which can be easily verified:

h is a function Because g a n d t h e members of A are. Dom h is transitive Because t h e domain of h is t h e union of {y} a n d t h e domains of t h e functions in A , a n d t h e union of t h e latter is a transitive set containing all t h e predecessors of y w.r.t. E . h satisfies t h e recursion equation on its domain O n y this holds by t h e definition of h. On elements z of t h e domain different from y , it is enough to note t h a t t h e y a r e in t h e domain of some function hl in A , which satisfies t h e recursion equation o n its domain by hypothesis (since it satisfies P ) . B u t t h e domain of hl is transitive, a n d thus hl is defined o n all predecessors of z . T h e n k 1 ( z ) = k ( z ) ,a n d

W).

h ( z ) = h ( z ) = g(z,h ( z ) )= g(z, 3. f is AFKP We already know t h a t

f is CfKP.Being total, it is also nFKP:

f ( Y ) # 2 * (34[f(Y) = u A u # 4.

IV.3 The Set-Theoretical Hierarchy

409

4. f satisfies t h e recursion equation T h i s is similar to w h a t we have proved at t h e e n d of p a r t 2. Given y, let h be a function satisfying P a n d defined on y. Its domain is transitive, a n d t h u s it contains all predecessors of y w.r.t. E . By t h e uniqueness o f f , h(y) = f^(y). But h satisfies t h e recursion equation on its domain, since it satisfies P , a n d t h u s

f(Y)

= h(Y) = d y , h(Y)) = d Y , i ( Y ) ) .

0

Course-of-values recursion on natural numbers gives t h e value of a function using a n y set of values for previous elements, a n d not only t h e value for t h e immediate predecessor. For sets, t h e set of previously generated elements c a n be seen as t h e transitive closure of t h e set, i.e. its downward closure under E .

Definition IV.3.11 The transitive closure Tc(z)of a set z is the smallest transitive set containing all elements of 5. Corollary IV.3.12 The transitive closure is a A p K Pfunction.

Proof. Let Tc(z)= z u

(UTc(y)). VEz

By t h e theorem, T c ( z )exists a n d it is AfKp. Moreover, it is really t h e transitive closure: 1.

2

c Tc(z)

By definition.

2. Tc(z) is transitive By induction on E . Suppose t h a t Tc(y) is transitive, for every y E z E t E T c ( z ) .By definition, there a r e two cases: 0

2.

Let

1 E z T h e n T c ( t ) is transitive, z E T c ( t ) & T c ( z ) ,a n d z 6 T c ( s ) .

t E U V E Z Tcb) T h e n t E Tc(y) for some y E z, a n d T c ( y ) is transitive. T h u s z E Tc(y) T c ( z ) ,a n d z E Tc(z).

c

3. z C z A z transitive 3 Tc(z) By induction on E . Suppose y

z

A

z

z transitive

3

Tc(y)

c z,

for every y E z. Given such a y, from z C z we have y E z , a n d y by transitivity. T h e n t h e induction hypothesis applies, a n d Tc(y) Thus Tc(y)) C z. Since z z by hypothesis, T c ( z ) z . 0

(uvcz

cz c z.

410

IV. Hierarchies and Weak Reducibilities

We can think of primitive recursion on t h e transitive closure as being t h e analogue of courseof-values recursion. T h e next result is thus t h e analogue and it shows t hat AFKp has sufficiently strong closure properties to of 1.7.1, allow for th e usual arithmetization results, in set-theoretical setting.

Corollary IV.3.13 Course-of-values recursion on E (Von Neumann [1923], [1928], Karp (19671) T h e class A y K P is closed u n d e r course-ofvalues recursion over E. Formally, let g be a total AFKPf u n c t i o n , a n d

T h e n f is total and AFKp.

Proof. T h e proof of IV.3.10is based on induction on E , i.e.

T h e same proof goes through, once we prove t hat a similar principle of induction on t h e transitive closure holds:

(V4KVY E T c ( z ) ) P b )

-

Wl

=+ (V'.)P(.).

Its hypothesis suggests t o prove not ( V z ) P ( z )directly, but rather (Vz)(Vz E T c ( z ) ) P ( z )

(from which t h e former follows, since z E T c ( { z } ) )We . proceed by induction on E. Given .z we have, by induction hypothesis,

(VY E z)(Vz E T c ( v ) ) P ( z ) , and we want t o prove (Vz E T c ( z ) ) P ( z ) If . z E T c ( z ) ,there are two cases: 0

zEz

Then, by th e induction hypothesis, (Vu E T c ( z ) ) P ( u ) ,and t hus P ( z ) holds by th e hypothesis of t h e principle.

.

z E Uyc*W Y )

Then, by t h e induction hypothesis, P ( x ) holds.

0

Exercises IV.3.14 a) Sum and product o n the ordinals are A f K p . (Hint: the order relation on the ordinals is induced by E.) b) T h e A F K pfunctions are closed under composition and m e definition. (Hint: by logical operations and quantifier manipulations.)

IV.3 The Set-Theoretical Hierarchy

41 1

The levels of the Set-Theoretical Hierarchy T h e classes of t h e Set-Theoretical Hierarchy share a number of properties with their analogues in the Arithmetical and Analytical Hierarchies.

Proposition IV.3.15 Closure properties (Levy [1965])

nZKp

1. R i s CEKP i f and only i f 1 R is R is i f and only i f 7 R is C Z K P

nZKp

2. AZKpis closed under negations 3. CZKp, IIZKP, and AZKP are closed under conjunction, disjunction, and

bounded quantification

4. for n 2 1, C Z K P is closed under existential quantification, and HZKp is closed under universal quantification 5. the universal quantification of a C Z K P relation is II:f:, tential quantification of a nZKP relation is c,G,KP.

and the exis-

Proof. Everything easily follows from logical operations and quantifier manipulations, as in IV.1.8. 0 T h e closure properties just proved can be dealt with in formal theories for Set, Theory, like G K P , because they only require formulas manipulations. T h e remaining properties need instead the existence of elements, and hence they will be proved not for theories, but for models.

Theorem IV.3.16 Enumeration Theorem (Kripke [1964], Levy [1965], Platek [1966]) For any standard model A of G K P , and each n,m 2 1, there i s a n m + 1-ary C: relation that enumerates over A the m - a r y C! relations. Similarly f o r H.: Proof. We prove the result for n = 1. The remaining cases follow from it by induction on n, as in IV.1.9. First we introduce some general tools: 1. a AFKpcoding and decoding mechanism We want t o show how t o code and decode finite sequences of sets. The idea is t o use the AFKPnotions of ordered pair: (z,y) = z

e (3a E z)(3b E .)(a = {z} A b = {z,y}

and relative projections: (z)1 = (.)2

x

=y

@

(3Y E .)(. (32 E

.I(.

= (XlY)) = (Z,Y>).

A

z = {a,b}),

IV.Hierarchies and Weak Reducibilities

412

By courseof-values recursion (since t h e ordered pair of two sets is two levels higher than its components) we can define

0 (21,.

f

. ,%+I)

=

0

=

((z1,.

. . ,x 7 J r ~ T l + 1 ) -

It is then immediate t o obtain, again by courseof-values recursion, a AFKp predicate telling whether a set is a coding sequence, a n d AFKp functions giving t h e length of a coding sequence, a n d t h e components of a n n-tuple. Note t h a t the Axiom of Infinity is not needed in t h e coding procedure: we use th e natural numbers individually, but we never need to consider their set. Recall t hat t h e expressions z E w, z C w ,z = w a r e all A,""p, even without t he Axiom of Infinity.

2. a A f K p satisfaction predicate for A g K P formulas By using th e coding and decoding mechanism, a n d t h e possibility of d e ing course-of-values recursions, we can now proceed as in usual arithmetizations. In particular, we can define a A Y K p satisfaction predicate 7 ( e ,z,y), meaning: t h e formula coded by e , with inputs coded by x (i.e. when its free variables 7 1 ,~7 1 2 , . .. are substituted, in a n orderly fashion, by th e components of x), is t r u e in t h e transitive closure of y.

7 is AfKP because t he definition of t r ut h in a standard structure is a courseof-values recursion over E (in general, it would only be a courseof-values recursion over the membership relation of t h e structure). Note t h a t what we have here is a AfKP satisfaction predicate for A $ K P formulas, since t he effect of interpreting a formula over a (transitive) set is t o bound all quantifiers over t h e set.

3. a C F K P universal predicate T h e definition of a universal C f K P predicate for C fKP m-ary relations is now immediate:

W(e,z1,. . . ,xm) (%)[a is transitive A 2 1 , . . . ,z,

E aA

T ( e ,( 2 1 , . . . , ~ , ) , a ) ] .

Note t h a t we only dealt with relations without parameters, b u t t h e case of relations with parameters can be treated similarly: all parameters can be coded into one, and thus m-ary relations with parameters are just m 1-ary relations without parameters, in which one variable has been substituted with a constant.

+

IV.3 The Set-Theoretical Hierarchy

413

Let now A be a standard model of G K P , and consider

WA(e,xl,... ,x,)

@

( A , E)

W(E,%,. . . ,Z,),

i.e. interpret W over A. This is obviously a Cf predicate, and we want t o show that it enumerates the m-ary Cf predicates. If cp E E f , let u l , . .. ,urn be its free variables, and z1, ... ,z, be in A . It is enough t o show that if cp(z1,.. . ,)z, holds in A, it also holds in some transitive set a E A (this is called &-reflection). Then, if e codes cp, A satisfies W(e,xl,. . . ,x,) by definition, as wanted. Suppose cp(zl,.. . ,zm) holds in A . Since cp is C f , there is @ without unbounded quantifiers, such that cp(x1,.. . ,x,) ++ ( 3 y ) @ ( z l , .. . , x m , y ) . To say that, for xl,.. . ,z, E A , cp(x1,. . . ,zm) holds in A , means that there is y E A such that @(zl,. . . , x,, y ) is true over A . But since @ has no unbounded quantifiers, this is true in the transitive closure of ( ~ 1 , .. . ,,z , y} as well, which is a member of A (by the closure properties of A , which is a model of G K P ) . 0 We can rephrase what we proved as follows. Since t o interpret a formula of any complexity over a set turns it into AFKPform, by bounding the quantifiers, the global satisfaction predicate over a set is AFKP. The satisfaction predicate for CZf: formulas over a class is CZfT. Of course, there is no definable notion of global satisfiability over a class, by Tarski's Theorem (see p. 166).

Xt relations. I f A is Q standard model of GKP, W A is the enumeration predicate for C;" m-ary relations, and R is an m + 1-ary relation, there is e E A such that

Exercise IV.3.17 The Fixed-Point Theorem for

. . ,zm)* R ( e ,X I , . . . ,z,,,).

W A (e,z1,.

(Hint: there is a A$ analogue of the %-Theorem, by standard methods. Moreover C;" relations are closed under substitution of A? functions, because

* (3Y"

P(,'lf(O>

=I(.')ApP(.",Y)l,

and C;" is closed under conjunctions and existential quantifications.)

Theorem IV.3.18 Set-Theoretical Hierarchy Theorem (Kripke [1964], Levy [1965], Platek [1966]) For a n y standard model A of GKP, the SetTheoretical Hierarchy over it does not collapse. More precisely, f o r a n y n 1 the following hold:

>

1. C i -

IIt # 0, and hence A:

# 0, and C t u KAc A!+, .

2. IIt - E t 3.

c Ck hence A t c II$

Proof. By diagonalization and IV.3.16, as in IV.1.13.

I3

414

IV. Hierarchies and Weak Reducibilities

7l.F and the Arithmetical Hierarchy We have noted that the Axiom of Foundation allows a representation of the transitive closure of a set as a well-founded tree, describing t h e set-theoretical build-up of the set from the empty set. We now analyze t h e sets whose associated tree is finite.

Definition IV.3.19 X 3 is the set of hereditarily finite sets, i.e. the smallest class A of sets such that: 1. ~ 2. if

E A 51,.

. . ,x,

E

A then {XI7 . .. x,} E A.

Note the difference between being finite, i.e. having only finitely many elements (like {w}, that consists of only one element), and being hereditarily finite, i.e. having only finitely many elements, each of which is hereditarily finite (a definition by course-of-value recursion). In other words, a set is in XF if and only if its transitive closure is finite.

Proposition IV.3.20 XF is the smallest transitive model of G K P . Proof. We first verify that NF is a transitive model of G K P : 1. transitivity The tree representation of an element of x is a subtree of the representation of x,and it is finite if this is.

2. extensionality and foundation Automatic from transitivity.

3. pair T h e tree representation of {x,y} consists of a vertex on t o p of t h e tree representations of x and y 7and thus it is finite if these are.

4. union

T h e tree representation of u x consists of a vertex on top of t h e tree representation of t h e elements of t h e elements of x, and thus it is finite if there are only finitely many of these trees, each of them finite.

5. separation T h e tree representation of a subset of x is a subtree of the representation of x, and it is finite if this is.

IV.3 The Set-Theoretical Hierarchy

415

6. collection Suppose (b’xE a ) ( 3 y ) c p ( x , y )holds in X 3 . Since n is finite, it has only finitely marly elements ~ 1 , . . ,xn. For each of t h e m , t h e r e is a set y l in X 3 such t h a t cp(x,,yi)holds. By pair a n d union, which we have already verified, b = {yl,.. . , y n } is in ‘HF, a n d thus (Vx E n)(3y E b)cp(x,y) holds in ‘H3. We now verify t h a t ‘HFis t h e smallest transitive model of G K P . Suppose A is such a model: t h e n i t contains 8, a n d it is closed under pair a n d union. We want to show t h a t ‘HF5. A . T h i s is easily seen by induction, since each element z E ‘HF is obtained from t h e emptyset, by finitely many applications of pairing a n d union.

Note t h a t t h e natural numbers, represented in set-theoretical t e r m s (as finite ordinals), a r e all in ‘HF, by induction:

O=8

and

n+l=n~{n}.

Exercises rV.3.21 a) 7 - U = V,. b) 7-V is the smallest model of Z F C with the Axiom of Infinity replaced b y its own negation. (Hint: the power set of a finite set is finite. Choice is trivial, since the elements of W are all finite. Since w is not in ‘HF, the Axiom of Infinity does not hold. Its negation, that every set is finite, holds because if x E 7-V then any function from z to some natural number is already in W . )

T h e reason why we are particularly interested in ‘H3is t h a t , for sets of natural numbers, set-theoretical definability over it coincides ( a n d not only g l o l d l y , b u t level by level) with arithmetical definability. T h i s provides a n alternative, set-theoretical way of seeing Recursion Theory, a n d it is t h e s t a r t i n g point for some interesting generalizations (see p. 421).

Theorem IV.3.22 Set-theoretical definability of the Arithmetical Hierarchy (Ackermann [1937]) Let A C_ w. Then: 1. A E

‘HFi f and only

2. A is definable over f o r n 2 1:

if A is finite

‘H3i f and only i f A is arithmetical. More precisely, AEA;-’ AEC?’

AEA:

e AEC:.

Similarly f o r relations, of a n y niimber of variables.

IV. Hierarchies and Weak Reducibilities

416

Proof. T h e first assertion is easy to see: a n hereditarily finite set is, in particular, finite; a n d a finite set of natural numbers is hereditarily finite, because so a r e t h e natural numbers (as set-theoretical objects). T h e proof of t h e second assertion is more cumbersome, arid it a m o u n t s to show t h a t we can translate, by preserving t h e logical complexity, arithmetical assertions into set-theoretical ones, a n d conversely: 1. translation from Arithmetic t o Set Theory We already know how t o interpret natural numbers in set-theoretical terms. T h e n number quantifiers can be easily t u r n e d into set quantifiers:

(3x)(P(x) (Y.)(P(.)

--

( 3 x ) ( xE w A Cp(x))

(W. E w

--+

Y W .

T h e expression x E w is AFKp,hence A:=, a n d t h u s it does not increa3e t h e complexity of t h e rriatrix. I t only remains t o translate recursive matrices, i.e. graphs of recursive functions, into AFKr predicates. We refer to t h e characterization of recursive functions given in 1.1.8. S u m , product, a n d composition have already been dealt with in IV.3.14, while identities a n d equality a r e trivially AfK? For p-recursion, let

f ( q = CLYR(Z,Y). Then

f(z)= y

@

R(z,y) A (‘d~< y ) i R ( i ? , 2).

If R is A p K Pthen so is t h e graph o f f , by t h e closure properties of AFKP, a n d t h e fact t h a t bounded number quantifiers translate into bounded set quantifiers (because t h e order relation on ordinals is iiiduced by t h e membership relation). Note that, t h e whole argument does not require t h e exist,erice of w. If

w were present (i.e. if we worked with a model of G K P plus infinity), all t h e translations of arithmetical fornnilas would simply become Ao, because t h e number quantifiers would then be translated into bounded set quantifiers (see p. 419 for more on this point).

2. translation from Set Theory to Arithmetic First of all we have to interpret members of 7-t.F as natural numbers, a n d this can be done by induction 011 t h e construction of 7-t.F:

IV.3 The Set-Theoretical Hierarchy

417

(we suppose all t h e z, distinct, since a set is determined solely by its elements). T h i s simply amounts to using canonical indices (11.5.13) hereditarily, by inductively decomposing t h e exponents in t h e binary decomposition of a number. Now set-theoretical quantifiers c a n be turned into number quantifiers:

(3z)(P(x) @

*

(VX)(P(.)

(wf4f-w (W(P(f-'(74).

I t remains to be proved t h a t t h e translations of AZF formulas a r e recursive. By t h e parallel closure properties of AFKP formulas a n d recursive relations, this reduces to show how to deal with membership a n d constants. For t h e former, note t h a t E

Y

* f(.)

E Df(v),

which is a recursive relation. To deal with constants, note t h a t a m o n g t h e values off there a r c some t h a t naturally correspond to t h e set-theoretical integers :

do) = 0 g ( n + 1) = g(n)+29(")

+

(because 0 = 0, a n d n 1 = n U {n}). Now g can be thought of as a function b o t h from 7l.F to w ,a n d from w to w. In t h e latter case, it is a recursive function. Arid g can be used to substitute occurrences of set-theoretical natural numbers with occurrences of t h e corresponding natural numbers coding t h e m as sets. 0

We have proved a correspondence between definability on Arithmetic a n d

X.F, for relations on natural numbers. T h i s suffices for our recursion-theoretical purposes, b u t there is more to it. It can actually b e shown t h a t , for s t a t e m e n t s a b o u t natural numbers, P A is equivalent to Z F C with the Axiom of Infinity replaced b y its negation (which is equivalent to V = 'H3) (Ackermann [1937]). I n other words, t h e translations provided in t h e proof above a r e actually faithful interpretations of t h e stated theories into one another (where interpretation means t h a t provable statements a r e translated into provable statements, a n d faithfulness t h a t no translation is provable unless its original version was already provable). Note t h a t a symmetric role is played by induction a n d foundation, which is t h e reason to consider Peano Arithmetic, a n d not weaker systems. T h e absence of t h e Axiom of Infinity is enough for t h e faithfulness of t h e translation of P A into Set Theory. T h e substitution of t h e Axiom of Infinity

418

IV. Hierarchies and Weak Reducibilities

with its negation is instead crucial to prove t h e faithfulness of t h e translation of Set Theory in P A (since otherwise ( V X ) ( XE XF),which is equivalent to t h e negation of t h e Axiom of Infinity, is not provable in Set Theory, while i t s translation, which amounts to ( V X ) ( XE u), is provable in P A ) .

Absoluteness and the Analytical Hierarchy Since there is no single privileged standard s t r u c t u r e for Set Theory, we will have to interpret t h e formulas on t h e various structures. T h e problem is t h a t t h e s a m e formula could be t r u e in some model of G K P a n d false in some other, t h u s not having a n absolute meaning. As a n example, consider t h e set zEbw

x E a A ( V z ) ' p ( ~z, ) ,

obtained from a by separation. Suppose 'p has no quantifier. For a n y A such t h a t a E A , there is a set b A obtained by interpreting t h e definition of b over

A: z E b A -35 E a f'

A A (b'z E A)cp(Z, Z ) .

If A is transitive then a n A = a , so X

E bA @ X E aA('dz E A ) c p ( ~ , z ) ,

Rut (Vz E A ) y ( z ,z ) could hold even if ( V Z ) ~ (zX) ,does not. T h u s b b A , a n d ' A believes b A is b', while this is not necessarily so. Similarly, by changing t h e universal quantifier into a n existential one, we could have b A b, while for more complicated formulas there is n o simple relationship between t h e t r u e b a n d t h e set b A , t h a t A believes to be b. T h i s is of course a n unpleasant situation, introducing a n element of relativity in Set Theory: on one side we have t h e 'real' sets, on t h e other their interpretations over given models, with no apparent connection between t h e m . T h e situation is not as disruptive as it might seem at first sight, since a number of formulas t u r n o u t t o have a n absolute meaning, in t h e sense of defining t h e s a m e set on every model.

Definition IV.3.23 (Godel [1940])A formula is absolute for a class of structures i f it has the same truth value in each structure of the given class. T h e next result isolates a class of formulas t h a t a r e absolute, a n d i t is q u i t e useful in applications. T h e notion of s t a n d a r d model is used in a crucial way, t h u s providing another reason to restrict our attention to such models.

Proposition IV.3.24 AT formulas are absolute f o r the standard models of T .

IV.3 The Set-Theoretical Hierarchy

419

Proof. Fix a transitive model A of T : we show t h a t t h e t r u t h value of any

AT formula interpreted over A is independent of A , a n d it coincides with t h e t r u t h value of t h e formula in t h e universe V of sets. First of all note t h a t , over elements of a transitive set A , membership is absolute: indeed, if z E A then z n A = z, a n d t h u s z E z has t h e s a m e meaning over A a n d over V . This shows t h a t , in particular, bounded quantifiers preserve absoluteness a n d thus, by induction on their complexity, A: formulas a r e absolute for standard models of T . Suppose now cp is AT. We want to show t h a t cp is t r u e over A if a n d only if it, is t r u e (over V ) . Let $1 a n d $2 be AT formulas such t h a t , in T , cp(.’)

-

P!/)@l(Z,Y)

-

(VY)+2(Z,Y).

Suppose cp(Z), i.e. (EIy)+l(Z,y), is t r u e over A . T h i s means t h a t , for some y E A , +l(Z,y) is t r u e over A . But this is a AT formula, which is absolute. T h e n @1(5,y), a n d hence (3y)@l(Z,y) a n d (p(Z),a r e true. Suppose now cp(Z), i.e. (Vy)@2(Z,y), is true. T h e n @2(?,y) is t r u e for all y , in particular for all y E A . T h u s (Vy)+2(Z,y), a n d hence cp(5),a r e t r u e over

A.

0

In t h e Arithmetical Hierarchy quantifiers range over w. For a n y model A of G K P a n d t h e Axiom of Infinity, w E A . T h e n number quantifiers can be interpreted as bounded set-theoretical quantifiers, a n d arithmetical relations are absolute, being translated into A; formulas (when s u m a n d product a r e replaced by their A f K pdefinitions, see IV.3.14.a). In t h e Analytical Hierarchy quantifiers range also over P(w), which is only a I I ? ~ ‘ object: z = P(w) H Vy(y E z H y w).

c

P(w) i s n o t absolute, a n d hence not A:”‘: by absoluteness of y C w , i t s interpretation over a transitive model A of G K P is P(w) n A , a n d t h u s it varies with A (see also IV.4.27.c). T h i s means t h a t function quantifiers do not automatically translate into bounded quantifiers, a n d analytical relations a r e not, automatically absolute. But relations in t h e first two levels a r e , as we now see.

Proposition IV.3.25 (Mostowski [1949])II: relations are absolute for standard models of G K P containing w (i.e. models of Z F - ) . Proof. By t h e First Representation Theorem for IIi sets (IV.2.15), A is II: if a n d only if t h e r e is a recursive sequence {Tz},E,of recursive trees, such t h a t z E

A ts T, is well-founded.

IV. Hierarchies and Weak Reducibilities

420

Rut we have already noted t h a t recursive relations (being arithmetical) a n d well-foundedness (being A f K P )a r e absolute. T h e n so is A . 0 Note t h a t t h e proof shows t h a t every ni relation is actually A, over G K P plus infinity. Similarly for Ci relations, by taking negations. Exercise IV.3.26 n: formulas are n o t absolute f o r standard models of Z F C minus infinity. (Hint: W is a model for it, but the relations over w definable over it are all arithmetical.)

Theorem IV.3.27 (Shoenfield [196la])Ci relations are absolute f o r standard models of G K P containing all countable ordinals. Proof. By t h e relativized version of t h e First Representation T h e o r e m for ni sets, A is C; if a n d only if there is a recursive sequence { T , , f } z c wof trees uniformly recursive in f , such t h a t zE

A

H

(3f)(T,,f is well-founded).

Since well-foundedness of T,,f is equivalent to t h e existence of a n order-preserving m a p from T,,f t o t h e countable ordinals (because t h e trees a r e countable), zE

A

tj

(3f)(3g)(g : T,,f

+ w1

is order-preserving)

Now we can reproduce t h e proof of t h e First Representation T h e o r e m IV.2.15 (by looking at t h e first places in which t h e matrix fails), a n d get uniformly absolute trees R,,f,s on w x w1 such t h a t

x E A tj R,,f,, is not well-founded (since A is now defined b y existential quantifiers, in place of universal ones). T h u s , if M contains all countable ordinals, A is absolute for M . 0 Note t h a t t h e proof shows t h a t every relation is actually of t h e form (Y ranging over countable ordinals, a n d ‘p A, over G K P plus infinity.

(3a countable)cp, with

Exercise IV.3.28 Ci formulas are not absolute f o r standard models of Z F C . (Hint: the formula translating ‘the set X w codes a countable transitive model of ZFC’ is A:, by arithmetization. ‘There exists a set X coding a transitive model of ZFC’ is thus a true Ci formula, which is not true in the least countable transitive model of Z F C . Note that this reasoning requires the existence of a standard model of Z F C , and thus it is not formalizable in Z F C . For the weaker result relative to standard models of Z F C - only, i.e. without the Power Set Axiom, such a n assumption is not necessary.)

IV.3 The Set-Theoretical Hierarchy

Admissible sets

42 1

*

We have worked with the theory G K P because we wanted t o have structural results for all levels of t h e Set-Theoretical Hierarchy. But the full power of G K P is needed only for the full results, and it is possible t o refine G K P , and isolate what is needed to get the structural results for t h e first (or, more generally, t h e n-th) level only. T h e Kripke-Platek system K P (Kripke [1964], Platek [1966]) has t h e same axioms of G K P , with separation and collection limited t o A0 formulas, and it can be seen as a kind of constructive Set Theory. This theory is strong enough to prove (separation for A, formulas, collection for C1 formulas, and) t h e closure properties of t h e first level of the Levy’s Hierarchy. T h e transitive sets which are models of K P (i.e. the standard models) are called admissible sets, and can be seen as domains suitable for a theory of C1 relations and functions analogous t o that of r.e. relations and partial recursive functions, and hence for a Generalized Recursion Theory on sets. Note that XF is the smallest admissible set, and thus the usual notion of recursiveness is a special case of recursion on a n admissible set (by IV.3.22). Moreover, Gordon [1968] has proved that on a n admissible set the notion of recursiveness coincides with t h a t of search computability (see p. 204). This should not be taken t o mean that the notion of admissibility is either sufficient or necessary for a n abstract analogue of all parts of Recursion Theory: Simpson [1974] and Harrington (see Chong [1984]) have shown t h a t there are admissible sets which do not admit a positive solution to the analogue of Post’s Problem, while F’riedman and Sacks [1977] have extended a good deal of Recursion Theory, including a positive solution t o the analogue of Post’s Problem, t o special nonadmissible sets. T h e full power of G K P is not always avoidable, even in the study of t h e first level of t h e Set-Theoretical Hierarchy. A crucial example is the notion of well-foundedness, which is ApKP but not Afp, and thus is not absolute for admissible sets. What fails here, since well-foundedness is IIfp by definition, is the possibility of carrying on recursion on well-founded relations (the so called @-property,Mostowski [1959]), and thus t o provide the C r p form: we have noted t h a t t h e justification of recursion on well-founded relations requires some form of separation. A,-separation, provided by admissibility, is not enough, although C1-separation is. In particular, the Collapsing Lemma IV.3.5 is not provable in K P , although its corollary is (because its proof requires only a course-of-value recursion on E). For a development of t h e theory of admissibility see Barwise [1975] and Fenstad [1980]. T h e implications of the notion of admissibility for t h e study of P(w) will be dealt with in Volume 111.

422

IV.4

IV. Hierarchies and Weak Reducibilities

The Constructible Hierarchy

T h e Analytical Hierarchy is immensely extended a n d i t contains, already at low levels, all sets of natural numbers naturally occurring in practical considerations. Nevertheless it is still countable, a n d t h i s h a s to be t r u e of all t h e hierarchies t h a t simply stratify t h e relations definable in s o m e fixed countable language, including t h e universal language of Set Theory. If we want to overcome this defect, we have to allow for a n extension of t h e notion of definability. O n e way to do this is by transfinitely iterating definitions, a n d we pursue t h i s p a t h here.

The Constructible Hierarchy At t h e t u r n of t h e century various mathematicians began t o feel uncomfortable with t h e development of Set Theory. T h e center of t h e dispute was t h e Power Set Axiom t h a t , in one of its simplest applications, allowed consideration of t h e class of all sets of natural numbers as a completed totality. T h e discovery of paradoxes added ground to t h e objections, a n d one possible way o u t was seen in a strong form of t h e vicious circle principle, according to which a totality cannot contain members t h a t a r e only definable in t e r m s of it (PoincarC [1906], Russell [1908]). As a consequence, only predicative definitions, t h a t define objects without referring to sets already containing t h e m , would be accepted. Contrary to t h e form of t h e vicious circle principle considered o n p. 399, which is consistent with usual mathematical practice, a n d is actually a consequence of t h e Axiom of Foundation, this strong form would permit only a limited development of mathematics. See e.g. Weyl [1918] for what c a n be saved of analysis, a n d Godel [1944], [1964] for discussion. T h e idea of a predicative iterative approach, in which o n e would s t a r t with easily definable a n d graspable sets, a n d would add at each step only t h o s e sets t h a t were definable by using t h e previously obtained ones, is however worth pursuing. T h e decision of when (i.e. at which ordinal) to stop t h e iteration process is crucial. If we really were interested only in t h e predicatively definable sets, t h e n we should allow only for predicatively definable ordinals: t h i s requires a n analysis of t h e notion of predicativity, a n d will be considered in Volume 111. Here we t a k e a more generous s t a n d , a n d allow for a n y number of iterations. By so doing we lose t h e property of predicativity, a n d we can think of the constructible hierarchy as consisting of those sets which are predicatively definable modulo the ordinals. T h i s hierarchy should not be taken as exhausting t h e whole universe, b u t rather as a kind of minimal model (see IV.4.7). T h e constructible hierarchy also extends ideas of Hilbert [1926], w h o tried to prove t h e Continuum Hypothesis by considering t h e generalized recursions (using higher-type objects) needed t o generate all functions of natural num-

IV.4 The Constructible Hierarchy

423

bers, and attempted a proof to show t h a t they could be reduced t o transfinite recursions on ordinals up t o w1. Gbdel’s improvements on Hilbert’s tentative are of two kinds: he uses all ordinals, instead of only countable ones, and firstorder definitions, instead of recursions (which correspond only t o onequantifier definitions).

Definition IV.4.1 (Godel [1939])T h e Constructible Hierarchy is described, in terms of ordinals, as follows:

Lo La+1 Lp L

= 0 = def(La) = Ua
where def(x) is the set of subsets y of x which are definable (with parameters) over x. Ifx E L then x is constructible. There are alternative ways of presenting the successor steps in t h e construction of L . First of all, we can allow parameters or not. T h e reason is that if a set is definable over La with parameters, it is also definable over it without parameters, by hereditarily substituting the parameters with their definitions (in t h e sense t h a t the definition of a parameter may involve other parameters, which have to be discharged as well). Note that no circularity is involved, because t h e definition of a n element of L, can only use parameters from Lg with ,l3 < a , and thus the ordinals are decreasing, although an ordinal greater than a may be needed t o carry out t h e induction. Another way t o define L (Godel [1940]) relies on an analysis of t h e operation def (x), and on the isolation of finitely many (eight) operations t h a t explicitly produce, by composition over the transitive closure of x , any set definable over x . This approach effectively reduces the infinitely many formulas of t h e language of Set Theory t o a finite set of operations, and it allows a finite axiomatization of Set Theory, based on the concept of class (the Von NeumannGodel-Bernays system N G B , see Fraenkel, Bar-Hillel, and Levy [1958]). Finally, it has turned out that the levels L, of the constructible hierarchy are well-behaved only for limit ordinals, while in general they are not closed under very natural functions (like ordered pairing) that, although obviously constructible, increase levels. For a finer analysis, t h e constructible hierarchy has been substituted by Jensen Hierarchy (Jensen [1972]), whose levels J , are extensions of L a , and possess the closure properties that only t h e limit levels of L have. T h e levels of the two hierarchies coincide exactly at those stages a such that w . a = a . See Devlin [1984] for a treatment.

Axiom IV.4.2 T h e Axiom of Constructibility is the assertion V = L that every set is constructible, i.e. (Vx)(x E L ) .

IV. Hierarchies and Weak Reducibilities

424

Here we develop t h e study of L only for what concerns our immediate interest, i.e. t h e study of subsets of w. General references for constructible sets a r e Godel [1940], Mostowski [1969], a n d Devlin [1984].

The levels of the Constructible Hierarchy T h e first few levels of L have already been considered: since def (x)= P ( x ) for a n y finite set x, we have L, = V, for a 5 w. In particular, L, = ‘H3. Some of t h e properties of these first levels generalize to every level.

Proposition IV.4.3 (Godel [1939]) L, is transitive, and i t has finite cardinality i f a < w,and the same cardinality as a otherwise. Proof. L, is transitive, by induction on a : 0

a=o Obvious.

0

cr=p+1 Let x E y E La. By definition, y C Lp, a n d t h u s x E Lp. By induction hypothesis Lp is transitive, a n d t h u s x Lp. T h e n x E Lp+l because x is definable over Lp, by t h e formula cp(z) @ z E x (in which x a p p e a r s as a parameter).

0

a limit T h e union of transitive sets is transitive.

L, is finite, by induction on n E w ,because t h e power set of a finite set is finite. Suppose then cr 2 w:we show by induction on a t h a t L , h a s t h e s a m e cardinality as a. 0

cr=w L, is countable because all t h e L, a r e finite.

0

cy=p+1 T h e cardinality of Lo+, is equal to t h e cardinality of t h e set of formulas with parameters in Lp, a n d hence to t h e cardinality of Lp (because t h e number of possible formulas is countable, a n d p 2 w). B y induction hypothesis this is t h e cardinality of p, a n d hence of p + 1 = a .

0

cr limit L, is t h e union of cr many sets, each of cardinality at most t h e cardinality of cr. T h u s it has cardinality at most equal to t h a t of a (by well-known properties of cardinals, following from t h e Axiom of Choice). 0

IV.4 The Constructible Hierarchy

425

Proposition IV.4.4 Hierarchy Theorem (Gijdel [1939]) 1. a

5 p* L ,

Lp

2. a < p * L , C L p .

Proof. Note t h a t L, E L,+l, being definable by x = x over L,. To show t h a t L, G Lp if a 5 p, we may suppose a < /3 (since t h e case a = p is trivial). And, if a < p, it is enough to show t h a t L , E Lp, because Lp is transitive (and thus its elements a r e contained in it). We thus prove, by induction on p, t h a t L , E Lp, for any cr

< p:

p=O Obvious.

P=y+l If a < p then either a = y, a n d thus L , = L, E LTil = Lp, o r a < y, a n d thus, by induction hypothesis, L , E L, E L,+1: by transitivity, La G L,+l.

p limit If a < p, there exists y such t h a t a < y < p. T h e n L , E L, by induction hypothesis, a n d L, C Lp by definition of Lp for p limit. T h u s L , E Lp. To show t h a t t h e hierarchy is proper it is enough to show, by induction on a , t h a t a E L,+1 - L,. Suppose t h a t , for every p < a , p E Lp+, - Lp: t h e n (Vp < a)(pE L,) (because, by t h e first p a r t of t h e proof, if p < a then Lp+l G L,), a n d a C L,. Now:

a@La

Otherwise, by definition of L,, ct C Lp for some /3 contradicting t h e induction hypothesis.

< a , a n d t h u s p E Lp,

a E L,+l a C L,, a n d cr @ L,. Moreover, t h e ordinals in L , are closed downward (because L , is transitive). This means t h a t cr is t h e set of ordinals in L,, a n d thus it is definable over it by t h e formula ‘x is a n ordinal’. T h e n c t E L,+1. 0

The structure of L T h e next result shows t h a t t h e definition of L, is absolute, a n d t h u s i t has t h e s a m e meaning in every model of Z F - .

Proposition IV.4.5 (Takeuti [1960], Karp [1967])

IV. Hierarchies and Weak Reducibilities

426

1. La is a AfF- function total on the ordinals 2. x E L , is A;"-,

3. x E L is

as a relation of x and

N

zfF-.

Proof, L, is defined by recursion on t h e ordinals, a n d t h u s it is enough to verify t h a t t h e recursion cases a r e AfF-. T h e only nontrivial case is t h e successor step. B u t to say t h a t x La is definable over L, means t h a t t h e r e is a formula (this introduces a n existential quantifier, which is b o u n d e d over t h e Af F- set of formulas, which is a set by t h e Axiom of Infinity') such t h a t x is t h e set defined by it over L , (and t h e satisfaction predicate is A f K P ,as in IV.3.16). T h e n x E L, is AfF-, a n d x E L is C f F - because

xEL

@ ( ~ N ) ( Nordinal

A

x E La).

0

Note t h a t L , is absolute for s t a n d a r d models of Z F - , being but L is not, being only C f F - . Given a s t a n d a r d model A of Z F - , if N E A t h e n L, E A , by replacement. T h u s L in A becomes UaEA L,, which is L itself if A contains all ordinals. In other words, L is a,bsolute f o r standard models of Z F - containing all ordinals (Shepherdson [1951]). O n e important tool for later proofs is t h e following.

Theorem IV.4.6 Reflection Principle for L (Levy [1960], Montague [196l]) Given a formula cp with n free variables, f o r every ordinal N there exists p 2. N such that, whenever X I , .. . ,x, E L g ,

L

k cP(x1,.. . , X T L )

@

Lg

t=

cp(X1,.

. . ,GI).

Proof. Let cp have no occurrence of V (by possibly replacing all occurrences of V with -&), a n d let $ 0 , . . . , $, be t h e finitely many subformulas of cp. Define a sequence of ordinals by starting with Po = a , a n d letting pTL+,be t h e least ordinal y > pn such t h a t , for every sentence ( 3 u t ) $ J k ( u , with ) constants in Loo, a n d t r u e in L , $ J k ( a zis ) t r u e in L for some a, E L,. Note t h a t Pn+l exists, by t h e Axiom of Replacement. Let p be t h e 1.u.b. of {,BTl}nEw. I t is now immediate to check, by induction on t h e length, t h a t if $ is any instance of a subformula of cp (in particular cp itself) with constants from L g , t h e n 11) is t r u e in L if a n d only if it is t r u e in Lp. T h e atomic case holds by absoluteness, propositional connectives a r e trivially handled, a n d t h e case of 'The Axiom of Infinity could be avoided, thus replacing Z F - by G K P throughout this section, if the alternative approach to constructibility by Godel's functions, quoted after IV.4.1, were used.

IV.4 The Constructible Hierarchy (existential) quantification holds by construction.

427 0

Note t h a t t h e Reflection Principle has nothing much to do with L , a n d it works in general for any class which is t h e union of a n increasing hierarchy of sets defined o n all ordinals, such t h a t t h e limit levels a r e defined as t h e union of t h e previous ones. I n particular it works for V , a n d it shows t h a t in Z F we c a n find a model (with a V,, a n d hence a set, as universe) for a n y finite set of theorems of Z F , a n d t h u s prove t h e consistency of any finite part of Z F . By Godel's Second Theorem (p. 169), Z F i s n o t finitely axiomatizable (McNaughton [1954], Montague [1961]). In other words, collection a n d separation a r e not reducible, like foundation was, to a finite set of statements. As we have quoted, t h e r e exists a finitely axiomatizable set of axioms N G B for Set Theory, based on t h e notion of class (see F'raenkel, Bar-Hillel, a n d Levy [1958]). T h e next result characterizes L in set-theoretical terms. We consider t h e full theory Z F , instead of j u s t Z F - , not only because t h e stated result is stronger, b u t because we need to consider t h e Power Set Axiom in t h e next subsection.

Theorem IV.4.7 (Gijdei [1939])L is t h e smallest standard m o d e l of Z F containing all t h e ordinals.

Proof. We have already noted t h a t every model of Z F - containing all t h e ordinals must contain L , by absoluteness of t h e definition of La. Since L contains all t h e ordinals, because a E La+, by t h e proof of IV.4.4, it remains to prove t h a t L is a model of Z F . T h e argument is in four p a r t s , exploiting different properties of L in t h e verifications of t h e axioms. 1. extensionality and foundation T h e y a r e automatically satisfied, because L is transitive. 2. infinity Since w is a n absolute object, it is enough to show t h a t i t is in L. B u t

w E L,+l. 3. separation Suppose u E L , a n d 'p is a formula with parameters in L . We want to show t h a t t h e set x defined by Z E X

e L+ZEUA~(Z)

is in L. T h e r e exists a such t h a t La contains b o t h a a n d t h e parameters of 'p, a n d hence t h e set x,

IV. Hierarchies and Weak Reducibilities

428

is in L , being in IJa+l. But there is no reason to believe t h a t x = x,, since p may have quantifiers, t h a t mean different things when interpreted over L a n d over L,. To straighten this o u t , we apply t h e Reflection Principle IV.4.6: let p 2 a be such t h a t , whenever z E Lo,

Thus, if z E ~p

@

Lp

z E U Av ( z ) ,

s

we have xp n Lp = x n Lp. But, since a E La Lp, a is a subset of Lp by transitivity, a n d t h e n so is x. T h u s x p n Lp = x, a n d x E Lg+l. 4. large sets existence T h e remaining axioms a r e all of t h e same type: given certain sets, they produce sets bigger t h a n them (in contrast to t h e Separation Axiom, t h a t isolates subsets of given sets). T h e y a r e treated in t h e s a m e way, a n d t h e proofs t h a t they hold in L all follow from a single fact: t h a t each subset L of L is contained in some L,. T h i s is easily seen to be true: if a then (Vx E a)( 3o) ( xE L a ) , a n d b y collection (or replacement) t h e r e is a such t h a t (Vx E a)(. E L a ) , so t h a t a C La.

pair T h e set b is t h e pair {x,y} if

(VZ E b ) ( z = x V z = y), which is a n absolute definition. Given x , y 6 L , we must then show t h a t { x , ~ E} L. {x,y} exists by t h e Pairing Axiom in V , a n d it is contained in L. T h e n it is contained in La, for some a . Since it satisfies t h e definition above over L a , it is in La+l.

union Similar. power set T h e set b is t h e power set P(a) if (V'Z)(Z E b

-

z & a),

which is not a n absolute definition. T h u s b is t h e power set of a in L if it satisfies this definition o n L , i.e. (by absoluteness of inclusion)

(VZ E L ) ( z E b

z

C a),

IV.4 The Constructible Hierarchy

429

which defines P(a) n L over V . Given a E L , we must t h e n show t h a t P(a) n L is in L. ? ( a ) n L exists by t h e Power Set Axiom in V , a n d it is contained in L. T h e n it is contained in L a , for some a. Since it satisfies t h e definition above over L a , it is in La+l.

collection If a E L , a n d

L

i= (Yz E a > ( & ) c p ( z d h

we need to find b E L such t h a t

L

i= ( V X

E aH3Y E b)cp(z,y).

The hypothesis means t h a t

(YE E a n L)(% E L ) ( L i= cp(z:,y)). By collection in V , there is c

L such t h a t

(t/.E an L ) ( 3 E c ) ( L

t=

cp(.lY))

T h e n c C La for some a,a n d hence (Ys E a n L ) ( 3 E~ L a ) ( L

+

~ ( z i ~ ) ) .

Since La E L , it is enough to let b = La to have

(Yz E an L)(3Y E b ) ( L I= cp(z,y)), a n d hence

L

l= (Yz E a ) ( $

E b)cp(X,Y).

0

Corollary IV.4.8 L is a model of V = L . Proof. We have to prove

L

+ (Y.)(X

E L).

Since L is absolute for models of Z F - containing all ordinals, a n d L is such a model, this amounts to showing t h a t

(Vz E L ) ( z E L ) , which is trivially true.

0.

T h e corollary is not only interesting for its own sake, b u t also in applications: to prove t h a t some fact holds in L , i t is enough to show t h a t it is provable

in Z F plus V = L .

IV. Hierarchies and Weak Reducibilities

430

Corollary IV.4.9 (Kreisel [1956], Shoenfield [196la]) If 'p is a X i sentence of Second-Order Arithmetic provable in Z F plus V = L, then 'p is already provable in Z F alone. Proof. Let cp H ( 3 A ) + ( A ) ,with $ E n?j. If cp is provable in Z F plus V = L t h e n it is t r u e in L , i.e. t h e r e is A E L such t h a t + ( A ) holds in L. By a b s o l u t e ness of IIk formulas for I, (IV.3.27) + ( A ) must t h e n be true, a n d t h u s 'p holds. But t h e whole reasoning took place in Z F , since only Z F is needed to prove IV.4.12 a n d IV.3.27. T h u s 'p has been proved in Z F alone. 0 T h e result is best possible since, by IV.4.22, ( V A ) ( AE L ) is a IIi s t a t e m e n t t r u e in L (because L satisfies V = L ) , b u t not provable in Z F C (because its negation is consistent with Z F C ) . T h e usefulness of t h e corollary is quite evident: V = L does not prove any new sentence of Second-Order Arithmetic, a n d t h u s it c a n be used freely when trying to prove such a sentence in Z F alone. T h e s a m e of course holds for any of t h e consequences of V = L , like t h e Axiom of Choice a n d t h e Continuum Hypothesis, proved below. Actually, in t h e latter cases t h e result is not t h e best possible, a n d it can be improved to hold for bigger classes of sentences (see Platek [1969]).

Theorem IV.4.10 (Godel [1939], Karp [1967]) 1. There exists a A f F - well-ordering
2. There exists a C f F - well-ordering < L of L .

Proof. W e define a AgF well-ordering


of L a , by recursion on t h e ordinals.

1. c r = o Since Lo = 0,
2. a = @ + l By induction hypothesis we have

H

( X E

L p A y € LpAx
(xELpAYELp+l-Lp)V ( x E Lp+1- Lp A Y E Lp+1 - L p A t h e first formula defining x over Lp precedes t h e first formula defining y over L o). 3.

ct

limit

43 1

IV.4 The Constructible Hierarchy

By definition, if a < p then
n<m

0

n = m a n d cp precedes q!J (as formulas) n = m, cp = q!J and, for the first i such t h a t ai

# bi, ai

Again t h e conditions in this definition are A;"-, ordering is AHF-. Then X
ej

a n d thus t h e whole well-

(3CX)(ZELaAyELaAX
is C ; F - . Note that, for each a ,

0

Exercises rV.4.11 a) Zf L , is a model of Z F - , the well-ordering <, of L, has length a. (Hint: by replacement, L , is closed under the rang function of <,

f(Y)

=. P {( . )

+ 1: Z

Y),

and thus sup{f(z) : z E L , } 5 0. Conversely, since f is one-one, f-' is a function with range L,, and if the length of <, were P < 0, then the range of f -' would be in L,, i.e. L , E L,, contradiction.) (Hint: in this case b) I f V = L then Q is

+ cL y) and thus < L is also

z = Y v ( ~ c x ) (Ez L , A Y E L , A Y

<, z),

nfF-.)

One of t h e original reasons t o introduce L was t o prove t h e following fact, whose consequence is t h a t the Axiom of Choice is consistent with Z F (Gijdel [1938]), since it holds in a model of Z F . Note that also V is a model of Z F , but there is no obvious way t o show that it satisfies t h e Axiom of Choice, t h e difficulty being in well-ordering t h e power set of a given (well-ordered) set. Corollary IV.4.12 L is a model of Z F C plus

V = L.

IV. Hierarchies and Weak Reducibilities

432

Proof. By IV.4.7 and IV.4.8, it only remains t o prove t h a t t h e Axiom of Choice holds in L. T h e theorem just proved shows t h a t L is well-orderable, a n d thus th e Axiom of Choice holds if V = L. But L is a model of Z F plus V = L , a n d hence t h e Axiom of Choice holds in L. 0 T h e fact th a t L is a model of Z F C has a n important consequence: all theorems of Z F C are true in L . In particular, it is possible t o carry on inside L t h e theory of cardinals, which is largely based on t h e Axiom of Choice. Of particular interest for us is w f , t h e analogue of t h e first uncountable cardinal, which is t he least ordinal N such t hat no one-one function f : N 4 w exists in L. In other words, ' L believes wf is w1'. Clearly w,"5 w1: if there is no function with certain properties in V , there is none in L either. The assertion wf = w1 is consistent with Z F C , since so is V = L (by IV.4.8). But also the assertion wf < w1 is consistent with Z F C (Cohen [1966]), by cardinal collapsing (see Volume 111). It also follows from large cardinals hypothesis (Rowbottom [ 19711).

Constructible sets of natural numbers We have introduced t h e Constructible Hierarchy not for its own sake, but to study subsets of w.We thus turn now to our real interest, t h e set P(w)n L of constructible sets of natural numbers. Since P(w)n L L , there is N such t h a t P(w)n L L,, a n d t hus there is a stage after which no subset of w is ever generated. We now improve on this, by exhibiting such a n a (which will t ur n out t o be t h e best possible).

c

Theorem IV.4.13 (Godel [1939])P(w)n L

cLw~.

Proof. T h e theorem reduces t o prove, in Z F C , t h a t P(w)n L C L,, . Since L is a model of Z F C , t h e result also holds in L. To see what this means, recall t h a t P(w),th e power set of w ,means P(w)n L in L (see t he proof of IV.4.7). L is absolute for L , because L is a model of Z F - containing all ordinals. And wt is, by definition, t he w1of L.Thus P(w)nL G L,, means P(w)n L 2 L,: , when interpreted inside L , and this is t he statement of t h e theorem. We thus turn t o t he proof of P ( w ) n L 5 L,,, which roughly consists of the following. Let A C w be constructible, i.e. A E L . Then A E L,, for some N : we want t o show t hat N may be supposed to be countable. To achieve this, we use the Lowenheim-Skolern Theorem of logic (for t h e reader not acquainted with it, we prove in 2 below t he special case we need). Its essence is to cut out, from a model of a sentence, a countable model, leaving unchanged a given countable subset. Here we know t hat A E L,, and thus we only have t o find a sentence 'p whose models are exactly t h e sets of t h e kind L, (this is done in 1

IV.4 The Constructible Hierarchy

433

below). T h e n L, is a model of cp, a n d it contains { A } : t h e Lowenheim-Skolem Theorem produces a countable model of cp containing { A } , which must t h e n be of t h e form L,, with cr countable (because, by IV.4.3, L , is countable if a n d only if cr is). I t only remains to find cp, a n d prove t h e Lowenheim-Skolem Theorem.

1. There is a sentence cp such that if a set M is a standard model for cp, then M = L, for some (limit) a. By absoluteness of L,, if M is a s t a n d a r d model of Z F - t h e n U a E ML , is L in M , a n d it is contained in it. Let N M be t h e smallest ordinal not in M : if N M is limit t h e n , by definition of L , U n E ML, = L,,, a n d t h u s L,, M . If V = L also holds in M , i.e. M is equal to L i n it, t h e n L,, = M . We can t h e n let cp be t h e conjunction of t h e following: 0

t h e finitely many axioms needed to prove t h a t t h e notions of ordinal a n d of L, a r e absolute for s t a n d a r d models of t h e m t h e finitely many axioms needed to prove t h a t a n y model of t h e m is closed under L , (as a function of a )

0

t h e finitely many axioms needed to prove t h a t t h e r e is n o greatest ordinal

0

V=L.

2. Given asentencecp, aset M which isamodelforit, and acountablesubset X of M , there is a countable model MI of cp such that X C_ M' C_ M (Lowenheim [1915], Skolem [1920]). T h e proof is a variation of t h e proof of t h e Reflection Principle IV.4.6. As there, t h e only important thing is to provide t h e witnesses for t h e existential sentences t r u e in M . Let cp have no occurrence of V (by possibly replacing all occurrences o f V with 4a) nd, let $0,. . . ,$- be t h e finitely many subformulas of cp. Define a sequence of subsets of M , s t a r t i n g with MO = X , a n d letting M,+1 be a countable subset of M containing M , a n d such t h a t , for every sentence (3v,)$k(vi) with constants in M , a n d t r u e in M , $ k ( u i ) is t r u e in M for some a, E M,+I. Let M' = UnEw M,. First of all we have to prove, by induction on n, t h a t M,+1 exists, i.e. t h a t t h e r c is a countable structure as wanted. MO is countable by t h e hypothesis on X . Suppose M,, is countable: then t h e r e a r e only countably many possible sentences with constants in M, to consider. Since we only need to add o n e witness for each of them, we c a n choose Mn+l as a countable subset of M .

IV. Hierarchies and Weak Reducibilities

434

It is now immediate to check, by induction on t h e length, t h a t if ?1, is any instance of a subforinula of cp (in particular cp itself) with constants from M I , t h e n ?1, is t r u e in M if a n d only if it is t r u e in MI. T h e atomic case holds by absoluteness, propositional connectives a r e trivially handled, a n d t h e case of (existential) quantification holds by construction. Fact 1 refers to standard models, a n d this requires a small p a t c h u p of t h e w such t h a t A E L,, we may supsketch given at t h e beginning. Given A pose cr w limit: then we have a model L , of cp, containing w U { A } . By t h e Lowenheim-Skolem Theorem (fact 2), t h e r e is a countable model with t h e s a m e properties, b u t we cannot yet conclude t h a t this is a countable L,, because it is not necessarily a standard model (an assumption used in t h e proof of fact 1 above). B u t we can apply t h e (corollary of the) Collapsing L e m m a IV.3.5, a n d find a standard structure isomorphic. to it, a n d hence satisfying t h e s a m e sentences, cp in particular. Now we do have a s t a n d a r d s t r u c t u r e satisfying cp, still containing A (because w U { A } is a transitive set, since A w , and t h u s it is not changed by t h e collapsing function). T h e n t h i s model must be a countable L a , a n d cr is countable itself by IV.4.3. 0

>

O n e of t h e original reasons to introduce L was to prove t h e following fact, whose consequence is t h a t the Continuum IIypothesis is consistent with Z F C ( G d e l [1938]), since it holds in a niodel of Z F C . T h e reason why t h e Continuuni Hypothesis follows from V = L , while it is independent of Z F C , is t h a t t h e former completely specifies t h e extension of t h e power set operation (left undetermined by t h e latter), by determining which subsets of a given set a r e available, namely only those t h a t can be predicatively defined from t h e ordinals. By providing a sort of minimal interpretation for t h e power set operation, V = L limits t h e number of subsets of a given set to t h e least possible value.

Corollary IV.4.14 The Continuum Hypothesis holds in L. Proof. We have proved t h a t P ( w ) n L & L w l . If V = L t h e n t h i s nieans t h a t t h e power set of w has at most t h e power of L,, , which is w1 by IV.4.3. By Cantor’s Theorem (11.2.1), t h e cardinality of t h e power set of w must be uncountable, a n d hence at least w l . T h u s it is exactly w1. T h i s shows t h a t t h e Continuum Hypothesis follows from Z F C plus V = L ( t h e Axiom of Choice is needed for cardinal arithmetic, in particular in t h e proof of IV.4.3), a n d by IV.4.12 it then holds in L . 0 We will see in Volume I11 t h a t all the following possibilities are consistent with Z F C : 1. P(u)

L , which follows frotii V = L .

IV.4 The Constructible Hierarchy 2.

435

L and P(w) n L countable, by collapsing wf (this also follows from large cardinals hypothesis, by Rowbottom [ 19711).

P(w)

L and P(w) n L uncountable, by starting with a model of t h e Continuum Hypothesis, and building a generic extension of it t h a t preserves cardinals.

3. P(w)

By th e proof of t h e corollary, P(w) n L is not contained in La for a < w:, because otherwise t h e power set of w would be countable in L , contradicting Cantor’s Theorem in L . T hus {P(w)r l L Q } Q < was~ a, hierarchy for P(w)n L , has the smallest possible length. We now show that t h e hierarchy is not proper.

Proposition IV.4.15 (Putnam [1963])There exists a new subset o f w is generated in L at stage a -k 1, i.e.

< wf such that no

P ( w )n (LQ+l - L a ) = 0. Proof. Let @ be t h e formula

(ga)[P(w) n (La+1 - La) = 01. By IV.4.13,

P(w) n (Lw;+,

- Lw;) = 0

a n d thus, by absoluteness, $ is true in some Lg with p limit (e.g. let p be wf w). By t h e same argument of the proof of IV.4.13 (using t h e formula cp whose models ar e all limit levels of t he constructible hierarchy, t h e LowenheimSkolem Theorem, a nd t he Collapsing Lemma), there is a countable p with t h e same properties. Since $ holds in Lg, there is a n ordinal a < ,f3 such t h a t

+

P(w)

(LQ+l

-

= 0,

a n d a is countable, because so is p. This is not enough yet, because we actually want a < w f ,i.e. constructibly countable. To get this stronger version we just need to note t h a t t h e LowenheimSkolem Theorem has been proved in Z F G , and hence it holds in L. It s version in L sounds as follows: Given a sentence (p, a set M E L which is a model for it, and a subset X E L o f M which is constructibly countable, there is a constructibly countable model M ‘ E L of cp such that X 5 M’ 2 M . By using t h e argument of IV.4.13 with this version, which is applicable because t h e starting model is Lg, which is constructible, we obtain a constructibly countable /3 as wanted. 0

IV. Hierarchies and Weak Reducibilities

436

Since t h e hierarchy { is obviously proper, by IV.4.4, t h e meaning of t h e last result is t h a t , at t h e stages at which no new subset of w is generated (called gap ordinals), new sets appear, t h a t will later be used to define new subsets of w. T h i s shows t h a t t h e definition of P(w) n L is intrinsically settheoretical. Since V = L is consistent with Z F C , a n d t h u s P(w)n L could be t h e t r u e power set of w,set-theoretical methods are essential in the analysis of the continuum, i.e. in Classical Recursion Theory as defined on p. 1. T h i s adds to t h e remarks made by Sacks in t h e Foreword. T h e study of gap ordinals has been pursued in various directions, some of which a r e treated in Volume 111. O n o n e side, t h e possible lengths of gaps have been investigated: it turns out t h a t there are gaps of any length less than

wf,and the gaps are distributed in a very orderly fashion, according to their lengths (Marek a n d Srebrny [1974]). O n t h e other side, a characterization of g a p ordinals has been obtained: N is a gap ordinal i f and only i f La+,

+

ck

uncountable,

in t h e sense t h a t there is no one-one function in L,+1 from cr to w (Boolos a n d Putnani [1968], Jensen [1972]). T h e condition is obviously sufficient, because such a function induces a well-ordering of w of length a , t h a t cannot be in L , since a is not. T h e converse is a strengthening of IV.4.13, a n d it is obviously necessary, if L,+1 is a model of enough Set Theory: in this case IV.4.13 holds in it, a n d t h u s all constructible subsets of w which a r e in L,+1 must be already in L L,+, , i.e. must be constructed at stages countable in L,+1. T h u s , by w1

absoluteness,

P(w)n ( L + I

-

J+"+l)

= 0.

And if N is not countable in L a + ] ,wf"'l 5 a . Note t h a t t h e g a p could be quite large, depending o n how much of Set, Theory is modelled by La+*,since L,+l could prove t h e existence of various cardinals (inside it) greater t h a n w~"". Exercises IV.4.16 a) T h e gap ordinals are &. (Putnam [1963]) (Hint: there is a gap above any given constructibly countable ordinal.) b) For any ordinal y admitting an absolute definition, there are gaps of length y. (Putnarn [1963]) (Hint: consider a y instead of CI 1 in the proof above.)

+

+

One might think of mimicking t h e definition of L in a n autonomous way, by relying only on subsets of w.

Definition IV.4.17 (Kleene [1959b]) The Ramified Analytical Hierar-

IV.4 The Constructible Hierarchy

437

chy i s described, in t e r n s of ordinals, as follows: RAo

RA,+1 RAP RA

= the arithmetical sets = defz(RA,) = RA, (p limit) = U,RAa,

u,.+

where def2 (x)i s the set of subsets of w which are definable in Second-Order Arithmetic with set quantifiers restricted to x, and parameters in x.

RA will be studied in Volume 111. Leeds a n d P u t n a m [1974] show t h a t parameters c a n be avoided in its definition. By cardinality reasons, RA breaks down at an ordinal p0 (since if each step adds a new subset, t h e subsets of w run o u t in at most a continuum of steps). Boolos a n d P u t n a m [1968] show that,, when account is taken of the different starting points, the Ramified Analytical Hierarchy coincides, up to p0 and level b y level, with the Constructible Hierarchy restricted to P(w). I t is thus not surprising t h a t p0 is the first gap ordinal, i.e. t h e first point in which sets not in P(w) become essential for t h e definition of new subsets of w. Exercises rV.4.18 a ) Po is countable. (Cohen [1963a]) (Hint: by cardinality considerations, there is an ordinal (r such that RA,+, = RA,. As in IV.4.15, and by absoluteness of RA,, there is a countable one.) b) R A c A:. (Putnam 119641) (Hint: there is a A: predicate enumerating the members of RA. Thus the members of RA are uniformly A:. By diagonalizing the enumeration predicate, there is a A: set not in RA. To get the enumeration predicate, by part a) we may restrict attention to countable ordinals, and hence to well-orderings of w. The methods of IV.2.22 and IV.4.5 show that A E RA, is uniformly A:, whenever z varies over the elements of a well-ordering of w , used as parameter. Since A E RA if and only if there is a well-ordering of w such that A E RA, for some z in it, we have the Ci form, because being a well-ordering is a IIi condition. Similarly, A E R A if and only if, for every well-ordering of w which is sufficiently long, i.e. it has elements y and z such that z is the successor of y in it, and RA, = RA,, then A E RA, for some 2. This provides the Ill form, because being a well-ordering is now an hypothesis, and thus it becomes Ej.)

E?j sets We have studied t h e set P ( w ) n L as a totality, b u t we do not know yet which sets a r e in it. Of course, since V = L is consistent with Z F C , it might be t h a t all sets (of natural numbers) a r e constructible. But even if this were t h e case, this would not be provable in Z F C , because Cohen [1963] has shown t h a t it is consistent with Z F C t h a t P(w) L (see Volume 111).

IV. Hierarchies and Weak Reducibilities

438

It is thus a nontrivial problem to ask which subsets of w a r e provably in L , in t h e sense of being provable in Z F C t h a t they a r e constructible.

Theorem IV.4.19 (Mostowski [1949], Shoenfield [196la]) Ci U I I i

L.

Proof. Since w E L , a n d L satisfies separation, for a n y formula 'p t h e r e is a set n E I, such t h a t z E n e LbzEwA'p(z). T h i s of course does not mean t h a t every definable subset of w (in t h e real world, i.e. in V ) is in L, because 'p does not need to be absolute, a n d t h u s t h e set it defines in V is not necessarily a. But this certainly holds for t h e formulas whose meanings a r e t h e same over L a n d in t h e real world. By 1V.3.27, t h i s is t h e case of Ea formulas, because I, is a s t a n d a r d model of GK P containing all t h e ordinals. T h u s Ea sets a r e all in L , a n d t h e same holds for their complements in w ,i.e. t h e ni sets. 0 . T h i s result is t h e best possible: Jensen a n d Solovay [ 19701 have shown t h a t This also follows from large cardinals assumptions (Solovay [1967]). O n t h e other hand, Harrington [1974] has shown t h a t f o r any n such that 3 5 n 5 w, at is consistent with Z F C that P ( w )n L = A:. See Volume 111 for all this.

it is corlsistent with Z F C that P(w) n L is properly contained in A:.

Recall t h a t t h e Arithmetical a n d Analytical Hierarchies classify sets of natural numbers according to t h e complexity of their definitions over t h e s t r u c t u r e of Arithmetic. T h e difference between t h e two hierarchies is in t h e range of quantifiers, over w in t h e former case, a n d over P(w) in t h e latter. O u r original motivation for t h e introduction of t h e Levy's Hierarchy was to pursue t h i s trend, a n d allow for more general quantifiers, over sets in a given model. Our first example was X F ,a n d definability over it coincided, for subsets of w,with arithmetical definability (IV.3.22). We t u r n now to definability over L. Of course, every set in L is definable over i t , using itself as parameter, b u t this is not what we mean: in t h e Analytical Hierarchy we did quantify over sets, b u t did not allow t h e m as parameters ( n o because t h e proof of IV.3.22 shows t h a t elements attention is needed over 'H3, of XF a n d natural numbers are, in a sense, t h e s a m e thing). W h a t we really look for is t h u s definability without parameters over L.

Theorem IV.4.20 Definability of Ci sets (Takeuti and Kino [1962]) Let A 2 w. Then

A E Ci

tj

AE

El

without parameters.

Similarly for relatzons, of any number of variables.

IV.4 The Constructible Hierarchy

439

Proof. Let A E Ci. By t h e proof of IV.3.27, A is of t h e form (la countable)cp, with N ranging over countable ordinals, a n d cp A, over G K P plus Infinity. Since L is a model of G K P , a n d it contains all t h e countable ordinals, A is Ef without parameters. For t h e converse, s u p p m e A C_ w is Ef without parameters, i.e. 2 E

with

A

* L I= (3y)+(x,y),

+ E At. By definition of satisfaction, E A * (3Y E L ) ( L I=+(Z,Y)). 5

By definition of L , a n d absoluteness of A; formulas, zEA

-3 ( 3 @ ) ( 3 E /

a n d hence

xEA

LdLa

* ( W Lt=

I= +(Z,Y)),

(3Y)+(Z,Y)).

Using t h e Lowenheim-Skolem Theorem a n d t h e Collapsing Lemma as in t h e proof of IV.4.13, a n d with cp as there, zE

A

@ @)[a

countable transitive A a

cp A (3y)+(z,y)].

Suppose such a n a exists: being countable, ( a , € ) is isomorphic to a wellfounded s t r u c t u r e ( w , E ) , a n d t h e Collapsing Lemma applied to t h e latter r e produces t h e original n (because t h e transitive collapse is unique, a n d a is transitive). I n particular, if f is t h e collapsing function, z = f(f-'(z)), a n d (3y)+(z,y) holds on a if a n d only if (3y)+(z,y), with f(z) = z, holds o n ( w , E ) . Thus zEA

@ ( ~ E ) [ ( w , E )well-founded A

(W,E)

b cpA (3!4)(3z)(f(z)= xA+(z,lI))l1

where f is t h e collapsing function of ( w , E ) . I t only remains to compute t h e complexity of t h e last expression for A : 0

0

0

E is a binary relation o n w, t h a t can be coded by its characteristic function: t h e first existential quantifier is t h u s a function quantifier.

Well-foundedness is

nj, see e.g. IV.2.15.

T h e graph of t h e collapsing function, when t h e values a r e restricted to w, is A:, because it c a n be defined by recursion with clauses arithmetical in E , as follows. First of all, f(z) = 0 if a n d only if z is t h e (unique) element of w which does not have predecessors w.r.t. E . And f(z) = n 1 if a n d only if, for some y, f ( y ) = n, a n d z is t h e successor of y w.r.t. E (i.e. t h e set y U {y}, when membership in interpreted as E ) .

+

IV. Hierarchies and Weak Reducibilities

440 0

T h e satisfaction relation is A:, by arithmetization

T h e whole expression is t h u s El.

w. Then

Corollary IV.4.21 Let A

A E Ei

0

@

A E Cfyl without parameters.

Similarly for relations, of any number of variables. Proof. T h e first p a r t of t h e proof goes through because L,, is a model of G K P , a n d it contains all t h e countable ordinals. T h e second p a r t of t h e proof can be repeated as above, without appeal to t h e Lowenheim-Skolem Theorem, since every La with Q < w1 is countable. 0 We have proved t h e result for relations on w ,b u t little work is needed to extend it to relations on w a n d P(w): we j u s t have to show how to t a k e c a r e of set variables, a n d for this it is enough to prove t h a t t h e graph of t h e collapsing function f , when t h e values a r e restricted to P(w),is still A:. Recall t h a t , by definition o f f (see IV.3.5),

f(z) = {f(Y)

:Y E

wAYE

2).

Thus

f(z) = A

@

(VX)[. E

A

(3"

= f(9)A Y E

41.

T h e use of f on t h e right-hand-side is restricted to number values, a n d t h u s it is A: by t h e proof of IV.4.20. T h u s t h e whole expression is A:, because only number quantifiers a r e used.

Corollary IV.4.22 (GMel [1940], Mostowski, Addison [1959a]) The relations A E L and A < L B , for A and B subsets of w, are Xi. Proof. We know from IV.4.5, IV.4.10, a n d IV.4.7 t h a t x E L a n d x 5~ y a r e Cf. When restricted to P(w) they become relations of set variables, a n d t h u s t h e y a r e Eh. T h e complexity of P(w)nL computed above is best possible, since i f A E L is IIi then P(w) L , i.e. every subset of w is constructible: indeed, if A E L is IIi then ( 3 A ) ( A$! L ) is a Ei formula false in L , a n d b y absoluteness it is also false in V , which means t h a t P(w) 5 L. By IV.4.11.b, if V = L then < L is actually A:.-, a n d t h u s t h e well-ordering 5~ is actually Ah on P ( w ) . In particular, A B if A 5~ B . T h u s it is

IV.4 The Constructible Hierarchy

44 1

corisistent with Z F C that the A;-degrees are well-ordered, a n d no result contradicting t h i s can be proved in Z F C . T h e s t u d y of t h e structure of A;-degrees can t h u s be pursued along two different paths: one is to prove consistency results, t h e other to introduce new axioms a n d s t u d y t h e structure under them. We will follow b o t h paths, in Volume 111. Of course similar considerations hold for t h e A:-degrees, for any n 2 2.

3-IC and the Analytical Hierarchy We have seen in IV.3.22 t h a t t h e Arithmetical Hierarchy can be viewed as a set-theoretical hierarchy over t h e hereditarily finite sets. We now provide a similar interpretation of t h e Analytical Hierarchy.

Definition IV.4.23 X C is the set of hereditarily countable sets, i.e. the smallest class A of sets such that: 1. ~

E

A

2 if (Vn E w)(z, E A) then

(5,

: 71 E

w} E A.

A number of properties of 7i.F can be extended to XC, by similar proofs. E.g., a set is hereditarily countable if a n d only if its transitive closure is countable. Proposition IV.4.24 7iC is a transitive model of G K P containing all countable ordinals. Proof. The axioms of G K P can be checked in a way similar to IV.3.22. Moreover, note t h a t a n ordinal is transitively closed, a n d t h u s it is in ‘FIC if a n d only if it is countable. Since a countable subset of HC is a countable set with hereditarily countable members, it belongs to 7iC. In particular, since w E 7iC, P(w) 7iC. Note t h a t t h e smallest transitive model of G K P containing all countable ordinals is L,, , a n d t h u s L,, C 7iC (directly, if 5 € L,, t h e n , for some countable Q , r is in L a , which is transitive: then t h e transitive closure of z is contained in L a , which is countable). However, since it is consistent with Z F C t h a t s o m e subset of w is not constructible, we cannot prove t h a t L,, = 7iC. Exercises N.4.26 a) If V = L then W = L,,. (Hint: by €-induction. Suppose that (Vy E z)(y E L,,), and z is countable. Then z & I,,,, and z C L , for some countable a. As in IV.4.13, by starting from L , U (z}, if V = L then z E L,, .) 1)) W is a model of Z F C - pZus V = W . (Hint: choice follows from the following facts: a subset of ‘HC is well-orderable in V by the Axiom of Choice; a wellordering of

442

IV. Hierarchies and Weak Reducibilities

an hereditarily countable set is still such; well- foundedness is absolute for standard models of G K P . V = ?Chold3 because if x E ‘HC then any function from x into w is already in ‘HC.) T h e main reason for us t o consider ‘HC is t h e following analogue of IV.3.22, which provides a n alternative set-theoretical way of seeing t h e Analytical Hierarchy. O n e should note t h a t each level of t h e hierarchy o n IFIC corresponds to t h e next level of t h e Analytical Ilierarchy. T h i s is not accidental, a n d it h a s already been observed in t h e discussion o n p. 397.

Theorem IV.4.26 Set-theoretical definability of the Analytical Hierarchy. Let A 2 w. Then A is definable ouer ‘FIC without parameters i f and only i f A is analytical. More precisely, for any n:

A E AA+2 A E A,;: A E CA+, ts A E, :C :

without parameters without parameters

Similarly for relations, of any number of variables.

Proof. For n = 0 t h e proof of IV.4.21 goes through without, changes, using absoluteness of Ape formulas. Moreover, as noted after t h e proof of IV.4.21, t h e result holds for relations not only over w, b u t also over P(w).T h e cases 0 for n > 0 t h e n follow by adding quantifiers, because P(w)C

‘He.

To generalize t h e result for Ci to a n y level we have used in a crucial way t h e fact t h a t P(w) IFIC. T h e analogue could not be proved for L , a n d t h i s was t h e reason why we had to content ourselves with t h e absolute result for X i , in IV.4.20. Of course, t h e full result can be obtained also for L if we assume V = L (Takeuti a n d Kin0 [1962]), by t h e s a m e proof (or by IV.4.25.a). As it was already t8hecase for 1V.3.22, embedded in t h e proof of IV.4.26 a r e translations of Second-Order Arithmetic a n d Set T h e o r y on IFIC into o n e another. T h e translation of Second-Order Arithmetic to Set T h e o r y is stand a r d , while t h e other consists of seeing hereditarily countable sets as countable trees (and t h e n coding t h e m as subsets of w),a n d by interpreting equality as tree isomorphism. Peano Arithmetic was equivalent to Z F C with t h e Axiom of Infinity replaced by V = ‘H3:we now have t h a t Second-Order Arithmetic is equzualent to Z F C with the Power Set Axiom replaced b y V = ‘HC (Kreisel [1968], Zbierski [1971]). It should be noted t h a t by Second-Order Arithmetic we mean here t h e second-order version of PA plus t h e Axiom of Comprehension for analytical formulas (asserting t h e existence of t h e analytical sets), a n d a n Axiom of Choice which asserts t h a t if (Vx)(ElA)(p(x,A)t h e n there is a subset A of w x w such

IV.4 The Constructible Hierarchy

443

T h i s obviously t h a t , if A , is t h e section of A w.r.t. 2, then (b'z)cp(z,A,). corresponds to collection for sets, a n d it is needed ( G a n d y [1967a]) to model t h e Axiom of Collection. Second-Order Arithmetic, as well as subsystems of it obtained by variously restricting t h e Axioms of Comprehension a n d Choice, will be studied in Volume

111. Exercises W.4.27 a ) Levy Absoluteness Lemma. A Ef" formula with parameters in 'HC true in V is already true in 'HC. (Levy [l965]) (Hint: by LowenheimSkolem, Collapsing Lemma, and absoluteness, as in IV.4.13.) h) P ( w ) n L G L,, follows from Levy's Absoluteness Lemma and Af"-definability of L, (Karp 119671). This provides a slightly different and easier proof of IV.4.13. (Hint: if A G w and A E L the CP" formula (+)(A E L a ) with parameter A E 'HC is true in V. By Levy Absoluteness it is true for some countable ordinal.) c) P ( w ) is not A?". (Hint: otherwise the formula asserting its existence would be true in 'HC.)

Recursion Theory on the ordinals

*

T h e whole idea of constructibility rests o n t h e fact t h a t L is definable by recursion on t h e ordinals. Since usually t h e ordinals a r e defined within Set Theory, so is L. While investigating t h e theory of ordinals, Takeuti [1957] discovered t h a t it could be developed independently of Set Theory. It t h u s became clear that, if o n e could also develop independently a theory of recursion o n t h e ordinals, t h i s would allow a different approach to L . Takeuti [1960] carried o u t t h e task, by defining t h e notion of recursive function on t h e ordinals by schemata, in i i way similar to recursiveness on t h e integers. He then discovered t h a t Giitlel's result IV.4.13 could be recast in recursion-theoretical t e r m s by saying t h a t w, (and, more generally, any uncountable ordinal) was stable, in t h e sense of being closed under t h e recursive functions on all t h e ordinals. W i t h this, Recursion Theory was generalized both t,o t h e class of all ordinals, a n d to cardinals. Independently, a n d motivated by needs related to t h e theory of infinitary languages, Machover [1961] developed a n equivalent approach to recursion on cardinals, using systems of equations. T h e circle was closed by Takeuti a n d Kino [1962], when it was realized t h a t recursion on t h e ordinals was actually equivalent to E: definability. After discovering t h a t cardinals were appropriate domains for Recursion T h m r y , it was natural to wonder whether t h e strong closure properties of cardinals were somehow needed. Kripke [1964] a n d Platek [1966] answered t h e question by reversing t h e attack. T h e y relativized t h e previous approaches to any ordinal a , by defining t h e a-recursive functions (e.g. by schemata, using a scm-c.11 operator on ordinals less t h a n a ) . T h e n t h e y defined admissible

IV. Hierarchies and Weak Reducibilities

444

ordinals as t h e a’s closed under t h e a-recursive functions, a n d showed t h a t for t h e m all approaches a r e equivalent. In particular, for a n admissible ordinal a , &-recursiveness means ,Efa definability, a n d L, is t h e smallest s t a n d a r d model of K P of ordinal a . This relates effective Set Theory to Recursion Theory on t h e ordinals, a n d provides finer versions of various results of t h i s section. T h e first two admissible ordinals a r e w a n d wfk (see p. 385). While admissible sets t u r n out to be nice domains only for elementary Recursion Theory (see p. 421), niany deeper p a r t s of Recursion T h e o r y carry over to any admissible ordinal (see Chong [1984] for a detailed t r e a t m e n t ) . In particular, Post’s Problem always a d m i t s a positive solution (Sacks a n d Simpson [1972]). Admissible ordinals will provide, in Volume 111, a uniform way of describing a number of classes of subsets of w ,a n d will also be useful from a methodological point of view, for a better understanding of which properties of w a r e used in proofs of single results.

Relativizations

*

As for t h e Arithmetical a n d Analytical Hierarchies, t h e work d o n e for t h e Constructible Hierarchy c a n also be relativized to a given set A . T h e r e a r e two natural ways of doing t h k , corresponding to adding A as a constant or as a predicate.

Definition IV.4.28 (Hajnal [1956], Levy [196Oa]) 1. The class L[A] is defined like L, by starting with A U T c ( A ) in place of

0. 2. T h e class L(A) is defined like L by allowing, at successor stages, also parameters over A . From a set-theoretical point of view, t h e two ways a r e not equivalent. T h e first produces t h e smallest standard model of Z F containing A a n d all t h e ordinals, a n d i t does not necessarily satisfy t h e Axiom of Choice, without further assumptions on A . T h e second produces t h e smallest s t a n d a r d model M of Z F C containing M n A a n d all t h e ordinals. From o u r point of view, however, they a r e equivalent, since we only consider relativizations to sets A w.

Definition IV.4.29 A is constructible from B ( A ILB ) zf i t i s in L [ B ] . A i s constructibly equivalent to B (A B ) i f A 5~ B and B st A . Exercises W.4.30 a) If A is constructible, then A b) I J A < L B and B is constructible, so is A.

5~ B

for any

B.

IV.4 The Constructible Hierarchy Note t h a t relation.


is reflexive a n d transitive, and thus

445 EL

is a n equivalence

Definition IV.4.31 The equivalence classes of sets w.r.t. constructibility equivalence are called L-degrees, and ( D L ,5 ) is the structure of L-degrees, with the partial ordering 5 induced on them b y S L .

Of course, not much c a n be said a b o u t t h e structure of L-degrees in Z F C alone: the assertion that there is exactly one L-degree (containing all subsets of w) is consistent with Z F C , since so is V = L , a n d t h u s no result contradicting i t c a n be proved in Z F C . T h e s t u d y of t h e structure of L-degrees can t h u s be pursued along two different paths: o n e is to prove consistency results, t h e other to introduce new axioms, a n d s t u d y t h e structure under t h e m . We will follow b o t h paths, in Volume 111.

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Chapter V

Turing Degrees T h e first four chapters of this book introduced t h e basic notions a n d methods of Recursion Theory. It, is now time to p u t all this machinery to good use, a n d begin a systematic study of t h e cont,inuum from a recursion-theoretical point of view. While a great deal of Recursion Theory, as seen in previous a n d following chapters, has a niore limited scope a n d analyzes increasingly bigger, b u t always countable, subsets of P ( w ) ,this a n d t h e next chapters a t t e m p t a global attack, by trying to characterize t h e structure of P(w) in t e r m s of degrees. Here we s t u d y Turing degrees, by developing a paradigm t h a t will later be followed for t h e s t u d y of many other notions of degrees. T h e main results we will obtain a r e of two kinds:

Algebraic. We will look for results t h a t describe t h e algebraic s t r u c t u r e of degrees, as a partially ordered set. W e will ask natural questions a b o u t this structure, concerning: linearity, density, embeddability of partial orderings, ideals, nontrivial automorphisms, a n d so on. Despite t h e fact t h a t (contrary to t h e case of m-degrees) there is no complete characterization yet, in Section 7 we will be able to derive a number of interesting global results.

Computational. T h c first natural question here is t,he decidability of theory of degrees, i.e. t h e existence of a n effective method t h a t would tell, of any given sentence in t h e language of degrees, whether it is t r u e or not. We will be able to show in Section 7 t h a t such a method does not exist, a n d t h a t actually it is as difficult to decide first-order sentences a b o u t t h e order of degrees as it is to decide analytical sentences of arithmetic. In precise words, the theory of degrees is recursively isomorphic t o SecorzdOrder Arithrnetic. Knowing t h a t t h e theory is undecidable, we may look for partial decidability results: we will discuss t h e fact t h a t it is possible

447

448

V . Turing Degrees t o decide th e theory of two-quantifier sentences, b u t not t h a t of three quantifiers.

T h e division in sections is methodological and it reflects t h e different, increasingly more powerful tools used in t h e proofs. We start in Section 2 with t h e finite extension method, which is just a version of Baire Category, t h e relationship being analyzed in Section 3. We discuss t h e fact t h a t some results, like th e existence of minimal degrees, cannot be proved by finite extensions. We then introduce more powerful tools, namely t h e coinfinite extension method in Section 4, and t he tree method in Section 5. T h e work culminates in Section 7, where t h e global structure of T-degrees is investigated, a n d t h e quoted undecidability results are proved.

V.l

The Language of Degree Theory

We have defined t h e structure of Turing degrees in 11.3.3, and here we begin a close look at it. To improve readability, as well as t o follow common use notations, we adopt (in this chapter) a number of conventions: 1. degree will always mean T-degree

2. sets will be used as representatives for the degrees, a n d this is possible because a function and its graph are T-equivalent, a n d thus each degree contains a set

3. a set will be identified with its characteristic function 4. partial recursive functions with oracle A will sometimes be denoted b y

{e}*, in place of(p: 5 . we will use lowercase Greek letters for strings, i.e. for partial functions with finite domains (see V.2.1)

6. degrees will be denoted b y lowercase boldface letters 7. a < b w i l l m e a n a < b A a # b

8. a[b will mean a

b A b f a, i.e. t hat a a nd b are incomparable.

The join operator Since ( D , 5 ) is a partially ordered set, it makes sense t o talk about t h e 1.u.b. a n d t h e g.1.b. of a pair of degrees.

Definition V.1.1 Given a and b, then:

V . l The Language of Degree Theory

449

1. a U b (also called the join of a and b) is their least upper bound

2. a n b is their greatest lower bound.

We now show t h a t t h e 1.u.b. always exists, a n d it k induced by t h e disjoint s u m of two sets. T h e g.1.b. of two degrees may instead exist or not (V.2.16 a n d

V.4.7). Recall t h a t t h e disjoint s u m of two sets is defined as

A @ B ts {2z : z E A } u {2z + 1 : z E B } . Notice t h a t A @ B is obviously invariant under T-equivalence, a n d t h u s i t makes sense to consider it as a n operator on T-degrees.

Proposition V.1.2 Given a and b, a U b is the degree of A @ B , for any A E a and B E b. Proof. Clearly A , B z E A

A @ B , since


* 22 E A @B

and

zE

B H 2 z t 1 E A @ B.

And if A , B C, let A = 'pz a n d B = p'.: Then A @ B t h e function partial recursive in C such t h a t 'p,"(z)

if z = 2 z if z = 2z

T h u s A @ B is t h e 1.u.b. of A a n d B w.r.t. < T . Notice t h a t the j oi n operator is definable in aU

< cA b <

b = c t a~

'p(z), where 'p is

+ 1. 0

(V, <), as

c A ( V d ) ( a5 d A b

5

d + c < d).

We can t h u s freely add U to t h e structure (P, 5 ) simply recalling, when needed, t h a t one universal quantifier is needed to express it. We can also introduce a more general notion of join operation, as follows:

Definition V.1.3 Given a countable family { A n } n c , , with I

e n E ~ A= n { ( n , ~: z) E A,

A nE

w, then

Z}.

T h i s notion is going to be useful, b u t we should be aware of t h e fact t h a t i t is not invariant under ?'-equivalence. Exercises V.1.4 (Kleene and Post [1954]) a) I f I = { a , b } then (@,,CIA,,)is recur-

sively equivalent to A , @ A*. b) If I is finite then the degree of (enEIAn) is uniquely determined by the degrees of the A, 's, but this fails i f I is infinite. (Hint: let A be any set, and x E A,, e+TTLE A. Then the An's are all recursive, but (@,twAn)has the degree of A.) c ) If A,
V. Turing Degrees

450

The jump operator T h e fact t h a t K is a complete r.e. set,, a n d t h u s it embodies one existential number quantifier, is now generalized with t h e introduction of a degree operation t h a t corresponds t o number quantification.

Definition V.1.5 The jump of a set A ,is the relatiwization of K to A , defined as z E A’ H z E W,” H (z) 1 . By relativization of t h e properties of K (recall t h e general comments on p. 177), we get t h e following properties of t h e j u m p operator:

1. A Is r.e. in B if and only i f A

B’

From t h e fact t h a t A is r.e. if a n d only if A

K

(p. 320).

2. A S 1 A’ This is j u s t a particular case of 1 3. A’ i s not recursive in A From t h e fact t h a t K is not recursive (11.2.3). T h e next result exhibits a connection between Turing a n d o n e o n e reducibilities, through t h e j u m p operator.

Proposition V.1.6 A ST B if and only i f A’ 51 B’

Proof. If A S T B , from A’ r.e. in A we have A’ r.e. in B : by fact 1 above, t h e n A’ 5 , €3’. To show A 5~ B we j u s t have to prove t h a t both A a n d a r e r.e. in B or, by fact 1 above, t h a t A S1 B’ a n d A R’. Both follow from A‘ < I B’ by transitivity, t h e former because A 5 , A’, t h e latter because 2 51A’ (since 2 is r.e. in A ) . 0 In particular this shows t h a t t h e j u m p operator is invariant under T-equivalence, a n d thus it induces a n operator on degrees.

Definition V.1.7 (Kleene and Post [1954]) The jump a’ of a is the degree of A’, for any A E a, and ( P , 5 ,’) i s the .structure of Turing degrees, with the partial ordering 5 and the jump operator ’. Usually we simply write 2) when we refer t o t h e s t r u c t u r e of degrees without j u m p operator, a n d P’ when we include t h e j u m p operator. We will see in XI.5.17 t h a t the jump opemtor is definable in D .

451

V.l The Language of Degree Theory

Recall that, we defined 0 as the degree of the recursive sets, and 0' as the degree of K. This is consistent with the present, notation, since K: is by definition thc jump of the emptyset, and 0 E 0. The jump operator can be iterated, both on sets and on degrees.

Definition V.1.8 For any set A , the n-th jump A(") of A is defined inductively as follouis: A(O) = A A("+') = (A(7L))'. The n-th jump a(") of a degree a is the degree of A ( n ) ,for any set A E a.

Of course a(") can also be directly defined by induction, as follows:

=a

a("+l) = ( a ( " ) ) ' .

we will use 0(71)as our usual representative for o(") Exercises V.1.9 a)

0' K ~~

O(n+')

is C;+,-complete. (Hint: by iteration of the fact that

is Cy-complete.)

b) A set A is arithmeticfil if and only if it is T-rxducible to zf its d q r w is bounded by some O(").

a(") for some 71, i.e.

There is also a transfinit,e operation of j u m p , that can be defined using the infinite join of the finite iterations.

Definition V.1.10 (Kleene and Post [1954])For any set A , the w-jump A(") o f A is defined as

A(") = @ 7 L c w A ( n ) . For any degree a, the w-jump a(") of a is the degree o f A("), for any set

A E a. Exercises V . l . l l a) If A
First properties of degrees We now state some siitiple but basic facts about P and P'

Proposition V.1.12 (Kleene and Post [1954])A s a partzally ordered structurt. V zs a n uppersernzlaftzcc of cardznalzty 2No vrifh a least but no mammal element. Moreover, m c h element has 2Nu successors and a1 most countably many predecessors.

V. Turing Degrees

452

Proof. We have already noted t h a t t h e join operator provides with a least upper bound operator, a n d t h u s 2) is a n uppersemilattice. T h e least element is t h e degree 0 of t h e recursive sets. T h e r e can be n o maximal element, since t h e j u m p operator is a strictly increasing operator. Given a set A , for any set B such t h a t B ST A t h e r e is a number e E w such t h a t B 21 (pt,a n d thus there can be at most countably many sets T-reducible to A . T h i s means t h a t a degree has at most countably many predecessors, a n d it contains at most countably many sets. In particular, t h e r e a r e at least 2"O degrees. T h e r e cannot be more t h a n 2N0 degrees, because we c a n define a o n e o n e m a p from t h e degrees into P(w),by choosing a set from each degree. O n t h e other hand, t h e m a p B c) A @ B is a one-one m a p from P(w) into { X : A 5~ X } : this means t h a t t h e r e a r e at least 2"O sets to which A is T-reducible, a n d hence there a r e at least 2"O successors of a n y degree. T h e n there a r e exactly 2N0 ones. 0 Note t h a t 0, being t h e least degree, is definable in D as c = 0 tj (Vz)(c

5 z).

T h u s we can freely add 0 to t h e structure 2) simply recalling, when needed, t h a t one universal quantifier is needed to express it.

Definition V.1.13 The cone above a is the set

P ( 2 a) = { b : b 3 a}. T h e properties j u s t proved for 2) a r e obviously t r u e for P(> a) as well, a n d we may wonder whether t h e structure 2) is homogeneous, in t h e sense of being isomorphic (or at least elementarily equivalent, i.e. satisfying t h e s a m e first-order properties) to all of its cones. If it were so, this would justify t h e principle of relativization (p. 11.3). As it happens, t h i s actually fails (V.7.13), a n d t h u s relativization of a result has to be verified in each case.

A connection between t h e join a n d j u m p operators is given by t h e following result. Proposition V.1.14 (Kleene and Post [1954]) For any pair of degrees a and b, a < b = ~ a ' S b ' and a ' U b ' < ( u U b ) ' . Proof. We know t h a t if A proves t h e first part.


B then A' 51 B', a n d hence A'


B': t h i s

V . l The Language of Degree Theory

453

Since d b is t h e 1.u.b. o f a a n d b , w e h a v e a < d b a n d h e n c e a ' 5 ( U b ) ' . Similarly 'b 5 ( a U b)'. Since a' U 'b is t he 1.u.b. of a ' and b', it follows t h a t a ' U b' 5 ( a U b)'.

We will see (V.2.27) t h a t t h e result is best possible, since every possibility compatible with th e properties above can be realized.

The Axiom of Determinacy

*

We have argued above t h a t t he cardinality of P is 2 N o , by providing a oneone map from P t o P ( w ) , but this required t h e Axiom of Choice (we chose a set from each degree). Yates [1970] has proved t h a t t h e use of this axiom is not avoidable, since it is consistent with Z F (plus t h e Axiom of Dependent Choices) t h a t there is no one-one map from P into P ( w ) . We will see t h e proof in Volume 111, but now we introduce a new axiom t h a t implies t h e same result. Suppose I an d I1 are two players, t h a t alternately play integers:

I I1

a0

...

a2 a1

a3

...

Putting together their moves, we get a function f ( n ) = a,. To decide who is going t o win t h e play, we choose ahead of time a set A of total functions, a n d we say t h a t I wins if, at t h e end of t he game, t he function f is in A, and I1 wins otherwise. Thus any set A defines a game G(d). One of t h e two players might not only win, but even have a winning strategy, i.e. a way t o decide his or her moves so t h a t , independently of how t h e other player moves, he or she will win. E.g., if A is t h e set of functions with value 0 for odd arguments a winning strategy for player I1 simply consists of playing 0 when it is his or her turn. Of course only one of t he two players can have a winning strategy, but it is conceivable t hat there are games for which none has one.

Definition V.1.15 (Gale and Stewart [1953], Mycielski and Steinhaus [1962]) A winning strategy for player I in the game G(d) is a function W I Seq + w such that I wins the game if he consistently plays following the strategy. I n other words, any function f such that f ( 2 n ) = w l ( f ( 2 n - 1)) is in A, independently of the values o f f for odd arguments. A winning strategy for player 11 is defined similarly. The game G ( d ) is determined i f one of the two players has a winning strategy in at, and the Axiom of Determinacy is the assertion A D that every game G(A) is determined. T h e interest of t h e Axiom of Determinacy for Degree Theory lies in t h e next result.

454

V. Turing Degrees

Theorem V.1.16 (Martin [1968]) If A D holds, then every s e t of degrees either contains a cone or i s disjoint from a cone. Proof. Given a set A of degrees, let A’ consist of t h e sets whose degrees are in A. Suppose there is a winning strategy for I in C ( A * ) .T h e strategy, being a function from (sequence) numbers t o numbers, has a certain degree a: we show t h a t every degree b 2 a must be in A. Choose any function g in b, a n d let player I1 play according t o g (i.e. t he n-th move of I1 is g ( n ) ) . Let also player I play according t o t h e winning strategy. Since t h e moves of I are determined by a function of degree a,and those of I1 by a function of degree b, a n d a 5 b, t h e final outcome will be a function of degree b. But since I was following a winning strategy he wins t h e game, and thus t he outcome must be in A * : this means th at its degree, i.e. b, must be in A. Similarly, if player I1 has a winning strategy then all degrees above t h e one in which t h e strategy lies must be in 2,since I1 wins t h e game. This results has a number of interesting consequences for degrees. An astonishing one is th e following.

Theorem V.1.17 (Martin) If A D holds, every m a p f r o m degrees t o s e t s is constant o n a cone. Proof. Let F : P + P(w) be given. F ( z ) is a set of numbers, for each degree z.Let = {z : n E F ( z ) } : this is a set of degrees, a n d thus there is a degree a, which is t h e base of a cone contained either in or in &. T hi s means t h a t n is in F ( z ) either for all 2’s above a, or for none, and thus t h e behavior of F ( z ) on n is fixed on t h e cone above a,. Consider a degree a above all t h e an’s: it exists, because there are only countably many of them (e.g. take t h e infinite join of sets A , E a,). Then F niust be constant on t h e cone with a as a base, since this time its behavior is fixed for any n. 0 Corollary V.1.18 If A D holds, there is no one-one m a p f r o m P i n t o P(w). This contradicts the Axiom of Choice, which allows t o choose a set from each degree, thus producing a one-one map from P into P(w). In particular, t h e Axiom of D e t e r m i n a c y i s inconsistent with t h e A x i o m of Choice (Gale a n d Stewart [1953]). It should be noted t hat the proof of t h e theorem above used a weak form of t h e Axiom of Choice, by actually picking up representatives in countably many degrees. This weak form is actually a consequence of A D , and can t hus be freely used. T o see why, note t hat we know t hat (Vn)(3A)(A E a,). Consider t h e game in which I constantly plays n while I1 plays t h e characteristic function of a set A , an d such t hat I1 wins if A E a,. Player 1 does not have a winning

V.1 The Language of Degree Theory

455

strategy, since there is a n A E a, t hat can be played by 11. Then, by A D , I1 has a winning strategy, which produces a function choosing, for every initial move n of I, a set in a,. We might read t h e corollary as saying t h a t if A D holds there are more degrees than sets, b ut we probably should not, when Choice is not present, compare cardinalities by using one-one maps. If we use instead onto maps, these peculiarities disappear. Exercise V.1.19 Without using the Axiom of Choice, there are onto maps between 2, and P(w). (Hint: in one direction, send a set to its degree. In the other direction, build a tree of sets with pairwise incomparable degrees, see V.2.11, then send the degree of each branch to the branch itself, and all other degrees to a fixed set. This suffices, because P ( w ) is isomorphic to a tree.)

Another interesting consequence of V.1.16 is t h e following.

Proposition V.1.20 (Martin) If A D holds, there is a countably additive measure on V . Proof. Recall t h a t a countable additive measure is any function p defined on subsets of 2) such t h a t 1. p(0) = 0 and p(V)= 1

2. if X

Y then p ( X ) 5 p(Y)

4. if {X,},Ew is a collection of pairwise disjoint sets of degrees, then nEw

nEw

It is then enough to define

P(A) =

{

1 if A contains a cone 0 otherwise.

T h e only condition t hat requires some arguing is t h e last one. There are two Case:

some X,, contains a cone = 1. Moreover no X , with Then so does U n E w X n ,and p(U,,,Xn) n # no contains a cone, because t he intersection of two cones is not empty, while t he X , a r e disjoint. Then CnEw p ( X n ) = p(X,,) = 1.

V. Turing Degrees

456

no X , contains a cone Then p ( X , ) = 0. Moreover contains a cone for each n, by V.1.16, and then so does their intersection, as in V.1.17. But

x,

xnEw

nEw

and thus

UnEwX ,

nEw

does not contain a cone: hence p ( U n E wX,) = 0.

0

Since t h e countable ordinals are represented by well-ordered sets of natural numbers, it immediately follows that i f A D holds then N1 is a measurable cardinal (Solovay), by giving a set d of countable ordinals the same measure as t h e set of degrees containing well-ordering with ordinal in d. One might wonder why we should accept a n axiom like A D . T h e fact is that, as we will prove in following chapters, various special cases of it can be proved in Z F C , and thus some versions of t h e results proved above simply hold. More precisely, Martin [1975] has showed in Z F C t h a t every Borel game is determined: thus every Borel set of degrees either contains a cone or it is disjoint from a cone, and every Borel map from ZY t o sets is constant on a cone. We will investigate t h e effect of restricted versions of A D on Recursion Theory in Volume 111.

v.2 The Finite Extension Method Degrees are represented by sets, and thus existential properties of degrees can be proved by building appropriate sets. Since building sets by successive approximations will be our main concern in this chapter, we set u p a n efficient notation t o deal with t h e problem. Recall that a set is identified with its characteristic function, which is just a sequence of 0’s and 1’s. Definition V.2.1 A string is a partial function u : w -+ ( 0 , l ) with finite domain. If the domain of u is an initial segment of w, we call u a n initial segment, and let lul be its length. Given two strings u and u’, then:

1. u’ is a n extension of u i f it extends it as a partial function, i.e. u ( z )1 =+ d ( z )1 A u ( z ) N

u’(z).

2. u and u’ are incompatible i f they differ on some argument on which they are both defined.

457

V.2 The Finite Extension Method

Initial segments will be identified, when needed, with sequences of 0's and l's, or with their sequence numbers. T hen o * T will be t h e juxtaposition of t h e two sequences CT and T , and h ( a ) t h e length of CT. Strings ar e finite objects, and thus they can be coded by natural numbers. In particular, there is a canonical ordering of strings, t o which we will implicitly refer when talking of 'smallest string satisfying a given property'. Recall t h a t , by compactness and monotonicity (11.3.13), convergent computations with oracle A can be approximated by convergent computations having as oracle some finite approximation of A , which can then be coded by a string. In other words, using t he notations for this chapter,

{elA(.) = Y

-

(30

G A ) ( { e } " ( x )2: Y).

It is exactly because of these continuity properties t hat we will be able t o prove results about degrees.

Incomparable degrees Looking at t h e partially ordered structure V ,the first problem t h a t comes to mind is whether t h e order is really partial. T h e next result shows t h a t this is t h e case, a n d it introduces our first method of proof.

Theorem V.2.2 (Kleene and Post [1954]) There are two incornparabze degrees.

Proof. We want t o build sets A and B such t h a t A g~ B a n d B g~ A . T h e proof consists of two steps: first we break down these two global conditions into infinitely many local conditions, a n d then satisfy each of them by a finite action. Note t h a t A ST B means t hat , for some e , A 21 { e } B . Similarly for B 5~ A . We can thus rewrite t h e global conditions as the sequence: R2e

:

~2e+1

:

A $ {e}B B $ {e}".

T h e construction of A and B is by finite initial segments. We will let A = UsEw CT, and B = UsEwT,. At each stage of t h e construction we will take care of one requirement, once and for all. We begin by setting CTO = TO = 8. At stage s + 1, suppose o, a n d T . are given.

If s = 2e then we satisfy Rze Let x be t h e first element such t hat a s ( x )does not converge: this means t h a t we have not yet decided whether x has t o be in A or not. We will decide this now, and we will use x t o witness t h a t A $ { e } B . In other

V. Turing Degrees

458

words, we will arrange t h a t A ( $ ) $ { e } B ( x ) :t h e idea is obviously to diagonalize, i.e. to make A on x different fronr {e}B on x. B u t since we have not yet defined B , we do not even know whether { e } B ( x )converges. W h a t we do know, by compactness, is t h a t if it will converge in t h e end so will {e}.(x), for some T B. Since T~ B b y construction, if such a T will exist t h e n it will be compatible with T~ (since B will extend both). We may also suppose, by monotonicity, t h a t T will actually extend rs. At this point we may t u r n things around, a n d see whether there is any string T 2 T~ such t h a t {e}+(x) converges. If such a string does not exist, we know t h a t {e}”(x) will be undefined, a n d since A ( x ) will certainly be defined (being a total function), i t does not matter what we do. E.g., let 0,+1 be t h e smallest initial segment extending us a n d defined on x. Since nothing h a s to be d o n e on B , let TS+I = 7.. If however such a T exists, then i t might be t h e case t h a t { e } B ( x )will converge, a n d we want to dispose of this case too. W h a t we can do, since we a r e also building B , is to insure t h a t B will extend T : then {e}’(x) = {e}‘(z) by monotonicity. To insure this, we only have to let T ~ = + T~. But now we have to be careful with A: we know t h a t {e}”(z) converges, a n d thus we want A ( x ) different from it. T h e n we let o ~ be + ~ t h e smallest initial segment extending us, a n d such t h a t

If s = 2e + 1 then we satisfy &+I T h e construction is t h e same in this case, simply with t h e roles of A a n d B interchanged. Notice t h a t there is no reason to define t h e strings as initial segments, except for being sure t h a t a set is obtained as their union. If we did not do this, t h e n we could just say t h a t A is any set extending UsEwu s ,a n d similarly for B . Also, t h e only reason to specify t h a t we choose particular z’sa n d particular o’s, when many possibilities a r e open, k simply to have t h e construction as effective as possible: this is going to be useful in t h e corollary below, where a n evaluation on t h e complexity of t h e construction is needed. 0

We can distinguish between global results, t h a t hold for t h e whole struct u r e P, a n d local results, t h a t hold instead in t h e degrees P ( 2 u) below a given degree a. Not all results can be localized below a given nonrecursive degree, a n d those t h a t do may require more sophisticated proofs. B u t t h e simple analysis of a proof will usually provide a n upper bound to t h e noneffective parts

V.2 The Finite Extension Method

459

in i t , a n d hence a local version of t h e global result (although not necessarily t h e sharpest one).

Corollary V.2.3 There are incomparable degrees below 0’.

Proof. We only have t o show t h a t t h e sets A a n d B built above a r e recursive in

K. T h i s follows from t h e fact t h a t t h e only nonrecursive step in t h e construction is to decide questions of t h e form: ‘is there a string u‘ extending a given string converges, for a given z?’ T h i s is easily seen to be recursively enumerable: it is enough to dovetail t h e computations of {e)O’(z) for all strings u’ extending u (note t h a t this last condition c a n be effectively checked, because strings a r e finite functions). Since K is m-complete, each of these questions can t h u s be answered recursively in K. 0 u a n d such that {e}‘’(x)

Exercises V.2.4 a) The existence of incomparable degrees follows from 2 ” O # N1, by cardinality considerations. (Myhill [196l])(Hint: each degree has only countably many predecessors, and thus a linear ordering of degrees must have cardinality a t most N1. But there are 2 ” O degrees.) ti) The existence of incomparable degrees can be proved by set-theoreticd considerations. (Kreisel) (Hint: the existence of incomparable degrees is absolute for standard statement. Thus it is enough to show that it models of Z F - by IV.3.25, being a holds in a model of Z F ~. By forcing, there is such a model in which the Continuum Hypothesis fails and thus, by part a), in which there are incomparable degrees.) c) The Continuum Hypothesis is equivalent to the assertion that there is a wfinal chain of degrees of order type N1, where cofirial means that each degree is bounded by some element of the chain. (Hint: the downward closure of such a chain still has cardinality HI,because each degree has a t most countably many predecessors.)

Embeddability results Having shown t h a t D is not linear, one immediately wonders a b o u t i t s possible complexity as a partial order. This can be measured by t h e quantity of partial orderings t h a t can be embedded in t h e structure. We show in t h i s subsection t h a t , in t h i s sense, 2) is q u i t e complicated. We could proceed by b r u t e force a n d build, given any countable partial ordering, a set of degrees isomorphic t o it. T h e constructions of these sets would not differ very much, a n d most of t h e work would be to reproduce t h e given partial orderings. We will instead isolate t h e recursion theoretical ideas in j u s t one result, which will produce a kind of universal countable partial ordering, in which all t h e others a r e embedded. T h e idea consists in building degrees which a r e very independent, one from t h e others, in t h e sense t h a t none is recoverable from t h e remaining ones: t h e universal partial ordering will t h e n be given by all their possible combinations. T h e notion of independence is captured by t h e following definition.

460

V. Turing Degrees

Definition V.2.5 (Kleene and Post [1954]) A set { A n } n Eis~ recursively independent if, for each n E I ,

A,, $I@{Arn: m E I

A

m # n}.

Exercises V.2.6 a ) The set { A } is recursively independent zf and only if A is n o t recursive. b) { A , B } is recursively independent if and only i f A and B are incomparable. c) If { A , ) , ~ I is recursively independent then the A, are mutually incomparable, but not conversely. (Hint: if { A , B , C } is recursively independent, then A @ B , B @ C , and A@C are mutually incomparable but not independent, since e.g. A @ B is reducible to ( A @ C) @ ( B @ C).)

Proposition V.2.7 (Kleene and Post [1954]) There exists a countable, recursively independent set.

Proof. T h e proof differs only in t h e details from t h a t of V.2.2. We have to build a countable sequence {An},,Ewof sets, a n d we j u s t build a giant set A t h a t p u t s t h e m all together. We will t h e n let z E A,, e ( n , z )E A ,

so t h a t A c a n be thought of as t h e infinite join of t h e An's, a n d each A , c a n be thought of as t h e n-th column of A . T h e requirements are:

We will build A by finite initial segments u s . We s t a r t with uo = 0. At s t a g e 1, let us be given. If s = ( e , n ) 1, t h e n we a t t a c k R(e,,,). We choose a number ( n , z )such t h a t us is not yet defined on it, so t h a t in particular membership of z in A , has not yet been decided. Note t h a t in t h e e n d @,+,A, will be equal to A except for t h e n-th column, which will not contain any elements. T h e n we look for a string u such t h a t : s

+

0

+

u is 0 on t h e elements of t h e n-th column, i.e. t h o s e of t h e form (n,z ) u extends us when this is defined, on elements which a r e not o n t h e n-th column

If such a string does not exist then {e}@m#nAm cannot converge on ( n , ~ ) , a n d we let us+]be any initial segment extending os a n d defined on ( n , z ) . Otherwise, we t a k e o n e such string u , a n d define us+l as a n y initial segment which extends u s ,extends u on elements not on t h e n - t h column, a n d a8+i((n,z))= 1 - {e}'((n,z)).

V.2 The Finite Extension Method

461

As we announced, t h e result is enough to prove all t h e embeddability results we need. Exercise V.2.8 There is a recursive partial ordering in which all the countable partial ordering are embeddable. (Mostowski [1938]) (Hint: take the pairs of rationals, ordered as (z, y)

< (z‘, y’)

* x < z’ A y < y’.

This a planar version of the fact that every countable linear ordering is embeddable in the rationals. A direct construction is also possible, by having a t any finite stage a finite approximation, and extending it by adding a new element in each possible way consistent with the partial ordering requirement.)

Theorem V.2.9 (Sacks [1963])A n y countable partial ordering is embeddable in t h e degrees (below 0‘).

Proof. By t h e exercise, i t is enough to show t h a t a n y recursive countable partial ordering 5 is embeddable in t h e degrees. We use t h e recursively independent set {An}nEw,a n d associate to each element a in t h e domain of 5 t h e set en5 ,A, : (n,x) E B, e n 5 a A x E A,. It remains to show t h a t

a5b

* Ba S T Bb,

so t h a t t h e structure of t h e degrees of t h e B,’s is isomorphic to

if a 5 b then B, ST Bb Note t h a t , since 5 is transitive, B,

T h u s B,

ST Bb, because 5

5:

C Bb in this case, a n d t h u s

is recursive.

if B, ST Bb then a 5 b Suppose a $ b: then each element t h a t Bb ST ( @ n # a A n ) :

TL

5 b is different from a. I t follows

But A , ST B,, since a 5 a , a n d t h u s it cannot be B, ST Bb otherwise, by transitivity, A , ST (@,+,A,), contradicting t h e recursive independence of t h e An’s.

V. Turing Degrees

462

If we let A , be t h e set constructed in V.2.9, then BnEwAnis recursive in

K ,a n d hence so a r e all t h e B,'s. T h u s 5 is actually embeddable in the degrees below 0'.

0

A sentence in t h e language of P is a one-quantifier sentence if i t is logically equivalent to a sentence in prenex normal form with a prefix consisting of quantifiers of t h e same type, a n d a matrix consisting of a Boolean combination of atomic formulas of t h e form 2 5 y. Corollary V.2.10 (Lerman [1972]) T h e one-quantifier sentences of D a n d D ( 5 0 ' ) a d m i t a decision procedure. Proof. I t is enough to decide t h e existential sentences. B u t such a sentence simply asserts t h e existence of finitely many elements q ,. . . ,z, in a certain order relationship. Since a n y countable (and hence a n y finite) partial ordering is embeddable in P,t h e sentence is t r u e in 2) if a n d only if it is consistent with t h e fact t h a t 5 is a partial ordering. 0 Exercises V.2.11 Independent sets of degrees. Since the infinite join is not a degreetheoretical operation, a set of degrees A is called independent if no element in it is bounded by the 1.u.b. of some finite subset of A. a) There is a countable independent set of degrees. (Kleene and Post [1954]) (Hint: see V.2.7.) b) There is a n independent set of degrees of cardinality 2N0.(Sacks [196l]) (Hint: build a tree of sets, each branch of which is not recursive in any finite join of different branches.)

Sacks [1961] has proved t h a t every partial ordering with o n e of the following

properties is embeddable an P:

1. cardinality 2N0 a n d f i n i t e predecessor property 2. cardinality N, and countable predecessor property

3. cardinality 2'O,

countable predecessor property a n d N1 successor property.

In particular, t h e last two conditions a r e equivalent a n d best possible, if t h e Continuum Hypothesis is assumed. T h e best absolute result would clearly be to prove t h e embeddability of any partial ordering with 4. cardinality 2"O a n d countable predecessor property

b u t it is unknown whether this holds.

V.2 The Finite Extension Method

463

The splitting method We have completely decided t h e one-quantifier theory of 2). We thus start to deal with two-quantifier sentences, that will be discussed in general on p. 490. A typical problem is to get a degree incomparable with a given one. Here t h e game is more subtle than t h e one in V.2.2: there we could build both sets, while here one of them is given t o us and we have no control over it. T h e technique used in the solution t o the problem is going t o be extremely useful, a n d will be exploited in many different situations. T h e idea is natural, and it is a refinement of t h e one already used: since we control only one of t h e two sides, but we still need two possibilities t o be able t o diagonalize, we look for both of them on t h e side we control.

Definition V.2.12 Two strings u1 and u2 are a n e-splitting if, for some x, {e}ul(x) and {e}uz(x) both converge, and are diflerent. I n this case we say that u1 and u2 e-split on x. Theorem V.2.13 (Kleene and Post [1954]) Given any nonrecursive degree b, there is a degree a incomparable with it. Proof. Let B be nonrecursive: we want A such that R2e R2e+l

: :

A + {e}B B?qe}A.

T h e two types of requirements look t h e same, but are very different: B is given ahead of time, while A is constructed. Thus they require different actions. As usual, we will let A = U s E w u sWe . start with 00 = 8. At stage s 1, let us be given.

+

If s = 2e then we satisfy This is done by t h e method of the last subsection: we choose x on which us is not yet defined, and we extend us t o a us+] such t h a t

Note t h a t we simply ask whether { e } B ( x )converges, instead of asking, as before, whether we may make it converge (because B is given). 0

If s = 2e + 1 then

we satisfy R2e+l We now see if there are e-splitting extensions of u s , i.e. if we have two possible choices for {e}A, on some element x. If this the case, choose T~ and ~2 extending us and e-splitting on some x. Then {e}.'(x) and

V. Turing Degrees

464

{e}T2(z) are both convergent, and one of them must be different from

B ( z ) . Let o,+1 be -r,,where If such strings d o not exist, let o,+1 = 0,. We still have t o argue t h a t R2e+l is really satisfied, also in t h e case t h a t e-splitting extensions of o8 d o not exist. Here is where t h e nonrecursiveness of B c o m e into the game: we claim that, in this case, {e}A is either not total or recursive, a n d thus it must be different from B, which is total a n d nonrecursive. Suppose {e}A is total: by compactness, given any 2 there must be a string T C A such t h a t {e}.(z) gives t h e right value. We can obviously suppose, by monotonicity, that T extends 08.But since there are n o e-splitting extending (T, it must be t h e case that, as long as {e}T(x) converges, t h e value is unique. This then suggests a recursive method to compute {e}A: given z, dovetail t h e possible computations {e}T(z), for all strings T extending us. T h e first converging one gives the right value. 0 From a topological point of view, the difference between t h e previous proof a n d the pure extension method of the last subsection is t h a t the splitting method requires something more than simple continuity, since we use t h e fact t h a t when a functional is constant on a n open set (i.e. there is no splitting above a given string), then its value is recursive. Notice also that a simple analysis of the proof does not give t h e result for degrees below 0’, since if 0 < b < 0’ we only get a 5 &‘ 5 0”. T h e result still holds below 0’, but a different proof is needed. T h e theory of degrees below 0’ thus requires a separate study, which we will undertake in Chapter XI. Exercises V.2.14 a) Given a countable set of nonrecursive degrees, there is a degree incomparable with every element of the set. (Shoenfield [1960]) (Hint: the requirements to satisfy are still countably many.)

b) Every maximal antichain of degrees is uncountable. (Hint: use part a) and Zorn’s Lemma.) c ) Every maximal independent set of degrees is uncountable. (Hint: every countable independent set of degrees can be extended.)

T h e next definition introduces two notions of minimality, one for pairs a n d one for single degrees, which are going to be useful in t h e future.

Definition V.2.15 1. Two degrees a and b f o r m a minimal pair if they are nonrecursive and their g.l.b. is 0, i.e. (Vc)(c< a A c < b

+

c=O)

465

V.2 The Finite Extension Method

2. A degree a is minimal i f it is nonrecursive and there is no degree between 0 and a, i.e. (Vc)(c< a C=OVC= a). Clearly two distinct minimal degrees form a minimal pair, b u t a minimal pair might consist of nonminimal degrees. T h e existence of minimal degrees is proved in Section 5, a n d it requires a different method of proof. W i t h t h e present tools we can however prove t h e existence of minimal pairs.

Proposition V.2.16 There exists a minimal pair of degrees. Actually, each nonrecursive degree is part of a minimal pair.

Proof. Let B be nonrecursive: we want A such t h a t

T h e first kind of requirement ensures t h a t A is not recursive, while t h e second ensures t h a t a n y set recursive in both A a n d B is recursive. T h e y a r e satisfied as usual, t h e first by diagonalization, t h e second by t h e splitting method. T h e idea to satisfy R2(e,i)+lis t h e following. First one tries to make t h e requirement vacuously true, by having convergent computations such t h a t

T h i s can be done, at a given stage n, by looking for e-splitting extensions of IT,. If they do exist, we choose t h e one t h a t produces a disagreement with

{i}B

o n t h e x for which t h e t w o strings e-split. If they do not exist, we c a n argue t h a t then is recursive if total, as in V.2.13. Note t h a t in t h i s case t h e hypothesis t h a t B is nonrecursive plays no role in t h e proof, a n d it is needed only to satisfy t h e definition of minimal pair. 0

Corollary V.2.17 There exists a pair of degrees with greatest lower bound. Exercises V.2.18 a) Two nonrecursive sets A and B form a minimal pair i f , for each a , {u}" N {u)" N c =+c recursive. (Posner) This slightly simplifies the presentation of requirements. (Hint: choose an element z in one of the two sets but not in the other, say z E A - B. Consider

{u'"

Then {u}"

= {e}"

and {u}"

2:

{ {z}*.)

{e}"

{i}"

x

ifz E otherwise.

V. Turing Degrees

466

b) There is a set of 2 ” O degrees, such that each pair from it is a minimal pair. This will be greatly improved in V.5.12, which however requires a more difficult proof. (Hint: build a tree of sets such that any two branches are a minimal pair.) T h e proof of t h e next result is quite clever, a n d it manages to diagonalize against uncountable sets of degrees by a counting trick. It is a n example of t h e method used by Sacks 119611 to obtain t h e results on uncountable ernbeddings quoted on p. 462.

Definition V.2.19 A s e t o f degrees is: 1. a chain if it i s linearly ordered

2. a n antichain i f its m e m b e r s are m u t u a l l y incomparable n o n z e r o degrees. T h e condition of nonrecursiveness for members of a n antichain is imposed only to avoid t h e trivial case (0).

Proposition V.2.20 (Sacks [196l]) 1. E v e r y countable chain of degrees is extendable, a n d t h u s e v e r y m a x i m a l chain h a s cardinality N1.

2. E v e r y antichain of cardinality less t h a n m a x i m a l antichain h a s cardinality 2”o.

2“O

i s extendable, a n d thus e v e r y

Proof. T h e first p a r t is easy: given a n y countable set {An}ncu,t h e set (QnEwAn)’has degree strictly above all t h e degrees of t h e An’s. Since every degree h a s at most countably many predecessors, t h e maximal length of a chain is N1. For t h e second part, let A be a set of degrees of cardinality 2 N o ,a n d such t h a t any pair in it is a minimal pair (it exists, by t h e last exercise above). Let B be any set of nonrecursive degrees of cardinality less t h a n 2”O. Note t h a t 0

0

t h e downward closure of B has cardinality less t h a n .LN0, since each member of B has at most countably many predecessors for each member of B there is at most one member of A above it, since any degree below two distinct members of A must be recursive, a n d hence not in B.

T h e n less t h a n 2N0 members of A, a n d hence not all of t h e m , a r e comparable with some member of B. 0

We have noticed in V.2.14 t h a t maximal antichains a n d maximal independent sets of degrees a r e uncountable. We have j u s t strengthened t h e first result,

V.2 The Finite Extension Method

467

by showing t h a t maximal antichains actually have cardinality 2'O. A similar strengthening of t h e second result is impossible, since the assertion that maximal independent sets of degrees have cardinality 2O ' is independent of Z F C (Sacks [1961], Groszek a n d Slaman [1983]).

Forcing the jump After a first s t u d y of some elementary properties of P,we now we t u r n o u r attention to D'. To control t h e behavior of t h e j u m p operator a new idea is needed, which will be variously mixed with different methods. We first present i t by itself, in t h e proof of t h e next result.

Proposition V.2.21 (Spector [1956]) The j u m p operator is not one-one.

> 0 such t h a t a' = 0'. To get A nonrecursive we diagonalize, while to have A' ST K: we want to decide, for each e , whether { e } " ( e ) 1. T h e requirements a r e then:

Proof. I t is enough to get a

Rae R2e+l

: :

A $ {e}

decide whether { e } A ( e )1

Note t h a t we do not consider t h e condition K ST A', which is automatically satisfied because 0 S T A , a n d hence K = 0' 5~ A'. As usual we build A by finite initial segments, starting with 00 = 8. At stage s 1, let os be given.

+

If s = 2e then we satisfy R2e Let x be t h e first element on which os is undefined, a n d let smallest initial segment extending os a n d such t h a t Us+l(X)

=

os+l be

the

otherwise.

If s = 2e + 1 then we satisfy See if t h e r e is a n initial segment o extending

us such t h a t { e } " ( e ) 1. If so, Ict o be t h e smallest one, a n d o,+~= o. Otherwise, let osfl= os.

By construction A is not recursive. Moreover,

e E A'-s { e } A ( e ) l H { e } u z c + z ( e ) ~ a n d , since t h e construction is recursive in

K , A'


K.

0

Corollary V.2.22 T h e j u m p operator is never one-one on its range.

V. Turing Degrees

468

Proof. By relativization, given any C we get A such t h a t A $ T C a n d ( A @ C)' ST C'. B u t t h e n A @ C is a set with degree d f i e r e n t from C, a n d j u m p C'. 0 Despite the absolute simplicity of t h e proof j u s t given, t h e ideas involved in it a r e quite deep. T h e argument used is a forerunner a n d miniaturized version of t h e forcing method, introduced by Cohen [1963] to prove t h e independence of t h e Continuum Hypothesis, a n d which will play a major role in t h e next volumes. Its main idea is to approximate truth b y f i n i t e i n f o r m a t i o n , although here we only consider one-quantifier sentences (through t h e j u m p operator which, as we know, corresponds to one quantifier). Note t h a t it is always t h e case t h a t , whenever a computation with oracle A converges, t h e n it does because of a finite a m o u n t of information a b o u t A . But t h e r e is n o reason t o believe t h a t t h e same holds for divergent computations as well, as t h e following example shows: if ( 3 z ) ( xE A ) tndefined otherwise.

{

W h a t t h e proof given above accomplishes is to build a set A such t h a t n o t o n l y

t h e convergence, but also t h e divergence of a n y computations u s i n g A as a n oracle c a n be determined b y a finite a m o u n t of i n f o r m a t i o n o n A . Such sets a r e called 1-generic, a n d will be studied in Section XI.2. T h e y a r e particularly useful in t h e examination of D(
UeEw

{e}'(z)

-u(z)

A.

Now some extension of ue defined on e is contained in A , and hence makes { e } converge on e , and some is not, and makes { e } diverge on e, contradicting the choice of u e . )

b) There is a recursively independent set of degrees with j u m p 0'. (Hint: combine V.2.7 with the proof above.)

A natural question a b o u t t h e j u m p operator concerns its range, a n d it is answered by t h e next result. Theorem V.2.24 Jump Inversion Theorem (Friedberg [1957b]) T h e range of t h e jump operator is t h e cone D ( 2 0 ' ) . Proof. We have already noted t h a t u' 2 0', for any a. To get t h e converse, let C be a set such t h a t K ST C: we want to get A such t h a t A' ET C. T h i s

V.2 The Finite Extension Method

469

splits into two separate conditions: A' ST C, which will be satisfied as above, a n d C ST A', which will be achieved by coding C into A . T h e two strategies will be pursued alternately. As usual we build A by finite initial segments, starting with uo = 8. At stage s 1, let u, be given.

+

0

0

If s = 2e then we see if there is a n initial segment u 2 u, such t h a t { e } ' ( e ) 1. If so, we let u,+1 = u for the smallest such, and otherwise we let u,+1 = 0,. If s = 2e

+ 1, we code t h e e-th element of C into A:

T h e construction is recursive in C: t h e first step is recursive in K ,which is recursive in C by hypothesis, and the second uses C directly. Thus, since

e E A'

@

{e}"ze+l(e) 1,

we have A'
c

and C 5~ A @I K. But A @ K ST A' (because A ST A' and hence C S T A'. 0

K ST A'), a n d

Notice t h a t t h e least possible jump of a degree a is a ' = a U 0', since a U 0' 5 a ' always holds. Not every degree realizes t h e least possible jump, e.g. any degree c 2 0' does not (since then c U 0' = c ) , but t h e proof above actually shows t h a t every degree c 2 0' is the jump of a degree realizing the least possible jump. Exercise V.2.26 For any c 2 0' them are infinitely many degrees with jump c. (Hint: relativize V.2.23.b, and apply the Jump Inversion Theorem.)

T h e j u m p operator by itself is not very problematic: using t h e previous exercise, it can be shown t h a t the first-order theory with equality of ( P , ') is decidable (Jockusch a n d Soare [1970]). But adding t h e jump to V is a different story, and even t h e one-quantifier sentences of V' a r e not known to be decidable. Our last immediate goal is t o determine t h e possible behavior of t h e j u m p operator. To provide some necessary counterexamples, we first prove a result.

470

V. Turing Degrees

Proposition V.2.26 (Spector (19561) For any c grees a and b such that a U b = a‘ = ‘b = c.

2 0‘

there exist two de-

Proof. We modify t h e proof of t h e J u m p Inversion Theorem. Let C such t h a t K: S T C be given: we want two sets A a n d B such t h a t A’ ET B’ ET C , a n d A @ B 31. C. T h e last condition involves t h e only new idea in t h e proof, a n d consists of building A a n d B by initial segments of t h e s a m e length, a n d to let t h e two sides have t h e same value only when coding elements of C , so t h a t C will be recoverable from t h e two sets together. a n d we s t a r t with 00 = TO = 8. At We let A = U s E w uaSn d B = UsEwrs, stage s 1, let us a n d T~ be given a n d of t h e Same length.

+

0

If s = 3e then we see if there is a n initial segment u 2 us such t h a t {e}‘(e) 1, a n d let us+l be t h e smallest such string if o n e exists, a n d us otherwise. Moreover, we also extend T~ o n t h e new elements x on which us+l has been defined, if any, by letting

so t h a t us+l a n d 0

0

~ , + 1

differ on these elements.

+

If s = 3e 1 we do t h e same, interchanging t h e roles of A a n d B (and hence of t h e 0 ’ s a n d 7’s). If s = 3e

+ 2 then we code C ( e ) ,b y letting

Note t h a t this is t h e only step where t h e two sides receive t h e s a m e value.

As before, we have

To recover C ( e ) ,we only have to look at t h e e-th place in which A a n d B agree, a n d this is recursive in A @ B . So

a n d hence t h e sets A @ B a n d C a r e T-equivalent.

Recall t h a t V. 1.14 provided some necessary conditions for t h e behavior of t h e j u m p operator. We can show now t h a t t h e y a r e best possible.

V.3 Baire Category

*

47 1

Proposition V.2.27 (Spector [1956],Shoenfield [1959])Every possibility compatible with a n y of the properties

a
and

a’Ub‘<(aUb)’

is realized. Proof. T h e following are t h e possible cases: 1. j u m p s of comparable degrees Let a b. Then we can have:

<

0

0

a‘ = b‘ by t he proof of V.2.21. a‘ < lf, if a = 0 and b = 0‘.

2. j u m p s of incomparable degrees Let a1 b. Then we can have: 0

0

0

a‘ = b‘ by V.2.26, since two degrees with j u m p c are strictly below it, and if they join t o it then they must be incomparable (otherwise their join would be one of them). a‘ < b‘, by having a degree b incomparable with a = 0’ a n d with j u m p greater t han 0”. Note t h a t a degree realizing t h e least possible j u m p cannot be above 0’, and thus we only need t o avoid b < 0‘. This is automatic if b‘ > Of’, and thus it is enough t o apply t he J u m p Inversion Theorem, and get b such t h a t ‘b = 0”’. a’( b‘, by taking two degrees whose jumps are incomparable, which exist by t he J u m p Inversion Theorem (since incomparable degrees above 0’ exist, e.g. by relativization of V.2.2).

3. distributivity of j u m p over join 0 0

V.3

a’ u b‘ = ( a u b)’ = 0’’if a = 0 and b = 0’. a’ U b‘ < ( a U b)’ for the degrees in V.2.26. 0

Baire Category *

In Section 11.1 we looked at t h e class P of partial functions, from a topological point of view. Here we d o t h e same for t h e class of total functions, a n d we will distinguish t h e class of functions (w“,t he Baire space) from t h e class of characteristic functions or sets (P(w),the Cantor space). There a r e various ways of introducing a topology on these sets, but t he induced topology will be unique, an d thus particularly stable. As a reference on topology t h e reader can consult Kelley [1955].

V. Turing Degrees

472

Topologies on total functions Natural topologies on ww and P ( w ) are t h e following:

1. the topology induced by the topology of P T h e open sets are t h e intersection of open sets of P with w" or P ( w ) . Thus a basic open set is determined by a finite initial segment, see p. 186. 2. the product topology of the discrete topology on w or ( 0 , l ) Here every subset of w is open, a n d a basic open set on ww is a product

where t h e Ai's are nonempty open sets. Thus t h e basic open sets are unions of sets determined by finite initial segments, namely t h e finite initial segments u of length n + 1 such t h a t u ( i ) E Ai.

3. the topology induced b y the following metric:

Thus t h e distance between two functions is determined by t h e smallest argument on which they differ, and the factor '+l' takes care of t h e possibility t h a t they differ on 0. T h e basic open sets are t h e balls relative t o t h e metric, and again they are determined by finite initial segments. 4. the topology induced by the order topology on the r e d s Here we use t h e canonical homeomorphism between ww a n d t h e irrationals between 0 and 1, which can be produced in two different ways. One is t o give t h e direct map 1

----

For t h e other, first note that there is a one-one map from ww t o P ( w ) , given by f 1 . .. 100...01 I . . . loo-.. f(0) + 1 f ( 1 )

+ 1 f(Z) + 1

Then consider t h e reals between 0 and 1, in binary representation. Each such real may be written as x i c 1 2 - ( a + 1 ) , for some I w. T h e binary irrationals are those reals between 0 and 1 t h a t cannot be so represented with a finite I . T h e composition of these two maps gives a homeomorphism between ww and the binary irrationals. These a n d t h e irrationals are homeomorphic, because binary rationals a n d rationals are

V.3 Baire Category *

473

both countable and dense in [0,11, so there is a o n e o n e order preserving correspondence between them, which may be extended in a unique way to a homeomorphism of [0,1] with itself. T h e restriction of it t o t h e binary irrationals is a homeomorphism with the irrationals. Since all these topologies are equivalent, we can use them interchangeably, but we will mostly continue to use the topology induced by the topology of P , whose basic open sets are determined by finite strings.

Proposition V.3.1 ww is separated, but neither compact nor connected. P(w) is separated and compact, but not connected. Proof. For separation, given two different functions f a n d g , let x be t h e smallest point on which they differ. Then t h e two open sets determined by f(x 1) and g ( x 1) are disjoint, and separate f and g . This also proves t h a t t h e two spaces are not connected. P(w) is compact by Konig’s Lemma (see V.5.23), but ww obviously is not since, e.g., t h e open sets determined by the finite initial segments ( n ) form a disjoint covering, from which no finite one can be extracted. 0

+

+

Corollary V.3.2 ww, P(w), and [0,1] are pairwise not homeomorphic.

Proof. T h e first two are not homeomorphic because one is compact but t h e other is not. And they are not homeomorphic t o [0,1], because they are not connected, b u t t h e reals are. 0 Exercises V.3.3 a) A set A ww is compact i f and only it is closed and bounded, i.e. there is a function f such that, for any g E d, (Vn)(g(n)5 f ( n ) ) . (Hint: use Konig’s Lemma for finitely generated trees.) b) A set A ww is clopen if and only i f , for some well-founded tree, A is the

c

union of the open sets determined by the branches of the trre. c) A set A P ( w ) is clopen i f and only zf, for some tt-condition u,A is the class of sets satisfying u. d) Both ww and P ( w ) are dimensionless, in the sense that any finite power of t h e m is homeomorphic t o the original space, but [0,I] is not. (Hint: by taking one internal point away we disconnect [0, 11 but not [0, 11 x [0, 11.)

Comeager sets T h e reason we introduce t h e topological approach t o total functions is given by t h e next notions, which will have a n immediate bearing on t h e methodology of Degree Theory.

Definition V.3.4 (Baire [l899])A set A in the Baire

OT

Cantor spaces is:

V. Turing Degrees

474

1. dense i f its closure, i.e. the smallest closed set containing it, is the whole space (alternately, i f its complement does not contain any open sets) 2. corneager i f it contains the intersection of a countable family of open

dense sets 3. meager i f its complement is corneager Exercises V.3.5 a) The intersection of a countable family of corneager sets is still corneager. b) The superset of a wmeager set is wmeager. c ) The wmeager sets form a filter an the lattice of subsets of the Baire or Cantor spaces under inclusion, and the meager sets form an ideal.

Proposition V.3.6 Given a set A in the Baire or Cantor spaces, then

1. A is dense i f and only i f

2. A contains an open dense set i f and only i f

Proof. In t h e first part, t h e right-hand side says t h a t no basic open set is contained in 2. This is clearly equivalent t o density, which means t h a t no open set is contained in 3. For t h e second part, let A 2 B,and B be open dense. We just proved t h a t , given u , there is g 2 u in B. Since B is open, there is a basic open set containing g, and contained in B. Let u1 be t h e string determining it: then g 2 0 1 , a n d any function extending 01 is in B. If r is t h e smallest string containing both u and 0 1 , which exists because they are comparable (being b o t h contained in g), then any function g 2 T must be in B,because g 2 u1. T h e condition is thus satisfied for B,and hence for A. Conversely, suppose (v~)(+u

2 o)('Jf 2 ~ o ) ( fE A).

Then t h e open dense set

UCf f 2 TJ U

is contained in A.

0

We now prove the basic result of this subsection, by t h e same finite extension method used t o prove the results on degrees in t h e last section.

V.3 Baire Category *

475

Theorem V.3.7 Baire Category Theorem (Baire [l899]) A comeager set is not empty. Proof. If A is comeager there is a sequence of open dense sets such that d 2 d,,. It is enough t o prove that d,,)# 0. We build a function f E d,,,by initial segments. Let uo = 0. At stage n + 1, let u,, be given: we ensure that f E d,,. Since d,, is open dense, there is a string u 2 u, such that all functions extending it are in d,,: it is then enough to let a,+] be any initial segment extending u . Since in t h e end f = UnEwa,, we will have f 2 u,+1 a n d hence f E d,,. 0

nnEw

nnEw

(nnEw

Corollary V.3.8 A comeager set is not meager. Proof. Note that the intersection of two comeager sets is comeager. If a corneager set d were meager, A would be comeager and hence not empty, contradiction. 0

flz

Note t h a t it is essential that we are considering only countably many dense open sets: the intersection of 2N0 open dense sets is not necessarily nonempty ((f)is open dense, being t h e union - of t h e open sets determined by t h e strings not contained in f ,but {f} is obviously empty). If the Continuum Hypothesis is assumed, this settles the question. But if t h e Continuum Hypothesis fails then there are uncountable cardinals below 2 ’ O , and we may ask whether the intersection of less than 2’O open dense sets is nonempty. T h e positive answer is known as Martin’s Axiom, and it is independent of Z F C (Martin and Solovay [1970]). For some of its interesting set-theoretical consequences see, e.g., Jech [1978] and Levy [1979].

nfEww

Exercises V.3.9 a) A comeager set is dense. (Hint: it is enough to show that it intersects any nonernpty basic open set. The proof is as above, only starting with 00 determining the given basic open set.) b) A corneager set has cardinality 2N0. (Hint: build a binary tree, each time extending a given string in two incomparable ways, and then continuing separately on each of them.) c) A countable set is meager. (Hint: it is enough to show that any singleton {f} is meager. Since comeager sets are closed under supersets, it is enough to find a comeager set not containing f, so that (f) is corneager. For every cr there is T, 2 cr such that every g 2 T, is different from f . Let A be the union of the basic open sets determined by T,. By definition A is open dense, hence corneager, and does not contain f.) Exercise V.3.10 Banach-Mazur games. Given A, consider the following game. The two players play strings, and each must play a proper extension of the last move.

476

V. Turing Degrees

Player I wins if and only if the union of the moves is in A. Then A is corneager if and only i f player I has a winning strategy, and A is meager i f and only i f player I I has a winning strategy. (Oxtoby [1957]) (Hint: suppose I has a winning strategy, and let ,4o be the basic open set determined by the first string played by I, according to the strategy. For any possible response 7 of I1 consider the response LT, of I, according to the strategy, and let A1 be the open set obtained as the union of the basic open d,,, sets determined by the crT, and so on. The An’s are open dense and A 2 because I uses a winning strategy.)

nnEw

T h e intuition about comeager and meager sets is t h a t they are, respectively, very large and very small. T h e next result accords with this intuition, and tells t h a t almost all sections of a large (small) set in a product space must be large (small).

Theorem V.3.11 Kuratowski-Ulam Theorem. Given A A g P ( u ) x P(w), let f E d g ++ ( f , g ) E J1

ww x ww or

be the section of A at g. Then: 1. ifd is comeager, the set { g : A, comeager} of its comeager sections is comeager 2. ifd is meager, the set { g : A, meager} of its meager sections i s comeager.

Proof. T h e second part follows from t h e first, by taking complements. Let then d be comeager: A contains r)nEw.G,, where t h e An’s are open dense in t h e product space. T h e comeager sets are closed upward under inclusion, a n d thus it is enough t o find a comeager set of functions g for which A, is comeager. Since A, 2 we look for g’s such t h a t ( A ) ,is open dense. By definition of product topology (A),is open for any g, a n d we would want (Va)(37-2 a ) ( V f 2 T)[fE (&)91,

n,,(.G,),

and hence

(v0)(37-2 a)(vf 2 T ) [ ( f l g ) E .An].

If this holds for every n, then A, is comeager. Let us consider t h e set of g’s for which this holds:

We see whether L3 is comeager. If

V.3 Baire Category

*

477

t h e n f3 = n,,u.O(,,,u),b u t t h e f3(n,u) a r e not necessarily open, because we use all of g in thelr definition. T h u s 13 is not necessarily comeager, b u t a small variation of it will do. Let E c ( n , u ) -S( 3 ~ ) ( 2 3 a)(v'f ~ 2 ~ ) ( v2h O(Y)>[(/,

h) E

A].

Now C(n,ul is open (because union of basic open sets) a n d dense (because so is

A ) .If

n

c = c(n,ul n,o

t h e n C is comeager a n d , as above,

d, is comeager whenever g E C.

0

Exercises V.3.12 A has the Baire property if there is an open set U such that ( A- U ) U (U - A ) is meager, i.e. A differs from an open set by a meager set. Ifd has the Baire property, then the converse implications in the Kuratowski-Ulani Theorem hold. (Hint: suppose A is not meager, but {g : A, meager} is comeager. Since Ag differs from the open set U, by a meager set, U, must be meager for a comeager set of g's. But U is not empty, and it contains a basic open set: hence there is also a comeager set of g's such that U, contains a n open set, and thus is not meager. The intersection of these two comeager sets is not empty by the Baire Category Theorem,

contradiction.)

Baire Category and Degree Theory Going back t o t h e proofs of t h e results in Section 2, we notice t h a t t h e y all shared t h e following characteristic features. We had a set of requirements R, to satisfy, which may be identified with t h e class of sets satisfying t h e m . T h e general p a t t e r n of t h e proofs was to show t h a t , for each n,

T h u s we were actually showing t h a t , for each n, t,here is a n open dense set contained in h!,, (by V.3.6). T h e Baire Category Theorem could t h e n be applied to claim t h a t t h e intersection of requirements is not empty, a n d hence t h a t t h e r e is a set satisfying all t h e requirements, without further constructions. More precisely:

Proposition V.3.13 The Finite Extension Method (Myhill [1961], Sacks [1963]) Given a countable collection of requirements R.,, such that (Yo)(+

the set

2 a ) ( V A 2 T ) ( A E &),

nnEwR,, is comeager (and hence nonempty).

V. Turing Degrees

478

T h e categorical approach is useful for a methodological analysis of t h e finite extension method. Precisely, from it we obtain t h at : 1. requirements can be taken care of separately, b y showing that each of them

is dense 2. we can freely combine constructions known to be performable separately, as long as the global list of requirements remains countable For example, suppose we want t o prove t ha t , given a countable sequence {Ao,A l , .. .} of nonrecursive sets, there is a set B incomparable with all of them. Then we only have t o prove t hat , for a given e a n d a given nonrecursive set C ,

(Vo)(+)(VA 2 7 ) ( A74 {e}") (Vo)(+)(VA 2 .)(C $ {e}"). We know th a t we can d o this, by t he work done in Section 2. It thus follows th at, for a fixed nonrecursive set C , t h e set

is comeager, and hence so is

being a countable intersection of comeager sets. In particular, t h e set is not empty by Baire Category Theorem. Note t h a t we could even avoid t h e consideration of t he condition A $T C , because there are only countably many sets recursive in C: t h e set { A : A $T C} is thus automatically comeager.

3. we cannot use the finite extension method t o produce sets belonging to a meager class This follows from t h e fact t hat t h e finite extension method produces sets in a comeager class, and t h e intersection of two comeager classes is nonempty, by t he Baire Category Theorem. In particular, if we can use th e finite extension method to build a set satisfying certain requirements, then we cannot use t h e same method t o build a set not satisfying t h e same requirements. 4. a game-theoretical approach to Degree Theory is possible, via Banach-

Mazur games This follows from V.3.10, and has been developed by Yates (19761. See also p. 495.

V.3 Baire Category *

479

We can reforniulate a number of results proved before in terms of category notions, with t h e convention t hat a set of degrees A is comeager or meager if such is t h e class of sets whose degree is in A. This makes sense because a degree contains only countably many sets, and thus it is a meager class.

Proposition V.3.14 The following sets of degrees are comeager: 1. { a : a is incomparable with a &ed

nonrecursive degree}

2. { a : a is the 1.u.b. of two incomparable degrees}

3. { a : a is the 1.u.b. of a minimal pair}

4. { a : a

realizes the least possible jump}.

The following sets o f degrees are meager:

5. { a : a is comparable with a fied nonrecursive degree} 6. { a : a is a minimal degree}.

Proof. T h e proof of V.2.2 builds two sets A and B with incomparable degree: this can be seen as t h e construction of a single set A @ B which is t h e least upper bound of two incomparable sets, a n d proves 2. Part 6 follows from this, since a minimal degree cannot be the 1.u.b. of two incomparable degrees. Similarly, t h e other parts follow from V.2.13, V.2.16 and V.2.24. 0 Note t h a t this shows, in particular, t hat degrees comparable with a given nonrecursive degree, as well as minimal degrees, cannot be built by the finite extension method. T h e work done so far also allows t o decide whether any quantifier-free question about jumps, I.u.b.’s a n d g.l.b.’s of degrees holds for a comeager set of degrees or not.

Proposition V.3.15 (Stillwell [1972])The theory of degrees with subformulas containing a term t l n t2 thought of as pre&ed by ‘tin t2 exists’, and with the quantifiers V and 3 interpreted, respectively, as meaning ‘for a comeager set of degrees’ and ‘-b’, is decidable. Proof. For simplicity, we will say ‘almost always’ t o mean ‘for a comeager set of n-tuples of degrees’. Note t h a t terms are obtained from variables and 0 by using n, U, a n d ’. By induction, we show t h a t for every term t there are degrees ai occurring in t such th at, for some m,

t = al u

-.- u a,,u o ( ~ )

V. Turing Degrees

480

is almost always true. T h i s reduces every t e r m to a normal form. 1. ( ~ ~ U . . . U U , U O ( P ) ) U ( ~ ~ U . . . U ~ , U O ( ~ ) a1 U U a, U b l U * . U b, U 0

-

..-

/=

max(p9q)).

T h i s always holds.

-

-

2. (alU- .Ua, U O ( P ) ) n ( b l U. -Ub , U O ( Q ) ) = c l U - .U c, almost always, where f

UO(m'n(P9Q))

{ c i , . . . ,~,}={ai,...,~,}n{bi,...,bn}.

First note t h a t it is always possible to rearrange terms, a n d possibly introduce a new O(m'n(P*q)) (which has no influence on U) in such a way to have, for some d a n d e, a1 u bl U

... lJa, . - u bn

u o(p) = U O(Q)

.

u . .u c, u o(m'n(p,q)) I u d

(cl

= (c1 U

0

-

U c, U O("'"(p~Q)))

U e.

It is t h e n enough to show t h a t almost always, given a a n d d, is ( a u d) n ( a u e)

= a.

And this is j u s t t h e relativization to a of t h e fact t h a t almost every degree d is p a r t of a minimal pair (V.2.16).

--

-

3. (al u * u a, u o(P))' = al u * u a, u o(P+') almost always. ' = aUO' a n d , by relativization to b, almost always First, almost always a ( a U b)' = a U b'. T h e n , by induction, almost always a u O(P) = a ( P ) . Finally, almost always a1U...Ua,

uo(p)

=(alU...Ua,)(P),

a n d hence

T h e decision procedure now follows easily, since every formula with free variables is 0,l-valued, being satisfied either by a comeager set of degrees or by a meager one. Precisely: 1. t l 5 t2, with t l a n d t 2 terms After putting t h e two terms in normal form, t l 5 t 2 holds if a n d only if all t h e variables of t l appear in t2, a n d t h e exponent of 0 in t l is not bigger t h a n t h e exponent of 0 in tz.

V.3 Baire Category

2.

*

481

7+

T h e complement of a meager set is comeager, a n d t h e complement of a comeager set is meager.

3. + A v T h e intersection of two comeager sets is comeager, and t h e intersection of any set with a meager set is meager. 4. Vx+(x) This follows from t h e Kuratowski-Ulam Theorem, by t h e interpretation of t h e universal quantifier. 0

We should not expect too much from this decision procedure, since practically every interesting question we may ask will involve real quantifiers as well. E.g., it is tr u e t hat almost every degree has no minimal predecessor (see V.3.17), but we cannot express this sentence in t he above language, since t h e notion of minimal degree requires a true universal quantifier.

Meager sets of degrees T h e results quoted above were simply old facts rephrased in categorical terms. We now prove a theorem whose very statement genuinely requires t h e notions introduced in this section. T h e plan of t he proof can be read independently, but its implementation relies on methods and notations t hat will be introduced in Section 5.

Theorem V.3.16 (Martin [1967]) If A is a downward closed, meager class of degrees then the upward closure of A - ( 0 ) is still meager. Proof. We want to build A by finite extensions, in such a way t o have

This condition can be broken down in t h e requirements

Re : (e)" total Let

00 =

0. At stage e + 1, let u,

(e)" @ A - ( 0 ) .

be given: we attack Re.

1 . If there is u 2 ue such t hat (3Z)(VT 2 m e ) W

t)

then we let oe+l = 0. This ensures t hat if A 2 total.

ue+l

then (e)" is not

V. Turing Degrees

482

2. If there is a (T 2 oe with n o e-splitting extensions, again let This ensures t h a t if A 2 0,+1 then { e } A is recursive.

ue+l

=

(T"

3. Suppose now t h a t :

0

(yo2 ae)(yX)(37 2 g)({e}T(z>1) for all (T 2 (T, t h e r e a r e e-splitting extensions of

0.

It will be useful in t h e following to picture {e}' as a finite string, with t h e property t h a t (T 0' {elu c { e l u ' .

*

T h e proof goes roughly as follows. We build a n order-preserving recursive m a p Q, from strings to strings such t h a t

( y 2~ @ ~ ( { e } ~ ~ )2) (m3e )0[ . = @({e}')]. This gives a homeomorphisin between t h e set {CP({e}") : (T 2 (T,} a n d t h e whole space of strings. Since A is meager, there is a n open set contained in 3, hence there is u 2 u, such t h a t

2 4(@WA) @ 4. But CP is recursive, a n d so Q , ( { e } A ) downward, hence

ST

{e}A.

Moreover, A is closed

( y A 2 ~ ) ( { e@}4 ~, and

Re is satisfied.

T o finish t h e proof, it remains to build Q,. We first build a n admissible triple (i.e. a uniform tree, see V.6.2) with t h e following properties: if TO(.) is t h e set of strings extending oe,of length g(n+ l ) , a n d agreeing with f~ on t h e g ( n + l ) ) ,a n d T,(n)is defined similarly using f ~t h e, n whenever interval [g(n), (TO E To(n) a n d 01 E Tl(n),(TO a n d (TI e-split. T h e construction is t h e s a m e as in V.6.5, only we have to consider every string extending u, a n d of length g ( n ) , when we build t h e n 1 level (instead of only t h e strings of length g ( n ) which a r e on t h e tree, as we did in V.6.5). We now define as follows. If CP(p)(m)converges for all nz < n a n d n < Ipl, then:

+

undefined

if p 2 {e}. for some o E To(n) if p 2 {e}' for some 0 E Tl(n) if pl{e}' for every u E To(n)u .I'I(n) otherwise,

V.3 Baire Category *

483

where pl(e}" means t h a t Ihe two strings a r e incompatible. By definition @ ( p ) is a n initial segment, a n d @ is order-preserving a n d singlevalued (if o E TO(.) a n d 0' E Tl(n)t h e n {e}" a n d {e}"' a r e incompatible). T h e third condition in t h e definition ensures t h a t @ ( p ) ( n )is defined whenever p is long enough. More precisely, if @ ( p ) ( n )converges for all m < n , then so does @ ( p ) ( n )whenever ,

Hence, in general, @ ( p ) ( n )converges if IPI 2 sup(l{e}"l

:0E

(J(To(m)u T1(m))}. m
T h i s also shows t h a t , if we choose our admissible triple in such a way to have I{e}"l 2 g(n) when o E To(n)U T l ( n ) (which is possible because, for each is total a n d so supo I{e}"l = m). T h e n @ ( { e } " ) ( n )1 for all A 2 oe, 0 E TO(.) U Tl(n),a n d

Fix now o 2 oe in some To(n)U Tl(n). We show t h a t @ gives t h e homeomorphism we wanted between { @ ( { e } p ): p 2 0) a n d t h e whole space. Given T 2 @ ( { e } " ) ,let p be t h e string of length g(17-1) such t h a t p E T + t ) ( m for ) In1 5 m <

17-1.

T h e n @ ( { e } p ) ( m )= ~ ( mby) t h e considerations above, a n d I@({e}p)l = T h i s proves t h e claim. 0

17-1.

T h e theorem may be seen as a further step in t h e determination of t h e classes A of degrees for which we can always find a degree incomparable with every element of { A } - 0. We showed in V.2.20 t h a t t h i s is possible if A h a s cardinality less t h a n 2No,a n d t h e theorem j u s t proved gives a solution also for some classes of cardinality 2N0.

Corollary V.3.17 The class of degrees without minimal predecessors is comeager, and hence nonempty. I n particular, the structure 2, is not atomic. Proof. T h e class of minimal degrees plus 0 is meager, a n d obviously closed downward. T h u s t h e upward closure of t h e class of minimal degrees, i.e. t h e class of degrees with minimal predecessors, is still meager. 0 T h e existence of degrees without minimal predecessors can also be proved by initial segments results, see V.6.16.c.

484

V. Turing Degrees

Measure Theory and Degree Theory

*

In place of using Baire Category we might have used measure theory, by considering Lebesgue measure on P(w) or, equivalently, t h e product measure of t h e measure on (0,l) given by ~((0)) = ~ ( ( 1 )= ) T h e r e is a well-known analogy, according to which meager a n d comeager sets correspond, respectively, to sets with measure 0 a n d 1 (although a comeager set c a n have measure 0). T h e measure-theoretical approach does not seem to have t h e s a m e n a t u r a l bearing on Degree Theory t h a t Baire Category does, a n d this is reflected in t h e fact t h a t proofs of results tend to be more complicated. Usually, however, t h e picture obtained by t h e two points of view is consistent, i.e. a set of degrees looks large in one case if it does in t h e other (Sacks (19631, Martin [1967], Stillwell [1972]). A notable exception is provided by t h e analogue of t h e theorem j u s t proved: Paris [1977] has shown t h a t the upward closure of a measure 0 downward closed set of nonzero degrees is not necessarily of measure 0 (see also K u r t z [1983] for a natural example, namely t h e set of 1-generic degrees), although the upward closure of the minimal degrees still has measure 0 .

i.

V.4

The Coinfinite Extension Method

We have proved in t h e last section t h a t construction principles m o r e powerful t h a n t h e finite extension method V.3.13 a r e needed, if we wish to prove results like t h e existence of minimal degrees. We introduce one of t h e m here, by approximating a set not by finite, b u t by coinfinite extensions.

Definition V.4.1 A coinfinite condition is a partial function 6 : w + w with coinfinite recursive domain. A coinfinite condition is recursive i f i t is recursive as a partial function. T h e notions of extension a n d compatibility for coinfinite conditions a r e t h e usual ones for partial functions. Recursive coinfinite conditions a r e t h e next logical step after finite strings (which a r e j u s t particular recursive coinfinite conditions, with finite domain). I t t h u s makes sense to see whether we c a n build, by using t h e m , sets which we cannot build by t h e finite extension method V.3.13. For example, we will see in Section 6 t h a t we can build a minimal degree by recursive coinfinite extensions. Nonrecursive coinfinite conditions a r e also useful because they allow coding a given set in a set we a r e building, in j u s t one step. I n this section we will deal with applications of this kind. T h i s is also one direct way to prove relativized results above given nonrecursive degrees, which cannot be proved b y t h e finite extension method V.3.13 (by V.3.14).

V.4 The Coinfinite Extension Method

485

Exact pairs and ideals By building a minimal pair we have proved t hat some pairs of degrees have g.1.b. We will now prove t h a t there are pairs without g.l.b., by generalizing t h e notion of minimal pair.

Definition V.4.2 (Kleene and Post [1954]) Two degrees a and b f o r m an exact pair f o r a set of degrees C if 1. both a and b are above all degrees in C , i.e. (ye€ c ) ( c5 a A c

5

b)

2. any degree below both a and b is also below some degree in C , i e . x

5

aAx

5

b+(3cEC)(x

5

c).

Theorem V.4.3 Spector Theorem (Kleene and Post [1954], Lacombe [1954], Spector [1956]) Every countable set of degrees in which every pair of elements is bounded has a n exact pair. Proof. Given {Bn}nEw we want to build two sets A a n d B such t h a t 1. (vn)(BnS T A , B )

2. C


A ,B

* C
for some n.

Actually, since any pair of B,’s, and hence any finite set of them, is bounded by some B,, we can replace t he second condition with t h e weaker one:

3. C

ST A , B

+ C ST (@m
T h e second condition looks like t he requirement for minimal pairs, except t h a t here we require a set recursive in both A and B t o be not outright recursive, but only recursive in some element of t h e given set. We thus extend t h e proof of V.2.16 in t h e natural way. First we get one upper bound for free, simply by letting B = @nEwBn. We then build A by coinfinite approximations 8,. To ensure t h a t t he first condition is satisfied, we require

c,

:

B,STA.

This is easily obtained, by coding Be into the e-th column A, of A , with at most finitely many exceptions. Using coinfinite conditions will make this step trivial, an d it will be possible to satisfy C, in just one step. Since A, and Be will differ only finitely they will have t h e same degree, and then A = @nEwAn will also be a n upper bound for {Bn}nEw.

V . Turing Degrees

486

T h e finite modifications are needed t o ensure t he third condition, with t h e relative requirements

Re :

N

{e}B=C

+ C 5~ (

C B ~ ~ for ~ B some ~ )n ,

(note th a t we use V.2.18.a t o simplify t h e presentation of t h e requirements). They will be satisfied as in t he minimal pair construction. We start with QO = 8. At each step e 1, given Qe (see Figure l ) , we simultaneously attack Re a n d C,. As in V.2.16, we see if there are e-splitting strings uo a n d u1 which are compatible with Be. Note t h a t we still use finite strings, because they are always enough to determine convergent computations, although now we cannot look for extensions of Be, which is infinite, a n d thus we only look for strings compatible with it. If they exist, choose t h e one ui t h a t produces a disagreement with {e}” on t h e element for which 00 a n d 01 split. We then define

+

If no e-splitting compatible with

Qe

exists, we let if Qe(x) I if 2 = (e, z ) , otherwise.

{ Be(z) Qe(2)

Qe+l(x)

Be-1 I 1

I Qs

I I I

Figure V . l : Step e

+ 1 for Spector’s Theorem

V.4 The Coinfinite Extension Method

487

Let A = UeEwOe.By construction, O,, extends 8, infinitely on t h e e-th column, and finitely elsewhere. Thus, by induction, we always code Be in t h e e-th column of A , except for finitely many points. Hence, for almost every z , 2

E Be w (e,z) E A.

Thus Be 5~ A holds, and the coding requirement Ce is satisfied. It remains t o see that Re has been satisfied by the action taken at step e + l . This is vacuously true if there were e-splitting extensions as required, because we then defined Oe+l, and hence A , in such a way t o have ;t {e}B. And if they did not exist, then any string cr compatible with 8, and making {e}‘(x) converge must produce t h e right value. This gives a procedure recursive in 8, to compute the function {e}A, if total. But is defined only finitely outside the first e-columns, and the m-th column, for m < e, codes B, except on finitely many points. Thus ee is recursive in @m<eBm, and hence Re is satisfied. 0

ee

Note t h a t the exact pair f o r {Bn}nc,given by the proof is actually recursive i n (enEwBn)’. More precisely, step e + 1 is r.e. in the join of the columns coded in B e , and hence the exact pair is recursive in @ecw(@mieB,)’. In particular, even if ( e n t w B n ST ) K ,i.e. if the set is uniformly recursive in 0’, then t h e exact pair just obtained is only below 0“. We will prove a partial version of the theorem for degrees below 0’ in XI.3.18. However, the full analogue of Spector’s Theorem fails below 0’ for cardinality reasons (basically, there are many ideals but few possible exact pairs, see V.4.6). Note also that the only instance of the theorem that is provable by the finite extension method V.3.13 is the existence of exact pairs for { 0 } , when the theorem reduces t o the existence of minimal pairs, because if t h e set contains a nonzero degree then t h e theorem produces degrees comparable with it, and hence a member of a meager class.

If C is downward closed then a n exact pair u and b for C obviously determines it, since then C E C W C ~ U A b.C ~ There are thus two conditions that have some bearing for exact pairs, and we collect them in the next definition.

Definition V.4.4 A set of degrees is: 1. a n initial segment if is closed downward

2. a n ideal if it is closed downward and under joins 3. a principal ideal if it is a n ideal determined by a degree a, i.e. i t is of the f o r m D ( 5 u ) = { c : c 5 u}.

488

V. Turing Degrees

Corollary V.4.5 Every countable ideal is the intersection of two principal ideals. Proof. An ideal is closed under joins and thus, if countable, it has a n exact pair. Being closed downward, it is determined by it. 0 Exercise V.4.6 There are ideals of degrees below O‘, without exact pairs below 0‘. (Hint: let (An},,Ew be a recursively independent set recursive in K,see V.2.9. Then the ideals generated by { A n } n C ~ ,for any X , are distinct for distinct X’s, by recursive independence. Thus there are 2N0ideals, but only countably many degrees recursive in

K,and thus only countably many possible exact pairs.)

Greatest lower bounds and least upper bounds T h e notion of exact pair is useful for studying I.u.b.’s of sets of degrees, as well as g.1.b.k of pairs.

Theorem V.4.7 (Kleene and Post [1954], Spector (19561) There are two degrees below 0’ without g.1.b. Proof. Consider an infinite ascending sequence of degrees: by Spector’s Theorem it has a n exact pair. T h e degrees forming such a pair cannot have g.l.b., since any lower bound admits a n element of t h e chain above it (by definition of exact pair), and thus is not t he greatest lower bound (because t h e sequence is increasing). An infinite ascending sequence certainly exists (e.g. iterate t h e j u m p operator, starting from any degree). But if we wish t o get t h e exact pair below O‘, we have t o choose a chain {Bn}nEwsuch t h a t (@,EwBn)’5~ K ,because this is the bound we obtained from t he proof of Spector’s Theorem. For this it is enough t o build (see V.2.7 and V.2.21) a recursively independent set { A n } n E w such th a t (enEwAn)‘ < T K : then t he sets B, = @,<,A, form a strictly ascending sequence, by recursive independence of t h e An’s, a n d

Corollary V.4.8 P and P ( 1 0‘) are not lattices. Note t h a t t h e finite extension method V.3.13 produced a minimal pair, hence a pair with g.l.b., and thus it cannot produce a pair without g.1.b. However, it is possible to prove b y the finite extension method that P is not a lattice, as shown in t h e exercises.

V.4 The Coinfinite Extension Method

489

Exercises V.4.9 a) Any pair without g.1.b. is the exact pair of an infinite ascending chain. Thus the proof given above is in a sense the only possible one. (Hint: given a pair of degrees, consider an enumeration of the countable set of degrees below both of them and, for each n, consider the join of the first n degrees in the list.) b) A set bounding a pair without g.1.b. can be built b y the finite extension method. (Jockusch 119811) (Hint: build simultaneously an infinite ascending chain and an exact pair for it. More precisely, build a recursively independent set {Bn},ew, so that the sets Bo @ ... @ B, form an ascending sequence. Build also another set C,which will provide finite modifications of the B,’s, as follows:

An(5) =

{ $:):

ifxsn otherwise.

Then the degree of A , and B, is the same. Let

A

and

=

B =BnEwBn,

and make sure that A and B are an exact pair for the chain { B , ) , E ~ . ) c) The set of degrees Q such that Do(
Proposition V.4.10 Compactness for l.u.b.’s (Spector [1956]) 1. A chain of degrees has l.u.b. i f and only i f i t is eventually constant.

2. A set o f degrees has 1.u.b. i f and only if i t there is a finite subset of it whose j o i n provides a n upper bound for the whole set. Proof. Given a chain, consider a n exact pair for t h e ideal generated by it. T h e 1.u.b. of t h e chain is t h e g.1.b. of t h e exact pair, a n d if t h e chain is not eventually constant it cannot exist, as in V.4.7. Given a set A, it is enough to reduce it to a chain t h a t has 1.u.b. if a n d only if A has it. First note t h a t we may suppose A countable, otherwise it cannot have upper bounds. Let then { cq,, al, . . .} be a n enumeration of A, a n d consider t h e chain bo bn+l

--a0 =

bn U a n + l .

Now t h e chain { bo 5 bl 5 . . .} has 1.u.b. if and only if it is eventually constant, a n d thus A has 1.u.b. if and only if it has a n upper bound which is t h e 1.u.b. of a finite subset of it. 0

In particular, no infinite ascending sequence o f degrees has 1.u.b.

V. Turing Degrees

490

Exercise V.4.11 The existence of a set of degrees without 1.u.b. can be proved by the finite extension method V.3.13. (Kleene and Post [1954]) (Hint: since any countable partial ordering can be embedded in the degrees below O', by V.2.9, there is a set of degrees below 0' isomorphic to the rationals. If a subset of it has 1.u.b. then this must be below O', and hence only countably many subsets can have 1.u.b. But there are 2N0such subsets, since they correspond to the reals, as Dedekind sections.)

Extensions of embeddings We have shown in V.2.10 how t he truth of one-quantifier questions reduces t o embedding problems. Similarly, Shore [1978] and Lerman [1983] have noticed t h a t the truth of two-quantifier questions reduces to problems of extension of embeddings. T h e idea is t h a t t h e truth of ('dZ)(3g)cp(ZC',g can be decided if one knows whether (W[ D ( q ( m V ( K $1 7 +

for each of t h e finitely many D's describing a possible order relationship among t h e x's (i.e. a conjunction of atomic statements xi 5 xj or negations of them). Moreover, cp can be reduced (by writing it into disjunctive normal form) to a finite disjunction of descriptions of possible order relationships among t h e Z'S a n d y's. T h e general form of t h e problem we want t o st udy is thus t h e following: given a n embedding of a partial ordering P in I), and a n extension R of P , can we extend t h e embedding of P t o a n embedding of R? We cannot expect t o be able t o do this in general, since there are three restrictions we have already encountered: 1. I ) has cardinality 2N0 and countable predecessor property. Thus appropriate bounds on t he cardinality of R are needed.

2. P is an uppersemilattice. Thus there is no hope t o embed in V any partial ordering which does not respect th e uppersemilattice structure of V . If P is already embedded then we d o not have t o worry about it, but we certainly have t o ask this of R. For example, given al, a2 and a3 such t h a t a3 < al U a2, we cannot introduce a new degree b which is above al a n d a2 b u t not above a3.

3. There are minimal pairs. Thus we cannot expect t o extend any given partial ordering by inserting new elements below given ones. For example, if al a n d a2 are a minimal pair, we cannot introduce a new degree b strictly between t hem and 0. We are thus led t o t h e following definition.

V.4 The Coinfinite Extension Method

49 1

Definition V.4.12 The partial ordering ( R ,CR)is a consistent extension of the uppersemilattice ( P ,C p ,UP) if: 1. R respects the uppersemilattice structure of P , i.e.

2. R is a n end-extension of P , i.e.

bER-PAaEP-

b&a.

T h e possible extensions of embeddings will always refer t o consistent extensions.

Theorem V.4.13 (Kleene and Post [1954])A n y embedding of a finite partial ordering P in 2) can be extended to a n embedding of any finite consistent extension R of P. Proof. Since R is finite, it is possible t o find a sequence of finitely many successive consistent extensions that add one element at a time, starting from P a n d ending with R . Thus we only need t o know how t o treat oneelement extensions. This amounts t o proving t h a t , given two finite sets {&}nEF and {Cm}m,G such t h a t no Cm is recursive in @,EpBn, we can build A such t h a t 1. for each n E F , B,

si-A

2. for each m E G, Cm $T A . This can easily be done, by t h e same methods of t h e proof of Spector’s Theorem. Since there a r e only finitely many sets t o code, the first conditions can be satisfied ahead of time, simply by setting

&(x)

21 Be(z)

if

e E F A x = (e,z ) .

To satisfy t h e second condition let 8, be given, and suppose we want to satisfy

for some m E G and e. We look for two strings which a r e compatible with 8, a n d e-split. If they exist, then we choose t h e one CT t h a t gives a value different from Cm on t h e element on which they split, and by letting BB+l = 8, U 0 we satisfy t h e requirement. If they do not exist then { e } A , if total, will be recursive in 8, as usual, and thus recursive in @ncpB, because, by induction, 8, is a finite modification of it. But since Cm is not recursive in @,EpBn, it

V. Turing Degrees

492 cannot be equal t o { e } A . 0 .

In the opposite direction, it is possible t o show t h a t t h e condition on R of being a consistent extension of P is necessary, not only for some P (as we have already argued), but also for any P. One part is trivial (since 2) a n uppersemilattice), but it holds also for the other one, since any finite uppersemilattice with least element P is isomorphic t o an initial segment of 2, (see p. 528), and thus some embedding of P cannot b e extended by inserting new elements b e low given ones. This completely characterizes t h e possible finite extensions of embeddings of finite uppersemilattices into P ,and it allows t o give a decision procedure for the two-quantzfier sentences of ’D (Shore [1978], Lerman [1983]). Schmerl has instead shown that the threequantifier theory of 2) is undecidable (see Lerman [1983]). Note t h a t only notational changes a r e needed in t h e proof given above (coding one set at a time), t o obtain one-element consistent extensions of countable embeddings (the requirements on the B,’s and Cm’s being now t h a t no Cm is recursive in any finite join of the B,’s). But not every consistent countable extension can be obtained by a sequence of one-element consistent extensions (e.g. when a n infinite descending chain is added above a given element), and some more work (sketched i n the exercise) is needed for t h e general case. Exercise V.4.14 Any embedding of a countable partial ordering P an D can be extended to an embedding of any countable consistent extension R of P. (Kleene and build sets Post [1954], Sacks [196l]) (Hint: this extends V.2.9. Given {Cn}nEw, (An},,Ew recursively independent over the Cm’s,and not introducing any new relationships among them, i.e. such that

An Cn

g~ ( C B m ~ w c m )CB (CBmfnAm) ST ( C B m E F C m ) CB ( C B m E w A m )

* cn ST

(@mEFCm),

for any finite set F. The second condition is ensured by the e-splitting method. If now P = { c , } ~ E ~ and R - P = {bn}ncw, and the degrees of the Cn’s are isomorphic to the ordering of the G ’ S , we let Bn = ( @ c m C b n C m ) @ ( @ b , E b n A m ) -

Note that the second condition above, together with the fact that R respects the uppersemilattice structure of P, proves the crucial part, namely that ck

bn

* ck gT Bn,

Indeed, suppose c k
(u{c,

V.5 The Tree Method

493

Sacks [1961] shows t h a t any embedding of a partial ordering P with cardinality less t h a n 2”O in P can be extended t o a n embedding of any consistent extension R of P such t h a t R - P is countable. He then uses this result to get t h e uncountable embeddings quoted on p. 462.

V.5

The Tree Method

Our knowledge of t h e structure 2) is beginning t o shape up, but we have not answered yet a n important question, regarding the density of t h e structure. We will now prove t h a t not only is P not dense, it even has minimal elements. To get this result we cannot use t h e finite extension method V.3.13, as we know, because t h e minimal degrees are a meager class. It would be possible to use recursive coinfinite extensions (see V.6.9), but first we prove the existence of minimal degrees in a simpler way, using a more powerful method. T h e idea of t h e finite extension method was t o build a n increasing sequence of strings un, and then take their union UnEw an. We may think of this as building a decreasing sequence of open sets T, = { X : X 2 u n } , a n d then taking their intersection T,. This naturally leads us t o consider more general sets T,.

nnEw

Definition V.5.1 (Shoenfield [1966]) A tree is a function T from initial segments to initial segments (which we will identify with squences of 0’s and 1 ’s), with the following properties: 1. T ( u ) ~ A T C3 U T(T)~AT(T)CT(O) 2. i f one of T ( u* 0 ) ,T ( u * 1) is defined, both are defined and incompatible.

T ( u* 1)

This way of looking at trees in not different from t h e one of IV.2.14: it only has a different emphasis. W h a t really matters in a tree T is always its range (which is still a tree in t h e sense of IV.2.14), but t o think of it in terms of a

494

V. Turing Degrees

function is a useful tool: we can simply talk of T ( u * 0) or T ( u * l ) , instead of 'the two smallest incompatible extensions of T ( u )o n t h e tree'. Similarly, to

state t h a t b o t h or neither of T(u*O),T(a*1) a r e defined is no loss of generality, since if e.g. we a r e only interested in t h e branch extending T ( u * 0), we may define T(u*1) as well, a n d t h e n let t h e tree grow afterwards only above T(u*O). A total tree, i.e. a tree which is total as a function from strings to strings, is nothing more t h a n a closed perfect subset of (0, l}w,in t h e terminology of Cantor [1883] (according t o which a set A C (0,l)" is perfect if t h e r e is n o f E A a n d a n open set 0 such t h a t Ond = { f } , i.e. i t has no isolated points). Intuitively, a total tree is just a set of branches or functions, none of which is

isolated. We say t h a t

1. A is o n T , or A is a branch of T , if T ( a )C A for infinitely many 0's 2. u is on T if it is in t h e range of T 3. T* is a subtree of T (T*C T ) if every u o n T* is also in T

4. T' is t h e full subtree of T above u if it consists of every s t r i n g o n T extending u. T h e concept of subtree is for trees what t h e concept of extension is for strings. T h e finite extension method consists of building a decreasing sequence (T,},c, of trees, where TOis t h e identity tree (consisting of all strings, also called t h e full binary tree), a n d Tn+l is a full subtree of T,. T h e method of trees is more general because it allows Tn+l to be a n y subtree of T,. T h e simplest application of t h e tree method will be with recursive trees, i.e. trees t h a t a r e recursive as (total) functions from strings t o strings ( a p p r e priately coded as numbers). We a r e really interested in t h e range of a tree, b u t t h e terminology is consistent: if a tree is recursive as a function, t h e n so is its range as a set (to know if u E T we generate t h e t r e e u p to a level in which all branches have length at least equal to t h e length of 0 , a n d see if u is on T at t h a t level). In later applications we will also deal with partial recursive trees, which need not be total: in this case t h e range will be r.e. T h e method of trees used above can be recast in t h e s a m e categorical framework of Section 3, by using a different topology t h a n t h e one generated by basic open sets determined by finite strings. More precisely, consider a n y set C of recursive trees, with t h e following properties: 0

t h e identity tree is in C

0

if T E C a n d

0

E T , t h e n t h e full subtree of T above 0 is in C.

T h e n C is t h e base of a topology 011 P(w) finer t h a n t h e original o n e dealt with in Section 3. W e can then say t h a t :

V.5 The n e e Method 1. A

495

C is C-dense if for any T E C there is S C T such t h a t S E A

2. A C P ( u ) is C-comeager if it contains t h e intersection of countably many C-dense sets.

A good deal of t h e theory of Baire Category can be developed in this generalized framework (Yates [1976]), in particular t h e Baire Category Theorem holds. Basic examples of C a r e given by t h e set of full trees (in which case we obtain t h e same notions as in Section 3), t h e set of recursive coinfinite conditions, a n d t h e set of recursive trees.

Hyperimmune-free degrees Dealing with t h e next notion will provide a useful warm u p with t h e tree method, before t h e real applications come. Moreover, t h e notion will be useful in extending results from T-degrees t o other types of degrees (see VI.6.18).

Definition V.5.2 (Martin and Miller [1968]) A degree a > 0 is hyperimmune-free i f for every A E a and f ST A , f is majorized by a recursive function, i.e. there is a recursive function g such that ( V x ) ( f ( x ) 2 g ( 5 ) ) . T h e name ‘hyperimmune-free’ is justified below. T h e definition has been given in a form which makes it easier to deal with it in applications. Exercises V.5.3 Hyperimmune degrees. (Martin and Miller [ 19681) A degree is hyperimmune if it contains a hyperimmune set, see 111.3.7. a) A degree is hyperimmune-free if and only if it is not hyperimmune. (Hint: one direction follows from 111.3.8. The other from the fact that if {a,,},,~~ enumerates A in increasing order, and f ST A is not majorized by any recursive function, then the set B defined as A @ : n E w } is in the same degree as A, because f ST A, and is hyperimmune. Otherwise, there would be a recursive function majorizing the elements of B in increasing order, and thus a recursive function majorizing f too, because f ( n ) 5 a,(,,).) b) The hyperimmune degrees are closed upwards, and the hyperimmune-free degrees are closed downward. (Hint: the second assertion holds by definition, the first follows from it by part a).) c) The set of hyperimmune-free degrees is meager. (Hint: we have to build a hyperimmune set by the finite extension method V.3.13,which is immediate by definition: given any string (T and an infinite disjoint strong array, there is a string extending (T such that all sets extending it intersect one element of the strong array.) d) Every nonzero degree comparable with 0’ is hyperimmune. (Hint: since 0‘ contains a hyperimmune degree, by part b) so do all degrees above it. To show that every nonzero degree below it does too, modify 111.3.13by using the limit lemma IV.1.17. Precisely, given A K nonrecursive, let A be the limit of g recursive. If f(z) is the smallest stage where g gives the right value of A ( y ) ,for all y 5 z, then f is increasing,

V. Turing Degrees

496

and hence its range B is recursive in A . By part b), it is enough to show that B is hyperimmune. If it were not then A would be recursive, because to know the real value of A ( s ) we just have to compute g at z, at the stage given by the z-th element of B.)

Since t h e set of hyperimmune-free degrees is meager, we cannot expect to build one member of it by t h e finite extension method V.3.13. B y using trees we can instead easily obtain t h e result. For later use we isolate t h e steps needed in t h e proof.

Proposition V.5.4 Diagonalization Lemma. Given e and a recursive tree T , there is a recursive tree Q T such that, for every A on Q, A $ { e } . Proof. Since T(0) a n d T(l) a r e incompatible, at least o n e of t h e m must disagree with { e } on some z. If T(z) is such, let Q be t h e full s u b t r e e of T above it. 0 Proposition V.5.5 Totality Lemma (Martin and Miller [1968]) Given e and a recursive tree T , there is a recursive tree Q 5 T such that one of the following holds: 1. f o r every A on Q, { e } A is not total

2. for every A on Q, { e } A is total and (V'n)(V(J)(l(Jl = n =+ { e } Q ( D ) ( ' n )l),

where

I(J[

is the length of u.

Proof. See if (30 E T)(~x)(VT 2 O ) ( T E T =+ { e } T ( s ) r ) .

If so, choose such a (T: for any string T extending it, { e } T is undefined o n a fixed element. It is then enough to let Q be t h e full subtree of T a b o v e (J. T h e n case 1 holds. Otherwise, for any string (T and any x there is a n extension T of (J t h a t makes {e}. defined on z. We can t h e n build a t r e e b y successive levels, a n d make { e } A converge on more a n d more elements. Precisely, let Q be defined as follows. First we s t a r t with

Q(0) = least T E T such t h a t { e } T ( 0 1. ) Inductively, given Q ( u )on T we know t h a t there is a n extension of it o n T , say T ( T )for some T , such t h a t { e } T ( T ) ( l o 1. l ) T h e n let

Q((J* 2) = T(T* z), for Z = 0,1, so t h a t they a r e two incomparable extensions of & ( a ) . 0

V.5 The Tree Method

497

Proposition V.5.6 (Martin and Miller [1968])There are hyperimmunefree degrees. Proof. T h e requirements on A are: h

e

: he+l

A${e} {e}" total =+- for some recursive g, ( V n ) ( {e }"( n )5 g ( n ) ) .

Define a sequence of recursive trees T, such t h a t Tn+l C_ T,,, a n d all branches of Tn+lsatisfy R,,. Precisely:

TO

= identity tree Tze+l = t h e Q of t h e Diagonalization Lemma, for T = Tze T z e + ~= t h e Q of t h e Totality Lemma, for T = T2e+l. Let now A E nnEwT,:i t exists because Tn(8) C Tn+1(8),since T, C_ Tn+1, a n d thus we just have to consider A = UntwTn(8). A is not recursive because it is o n T2e+l,a n d hence it is different from { e } . Suppose now {e}" is total. Since A is on T2e+2, it must be t h a t t h e second case holds in t h e Totality Lemma, hence

since { e } A ( n )is already defined at t h e n-th level of t h e tree, for each possible branch A . T h u s {e}" is majorized by a recursive function. 0 Exercises V.5.7 (Martin and Miller [ 19681) a) There is a hyperimmune-free degree below 0". (Hint: in the construction above, only twequantifier questions were asked, and thus the degree is less than or equal to 0". To show it is strictly below it, note that 0" is hyperimmune, because comparable with O'.) 1)) There is a n infinite ascending sequence of hypenmmune-free degrees. (Hint: by relativization, there is a hyperimmune-free degree above any hyperimmune-free degree.) c ) There are 2O ' hyperimmune-free degrees. (Hint: build a tree of sets, all of whose branches are of hyperimmune-free degree. This can be achieved by building a tree of trees, i.e. a function T from pairs of strings to strings, such that for each cr the function Tb(r)= T ( a ,T ) is a tree. The desired tree will be given by the function T,(Q)).Start by T0 being the identity tree. At any stage, instead of just building one subtree of the given tree, build two, above two incomparable strings. Precisely, given T,, apply the Diagonalization or the Totality Lemma, depending on whether 101 is 2e or 2r + 1, to the full subtree of T, above T,(O) to get 7b,o, and above T,(1) to get %*I.)

V . Turing Degrees

498

Jockusch [1969b] shows t h a t a hyperimniune degree contains a hi-immune set (while t h e converse does not necessarily hold), a n d strengthens V.5.6 by producing a degree without hi-immune sets. See Jockusch [1972] a n d Simpson [1977] for more on this topic. T h e results of this subsection show t h a t a characterization of t h e degrees containing hyperimmune sets cannot be simple. W i t h regard to o t h e r immunity properties introduced in Chapter 11, we quote t h e following results:

1. a immune u a > 0 (Dekker a n d Myhill [1958],see 11.6.13)

2 0” 3 a cohesive + a hyperhyperimmune (Jockusch [ 1969a1, [ 1973a1)

2. a‘

+ a‘ > 0‘

3. If a 5 O’, then a hyperhyperimmune u a cohesive e a’ = 0’’ (Cooper [ 19721).

Minimal degrees We t u r n now to a most important application of t h e tree method. We want to build a minimal degree, i.e. a nonrecursive set A such t h a t

C S T A =+ C recursive or C ET A . T h e proof we give is a simplification, d u e to Shoenfield [1966], of t h e original construction of Spector, which is given in Section 6 . When we constructed a minimal pair, we used t h e e-splitting method: if there was n o e-splitting then we got a recursive set, while in t h e opposite case we fixed one side, a n d diagonalized against it on t h e other, t h u s taking full advantage of t h e fact, of being able to work with two sets. Here we will have to get by with just one set, a n d this will complicate o u r life a bit, b u t t h e ideas will always be t h e same. T h e necded change is to use e-splittings in a global way.

Definition V.5.8 T is an e-splitting tree if, for everya, T(a*O)and T(cr*l) e-split. T h e interest of t h e notion comes from t h e following lemma.

Proposition V.5.9 (Spector [1956]) Given e, a recursive tree T , and A on T , if {e}” is total then: 1. if there is no e-splitting on T , {e}“ is recursive

2. if T is e-splitting, A I T {e}”.

V.5 The Tree Method

499

Proof. Since {e}A is total, given x we know t hat {e}A(x) converges, a n d A such t hat {e}'(x) converges, and gives t h e right thus there must be u value. Since A is on T , by monotonicity we may suppose u E T . If there is no e-splitting o n T , t o compute {e}A(x) is then enough to look for any string T on T such t h a t {e}T(x) converges, since its value must be equal to {e}'(x) (otherwise u an d T would e-split on x), and hence t o {e}A(x). Suppose now t h a t T is instead e-splitting. We show how t o generate increasingly long segments of A recursively in {e}A. Given T ( a ) A , since A is on T either T ( a * 0) or T ( O* 1) will be included i n A , and we have to decide which one. Being T e-splitting, there is a n x such t hat

c

c

with both sides converging. Only one of them can agree with {e}A(z), and this determines which of t h e two strings is contained in A . Precisely, T ( u * i) A if {e}T(~*i)(x) 21

c

Proposition V.5.10 Minimality Lemma (Spector [1956])Given e and a recursive tree T , there is a recursive tree Q C_ T such that one of the following holds: 1. for every A on Q, {e}A total d

2. f o r every A on Q, {e}A total

recursive

+A

Proof. T h e result follows from t h e previous lemma, if we just build Q with either no e-splitting on it, or as a n e-splitting tree. If there is a string o n T with n o e-splitting above it, take Q as t h e full subtree above it: then Q has no e-splitting. If any string on T has two e-splitting extensions (i.e. two strings on T which extend it a n d e-split), then we can build a n e-splitting subtree Q of T , by induction: given Q(a), let Q(a.0) arid Q(a*l) be two e-splitting extensions of it. 0 Note th at th e last lemma is a modification of the minimal pair construction.

If there is a string on T with no e-splitting extensions, we just take t h e full subt,ree above it, and have {e}A recursive for any A on it, as in t h e minimal pair construction. If any string on T has e-splitting extensions, take two of them, say 00 an d 01. Then, for some 2 , {e}uo(x) $ {e}ul(x), with bot h sides converging. Certainly {e}A(x), whatever A may be, disagrees with one of them, say {e}ux. Then A does not agree with ai. If 00 and u1 were t h e only possible beginnings of A , then A would have to agree with t h e other string. And we can force this to happen, by just discharging every other possibility, i.e. by making a n d u1 t h e first level of our subtree. And we can continue this, thereby building a n e-splitting tree such t hat any set A on it is then recursive in {e}A. a0

V. Turing Degrees

500

Theorem V.5.11 (Spector [1956])There exists a minimal degree.

Proof. We only have to satisfy t h e requirements

R2e R2e+l

: :

A $ {e} C N {e}A

* C recursive or A IT C.

We build a sequence of recursive trees, as follows:

T O = identity tree Tze+l = t h e Q of t h e Diagonalization Lemma, for T = T2e T2e+2 = t h e Q of t h e Minimality Lemma, for T = T2e+l. Then A =

UnEw T,(8) satisfies t h e requirements.

0

Having obtained minimal degrees by t h e t r e e m e t h o d , a n d knowing t h a t we cannot obtain them by t h e finite extension method V.3.13, it is natural to investigate t h e situation more thoroughly. More precisely, we would like t o know how much of t h e machinery j u s t introduced is really needed, in particular which methods a r e sufficient, a n d which a r e necessary. To t h e first question we will answer in V.6.9, where it will be shown t h a t recursive coinfinite extensions are suficienl, giving a sort of best possible answer. T h e second question is less precise, b u t we can argue t h a t total e-splitting trees are not necessary, because t h e y can be combined with totality requirements a n d automatically produce hyperimmune-free minimal degrees (see t h e exercises below). Now, hyperimmune-free degrees cannot be comparable with 0', b u t in XI.4.5 we will show t h a t there a r e minimal degrees below 0', which must t h e n be built by partial trees. In X1.4.3 we will also show t h a t , for t h e construction of minimal degrees below 0', partial e-splitting trees a r e not only sufficient, b u t also necessary (Chong [19791). Exercises V.6.12 a) There exists a minimal degree below0". (Spector [1956]) (Hint: in the proof above, only two-quantifier questions were asked, and thus the degree is less than or equal to 0".To show it is strictly below it, note that 0" is obviously not minimal .) b) There are 2N0 minimal degrea. (Lacombe) (Hint: build a tree of minimal degrees, as in V.5.7.c.) c) The diagonalization steps in the construction of minimal degrees are not needed. (Posner and Epstein [1978]) (Hint: similar to V.2.23, this time using trees T, which are either e-splitting or without e-splittings, and A on all of them. Another way is given by part b), because we can build a tree of sets satisfying all the minimality conditions, and only countably many of these sets can be recursive.) Exercises V.6.13 Hyperimmunefiee degrees. a ) There is a minimal, hyperimmune-free degree. (Martin and Miller [1968]) (Hint: combine the proofs of V.5.11 and V.5.6.)

V.5 The Tree Method

501

b) There is a hyperimmune-free d e g m that is not minimal. (Martin and Miller [1968]) (Hint: by V.5.7.b.) That there is a minimal degree which is not hyperimmune-free follows from the existence of minimal degrees below O‘, see XI.4.5. Exercises V.5.14 J u m p of minimal degrees. a) Not every minimal degree realizes the least possible jump. (Sass0 [1974]) (Hint: build a set A of minimal degree such that A’ $T A @ K. The requirements are treated by the following lemma: given e and a recursive tree T, there is a recursive tree Q T such that, for any A on Q, A’ $ {e)”@‘. The idea is that the question: ‘does A always go left at even levels of T?’ can be phrased as a question about A’, since it involves only one quantifier over A. Let a be such that (a}O(z)N 0 e+ CJ branches right at some even level of T.

By definition of jump, a E A’ e+ {a}”(.) 1. See if, for some T E T, { e } T @ K ( a )21 0. If so, to diagonalize we want a E A’, i.e. for every A we require {a}”(a) 1, and hence A must branch right at some even level of T. Take Q as the full subtree of T above T * 11, since T * 11 branches right twice, and one of T * 1 and T * 11 is at an even level. Otherwise, let Q be the subtree of T that branches left at every even level. Note that we use only even levels, because we still need to get a tree, and this is taken care of at odd levels.) b) Every degree above 0” is the double j u m p of a minimal degree. (Cooper [1973]) (Hint: note that if we also use the Totality Lemma in the construction of a set of minimal degree we can decide, recursively in the construction and hence in a”, whether {e}” is total or not, depending on which case applies in the lemma. Any question about the second jump of A or, equivalently, with two quantifiers over A, can be rephrased as a question about the totality of a given function recursive in A, and thus A” ST A @ 0”. We can now build a tree of T ST 0”, all of whose branches A have this property. If 0”
uuEc

A
and hence A @ 0”

ST C. Conversely,

c ST A @ T
A @ 0“,

and thus C GT A @ 0‘‘ % A’’.) Jockusch and Posner [1978] show that, for any minimal degree 0 , 0’ = (a U 0’)’, see XI.2.15. Cooper [1973] proves that actually every degree above 0‘ is the j u m p of a minimal degree, but this is much more difficult to prove, see Epstein 119751 or Posncr [1981] for a proof by recursive approximations, and Lerman [1983] for one by approximations recursive in 0’. Exercises V.5.15 Autoreducible sets. A set A is autoreducible if, for each z, the question ‘is x in A?’ can be answered recursively in A, without ever asking the

V. Turing Degrees

502

oracle about z. Thus every single element encodes redundant information, retrievable from the rest of the set. A nonrecursive degree containing only autoreducible sets is called completely autoreducible. a) Every m-degree contains autoreducible sets. (‘Ikakhtenbrot [ 19701) (Hint: consider A @ A. Then 22 and 22 1 give the same information.) b) An introreducible set (II.6.7) is autoreducible. (Jockusch and Paterson [ 19761) (Hint: let B be nonautoreducible. We build, by the finite extension method V.3.13, an infinite subset A of B such that B $T-A. To get B {e}” at stage n, given un look for an extension u of it such that a-’(1) & B, and there is z such that every T 2 u such that ~ - l ( l ) B satisfies {e}.(z) $ B ( z ) . Such a string must exist, since B is not autoreducible. Thus let un+l = on.) c) The set of completely autoreducible degrees is meager. (Trakhtenbrot [1970]) (Hint: build, by the finite extension method, a set A such that, for every e, there is z such that A ( z )$ {e}”-{”’(z).) d) There exists a complete2y autoreducible degree. (Jockusch and Paterson [ 19761) (Hint: build a set as in the minimal degree construction, with double e-splittings taking the place of e-splittings, i.e. using trees T such that, for any u, there are two distinct elements on which T(u * 0) and T ( u* 1) e-split. If T is doubly e-splitting, A is on T, and {e}” is a characteristic function, then {e}” is autoreducible: to compute {e}”(z) with oracle {e}” without using this value, first build large enough segments of A, with oracle {e}” but avoiding the particular value {e}”(z), which is possible with oracle A . ) thanks to the double e-splittings. Then compute Obviously, the completely autoreducible degree just built is minimal. The existence of minimal, not completely autoreducible degrees and of completely autoreducible, not minimal degrees will be proved in V.6.10.d and V.6.16.e. Jockusch and Paterson 119761 show that nonzero r.e. degrees and degrees above 0‘ are not completely autoreducible.

+

+

Minimal upper bounds

*

Exactly as we generalized t h e notion of minimal pair to t h a t of exact pair, we can extend t h e notion of minimal degree to that of minimal upper bound.

Definition V.5.16 A degree a is a minimal upper bound for a set of degrees C if

1. it is a strict upper bound to C, i.e. (Vc E C ) ( c < a) 2. there is no strict upper bound to

C below it, i.e.

( t / b ) [ b5 a A (VCE C ) ( c < b )

3

b = a].

A minimal cover for a degree b is a minimal upper bound f o r { b}, i.e. a > b such that there is no degree strictly between a a n d b.

degree a

V.5 The n e e Method

503

A strong minimal cover for b is a degree a strictly below it is bounded b y b, i.e. (Vc)(c

5 a+

c

5

>

b such that anything

b V c = a)

Both notions of minimal cover are plausible generalizations of t h e notion of minimal degree above a given set. Relativizations of t h e results about minimal degrees produce results about minimal covers, but not about strong minimal covers. For example, V.5.11 relativized shows t hat every degree has a minimal cover. O n t h e other hand, not every degree has a strong minimal cover (Shoenfield [1959]), e.g. 0' does not (by V.2.26 any degree above Of is t h e join of two strictly smaller degrees, and thus it cannot be a strong minimal cover of it, otherwise t h e two degrees would be below O f , and so would be their join). Jockusch [1981] shows t hat the set of degrees without strong minimal covers is commger, an d thus t h e class of degrees with a strong minimal cover is meager. It is not known whether t h e minimal degrees are included in it, i.e. if every minimal degree has a strong minimal cover.

Exercises V.5.17 There is a cone of minimal covers. (Jockusch I19731) (Hint: every rninimal degree has a minimal cover, and thus the set of degrees which are not minimal covers cannot contain a cone. By V.1.16, if A D holds then the set of degrees which are minimal covers must contain a cone. But this set is arithmetical, and thus only the Axiom of Determinacy for arithmetical sets is needed, which is provable in ZFC' by Martin 119751.) A direct proof of the result, not using Determinacy, will be given in XII.3.13. Spector's Theorem showed t hat any countable ideal has a n exact pair. T h e proof was a n extension of t he construction of minimal pairs, plus a coding method t h a t allowed us t o push t h e set constructed above given degrees. We now prove a similar result for minimal upper bounds, by extending t h e construction of minimal degrees. We also need a coding method, and this is provided by t h e next notion, iniplicit in t he proof of Spector's Theorem.

Definition V.5.18 (Sacks [1971])A recursively pointed tree T is a tree A whenever A is on T . which is recursive in all of its branches, i.e. T 51Exercises V.5.19 a) If T is recursively pointed, then T has branches of every degree above the degree of T . (Sacks [1971]) (Hint: given T 51A, consider B = UuCA T(cr).) T , and Q 51T , then Q is also &cursively b) If T is recursively pointed, Q pointed, and Q =T T . (Sacks [1971]) (Hint: Q is pointed because if A E Q T then Q 51-T ST A . If A is the leftmost branch of Q then T 51A by pointedness, and A ST Q by definition, so T ST Q.)

V. Turing Degrees

504

Proposition V.5.20 (Sacks [1971]) If T is recursively pointed and T ST A , then there is some recursively pointed tree Q C T such that Q ET A . Proof. We have t o code A in Q, and at every even level 2n we thin t h e tree down by just taking the right or left branch, according t o whether n is in A or not. Precisely, we define Q by induction. Given &(a)on T , let r be a string such t h a t Q(o)= T ( r ) ,which exists because &(a)is on T , by induction. Let Q(o * i) = T ( T* A(I.1)

* 2).

Then Q ST T @ A , so Q ST A (because T ST A ) . From any path B of Q and T itself we can recover A , and so A S T B (since T 51-B by pointedness). To have A 5~ Q it is enough t o choose any B ST Q, e.g. t h e leftmost branch. Finally, Q is pointed: given B E Q we can recover Q itself from B a n d T , by t h e uniformity of the construction, and again T ST B , so Q ST B . 0

Proposition V.5.21 (Sacks [1963]) Every countable set of degrees has a minimal upper bound. Proof. Let A = { m,al, . . . } be a countable set of degrees. We may suppose t h a t A is a chain, otherwise we can just consider t h e chain defined as bo

= a0

bn+l

= bn

U an+lr

whose minimal upper bounds are exactly those of A. We can also assume t h a t t h e chain has no greatest element, otherwise t h e result is obtained by taking any minimal cover of it. This assumption is made only t o avoid direct diagonalization against all t h e degrees in A. , T, is a Let then A , E an. We will build a set A E n n E w T nwhere recursively pointed tree of degree an. This automatically implies t h a t A is a n upper bound for A. To get minimality, we use the following extension of t h e Minimality Lemma. Given e and a tree T , there is a tree Q T such that one o f the following holds: 1. for every

A on Q, {e}" total

2. for every A on Q,

total

3

{ e } A ST Q

+ A ST {e}A @ Q .

This is proved by the proof of V.5.10,simply because when t h e relevant trees are not recursive then we have to take them into account in our computations. T h e construction is as follows. Let TObe a recursively pointed tree of t h e same degree as A0 (which can be obtained by starting from t h e identity tree, and applying V.5.20). Given T, recursively pointed of t h e same degree as A,, first get T C_ T, recursively pointed of t h e same degree as A,+l, by V.5.20,

V.5 The n e e Method

505

since A, 5~ A,+1. Then let T,+, be the Q of the Minimality Lemma stated above, which is still recursively pointed, and of the same degree as A,+1. If A E nnEwTn, suppose { e } A is total. By construction, there are two Cases:

{ e } A IT Te+1 Then { e } A ST Ae+l (since Te+lhas the same degree as A,+l), A is below some element of t h e chain, and it is not an upper bound to it.

A

S T { e } A CB Te+1 If { e } A is itself an upper bound t o the chain then hence A ST

T,+1

ST { e } A , and

Thus no upper bound t o the chain is strictly below A, and A is a minimal upper bound. Exercises V.5.22 a) Every countable set of degrees has 2N0 minimal upper bounds. (Sacks [1963]) (Hint: build a tree of minimal upper bounds.) b) Every countable ascending chain of hyperimmune-free degrees has a hyperimmune-free minimal upper bound (Martin and Miller [ 19681). Note that the result fails for countable sets in place of chains, because the join of hyperirnrnune-free degrees in not necessarily hyperimmune-free, see V.6.10.c.

Konig’s Lemma and II(:classes

*

In this subsection we consider again general trees as in IV.2.14 (i.e. as sets of sequence numbers closed under subsequences), and study their infinite branches. T h e classical result in this field is the following.

Theorem V.5.23 Konig’s Lemma (Konig [1926]) An infinite tree in which every node has only finitely many immediate successors has an infinite branch. Proof. Let T be such a tree. We define an infinite branch by induction. We start with 00. Given on with infinitely many extensions on T , let on+l be an immediate successor of on with infinitely many extensions on T . It exists because on has infinitely many extension on T , but only finitely many immediate successors. Thus at least one of them must have infinitely many extensions on T. T h e rest of this subsection is devoted t o an analysis of constructive versions of this result. We first investigate binary trees which, by definiteness, we define as sets of sequences of 0’s and 1’s.

Definition V.5.24 A class of sets is a ll! class if it is the set of infinite branches of some infinite recursive binary tree.

V. Turing Degrees

506

T h e reason for t h e n a m e is obvious: if T is a recursive tree, t h e n A is a n infinite branch of it if a n d only if ( Y X ) ( ? A ( XE) T ) . More generally, if P is a recursive predicate then t h e class of sets A such t h a t ( t l z ) P ( e ~ ( zis) )a @ class, since it is t h e set of infinite branches of t h e t r e e obtained by closing P under subsequences. class is never empty. As a fundamental Note t h a t , by Konig’s Lemma, a class: example, the sets separating two disjoint r.e. sets A and B f o r m a

C E SA,B e (Vz)[(z E A

-,z

E C ) A (z E B

-,z

$!

C)].

Explicitly, a recursive tree whose branches a r e exactly t h e members of SA,Bis t h e following:

* Seq(z)A z is correct u p to s t a g e ln(z),

z E TA,B

where ‘z is correct u p to stage s’ means t h a t for every i 5 l n ( z ) ,if i E A , then ( z ) i = 1, a n d if i E B, t h e n (z), = 0. In other words, we seal off a branch of T A , Bas soon as we discover t h a t it is incorrect. Note t h a t TA,Bhas a n infinite branch if a n d only if A a n d B a r e disjoint. Moreover, a n infinite branch of TA,Bis t h e characteristic function of a set separating A a n d B . We t h u s imniediately have:

Proposition V.5.25 Failure of the recursive version of Konig’s Lemma (Kleene [1952])There is an infinite recursive binary tree without infinite recursive branches. Proof. If A a n d B a r e recursively inseparable (11.2.5), t h e n TA,Bh a s n o infinite recursive branches. 0 I t is t h e existence of finite branches, i.e. t h e consideration of partial recursive trees in t h e sense of this section, t h a t produces t h e possibility of recursive trees without infinite recursive branches. If we restricted o u r attention to t o t a l recursive trees, t h e n t h e leftmost branch would be recursive. Instead, t h e leftmost infinite branch o f a partial recursive tree need not even be r.e., although it always has r.e. degree (see V.5.26 a n d V.5.33.a). Exercise V.6.26 There is an infinite recursive binary tree which has no infinite r.e. branch. (Jockusch and Soare [1972a]) (Hint: let S be Post’s - simple set (111.2.1l ) , and consider a disjoint strong array { D , ( 2 ) } z ~intersecting w S. Let

C is a norierripty rI: class because it contains its members are immune.)

3, and it has

no r.e. member because

Infinite binary trees without infinite recursive branches must be q u i t e fat:

V.5 The n e e Method

Proposition V.5.27 (Jockusch and Soare [1972]) A cursive members has cardinality 2"o.

q class without re-

Proof. I t is enough to show t h a t every isolated infinite branch of a recursive tree is recursive: then if there a r e no infinite recursive branches every branch splits, a n d t h e number of infinite branches is 2"O. Suppose A is t h e unique branch of T above u E T . To decide whether z E A , consider all strings of T above u a n d of length greater t h a n z. Generate T until all of t h e m except one die out. T h e only surviving one is contained in A , a n d is defined on z. T h u s u ( ~tells ) whether z is in A or not. 0 Exercise V.5.28 A II: class without recursive members is meager. (Jockusch and Soare [1972]) (Hint: a II: class is closed, since membership in 2 is determined by a finite initial segment, and thus by an open set contained in If ( V C T ) ( ~2Aa ) ( A g A) then ( V C T ) ( ~_>T u)(VA 2 7 ) ( A fZ A), and by V.3.6 2 is comeager. Otherwise, ( b ) ( V A 2 a ) ( A E A), and one such A is recursive.)

x.

Since t h e recursive sets do not provide, in general, witnesses for branches of a n y infinite binary tree, we introduce t h e following notion:

q

Definition V.5.29 A class of sets A is a basis for classes i f every rr(: class has a n element in A. A class of degrees is a basis i f so is the class of sets with degrees in it.

rr(: classes. First we show t h a t it will be

We now s t a r t a search of bases for impossible to find a single best answer.

Proposition V.5.30 The intersection of all bases for IIy classes is the class of recursive sets, which is not a basis. I n particular, there is no least basis. Proof. Every recursive set must be in every basis, because given B recursive t h e condition A = B defines a lIy class with B as t h e only member. To show that t h e intersection of all bases is t h e class of recursive sets, it is enough to show t h a t given A nonrecursive there is a basis B not containing A . We build 23 by putting a n element of each @ class in it. Given a class, there a r e two cases: either it has recursive elements, in which case we can p u t one in D,or (by t h e proposition above) it has 2N0 elements, a n d t h u s we can choose one different from A . 0 .

A simple analysis of t h e proof of Konig's Lemma provides t h e first result. Proposition V.5.31 Kreisel's Basis Lemma (Kreisel [1950]) An infinite recursive binary tree has a A! irlfinite branch.

508

V. Turing Degrees

Proof. Let T be a given infinite recursive binary tree. Given un E T , we have t o decide whether t o choose u,+1 as u, * (0) or an * (1). We see if (b'm

> n)(3r E T)(IrI= m

A

r 2 g,II.* (0)).

If yes, we let c,+1 = on * (0). Otherwise, on+l = u, * (1). Let A = UnEwun. Then A E T . Since the quantifier on r is bounded, because there are only finitely many strings of length m, and T is recursive, the question is @. It can thus be 0 answered recursively in K. Then A ST K ,and A E A!. Kreisel's Basis Lemma has been improved by Shoenfield [1960a], who proved t h a t there is always a branch of degree less than 0'. T h e next result is much stronger.

Theorem V.5.32 Low Basis Theorem (Jockusch and Soare [1972]) The degrees a such that a' = 0', called low degrees, f o r m a basis for @ classes.

Proof. We proceed as in V.2.21, on trees. Let T be a given infinite recursive binary tree. We want t o build A on T such t h a t e E A' can be decided recursively in K. Let To = T . Given T,, to decide whether e E A' we consider t h e set Ue = { u E T e : { e } " ( e ) t } . There a r e two cases:

1. We is infinite Then we can let Te+l = U,, and for every A on Te+l we will have e $ A'. Note t h a t U, is indeed a tree, being closed under subsequences (since if a string does not decide a computation, neither does any substring of it). 2. U, is finite Then we can let Te+l = T,. For every A on Te+l we will have e E A', since { e } " ( e ) is undefined for at most finitely many strings on Te+l,a n d thus it must converge for any string which is big enough. Since the case distinction is recursive in Ec, if A E neEwTe then A' ST And clearly A E T , because TO= T . 0

K.

Exercises V.5.33 (Jockusch and Soare [1972], [1972a])a) The r.e. degrees form Q b a s i s for IIy classes. (Hint: given an infinite recursive binary tree T , consider its leftmost infinite branch A. If

u E B w u E T A u is on the left of A

V.5 The n e e Method

509

then A ET B. Moreover, B is r.e. because if u E T is in B then we discover it by generating all strings of T of length up to n, for n big enough.) b) The r.e. d e p e s strictly below 0' do not form a b a s i s for II: classes. (Hint: the example given in V.5.26 consists only of effectively immune sets. They, by the proof of 111.2.18, cannot have r.e. degree strictly below d.) c) If b 2 0' then 0 and the degrees with jump b f o r m a b a s i s for rI: classes. (Hint: let T be an infinite recursive binary tree without infinite recursive branches. Since it has 2N0branches, it is possible to build a total subtree Q LT K all of whose branches force the jump, as in V.5.32. If B 5~ K then the branch determined by B, i.e. turning right at level n if n E B and left otherwise, has jump B.)

Proposition V.5.34 (Jockusch and Soare [1972])The hyperimmune-free degrees f o r m a basis for @ classes. Proof. We proceed as in V.5.6. Let T be a given infinite recursive binary tree. We want to build A on T such t h a t { e } A ,whenever is total, is majorized by a recursive function. Let To = T . Given Te,consider

U,Z = {o E T, : { e } ' ( x ) t } . There a r e two cases: 1. U z is infinite for some x Then we can let Te+l = U z for one such x , and for every A on Te+l we will not total. Note t h a t U z is indeed a tree, being closed under have subsequences (since if a string does not decide a computation, neither does a n y substring of it).

2. U z is finite for every x Then we can let Te+l = T,. For every A on T,+1 it is enough to find recursively a level n such t h a t , for all strings o of length n, {e}'(x) 1. Then

T h e results just proved for binary recursive trees can easily be seen to hold also for recursively bounded recursive trees, i.e. recursive trees such t h a t t h e size (i.e., t h e greatest component) of nodes on T of length n is bounded by f(n), for some recursive f . E.g., a binary tree is recursively bounded by 2. T h e results for recursively bounded recursive trees can be extended to finitely branching recursive trees, i.e. recursive trees such t h a t t h e number of nodes o n T of length n is finite. T h e reason is t h a t such trees a r e bounded by a function recursive in 0': to know a bound on t h e size of strings of level

V. Turing Degrees

510

n on T we can ask, for a n y m, whether t h e size of every string of level n on T is bounded by m. T h i s can be answered recursively in Ic, a n d we can inductively determine t h e smallest m for which such a sentence holds (which exists, because T is finitely branching). T h u s the results of this subsection hold for finitely branching recursive trees, when relatiuized to O f . E.g., V.5.32 shows t h a t a finitely branching infinite recursive tree has a n infinite branch with j u m p recursive in 0”. If t h e condition of being finitely branching is dropped, t h e n t h e situation changes radically (e.g., V.5.23 fails). T h e theory of infinite recursive trees with infinite branches can be developed in a way largely parallel to t h e o n e for binary recursive trees, when t h e notions of recursive set a n d Turing degree a r e replaced by those of hyperarithmetical set a n d hyperdegree. See Volume I11 for a treatment.

Complete extensions of Peano Arithmetic

A

Since t h e class of sets separating two disjoint r.e. sets is a nonempty so are:

class,

1. the set of consistent extensions of a given consistent theory 2. the set of complete extensions of a given consistent theory. Jockusch a n d Soare [1972] a n d Hanf [1975] provide converses to these examples in t h e spirit of 111.10.3, showing t h a t the class of degrees of members of a given IIy class coincides with the class of degrees of complete extensions of some (finitely axiomatizable) first-order theory. In t h e following we will restrict o u r attention to complete extensions of Peano Arithmetic, because they provide a possible description of t h e arithmetical world in accord with t h e partial b u t fundamental picture given by PA. In particular, we will t h u s be able to t a k e advantage of t h e power of P A in proofs, including t h e Induction Principle. T h e basic link with t h e subject of t h e last subsection is another basis result.

Theorem V.5.35 Scott Basis Theorem (Scott [1962]) If F is a consistent extension of PA, the sets recursive i n 3 form a basis for classes. Proof. Let T be an infinite recursive tree. To be able to choose a n infinite branch recursively in 3, we proceed inductively. Let 00 = 8. Given u, of length n, consider all its extensions of length n 1 o n T . Given a n y two of them, say TO and 7-1, we have to decide which one looks better as a n initial segment of a n infinite branch of T . T h e statement

+

&

(3m)(T0has

a n extension of length in on T , b u t

7-1

does not)

V.5 The Tree Method

511

is Ey (note t h a t the quantifiers on strings are restricted t o strings of a given length). If it is true then, for some m, so is TO

has an extension of length m on T , but

TI

does not.

But this is a true recursive sentence, which is then provable in in 3. Then so is $0. Similarly for $1

($

(3m)(-r1 has an extension of length m on T , but

PA, a n d hence

TO does

not).

Now $0 and cannot both be provable, because 3 is consistent (otherwise there would be a number m such that one of TO and TI has at the same time a n extension of length m, and no extension of length m). Thus T, looks better if $i is provable in .F, and otherwise TO and TI look the same. We can now compare all pairs of strings on T of length n 1 extending cn. Since there is an infinite branch on T , all strings which do not extend to an infinite branch are eliminated (when compared t o a string that does). All the remaining ones do extend to an infinite branch, and we can choose any of them

+

=cn+1.

0

T h e next result provides a converse to Scott Basis Theorem.

Theorem V.5.36 Characterization of the degrees of complete extensions of P.4 (Solovay) T h e following conditions are equivalent: 1.

a i s the degree of a consistent extension of P A

2. a is the degree of a complete extension

3. D ( < a ) is a basis for

ofPA

classes.

Proof. Clearly, 2 implies 1. Scott Basis Theorem shows that 1 implies 3. If 3 holds then there is a complete extension of P A recursive in a (because t h e set of complete extensions of P A is a class): that 3 implies 2 thus follows from t h e upward closure of the degrees of complete extensions of PA, which we now prove. Let 3 be a complete extension of P A recursive in a set C . It is enough t o build a tree of complete extensions of P A , recursively in F. Then t h e branch determined by C has the same degree of C . Let { ( P ~be}an~ enumeration ~ ~ of the sentences in t h e language of arithmetic. We start with F0 = PA. Given .Fv,we proceed i n two steps:

1. completeness Given vn,with n = ((71, we decide how t o consistently add t o 30one of

V. Turing Degrees

512 pn a n d -cpn.

As in V.5.35, we let

71,o

e

71,1

(3m)(nzcodes a proof of cpn in Fu, b u t no smaller m’ codes a proof of -cp,) (3m)(mcodes a proof of -pn in FU, b u t no smaller m’ codes a proof of cp,).

Since Fo is a finite extension of P A , it is a formal system, a n d t h u s $0 a n d $1 a r e Ey, a n d hence provable in F if true. Moreover, by consistency of F,they a r e not b o t h provable. Recursively in F we c a n see which of them, if any, is provable.

If $1 is provable in 3 t h e n we know t h a t $0 is not provable in hence t h a t -cpn is consistent with it. We can t h u s let

FA

Fu,a n d

= Fuu{-cp,}.

Otherwise, we can let

2. branching Since t h e sets of provable a n d refutable formulas of P A a r e a n effectively inseparable pair of r.e. sets (see III.10.11), t h e r e is a recursive function t h a t produces, given a disjoint pair ( A , B ) of r.e. sets extending t h e m , an element not in A U B . T h i s applies in particular to t h e sets of provable a n d refutable formulas of FA which is, by construction, a finite extension of PA. In other words, there is a n effective way to find a sentence 71, which is neither provable nor refutable from Fi.T h e n we c a n let FU*(O) FV*(l)

If

w-J{71,)

= = %J{-@}.

F ST C then Uucc3uis a complete extension of P A of t h e s a m e degree as

c.

~

0

T h e theorem shows t h a t t h e degrees of consistent or complete extensions of P A describe particularly simple bases for classes. A complete degreetheoretical characterization of them is not known, b u t from t h e basis results we already have we can easily derive a number of consequences, b o t h positive a n d negative. T h e latter show t h a t a complete extension of P A cannot be too simple, in various ways. T h e y generalize 11.2.17, which stated only t h a t a consistent extension of P A cannot be recursive.

V.5 The D e e Method

513

Proposition V.5.37 (Scott and Tennenbaum [1960], Jockusch and Soare [1972], [1972a]) A consistent extension of P A can have neither incomplete r.e. degree, nor minimal degree. B u t there are complete extensions of P A of low degree, as well as of degree 0'.

Proof. If t h e r e were a consistent extension of P A of r.e. incomplete degree, by Scott Basis Theorem t h e r.e. incomplete degrees would be a basis, contradicting V.5.33.b. Similarly, if there were a consistent extension of P A of minimal degree t h e n t h e minimal degrees together with t h e recursive sets would be a basis, contradicting t h e fact t h a t t h e members of t h e following nonempty @ class a r e neither recursive nor of minimal degree:

A@B E C

tj

( V e ) ( B ( e )$ { e } ( e ) )A ( V e ) ( A ( 2 e )74 (el(2e)) A ( V e ) ( A ( 2 e 1) $ { e } B ( 2 e

+

+ 1)).

T h e first t w o conditions imply t h a t both A arid B a r e not recursive, since they diagonalize against every recursive function (on fixed arguments). T h e last condition implies t h a t A $T B . T h u s A @ B cannot be recursive or of minimal degree. A n d C is fl because, e.g.,

B(e)

+ {eHe)

.3

M e )t

"

f {eHe>,

a n d to diverge on a given argument is a @ condition. T h e existence results follow from V.5.32 a n d V.5.33.a: t h e latter implies t h e existence of a complete extension of P A of r.e. degree, which must be 0' because all t h e r.e. incomplete degrees a r e ruled out by t h e first p a r t of t h e proof. 0 Jockusch a n d Soare [1972a] show t h a t the complete extensions of a n essentially undecidable formal system can have both incomplete r.e. degree and minimal degree. T h u s t h e role of P A (through t h e fact, used in t h e proof of V.5.36, t h a t t h e sets of t h e o r e m a n d of refutable formulas a r e not only recursively, b u t also effectively inseparable) is crucial. An i n t e r g t i n g way to analyze t h e behavior of consistent extensions of P A is to see which sets a r e weakly representable in them. Clearly, in t h e s t a n d a r d model exactly t h e arithmetical sets are. By 11.2.16, in every consistent formal system extending R exactly t h e r.e. sets a r e weakly representable. B u t for consistent extensions F of R which a r e not fornial systems t h e following might happen. If y weakly represents a set A in R, cp(Z) might become provable in F for some z E a n d t h u s cp represents only a superset of A in F,possibly a cofinite (and hence recursive) one. Moreover, since F is not a formal system, t h e set represented by y is not necessarily r.e.

x,

V. Turing Degrees

514

Proposition V.5.38 For every consistent extension 3 of PA, the class of sets weakly representable in 3 properly includes the recursive sets. Proof. Since, by 11.2.16, t h e recursive sets are actually strongly representable

in R,they remain strongly representable in every extension of it, a n d hence weakly representable in every consistent extension. Given a recursive enumeration {(P,},~~of t h e sentences in t h e language of arithmetic let, as in V.5.35 and V.5.36,

$o(n)

*

$l(n)

(3m)(mcodes a proof of cpn in PA, but no smaller m' codes a proof of -cp,) (3m)(mcodes a proof of 7 c p n in PA, but no smaller m' codes a proof of cp,)

By consistency of 3,$0 weakly represents a set A . If p n is a theorem of P A then $o(n) is a true Ey formula, which is then provable in 3. If -pn is a theorem of Pd then $ ~ ~ (isnprovable ) in F and hence, by consistency of F , $o(n) is not provable. Thus A separates theorems and refutable formulas of PA, an d thus it cannot be a recursive set (by 111.10.11). 0 We now provide some examples of complete extensions of PA, pathological from th e point of view of weakly representable sets.

Proposition V.5.39 (Jockusch and Soare [1972]) 1. There is a complete extension of P A in which no hypersimple set is weakly represent ab 1e. 2. There is a complete extension of

P A in which only A; sets are weakly

representable. 3. For any n 2 2 there is a complete extension of P A an which only A:+l sets are weakly representable, but no nonrecursive Cz or rr", set is.

4.

There is a complete extension of P A in which no arithmetical nonrecursive set is weakly representable.

Proof. By V.5.34 there is a complete extension 3 of P A of hyperimmune free degree. If A is weakly representable in 3,it is recursive in it. Thus it has hyperimmune-free degree, because t h e hyperimmune-free degrees are downward

closed (V.5.3). Thus 2 cannot be hyperimmune, and A is not hypersimple. Similarly, 2 follows from the Low Basis Theorem, which provides a complete extension of P A recursive in K , in which all weakly representable sets must then be A;.

V.5 The D e e Method

515

For t h e remaining results, we first show t h a t given countably many nonrecursive sets A , and an infinite binary recursive tree, there is A on it in which no A , is recursive. T h i s c a n be obtained inductively as usual, once we know how to get, given e , n, a n d a n infinite recursive tree T , a n infinite recursive subtree T’ of T such t h a t A , $ { e } A , for every A on T . T h e r e a r e t h r e e cases: 1. for some z, there are infinitely many u E T such that {e}‘(z) T h e n t h e set (0E

T:

1

(4T)

is a n infinite subtree of T (being closed under subsequences), a n d we can let it be T’. If A E T’ then { e } A is not total, a n d hence is different from

A,. 2. for some 5 , there is u E 7’ such that {e}.(x) is defined and different from A , ( z ) , and u has infinitely many extensions on T T h e n we can let T’ be t h e full subtree of T above 0.If A E T’ t h e n is different from A,. 3. otherwise T h e n we let TI = T , a n d show t h a t { e } A is recursive for any A on T , a n d hence different from A,. Given z, go t o a level n of T such t h a t {e}‘(z) 1 for every u E T of length 11: this is possible because we a r e not in t h e first case. If two strings give different values on z, then they cannot b o t h have infinitely many extensions on T , otherwise one of t h e m would give a value different from A,, a n d we would be in t h e second case. We c a n t h u s generate enough of t h e tree to discover which of t h e s t r i n g belong to finite branches, until only strings with t h e s a m e value remain. T h i s must be t h e value of { e } A ( z ) ,for any A E T . By letting { A n } n E w be a list of t h e arithmetical nonrecursive sets, we get p a r t 3. For p a r t 2 we only have to let { A n } n E be w a list of E: and to compute t h e complexity of t h e construction. T h e division in cases is based o n two-quantifier questions, a n d thus it is recursive in which accounts for t h e b o u n d n 2 2. 0

Ue,

a’’,

KuEera (19861 (see X.4.8) has shown t h a t part 2 cannot be improved as in p a r t 3, because there is no complete extension of P A in which only A: sets are

weakly representable, but no nonrecursive r.e. set is. For more on I l: classes a n d their applications to complete extensions of P A , see Jockusch a n d Soare [1971], [1972], [1972a], Jockusch [1974], [1989], KuEera

(19851, [1986], [1988], [1989].

V. Turing Degrees

516

V.6

Initial Segments *

Initial segments more complicated t h a n minimal degrees have been used in t h e original proofs of many of t h e global results of t h e next section. T h e y a r e now obsolete in this respect, since much simpler proofs have been obtained. Initial segments a r e still necessary for a complete algebraic characterization of t h e algebraic structure of D , as well as for some advanced p a r t s of Degree Theory (like t h e results quoted on p. 492). T h e techniques involved in t h e proofs a r e however mostly not recursion-theoretical, a n d outside t h e scope of our book. T h e reader is referred to Epstein [1979] a n d Lerman [1983] for detailed treatments. We will introduce only techniques which have other uses as well.

Uniform trees To obtain initial segments we need trees t h a t have more flexibility t h a n t h e simple ones used in t h e previous section. T h e following notion introduces some uniformities in our trees. Definition V.6.1 T is a uniform tree if, f o r every z = 0 , l a n d u,

1. IT(u)l depends only on 2. there is a unique

101

~ i ,depending

only on 1aI7s u c h t h a t T(o* i) = T ( a )* 7,.

A uniform tree is nothing more t h a n a tree in which, at each level, t h e strings immediately following each node a r e independent of t h e node itself, a n d have t h e s a m e length. A useful way of representing t h e situation is t h e following: t h e r e a r e three functions g (strictly increasing), a n d f ~f~, (left a n d right functions) such t h a t f~ a n d f~ t a k e values in (0,l) a n d are incompatible (i.e. they differ o n at least one argument) in every interval [ g ( n ) ,g(n 1)). T h u s t h e levels of t h e tree a r e determined by g, a n d a branch of t h e tree is simply a p a t h which at every node follows one of fL a n d fR, u p to t h e next node.

+

Definition V.6.2 (Spector [1956]) A n admissible triple is a triple g7 I,,,

f~ o f f ~ n c t i o n from s w into (0, l}, such that:

V.6 Initial Segments *

517

level n

+1

fR

level n

Our present task is t o build minimal degrees by uniform trees. By doing so we will actually reproduce Spector's original proof. What we do is to reprove the lemmas of the last section, this time building uniform trees instead of simple trees.

Proposition V.6.3 Diagonalization Lemma for uniform trees. Given e and a recursive uniform tree T , there is a recursive unaform tree Q C T such that, for every A on Q, A { e } . Proof. T h e same proof of t h e Diagonalization Lemma works, because any full subtree of a uniform tree is still uniform. 0 Proposition V.6.4 Totality Lemma for uniform trees (Martin and Miller [1968]) Given e and a recursive uniform tree T , there is a recursive uniform tree Q T such that one of the following holds: 1. for every A on Q, { e } A is not total 2. for every A on Q , { e } A is total and

(vn)(vO)(lal = n where

101

3

{e}Q(%)

11,

is the length of u

Proof. Seeif

(30 E T)(3z)(V.r2 O ) ( T

ET

{e}T(z)T).

V. Turing Degrees

518

If so, choose such a u: as in t he Totality Lemma, we let Q be t h e full subtree of T above u, which is uniform since T was. Then case 1 holds. Otherwise, we define Q by induction as follows. Q(0) = least Given Q(ai),for 1 5 i

T

E T such t h a t { e } 7 ( 0 )1 .

5 2n and ui string of length n, take:

Q(u1)* 7 1 E T Q(o2)* 71 * 7 2 E T etc.

such t h a t such t h a t

{e}Q("l)*T1 (n11 {e}Q(uz)*T1*Tz (41

Let T = T~ * . . . * 7 2 " : for each i we then have { e } Q ( u t ) * T (1.n It ) is now enough t o take two incomparable extensions po a n d p1 of T with t h e same length a n d such th at &(a,)* /.LO and Q(ui)* pl ar e on T for every i, which is possible because T is uniform, and let

Q(ai * 0 ) = Q(ai)* po

and

Q(u~ * 1) = Q ( 0 i ) * pi.

By definition Q is then uniform, since we extend all strings of a given level in t h e same way. 0

At this point we know how t o build hyperimmune-free degrees by uniform trees. To obtain th e same result for minimal degrees it is enough t o show t h a t we can handle, by uniform trees, the only case of t h e Minimality Lemma in which we a r e not taking full subtrees. Proposition V.6.5 (Spector [1956]) Given e and a recursive u n i f o m tree

T , if

1. every u on T has e-splitting extensions on T 2. (Vo E T)(V'z)(3.r2

D)(T

E T A (e}'('z)

1)

then T has a recursive e-splitting uniform subtree Q. Proof. Define Q inductively by t he following procedure. Suppose &(a,)is given, for any 1 5 i 2" a n d ai string of length n.

V.6 Initial Segments

*

519

By 1 there are TI and 7-2 such t h a t Q(u1)* 7-1 and Q(a1) * T2 are on T and e-split. Reproduce them above Q(u2):since T is uniform, the new strings are still on T . By 1 again, there are 73 and 74 such that Q(a2)* 7 1 * T3 and Q(a2)* T~ * 7 4 are on T and e-split, say on 5 . By 2 there is some ~5 such that {e}Q(u2)*T2*T6(x)Thus, for i = 3 or i = 4, Q(a2)* TI * ~i and Q(a2)* ~2 * 75 e-split on x . Reproduce TI * ~i and 7 2 * T5 above Q(a3),and so on. We thus get, at t h e end, two big strings T and T’ such that, for each i, Q(ai) * T and Q(ui)* T ’ a r e on T and e-split. By possibly extending one of them t o a string on T , we can actually find two such strings of the same length, and they provide the new level in Q. 0

I.

Theorem V.6.6 (Spector [1956]) I t is possible t o build minimal degrees by recursive uniform trees. Proof. We start with

TO= identity tree, which is clearly a recursive uniform tree. Given Tze,we let

T2e+l = the Q of the Diagonalization Lemma, for T = Tze. T o define T2e+2,let T = Tze+1and see if the condition 2 of the above proposition holds. If not, choose a string u on T for which it fails, and let

Tze+2= full subtree of T above u. Then {e}” is not total, for any A on it. If condition 2 holds, see if 1 does. If not, there is a string a with no e-splitting extensions, and we let

T2e+2= full subtree of T above u. Then {e}A is recursive if total, for any A on it. Otherwise, let

T2e+2= the Q of the proposition above. Then A ST {e}” if {e}” is total, for any A on it. If A is on all the T,’s, then it has minimal degree. 0 One might wonder why we should want t o build minimal degrees by uniform t r e e , since t h e proof is more complicated than the one given in V.5.11. There are two independent answers to this. T h e first is that this is a first step toward the construction of minimal degrees by recursive coinfinite conditions. T h e second is t h a t t h e proof just given can be modified and, taking advantage of t h e uniformities, turned into a proof of the existence of more complicated initial segments. These two applications are treated in the next subsections.

V. Turing Degrees

520

Minimal degrees by recursive coinfinite extensions W e close now t h e circle s t a r t e d in Section 4, by showing how coinfinite conditions a r e nothing else t h a n particular uniform trees.

Definition V.6.7 (Lachlan [1971]) T is a strongly uniform tree (or a 1-tree) i f it is uniform and, for all u, T ( u1: 0 ) and T ( u* 1) differ only on one argument (we say they are adjacent). Equivalently, we could define strongly uniform trees as admissible triples (V.6.2) satisfying t h e stronger condition t h a t t h e functions fL a n d fR differ, at each level, on exactly o n e argument. We can see t h e arguments on which t h e two sides differ as t h e uncommitted ones, a n d t h u s a strongly uniform t r e e defines a coinfinite condition

undefined

)O if ~L(z) = ~ R ( z= if f ~ ( x = ) fR(2) = 1 if f ~ ( x#) f ~ ( 2 )

Of course t h e s a m e translation would work for uniform trees as well. W h a t makes strongly uniform trees special is t h a t any set A extending 8 is o n t h e t r e e (since in each interval there is only one uncommitted point, t h e t w o branches f~ a n d f~ take care of all t h e possibilities). T h i s is not t r u e if t h e t r e e is only uniform (if in a n interval there a r e n uncommitted points t h e n t h e r e a r e 2n possible extensions, b u t only two of t h e m a r e o n t h e tree). Conversely, a coinfinite condition 8 defines a strongly uniform t r e e as follows. Let g enumerate, in increasing order, t h e elements on which 8 is undefined, a n d

Moreover, t h e translations j u s t given preserve recursiveness (recall t h a t a coinfinite condition always has recursive domain, a n d so g is always recursive), a n d t h u s recursive strongly uniform trees and recursive coinfinite conditions are

interchangeable. It is immediate t o note t h a t t h e Totality Lemma proved for uniform trees also works for strongly uniform ones (since we j u s t split a string at t h e very end), a n d t h u s it is possible to build hyperimmune-free degrees by recursive coinfinite extensions. To prove t h e analogue of t h e Minimality L e m m a requires instead much more work.

Proposition V.6.8 (Lachlan [1971)) Given e and a strongly uniform tree T , if 1. every u E T has e-splitting extensions on T

V.6 Initial Segments *

52 1

2. ('doE T ) ( ' d z ) ( 32~ O ) ( T E T A {e}.(z)

1)

3. T does not have strongly uniform subtrees without e-splittings then T has a strongly uniform e-splitting subtree Q.

Proof. Clearly condition 1 is redundant, and it follows from 3. We make it explicit only t o show where the new hypothesis is used. Since 2 holds, we may suppose t h a t ('d4(W(bl= 71. 3 {e}T'"'(4

L),

otherwise we apply V.6.4 first. Again we proceed by induction, showing t h e first two steps. Given u , by 1 there a r e T and T' extending it and e-splitting, say on z. We may suppose they are of the same length (otherwise extend t h e shortest), and t h a t for all strings p of that length, {e}p(z) 1 (otherwise go to a level high enough, by t h e initial observation). Since T is strongly uniform, there is a sequence T O , . . . ,T, of strings on it of t h e same length, each adjacent t o t h e . {e}TJ (z)1 for all following in t h e list, and such that T = TO and T' = ~ i Since j 5 i, and {e).O(z) $?{e}.l(z), two of these adjacent strings e-split on z. So u E T has e-splitting adjacent extensions. This shows how t o build t h e first level of t h e e-splitting uniform subtree. Now let 01 and up be given. We set up t o build t h e second level.

v'

xv

1. We first build a strongly uniform subtree of T above 02, as follows. Take a pair of e-splitting adjacent extensions of 02, and let it be t h e first level. Then consider t h e leftmost branch, take a pair of e-splitting adjacent extensions of it, and reproduce them on the rightmost one, and this is t h e second level. We go on by considering the leftmost branch at each level, finding an e-splitting adjacent pair above it, and reproducing it on every node of the same level.

V. Turing Degrees

522

2. Then we take this subtree and reproduce it, as it is, above o l . There must be a n e-splitting on this strongly uniform subtree of T, by condition 3. We want t o find one such t h a t t he two branches only differ on one element (since we are building a strongly uniform tree), a n d they also e-split above 02. Note t hat we know only t h a t t h e leftmost branches e-split above 02, a n d we thus make our search by staying on t h e leftmost branch of our subtree above 01. By methods we know, we may choose an e-splitting v a n d v‘ such t h a t )vI = /v‘I and: 0

v’ is on t he leftmost branch of t h e strong uniform subtree (given any e-splitting, say on x, it is enough t o wai; until, for a long enough segment v’ on t h e leftmost branch, {e}” (x)1: one of t h e original branches and v’ must e-split on x.)

0

0

v goes right on t he tree as late as possible (i.e. t h e common part of v and v’ is maximal) if x is t he element on which v and v’ e-split, a n d p E T , then

14 = I4 = I4 =+

{e}%)l.

Now take X which is like v , only it goes right one level after v does: X a n d v are obviously adjacent, and {e}’(x) N {e}”(x) by t h e choice of v (since th e common part of v and v’ is maximal). T hen {e}”(x) $ { e } ” ( x ) ,a n d X and v are an adjacent e-splitting above 01. By definition they are also a n (adjacent) e-splitting above 0 2 (since v’ lies on t h e leftmost branch). This shows how t o build t he second level of t h e e-splitting uniform subtree. T h e remaining levels can be built inductively, in t he same way. 0

Theorem V.6.9 (Lachlan [1971])It is possible t o build minimal degrees by recursive coinfinite extensions.

Proof. We use strongly uniform recursive trees, which we know t o be interchangeable with recursive coinfinite conditions. Let

TO= identity tree, which is clearly a strongly uniform recursive tree. Given T2e+l=

the

Q of t he

T2e,let

Diagonalization Lemma, for

T = Tze.

To define Tze+2, let T = Tze+1,and see if condition 2 of t h e above proposition holds. If not, choose a string u on T for which it fails, and let

7be+2= full subtree of T above o.

V.6 Initial Segments * Then { e } A is not total, for any not, let

523

A on it. If condition 2 holds, see if 3 holds. If

Tze+2 = a strongly uniform subtree of T without e-splittings.

Then { e } A is recursive if total, for any A on it. Otherwise, let T2e+2

= t h e Q of t h e proposition above.

Then A
Exercises V.6.10 a) The minimal degree built above is below 0“. (Lachlan 119711) (Hint: to ask whether there is a strongly uniform subtree without e-splitting is too complicated. But we can ask if the inductive process of building the e-splitting subtree in the proposition above does terminate or not. If yes, take the e-splitting subtree. If not, we know that there is a strongly uniform subtree without e-splitting, and thus we can search for an index of it.) b) There is a cone of degrees such that every element in it is the join of two minimal degrees. (Cooper [1972a]) (Hint: build a strongly uniform tree of minimal degrees recursive in 0”. Given C such that 0’‘ST C, choose A and B on the tree as follows. At level n, let x be the unique element on which the branches differ. Let A follow the branch that on x agrees with C , and B follow the other branch. Then A, B ST C @ 0” ST C , so A @ B ST C. And C
The three-element chain Recall t h a t an initial segment is simply a set of degrees closed downward. T h e set ( 0 , a}, with a minimal degree, is thus t h e simplest nontrivial initial segment, t h e two-element chain. T h e set of minimal degrees and 0 is a nontrivial initial segment of power PO. Since every degree has a minimal cover, there are degrees a and b such t h a t 0 < b < a,and there is no degree strictly between either 0 a n d b, or b and a. Hut there could be some degree c incomparable with b, and still below a. We now want to build a and b as above, but with t h e additional property t h a t

V. Turing Degrees

524

( 0 , b, a} is a n initial segment, i.e. t he only degrees below a are 0 a n d b. This is called a three-element chain. T h e idea is t o build A such t h a t t h e odd part of A , defined as

Od(A)= {X : 22 + 1 E A } , plays t h e role of t h e intermediate degree. Since Od(A) ST A automatically, t h e requirements on A are thus:

1. Od(A) nonrecursive

2. A g ~ o d ( A ) 3. { e } A total

+

{e}" recursive or {e}"

GT

Od(A) or { e } A ET A.

These are just generalizations of t he conditions for A being minimal, a n d in many respects so is t he construction, which employs uniform trees. T h e most crucial parts are the idea of forcing Od(A) t o be minimal (which allows us t o build just one set A , instead of t he separate sets A a n d B ) , and V.6.12 (which allows us t o make A not recursive in its odd part, a n d uses uniformity in a crucial way, thus forcing us t o use uniform trees in t h e construction of initial segments).

Proposition V.6.11 Given e and a recursive uniform tree T , i f for some u

T ( o * O),T(u* 1) disagree on their odd parts then there is a recursive uniform tree Q C T such that, for every A on Q, 0 4 A ) (4-

+

Proof. Since T ( u * O),T(a* 1) disagree on their odd parts, t h e odd part of one of them, say T(u * i), disagrees with {e}. Let Q be t h e full subtree of T above T ( u* 2 ) . 0 Proposition V.6.12 (Titgemeyer [1962])Given e and a recursive uniform

tree T , if for some u T ( u* 0 ) ,T ( u* 1) agree on their odd parts

then there is a recursive uniform tree Q A 74 {e}od(A).

T such that, for every A on Q,

Proof. Let z be such t h a t T ( u * O)(x) # T ( u * l)(z). Such a n z must exist, because T ( u * 0) and T ( u* 1) are incompatible. By t h e hypothesis on 0,it is not on their odd parts. See if

(3 2 T ( u* O ) ) ( T E T A { e } O d ( ' ) ( z )1).

V.6 Initial Segments

*

525

If not, let Q be the full subtree above T ( u * 0): if A is on Q, {e}od(A)is not total (hence it differs from A ) .

Otherwise, let TO be such a string, and let T~ extend T ( a * l )in t h e same way as TO extends T(o*O).Since T is uniform, TI E T . Now TO and TI have t h e same odd parts, since they extend in the same way T(a*O)and T ( a * I), which have t h e same odd parts. Then { e } o d ( T o ) (N x ) {e}od(T1)(x). But T O ( Z )# T ~ ( z and ), for i = 0 or i = 1 we have

(4

T,(z) # { e } o d ( T ~ )

Then let Q be t h e full subtree of T above

~ i . 0

From t h e two lemmas we know that we can diagonalize if we use uniform trees such t h a t T ( o* 0) a n d T ( u* 1) agree on the odds (i.e. on their odd parts) for infinitely many u’s, and disagree on t h e odds for infinitely many 0 ’ s . We will work, from now on, with trees which alternate agreements and disagreements on the odds. T h e lemmas just proved work for them too, since we are just taking full subtrees in their proofs. Of course lemma V.5.9 still holds, and we thus have conditions ensuring t h a t {e}A is recursive ( A o n a tree T with no e-splitting), or A ST { e } A ( A on a e-splitting tree). With a similar proof we get a similar lemma, t h a t takes care of t h e remaining case:

Proposition V.6.13 (Titgemeyer [1962], Hugill [1969]) Given e, a recursive uniform tree T , and A on T , i f is total then: 1. if on T there is no e-splitting which agrees on the odd, then {e}A S T Od(A) 2. i f whenever T ( u * 0) and T ( u* 1) disagree on the odds they e-split,

Od(A) S T { e } A -

V . Turing Degrees

526

Proof. In the first case we compute {e}". Given z, we know that {e}"(z) 1, and to get the right value it is enough to search for u on T such that {e}O(z) 1 arid Od(cr) C Od(A), and this is recursive in Od(A). In the second case we compute initial segments of Od(A), by induction. Suppose T ( a )such that Od(T(a)) Od(A) is given. If T ( u * 0) and T ( u * 1) agree on the odds, then both work (i.e. their odd parts are contained in the odd part of A). Otherwise, they disagree on the odds, and then they e-split, say on z. If {e}T("'i)(z) = {e}"(z), then T ( o* i) C Od(A). 0 Now we can prove the analogue of the Minimality Lemma.

Proposition V.6.14 (Hugill [1969]) Given e and a recursive uniform tree

T alternating agreements and disagreements on the odds, there is a recursive uniform tree Q C T alternating agreements and disagreements on the odds, such that one of the following holds: 1. f o r every A on Q, {e}" is not total

2. f o r every A on Q, {e}" is recursive 3. f o r every A on Q, {e}" ET Od(A)

4. for

every A on Q, A

IT{e}A.

Proof. If there are u on T and 5 such that (VT 2 O ) ( T E T 3 {e}r(z) t), then let Q be the full subtree above u. If A is on Q , then {e}" is not total. Otherwise, see if there is u with no e-splitting above it. If so, let Q be the full tree above it. If A is on Q, then {e}" is recursive (by V.5.9). Otherwise, there are e-splittings above any strings. See however if there is (T with no e-splitting extensions agreeing on the odds. If so, let Q be a uniform subtree of T above u , alternating branches agreeing on the odds and e-splitting branches. By hypothesis the e-splittings do not agree on the odds, and on Q there are no e-splittine agreeing on the odds. The conditions of V.6.13 are satisfied, and if A is on Q then A -T {e}". Otherwise, we can always find e-splitting agreeing on the odds. We want

Q e-splitting uniform tree, alternating agreements and disagreements on the odds. The level agreeing on the odds can be made e-splitting by hypothe sis. The level disagreeing on the odds can also be done, because we can take e-splittings, and then extend them to disagree on the odds. Let Q be such a n e-splitting subtree of T , which can be made uniform by the techniques of V.6.5. If A is on Q then A 5~ {e}", because Q is e-splitting (V.5.9). 0

Theorem V.6.15 (Titgemeyer [1962]) The three-element chain is embeddable as initial segment of V.

V.6 Initial Segments

*

527

Proof. W e already have all t h e ingredients, a n d t h e result follows by starting with

To = identity tree, which is a recursive uniform tree alternating branches agreeing on t h e odds a n d disagreeing o n t h e odds. T h e n we let: T3e+l T3e+2 T3e+3

= = =

t h e Q of V.6.11, applied to T = T3e t h e Q of V.6.12, applied to T = T3e+l t h e Q of V.6.14, applied to T = T 3 e f 2 .

T h e first two steps make Od(A) not recursive a n d A not recursive in O d ( A ) , so t h a t we do have one nontrivial degree below A . T h e last step makes s u r e t h a t t h i s is t h e only one. 0 Note t h a t in particular t h e proof produces a minimal degree with a s t r o n g minimal cover (V.5.16). Actually, with terminology as on p. 495, the set of min-

imul degrees with a strong minimal cover is comeager, for the topology induced b y recursive uniform trees (Sirnpson [1977]): it is enough to consider, given such a t r e e T Ia recursive uniform tree T* t h a t alternates branches agreeing on t h e odds with branches disagreeing on t h e odds, a n d such t h a t t h e odd p a r t s of t h e branches a r e exactly t h e branches of T . T h e proof above shows t h a t t h e set of degrees which a r e top elements of three-element chains is corneager, for t h e topology induced by t h e trees T'. T h e intermediate degrees, which a r e minimal degrees with a strong minimal cover, a r e then comeager for t h e topology induced by t h e trees T . It can then be argued, as in Section 3, t h a t minimal degrees without a strong minimal cover cannot be built b y recursive uniform trees. T h e existence of such degrees is an open problem, a n d t h e construction of o n e of t h e m would then require new methods of proof. Exercises V.6.16 More initial segments. a) The diamond is embeddable as initial segment of I ) , i.e. there is a degree with exactly two nontrivial degrees below it, a n d they a r e incomparable. (Sacks [1963]) (Hint: build A such t h a t E v ( A ) = {z : 22 E A } a n d Od(A) do the job. T h e only modification i n the proof of t h e three-element chain is t h a t O d ( A ) a n d E v ( A ) have t o be treated symmetrically and, whenever we had in V.6.15 branches t h a t simply disagreed on the odds, now we actually want them t o agree o n the evens. Note t h a t on trees on which all branches either agree o n the odds or agree on the evens, whenever there are e-splittings then we may find them agreeing o n the odds or o n the evens, by going by adjacent paths as in the first part of V.6.8, where now adjacent means agreeing on t h e odds or on t h e evens.) b) Every recursive linear ordering is embeddable as initial s q m e n t of D. (Hugill [1960]) (Hint: we may restrict ourselves t o infinite orderings 5 of w in which 0 is the least element, a n d 1 the greatest. Choose a set {Xn),,cw of disjoint, uniformly

V. Turing Degrees

528 recursive sets such that mdn

m4n

+ X,cX,

+ X, -X,

infinite,

and X O = 0, X I = w. We build A such that, if we define z E

X,(A)

++ the z-th element

(in order of magnitude) of X , is in A,

then the degrees of { X n ( A ) } , ~ , form an initial segment isomorphic to the partial ordering. Note that X o ( A ) = 0, X ] ( A )= A, and X,(A)
+

+

The initial segments of the degrees * T h e ideas involved in t h e proofs of t h e embeddings of t h e t h r e e e l e m e n t chain a n d t h e diamond can be pushed further, to prove t h a t every finite distributive lattice is embeddable as initial segment of D . T h i s relies on t h e fact t h a t if a finite lattice is distributive, then it is isomorphic to a sublattice of t h e power set of some finite set. It is then enough, given a finite distributive lattice, to find a partition of w into infinite, coinfinite recursive sets whose lattice s t r u c t u r e under inclusion mirrors t h e given lattice (like 8, Ev,Od a n d w did for t h e diamond), a n d build t h e t o p degree of t h e initial segment. W h a t is needed is some kind of representation theorem for t h e lattices, b u t no new recursion-theoretical ideas a r e involved. This theorem is d u e to Lachlan [1968a],b u t t h e approach just sketched comes from Yates [1972], who did it for Boolean algebras (which

V.6 Initial Segments *

529

require only t h e full power set of some set), and Epstein [1979], who got the representation. The same approach can be adopted for countable bottomed distributive lattices, with a new ingredient: if the lattice is not recursive, its representation via recursive subsets cannot be chosen ahead of time, and has t o be built along the way. This gives the result of Lachlan [1968a],and it is as far as t h e method can go: only distributive lattices can be treated this way. Nondistributive lattices are much more difficult t o deal with, and have been treated by Shoenfield, Lerman [1969], (19711, and Lachlan and Lebeuf [1976]. The final result for countable initial segments is the following: a countable partial ordering is isomorphic t o a n initial segment of 2, if and only af i t is an uppersemilattice with a least element. This completely characterizes the possible isomorphism types of P(5 a),since any degree has at most countably many predecessors. Namely, t h e principal ideals of 2) realize every possible countable uppersemilattice with least and greatest element. We will prove that V is not distributive without using initial segments, as a corollary of a n embedding result in the r.e. degrees, see X.6.27.b. 'I'reatments of the embeddings quoted so far can be found in Epstein [1979] and Lerman [1983]. Uncountable initial segments have been considered by Thomason [1970] and Rubin. Here it is not possible to assume the existence of a greatest element, and hence t o construct only the top degree. Thus the given uppersemilattice has t o be approximated, and a further obstacle is introduced by the fact that not every initial segment is extendible (e.g. not every degree has a strong minimal cover), and thus the approximations cannot simply be end-extensions. T h e method of forcing is used t o control the amalgamation of the various approximations, and Abraham and Shore [1986] have proved that a partial ordering with power

at most N1 is isomorphic to a n initial segment of V i f and only i f it is an uppersemilattice with a least element, and countable predecessor property. This conipletely characterizes the possible isomorphism types of linearly ordered initial segments, because a linear ordering with countable predecessor property must have power at most N1. This result settles the initial segments problem if the Continuum Hypothe sis holds, and it is the best possible result in Z F C , since Groszek and Slaman [1983] have shown that it is consistent with Z F C to have a n uncountable uppersemilattice (even of power Nz less than the continuum) with countable (even

finite) predecessor property, which is not embeddable in V as an uppersemilattice (and hence as a n initial segment). T h e characterization of initial segments is just one step toward a complete characterization of the algebraic structure of V . The next step would be t o

know which initial segments are extendable, but not even the very first step has been answered: it is still not known if every minimal degree has a strong

V. Turing Degrees minimal cover.

V.7

Global Properties

Until now we have studied first-order or local properties of degrees, mainly telling th at given configurations exist in t h e degrees, possibly extending given ones. We now take a different stand, a n d look at second-order or global properties of degrees: on t h e one hand we analyze t h e difficulty of t h e first-order theory of P (proving its undecidability a n d much more), a n d on t h e other hand we study its global algebraic properties. We will capitalize here on t h e work done so far, an d a cascade of interesting results will be obtained.

Definability from parameters What Spector’s Theorem accomplishes is t o define any countable ideal, by using two parameters. Indeed, if a and b are a n exact pair for a countable ideal 2, then cEZ@c
As we can see, th e formula t hat defines t h e ideal is always t h e same, and only t h e parameters change, for different ideals. This uniformity is quite important, because then we can avoid talking about ideals, replacing them by their exact pairs (since t h e dependence of t h e ideal on t h e exact pair is fixed). T hi s shows t h a t the first-order theory of P includes a n interpretation of quantification over countable ideals. We are now going t o extend this result by showing t h a t , in a little more complicated way, any countable relation is uniformly definable from a fixed number of parameters, which depends only on t h e number of arguments of t h e relation. We first prove a weaker result, which provides all t h e technical work.

Proposition V.7.1 (Slaman and Woodin [1986]) Every countable antichain is definable from finitely m a n y parameters in P,in a uniform way. Proof. Since a n antichain C = { C , , } ~ C ~ is made of pairwise incomparable degrees, th e idea t o define it is t o find a property P for which t h e given c,,’s are th e minimal solutions. We will use a n approach symmetric to t h e one adopted for Spector’s Theorem, namely we define two degrees a a n d b a n d we look at th e degrees z satisfying t he property P so defined:

P ( z ) w z # (zu a ) n (zu b). If we can make sure t hat t h e elements of C are t he minimal solutiorls of P , then 2 E

c

ts

P(z)A T @ ) ( z

< zA P ( % ) ) .

53 1

V.7 Global Properties

T h e trouble with this approach is t h a t we a r e asking too much, because being a minimal solution of P is a condition involving all degrees, a n d only countable information c a n be coded into a pair of sets. We t h u s relax t h e requirement a bit, a n d consider not all degrees, b u t only t h e countably many ones below any given degree c bounding C. T h e n we only need to have:

We will t h u s define C using three parameters a, 6, a n d c, of which c is simply a n y degree bounding every c,,. Given {Cn}7Ltw with C', E c,,, we c a n simply let C = BnEwCn.T h e conditions on A a n d B a r e t h e following: 1. c,, is a solution of P Since x 5 ( s U a ) n ( x U b ) always holds, weensure ( z U a ) n ( x U b ) x, for x in C. T h i s will be achieved by having, for every n, a set D, such that Dn ITCn @ A,Cn @ B b u t D, $T C,.

2. c,, is a minirnal solution of P For t h i s we need, for a n y x

x

#

5

c,

( x u a) n (zu b)

3

c,, 5 x , for some n.

Hence, for any X
( 3 D ) ( DITX @ A , X @ B A D

$T

X)

* C, S T X , for s o m e n.

As in Spector's Theorem, A a n d B will be built by coinfinite extensions 8, a n d 19,. We will have A = Ustw 9, a n d B = 19,. As usual, A a n d B will be thought of as m a d e of columns A, a n d B,, a n d corresponding columns in t h e m will differ only finitely (so t h a t their degrees will be t h e same). T h i s t i m e however it is D,,, rather t h a n C,, t h a t gets coded in t h e 71-th columns A,, a n d

Ustw

&. Since D, has to be recoverable from each of C, (3 A a n d C, @ B , we will use only t h e elements of C, to code information in A a n d B . Precisely,

Dn(x) =

{

:,,(T)

if x @ C,, otherwise.

T h e n Dn < T C , @ A by definition, a n d D,
V. Turing Degrees

532

Since we use coinfinite conditions, we will be able t o take care of all these requirements, for a fixed n, in just one step. Indeed, we can define

DA(z) =

1 - {e}cn(z)

if z is t h e e-th element of C,, a n d {e}cn(x)i otherwise.

DG obviously is not recursive in C,, and neither is any set differing only finitely from it (because each recursive function has infinitely many indices, a n d thus 0; differs infinitely often from every set recursive in Cn). It will then be enough to have A,, and hence D,, differ only finitely from D z . We now consider t h e minimality requirements. We will work with a fixed list {X,},,, of t h e sets recursive in C. Recall t h a t we want, for each m, D < T X ~ @ A , X , @ B A D ~ T X ,+ C n < ~ X m ,f o r s o m e n . T h e requirements are:

Re,, : {e}xm@A2: {e}Xm@B2: D A D

$T

X,

+ C, < T

X,,

for some n

(note that we use V.2.18.a t o simplify t h e presentation of t h e requirements). They will be satisfied as in t h e proof of Spector's Theorem. We start with 80 = 60 = 0. At each step s 1 we are given 8, a n d 29,, defined on t h e first s columns and on finitely many other points, a n d differing on each of t h e first s columns only finitely. We take care of t h e requirement Re,, if s = (e,m), and in any case we will ensure t h e needed diagonalizations on D,. There are three possible cases:

+

0

we can make {e}Xm@Anot total In other words, there is a string (T compatible with 8,, such t h a t (3y)(V~2 O ) ( T compatible with 8, =+ {e}xm@T(y) t).

Then take one such string, and let: e,+1 (XI

=

{

8, (z) if 8, (z) 1 o(z) if a ( z ) l D,*(z) if z = (3, z ) , otherwise.

dS+1is obtained similarly, by using 29, in place of 8,. 0

otherwise, but we can force a special kind of disagreement In general we would only look for two strings o a n d T , respectively cornpatible with 8, and d 8 , such that {e}X-eu(z)$ {e}X-@T(z) on some z. For reasons that will appear clear later, here we also request t h a t they agree on elements of C,, on t h e n-th column, for any n < s. If two such strings exist, take any of them a n d define 8,+1 a n d 8,+1 as above, using o a n d 8, for t h e former, a n d T a n d 29, for t h e latter.

533

V.7 Global Properties otherwise Then simply let

I9,+1

is obtained by using 19, in place of 8,.

It is clear th at in t h e first two cases t h e requirement Re,, is vacuously satisfied. We now have t o argue t hat it is so also in t he last case. Suppose t hat in t h e end we have {e}Xm@A2: {e}xm@B N D and D $T X,. We have t o prove t hat

C,


X,,

for some n.

Suppose there is no e-splitting compatible with 8,: then, as usual, we would have {e}Xm@A2: D ST X,, but this is impossible by hypothesis. T hus there a re e-splittings compatible with O,, and actually infinitely many such (because t h e same reasoning works above any given string). But we also supposed {e}Xm@A2: {e}Xm@B, which means t h a t t h e construction did not succeed in forcing a disagreement. Certainly t h e first case of the construction cannot have happened, because {e}Xm@A 2: D is a total function. Then it must have been t h e second step t h a t failed. Since 8, and 6, disagree only finitely, there are infinitely many e-splitting pairs, with one side compatible with 8,, and t he other compatible with 8,. Since we did not pick up any of them, it must be because none satisfies t h e additional condition of case two, and thus t h e two sides d o not always agree on t h e elements of C,, on t h e n-th columns, for some TL < s. We now show th at C, ST X,, for some n < s, in three steps:

1. for a fixed n < s, there must be infinitely many pairs of e-splitting strings, compatible with 8, and differing on some element of C, This is simply because t he same holds for some n < s, and there are only finitely many such n’s. Thus infinitely many disagreements must hold for t h e same n. We now fix this n .

2. there is an infinite subset of C, which is recursive in X , We show how to generate a n infinite ascending sequence of elements of C,. Suppose we have a n initial segment p of t h e characteristic function of such a subset. Simply look for an e-splitting u,0’ extending p , and compatible with 8,. As argued above, u and u’ must disagree on some element of

V. Turing Degrees

534

C, (unless their pair is one of finitely many exceptions, a case t h a t is effectively testable because t h e exceptions can be given in advance). B u t they can also disagree on some other element, not necessarily in C,,. To be able to sort out t h e right element, we only have to interpolate t h e two strings, by a sequence of strings 00,.. . , U , all extending p , making {e}bJ converge on t h e element on which u a n d U' e-split (which is possible, because we a r e not in t h e first case of t h e construction) a n d differing, each from t h e following one, on j u s t one element. Moreover, 00 = u a n d oi = u'. Since uo a n d ui e-split, there must be j < i such that u, a n d aj+l e-split. These strings c a n effectively be found, a n d now t h e y differ only on one element, which must t h e n be in C,. 3. Cn ITX m We now have a n infinite subset of C, recursive in X,. To be able to have all of C,, recursive in X,, we play a trick: we choose, from t h e very beginning, sets C,,'s which a r e introreducible (see 11.6.7), i.e. recursive in each of their infinite subsets. T h i s is possible, because every degree contains such a set (11.6.13). If C, is introreducible, t h e n the previous 0 p a r t already shows t h a t C, < T X,.

Note t h a t the parameters needed to define {C,},cwcan be obtained arithmetically in e n E w C n . Theorem V.7.2 Definability from parameters (Slaman and Woodin [1986]) Every countable relation is definable from finitely many parameters in V ,in a uniform way. Proof. We s t a r t by dealing with sets. Given C = { c ~ }we~would ~ ~ like , to spread it o u t t o get a set o f incomparable degrees, since we know how to define t h e latter. Choose c above every element of C (which is possible because C is of pairwise incomparable degrees not countable). Define a set A = {an}nEw introducing any new relation on degrees below c, i.e. such t h a t , for z 5 c, zUa,
@

a:SyAm=n.

A can be constructed by t h e methods of Section 4 (see V.2.9). We now spread o u t C by using any infinite subset A* = {af(n)}nEw of A, for a n y one-one total function f (for this part of t h e proof t h e set A itself would be perfectly sufficient, b u t t h e added generality will be useful later o n ) . Clearly A* is a n antichain (since so is A), a n d so is t h e set C' = {c,u a f ( n ) } n E w ,

by t h e choice of A (because c, U a f ( n )5 c, U af(,) only if f ( m ) = f ( n ) , a n d hence if m = n, f being one-one). T h e n b o t h A' a n d C' a r e definable with parameters, by t h e previous proposition. I t follows t h a t :

V.7 Global Properties

535

C is definable Indeed

x EC

tj

x

5

c A ( 3 a E A*)(xU a E C*).

Only t h e right-teleft implication has to be checked. Suppose t h a t x a n d x U af(,) is in C*: then

x U a f ( n )=

Cm

5

c,

Uaf(m),

for some m. By t h e properties of A it must be U af(n) = C, U af(n).

T h e n from x U q(,)5 c, U af(,) we have x C, 5 x , from which x = c,. T h u s c E C.

5

c,, a n d similarly

the m a p f* froin C to C' induced b y f is definable As for C, f * ( x ) = y ($ x E C A a : U y E C * .

We now t u r n to t h e definability of countable relations. Given a n n-ary relation R on degrees, we can first of all define t h e n projections of R,since t h e degrees in them a r e t h e relevant ones:

We can now take as c a degree bounding all t h e elements of Ul
V. Turing Degrees

536

B isdefinable, becausethedegrees f;(q)U...Uf;(z,,) a r e f i n i t e j o i n s o f members of A, a n d hence pairwise incomparable. And R is recoverable froni B , because t h e joins of elements of A a r e uniquely determined by their elements, a n d hence R(z1,. . . , z,,)'3 f;(zl)U...uf;(2,) Moreover, R is definable because t h e maps

fl

E

B.

a r e definable, a n d so is U.

0

Note t h a t in a sense t h e theorem is t h e best possible, since the parameters cannot be eliminated i n general: t h e relations definable in P a r e only countably many, b u t there a r e uncountably many countable relations. T h e r e are however many relations t h a t can be definable without parameters, as we will see.

The complexity of the theory of degrees T h e definability result of t h e last subsection allows for a quick proof of t h e following theorem, which completely characterizes t h e complexity of t h e theory of degrees.

Theorem V.7.3 Simpson's Theorem (Simpson [1977]) The first-order theory of P has the same degree (and actually the same isomorphism type) as the theory of Second-Order Arithmetic.

Proof. We prove t h a t t h e two theories have t h e s a m e m-degree by interpreting each in t h e other, t h u s providing faithful translations t h a t will preserve theorems. Since t h e translations will actually be one-one, t h e theories will have t h e s a m e 1-degree, a n d hence will be recursively isomorphic by 111.7.13. O n e direction is clear, since every formula a b o u t t h e ordering of degrees c a n be interpreted, in t h e natural way, as a formula a b o u t sets of integers. T h u s t h e theory of degrees is interpretable in Second-Order Arithmetic. For t h e converse, we want to show t h a t Second-Order Arithmetic is interpretable in P. A model of arithmetic is a s t r u c t u r e ( A , R , f l , f ~f 3, , a ) such that: 1. A is a countable set

2. R is a total ordering on A with first element a, successor given by f i , a n d such t h a t every element different from a has a n immediate predecessor 3.

a n d f3 satisfy t h e axioms of Robinson Arithmetic Q (p. 23), when interpreted as s u m a n d product, with f1 interpreted as successor.

fi

A standard model of arithmetic is a model in which R is a well-ordering. A s t a n d a r d model exists in t h e degrees, since a n y countable partial ordering is embeddable in boldmath V (by V.2.9). Moreover, given degrees 7i coding a

V.7 Global Properties

537

countable set of degrees (intended to be t h e universe A of a s t a n d a r d model of Arithmetic in P),we can say in a first-order statement of P t h a t given degrees Z code a relation a n d ( t h e graphs of) t h r e e functions on t h e set coded by Ti, satisfying t h e requirements for being a s t a n d a r d model of Arithmetic. T h i s is because Q has only finitely many axioms, a n d well-ordering can be expressed by replacing quantification over subsets of A by quantification over parameters t h a t define t h e m (since a subset is defined by a fixed number of parameters, in a uniform way). T h e n a sentence cp of Second-Order Arithmetic is t r u e if a n d only if t h e sentence cp* of P is, where cp* says t h a t there a r e degrees c?i a n d 2 coding a s t a n d a r d model of arithmetic in which t h e translation of cp (obtained, as above, by replacing quantification over subsets of A by quantification over parameters t h a t define t h e m ) holds. Since cp' can be effectively obtained from cp, we t h u s have a n 7n-reduction of Second-Order Arithmetic to t h e theory of 'D. 0

Corollary V.7.4 (Lachlan [1968]) T h e first-order theory of P is undecidable and n o t axiomatizable. Proof. So is t h e theory of (first-order) arithmetic.

0

Corollary V.7.5 (Jockusch and Simpson [1976], Simpson [1977]) T h e first-order theory of 2) is n o t absolute with respect lo models of Z F C containing all t h e ordinals. Proof. T h e r e is a sentence a b o u t t h e ordering of degrees which is t r u e in P if a n d only if P(w) 2 L , since t h e latter can be translated into a sentence of Second-Order Arithmetic by IV.4.22. 0 Lachlan obtained t h e undecidability of t h e theory of degrees as a corollary of t h e embeddability of every countable distributive lattice as a n initial segment of 2) (see p. 528), by using t h e undecidability of t h e theory of distributive lattices (and t h e Liiwenheim-Skolem Theorem, to be able to restrict attention to countable distributive lattices). Subsequently, Thoniason [1970] observed that, t h e result follows from t h e embeddability of t h e finite distributive lattices only, t h i s t i m e using t h e fact t h a t t h e set of sentences t r u e in all distributive lattices a n d t h e set of sentences false in some finite distributive lattice a r e recursively inseparable. In t h e first proofs of Simpson's Theorem, when only Spector's Theorem was available, only ideals (and hence initial segments) could be used to code sets a n d relations in P in a definable way, a n d more ingenuity was required. T h e original coding of Arithmetic by Simpson [1977] was very direct, a n d interpreted t h e natural number n as t h e degree O ( " ) , m+n as O("+"), a n d rn.n a5 O("'"). To define t h e set {O(n)}nEw Simpson hooked it u p to an initial segment with

538

V. Turing Degrees

controlled (double) j u m p , a n d t h e s a m e method was used for s u m a n d product. T h i s required only simple initial segments (namely chains), b u t t h e proof t h a t t h e coding worked was not straightforward (in particular, t h e j u m p operator had to be eliminated). An exposition of t h i s method is in Epstein [1979]. A much simpler coding, requiring more initial segments ( t h e countable distributive lattices) b u t less direct work, was devised by Nerode a n d S h o r e [1980]. T h e y noticed t h a t second-order logic on countable sets could be translated into t h e theory of countable distributive lattices with quantification over ideals, by first coding relations by graphs, a n d t h e n graphs by ideals of distributive lattices. T h e advantages of this method were simplicity, which allowed for improved calculations a n d sharper consequences, a n d generality, which permitted t h e extension of Simpson’s Theorem to a variety of different degree structures. An exposition of this method for t h e s t u d y of Turing degrees is in Lerrnan [1983], while t h e general result is proved in VI.4.7. T h e coding method we used is d u e to Slaman a n d Woodin [1986]. I t is easier t h a n all of t h e above, since it codes sets a n d relations directly, a n d t h u s it makes t h e translation of Arithmetic straightforward. It also h a s t h e advantage t h a t t h e only needed result (V.7.2) can be proved directly, a n d with very little machinery (only t h e coinfinite extension method), in particular avoiding initial segment constructions. Some additional work provides a substantial generalization of Simpson’s Theorem.

Theorem V.7.6 (Nerode and Shore [1980], (1980al)If C is an ideal of 2) closed underjump, thefirst-order theory of C has the same degree (and actually the same isomorphism type) as the theory of Second-Order Arithmetic with set quantifiers restricted to sets with degree in C. Proof. We refine t h e proof of V.7.3. As there, t h e translation of t h e theory of C into Second-Order Arithmetic with set quantifiers restricted to sets with degrees in C is immediate. For t h e converse, we first need to show t h a t we c a n pick u p a s t a n d a r d model of arithmetic in C. Certainly t h e needed configurations exist in t h e degrees below 0’ (by V.2.9), a n d hence in C. Moreover, also parameters 2 a n d Z defining a s t a n d a r d model of arithmetic exist in C, because t h e proof of V.7.2 provides parameters coding a given set arithmetically in its infinite join, a n d t h e methods of V.4.7 allow to choose t h e needed configurations not only below 0’, b u t actually with infinite join below i t . A simple translation of t h e relevant notions shows t h a t t h e r e is a first-order sentence of t h e theory of C t h a t says t h a t degrees 2 a n d Z code a model of arithmetic in C. But t h e method used in V.7.3 to show t h a t t h e s a m e holds

V.7 Global Properties

539

for s t a n d a r d models a5 well does not work here, since by quantifying over parameters in C we do not take care of all possible subsets, a n d hence we do not define a real well-ordering. To be a b l e to handle standard models we proceed as follows. Consider t h e standard part (corresponding to t h e set of integers) of a model. If t h e model is standard t h e n this p a r t exhausts its universe, a n d t h u s every proper initial segment of t h e universe is finite a n d has a least upper bound. Conversely, if t h e model is not s t a n d a r d then t h e standard part is a proper initial segment with n o least upper bound. T h u s t h e model is standard if a n d only if every proper initial segment of t h e universe has least upper bound. T h i s is still a second-order sentence, b u t it really needs only t h e standard p a r t of t h e model a n d t h e finite subsets of t h e universe. If we show t h a t these subsets a r e all coded by parameters in C when t h e universe is, then we can quantify only over parameters in C a n d still t a k e care of all t h e needed subsets. Finite subsets a r e easily handled, since C is a n ideal closed under j u m p , a n d t h e parameters coding a finite subset a r e arithmetical in its join. T h e s t a n d a r d p a r t can be enumerated arithmetically in Z a n d Z, because a n index relative to them of t h e first element (corresponding to 0) can be given for free, while t h e procedure to pick u p t h e index of t h e successor of a given element only involves a few quantifiers (needed t o express t h e order relation S T , a n d t h e successor relation). T h e parameters coding t h e standard p a r t a r e arithmetical in t h e enumerating function (which gives t h e infinite join) a n d , by transitivity, arithmetical in Zi a n d 1. T h i s already provides a translation of first-order arithmetic into t h e theory of C. To be able to handle set quantifiers as well, we need to know which sets can be coded by parameters in C, with respect to standard models of arithmetic in C. We claim they a r e exactly t h e sets whose degrees a r e in C,a n d this will finish t h e proof. First of all we can identify sets of natural numbers a n d sets of degrees contained in t h e standard p a r t of t h e model, by identifying t h e number n with t h e n-th element of t h e standard part. This preserves membership of t h e degree of t h e set in C, since t h e standard p a r t of t h e model can be recovyed arithmetically from the parameters coding it. Given a set B , parameters b coding it can be obtained arithmetically in it, by t h e proof of V.7.2. And given parameters b, t h e set B coded by t h e m can b e recovered arithmetically from t h e m , from t h e definition provided by t h e proof of V.7.2. Since C is closed under j u m p , B has degree in C if a n d only if there a r e parameters coding it in C , which is what we wanted to prove. 0 4

Note t h a t two ideals can be different b u t isomorphic, as t h e case of t w o diflerent minimal degrees illustrates. T h i s cannot happen for ideals closed under j u m p .

540

V. Turing Degrees

Corollary V.7.7 Two isomorphic ideals closed under jump are identical.

Proof. T h e proof above shows t h a t a n ideal closed under j u m p consists exactly of t h e degrees of sets which a r e coded by parameters in t h e ideal. T h e n two

isomorphic ideals closed under j u m p code exactly t h e s a m e sets, a n d m u s t t h e n be identical. 0 T h e corollary shows t h a t two different ideals closed under j u m p a r e not isomorphic. T h e theorem implies t h a t whenever t h e quantifications over sets with degrees in t h e two ideals have different power, t h e n t h e two ideals a r e also not elementarily equivalent. We provide immediately a first example, a n d many others will be given in following chapters.

Corollary V.7.8 (Jockusch [1973], Nerode and Shore [198Oa]) 2) is not elementarily quivalent to the degrees of A: sets, for any n 2 0. Proof. Clearly t h e degrees of A: sets a r e a n ideal closed under j u m p , since t h e A: sets a r e closed under number quantification. We c a n t h u s apply t h e theorem. T h e sets definable in Second-Order Arithmetic with set quantification restricted t o A: sets a r e all arithmetical in a n y enumeration of t h e A: sets, because set quantifiers can be replaced by number quantifiers over t h e indices of t h e enumeration. Since there is such a n enumeration which is A:*+, such sets a r e all A:,, , by IV.2.25. T h e complete E:+, is t h u s a n example of a set t h a t can be defined in Second-Order Arithmetic, b u t not by set quantifiers restricted to A: sets. T h i s Ls as far as we can go, since it will be proved in Volunie 111 t h a t the assertion that P is elementarily equivalent to the degrees of analytical sets is independent of Z F C .

Absolute definability We know t h a t every countable set or relation is definable in P from parameters. We a r e not completely satisfied with this, because each set or relation needs different parameters. We may think of uniformly defining sets a n d relations which a r e definable in Arithmetic simply by using their own definitions in Arithmetic, a n d interpreting t h e m over a fixed s t a n d a r d model of arithmetic in P (as in t h e previous subsection). T h i s uses only t h e fixed parameters needed to define t h e model a n d it is more satisfactory, in particular t h e parameters can be fixed in advance. In t h i s section we show how to get rid of these parameters too, for a wide class of sets a n d relations. T h e main results of t h i s subsection, as well as many others proved below a n d depending on i t , will be stated in a strong form, without a n y assumption. We

V.7 Global Properties

541

will freely use t h e following result, whose proof will be given only in XII.3.14, because it relies on methods and ideas t h a t do not belong t o this chapter (the construction of a minimal degree below Of among them).

Theorem V.7.9 Definability of the arithmetical degrees (Jockusch and Shore [1984])T h e set of arithmetical degrees is definable in V . However, t h e result just quoted is needed only to get t h e sharpest formulations of t h e main theorems of this section. T h e work done here is fruitful even without ever getting t o the proof of V.7.9, because what we prove can be reinterpreted in t h e following ways: 1. T h e set of t h e arithmetical degrees can be used as a parameter. T h e results so obtained are then not absolute, but t h e parameter used is fixed and natural. 2. T h e set of t h e arithmetical degrees can be defined in 2)' in a trivial way (see V.8.1). T h e results obtained can then be interpreted as results about V f, and as such they are absolute, although weaker than when stated for V alone.

T h e next result is t h e key t o obtain definability a n d other results.

Proposition V.7.10 (Shore [1982])For every standard model of arithmetic in the arithmetical degrees, the map taking a degree x above all the arithmetical degrees into a set (of natuml numbers in the standard model) of degree x is definable in V ,with parameters the degrees coding the standard model.

Proof. Fix arithmetical degrees coding a standard model of arithmetic. We want t o define in P t h e relation X is a set (of natural numbers in t h e standard model) of degree 2. Consider t h e set of degrees

X

= { z : t h e sets coded by parameters below I are all recursive in X } .

Being definable in Second-Order Arithmetic with the degrees coding t h e standard model as parameters, X is definable in V with t h e same parameters. If we showed t h a t d e g ( X ) is its least upper bound, then 2

=deg(X)

z is the 1.u.b. of X

would be t h e definition we are looking for.

V. Turing Degrees

542

To be t h e least upper bound of X , d e g ( X ) must be a n upper bound. But if we take a degree x and parameters below it coding a set, then we can recover t h e set recursively not in z , but only in z U a for some arithmetical degree a, t h e reason being th at we also need t o recover t he standard part of t h e model, a n d this involves a few quantifiers over t h e (arithmetical) parameters coding it, an d hence an arithmetical degree. To be sure t h a t a always stays below d e g ( X ) ,we then modify t h e definition of X into:

X = { z : ( V a arithmetical)(the sets coded by parameters below z U a are recursive in X ) } . Given z , there is a set of degree x coded by parameters below z U a,for some arithrnetical degree a. If x E X then z is recursive in d e g ( X ) ,and t hus d e g ( X ) is now a n upper bound of X . To show th a t it is t he least upper bound, it is enough t o show t h a t there a re two degreej in X which join t o d eg ( X ) . Note t h a t there is a fixed n such t h a t if ( V a arithmetical)[(% U a)(")< d e g ( X ) ] then z is in X . Indeed, below a U a one can code only sets of degree at most ( z U a)(n), for a fixed n depending only on t h e coding procedure. T hen it is enough t o find two degrees z1 and z2 such t h a t

x 1( n ) z~p ) = zlu z2 = d eg ( X ) . T h a t such degrees exist follows, by induction on n, from a relativization of V.2.26, together with t h e fact t hat d eg ( X ) is above all t h e arithmetical degrees, a n d in particular above O ( n ) . 0

Theorem V.7.11 Absolute definability (Harrington and Shore [1981], Shore [1981], Jockusch and Shore [1984]) A relation on degrees above all the arithmetical ones is definable i n P if and only i f it is definable in SecondOrder Arithmetic.

Proof. As usual, a relation can be definable in 2) only if it is definable in Second-Order Arithmetic. Let then R be such a relation on degrees z above all t h e arithmetical ones. First consider t h e relation R* on sets t h a t says t hat R holds for t h e degrees of its arguments: R* (XI,. . . X n ) 7

. . . ,d e g ( X n ) ) .

-3 R(deg(X1),

R* is still analytical and hence, given a standard model of arithmetic in P, it is faithfully translatable into a first-order formula 'p* of P with parameters t h e degrees coding th e standard model.

V.7 Global Properties

543

Then R ( z l ,. . . , 2,) holds if and only if for some standard model of arithmetic coded by arithmetical degrees, 'p* holds for some sets X I , . . . ,X , such that deg(X,) = 2;. Then R is definable in 2) because so are t h e set of the arithmetical degrees, the property of coding a standard model of arithmetic, and t h e map taking a degree z above all the arithmetical ones into a set X (of natural numbers in the standard model) of degree 2. T h e quantification over the degrees coding t h e standard model eliminates the explicit reference t o them. 0 We can actually avoid the restriction on degrees above all the arithmetical ones, with no additional work.

Corollary V.7.12 Let R be a relation on degrees which is invariant under join with arithmetical degrees, i.e. such that for every degree 2; and every arithmetical degree a; R ( z 1 , .. . ,z,) @ R ( q U al,.. . , z,U

a,).

Then R is definable in 2) i f and only i f it is definable in Second-Order Arithmetic.

Proof. It is enough to modify the definition of R given above by stating t h a t Xi is a set of degree z i U a;, for some arithmetical degree a;. 0 We now have a number of interesting examples of definable sets and relations. E.g., the sets of degrees of A:, analytical, and constructible sets are all definable in 2). By relativization, also the notions of A:-degrees and constructibility degrees are definable in 2). Similarly, individual degrees above all the arithmetical ones and definable in Second-Order Arithmetic are definable in 2). An example is O(y). By relativization, the w-jump operation is definable in 2). Lerman and Shore [19SS] have proved t h a t no degree a > 0 is definable i n 2) by a %-formula. Of course the definitions provided by t h e proof above are not very natural from a recursion-theoretical point of view, because they simply translate definitions from Second-Order Arithmetic. In the next section we will start a path that will be pursued all along the rest of t h e book, by finding natural recursion theoretical definitions of particular classes of (and relations on) degrees.

Homogeneity T h e fact t h a t every particular result about 2) seems t o relativize above any given degree, led to the following conjectures:

V. Turing Degrees

544

1. strong homogeneity (Rogers [1967]) For every degree a, t h e structures 2) a n d P(2a ) a r e isomorphic.

2. homogeneity (Yates [1970]) For every degree a, t h e structures D a n d D ( 2 a) a r e elementarily equivalent, i.e. they satisfy t h e same first-order formulas. T h e same relativization phenomenon which led to t h e homogeneity conjectures is t h e key to their disproval.

Theorem V.7.13 Failure of homogeneity (Shore [1982], Harrington

and Shore [1981], Jockusch and Shore [1984]) I f P ( 2 a ) is elementarily equivalent t o P,then a is arithmetical. Proof. Consider t h e formula p(z) defining t h e arithmetical degrees: when interpreted in 73 it defines t h e set A of t h e arithmetical degrees, while when interpreted in Do(?a ) it defines (by relativization of t h e proof of V.7.9) t h e set A a of t h e degrees arithmetical in a a n d above it. If P a n d P ( L a) a r e elementarily equivalent, then so a r e d a n d d a a n d hence, by V.7.6, t h e theories of Second-Order Arithmetic with set quantifiers restricted, respectively, to sets arithmetical a n d arithmetical in a. T h e n a is arithmetical, otherwise t h e sentence saying t h a t there is a nonarithmetical set would distinguish them (being false in t h e former a n d t r u e in t h e latter). 0

Corollary V.7.14 P and V ( 2 0 c w are ) ) not elementarily equivalent. Since homogeneity fails, relativized versions of results t h a t hold in 2) a r e not automatically true, and require proofs. Also, there a r e results t h a t simply fail to relativize. However this very result (saying t h a t not everything relativizes) does relativize to degrees b which a r e definable in Second-Order Arithmetic (because then we can define in Second-Order Arithmetic t h e formula saying t h a t all degrees satisfying cp a r e arithmetical in b ) . T h e exercises show t h a t it is consistent, b u t unlikely, t h a t t h e relativization holds in general. Exercises V.7.15 A cane of elementarily equivalent cones is a cone such that, for any a and b in it, the cones D ( 2 a) and D ( 2 b) are elementarily equivalent. The existence of such a cone would provide a homogeneous substructure of the degrees. a) If Projective Determinacy holds then there is a cone of elementarily equivalent cones. (Martin [ 19681) (Hint: as in V.7.18, using V.1.16 and cones instead of comeager sets. Note that the set of degrees a such that 'D(2a) satisfies cp is an analytical set, and thus only Projective Determinacy is needed, in place of full Determinacy.) b) If V = L then there is no cone of elementarily eguivalent cones. (Shore [1982]) (Hint: if V = L then the degrees above a and b are elementarily equivalent only if a and b are arithmetically equivalent. Indeed, with notations as in V.7.13, an exact

V.7 Global Properties

545

pair for the set of degrees satisfying ‘p defines d c in D(> c ) . The least such exact pair w.r.t. S L , which is definable in Second-Order Arithmetic, defines uniformly the same set in elementarily equivalent structures.) c) In a cone of elementarily equivalent cones, no degree can be definable an SecondOrder Arithmetic. (Hint: since V.7.13 relativizes to degrees definable in Second-Order Arithmetic, in any cone with such a base there is a cone not elementarily equivalent to it.)

To get a result t h a t fully relativizes we need to look at isomorphism, rather t h a n elementary equivalence.

Theorem V.7.16 Failure of strong homogeneity (Shore [1979], [1981], Harrington and Shore [1981], Jockusch and Shore [1984]) I f P ( > a ) i s isomorphic to P(2b ) , then a and b are arithmetically equivalent. Proof. Consider a n isomorphism carrying D ( 2 a) into D(>b ) . T h e image of a copy of t h e standard model of Arithmetic defined in t h e degrees arithmetical in a n d above a is carried into a structure isomorphic t o it, in t h e degree arithmetic in a n d above b. Indeed, a n isomorphism preserves definable properties, a n d thus degrees satisfying t h e formula defining t h e arithmetical degrees in D(2 a) a r e sent into degrees satisfying t h e same formula in P(2 b ) . Now t h e sets arithmetical in a a r e coded by degrees arithmetical in a,a n d their images in P(>b ) must code t h e same set (by t h e isomorphism), which is now arithmetical in b. T h u s a must be arithmetical in b. T h e converse holds similarly, a n d thus a a n d b have t h e same arithmetical degree. 0

A cone of isomorphic cones is a cone such t h a t , for any a a n d b in it, t h e cones P(>a) a n d P(> b ) a r e isomorphic. T h e existence of such a cone would provide a strongly homogeneous substructure of t h e degrees. Corollary V.7.17 There is no cone of isomorphic cones.

Proof. By t h e previous result, two cones can be isomorphic only if their bases a r e arithmetical one in t h e other. T h e n there a r e at most countably many cones isomorphic to a given one, a n d there cannot be any cone of isomorphic cones (because a cone has uncountably many elements). 0 We t u r n now to positive cases of homogeneity. We do not know whether there is a cone with nontrivial base which is elementarily equivalent to D ,or whether there a r e two isomorphic cones, b u t certainly there a r e lots of elementarily equivalent cones.

546

V. Turing Degrees

Proposition V.7.18 (Jockusch [1981]) There is a comeager set of degrees which are bases of elementarily equivalent cones, i.e. a comeager set such that if a and b are in it, the cones P(>a ) and P(>b) are elementarily equivalent.

Proof. Consider t h e first-order sentences of t h e language of partial o r d e r i n e . For each such sentence 9,consider t h e set of degrees a such t h a t P(> a) satisfies p. It is easy to prove (see V.3.15) t h a t this set is either meager or comeager. Let 4 be this set if i t is comeager, a n d its complement otherwise. Then is always comeager, a n d cp holds either in every P ( 2 a) or in none of them, for a in A,+,.Since there a r e only countably many sentences cp, t h e intersection A of all t h e 4 is still comeager. Moreover, t h e truth-value of cp in P(>a) is independent of a in A, for every cp. T h i s means t h a t t h e first-order theory of P( a) is independent of a in d. 0

4

>

Exercises V.7.19 a) There is a corneager set of degrees of elementarily equivalent principal ideals D(
Note t h a t Simpson’s Theorem has not been fully relativized to a n y degree a: i t is obvious t h a t the first-order theory of P(2 a ) is 1-reducible to the theory of Second-Order Arithmetic with an added predicate f o r a, uniformly in a, but it is not known whether the converse holds ( t h e proof of V.7.3 shows only t h a t t h e theory of Second-Order Arithmetic is 1-reducible t o t h e theory of D(>a),uniformly in a). If this were provable, a n d in a uniform way, t h e n it would follow t h a t P( a) a n d P(2 b ) a r e elementarily equivalent if a n d only if a = b, t h u s giving a final answer to t h e homogeneity problem.

>

Automorphisms T h e questions of t h e last subsection, about homogeneity a n d strong homogeneity, can be asked about t h e relationships of P not only with some cone, b u t also with itself. T h e relevant notions a r e t h e following.

Definition V.7.20 A map f : 2) + ?) is called:

1. an automorphism if it is an isomorphism that preserves the order, i.e.

V.7 Global Properties

547

2. an elementary map if it is preserves the first-order formulas, i.e. for any first-order formula cp,

T h e analogues of homogeneity and strong homogeneity then ask about t h e existence of nontrivial elementary maps and automorphisms. First of all, t h e two questions are equivalent.

Proposition V.7.21 (Slaman and Woodin [1986]) A map from P to is elementary if and only if it is an automorphism.

2)

Proof. An automorphism obviously is a n elementary map. For t h e converse, a n elementary map automatically preserves the order, and it is one-one because it preserves t h e equality relation. It remains t o prove t h a t any elementary map f is onto, i.e. that for any y there is x such that y = f(z). T h e proof of part 2 of the next theorem will show t h a t f is t h e identity on t h e cone above a degree a (e.g. O w ) . By V.7.2 there are degrees I coding a standard model of arithmetic, and 2 coding the graph of a function g enumerating (on t h e natural numbers of t h e model) t h e degrees below yU a. Moreover, there + is a first-order sentence with parameters y and a stating that I and d have t h e desired properties. Since f is elementary, this statement is true of f ( y ) , f(a), + fee), and f@). In particular, f(c) code a standard model of arithmetic, a n d frd) code a function t h a t enumerates (on the natural numbers of this model) t h e degrees below f ( y U a). But f ( y U a ) = y U a because f is t h e identity above a, a n d hence y is one of the degrees enumerated by t h e function coded by f@), say t h e n-th in t h e enumeration. Since f is elementary, it preserves all t h e relevant properties. In particular, it must be t h a t y$ t h e image via f of t h e n-th degree x enumerated by t h e function coded by d . This shows t h a t 0 y is in t h e range of f , as wanted. We can formulate t h e analogue of (strong) homogeneity as follows: t h e only automorphism of P is t h e identity. Algebraic structures without nontrivial automorphisms are called rigid. Although t h e rigidity of P has not been proved, all known results point in t h a t direction, and at least show t h a t the automorphisms of P are severely restricted.

Theorem V.7.22 Restrictions on automorphisms (Nerode and Shore [198Oa], Harrington and Shore [1981], Shore [1981], Jockusch and Shore [1984]) Every automorphism of P : 1. sends any degree into a degree which is arithmetically equivalent to it

548

V. Turing Degrees

2. is the identity on every degree above all the arithmetical ones, in particular on the cone above 0 ” .

Proof. Consider a n automorphism f : P 4 2). T h e first part follows from V.7.16 and t h e fact that f induces an isomorphism between t h e cones above x and f(x), for any x. For t h e second part, note that there is? copy of t h e standard model of arithmetic such t h a t t h e relation ‘the degrees b code a set of degree z’is analytical, and hence first-order definable for degrees x above all t h e arithmetical ones, by V.7.10. Then this relation is preserved by f and hence, for any coding a -+ set of degree 3 above all the arithmetical ones, f(b) code a set of degree f(x). But f is an automorphism, and hence t h e degrees 7; and f( b) must actually code t h e same set. Then x = f(x). 0 4

Actually something better can be achieved, by more direct calculations: any automorphism of 2, is the identity on a cone hawing a n arithmetical degree as a base (Nerode and Shore [1980a], Harrington and Shore [1981], Shore [1981], Jockusch and Shore [1984]). We now introduce a useful tool for t h e study of automorphism.

Definition V.7.23 An automorphism basis for 2) i s a n y set of degrees A such that the behavior of a n y automorphism is completely determined by its behavior on elements of A. Producing many automorphism bases is one way t o show t h a t there are few automorphisms. Another one is t o show that there a r e small bases. We will exhibit results in both directions. T h e first way t o obtain automorphism bases is to consider sets of degrees t h a t generate P under U a n d n.

Definition V.7.24 Given a set of degrees d, the set generated by A in P is the smallest set: 1. containing A 2. closed under joins

3. closed under g.l.b. ’s, whenever they exist.

Any automorphism of a partially ordered structure must preserve I.u.b.’s and g.l.b.’s, whenever they exist, and thus t h e behavior of a n automorphism on a set A completely determines its behavior on t h e set generated by it. In particular, if A generates P then A is a n automorphism basis.

V.7 Global Properties

549

Proposition V.7.25 (Jockusch and Posner [198l]) If A is a corneager set of degrees, then A generates P under U and n. More precisely, a n y degree can be represented in the f o r m (a1 u a2) n (a3

u a4),

with ai E A.

Proof. Let a be a given degree: by relativization of the minimal pair construction (V.2.16), given b we can get c such that ( a U b) f l ( a u c )

= a.

Fix b E A: then t h e set of such degrees c is comeager, and hence so is t h e intersection of this set with A. Thus we have b and c in A such t h a t t h e above equation holds. We only have t o represent any degree of t h e form a U d, with d in A, as t h e join of two degrees in A. Note t h a t , in general, A@D=TD@(AAD) for any A a n d D , where A A D is t h e symmetric difference (A - D ) U ( D - A ) . Moreover, if A is comeager then so is t h e set

AA = { A A D : D has degree in d}. We may then suppose t h a t the degrees b and c above are not only in A, but also in AA, a n d then t h e result follows. 13 Clearly a countable set cannot generate P,which is uncountable. But there are uncountable meager sets of degrees that d o generate P,as t h e next exercise shows. Exercise V.7.26 The minimal degrees generate 2). (Jockusch and Posner 119811) (Hint: given two sets A and B such that A ST B, we show that there are sets of minimal degree A41 and M2 such that B and A41 @ MZ have A as g.1.b. We then apply this to any B above A and 6”,which by V.6.10.b is the join of two minimal degrees, and have that any degree is generated by four minimal degrees. We extend the proof of V.6.10.b and build M I and M2 by recursive coinfinite conditions. There are two additional requirements:

A
The first one is satisfied by building a strongly uniform tree of minimal degrees, and letting M I be the branch that, at level n, agrees with A(n) on the unique element

V. Turing Degrees

550

on which the two branches disagree, while A 4 2 follows the other branch. The tree is built by stages, and infinitely many times a new level will be added. The second condition is satisfied by the usual minimal pair construction, with coinfinite conditions that satisfy the coding requirement: this accounts for the fact that, in absence of e-splittings, not outright recursiveness, but only recursiveness in A is obtained.)

A proper cone cannot generate t h e degrees, being closed under 1.u.b.k a n d g.l.b's, but can nevertheless be an automorphism basis. Proposition V.7.27 (Jockusch and Posner [1981])There is a comeager set of degrees which are bases of cones that are automorphism bases. Proof. It is enough t o find a comeager set A such t h a t , for any a in i t , D(>a) U D(>O(w))generates D .Since every automorphism is t h e identity above O("), then D(>a) must be a n automorphism basis. First note th at, for any b, t he set {a : ( b u a) n ( b u o ( ~ ) = ) b) is comeager (by relativization of t he fact t h a t t h e set of degrees which form a minimal pair together with O(") is comeager, see V.2.16). T hen t h e set of pairs a a n d b such th a t ( b U a) n ( b U O(") = b is comeager, a n d so is t h e set of degrees a such th at D(>a) U P ( > O ( " ) generates a comeager set (and hence D itself, by V.7.25). 0

V.8

Degree Theory with Jump

*

Although th e jump operator is known t o be definable in D (see XI.5.17), here we briefly discuss what happens when we add it to D ,thus obtaining t h e structure D'.Some of t he results are simply implied by results on t h e structure without jump, e.g. D' is obviously still undecidable, a n d recursively equivalent to Second-Order Arithmetic. Other results exploit t h e extra power provided by t h e jump operator, and produce both easier proofs a n d sharper definability results. For example, t he definability of t h e arithmetical degrees, which we had to postpone t o Volume I1 if working in D alone, is easily obtained in D'.

Proposition V.8.1 (Jockusch and Soare [1970])T h e set A of the arithmetical degrees is definable in V'. Proof. T h e natural definition of A is:

A = t he smallest jump-ideal.

V.8 Degree Theory with Jump

*

551

This is easily expressed in ID’, as:

z E d e ( V a ) ( V b ) [ ( V x ) (5 z a, b+ zf

5

a, b) =+

2

5

a,b]. 0

As we have already noted on p. 541, this result can be used t o turn the proofs of the results (about definability, homogeneity and automorphisms) given for ID under the assumption of the definability of the arithmetical degrees in ID, t o proofs of t h e same results for ID’, with the advantage of avoiding the proof of V.7.9. This use of the jump operator is conservative: the advantage of having easier proofs is paid by obtaining weaker results (since the same results are proved in a stronger structure). A more genuine use of the jump operator would be t o take full advantage of the added strength, and prove stronger results for the stronger structure. These require new methods and ideas, some of which we will introduce later on. Here we just quote t h e statements of the improved results: 1. definability Every relation on degrees above 0 ( 3 )which is definable in Second-Order Arithmetic is definable i n ID’ (Simpson [1977],Nerode and Shore [198Oa], Shore [1982]).

2. homogeneity If ID‘(> a ) is elementarily equivalent to ID’, then a has triple jump 0(3)(Simpson [1977], Nerode and Shore [1980a], Shore [19Sl]).

If ID’(?

a ) is isomorphic to ID’(> b), then a and b have the same triple jump (Feiner [1970], Yates [1972], Nerode and Shore [198Oa], Shore [1981]).

3. automorphisms Every automorphism preserving the jump is the identity on the cone above 0 ( 3 )(Jockusch and Solovay [1977], Richter (19791, Epstein [1979]). Some of the results are actually stronger than we stated, and do not require the full power of the jump operator, but only a small fraction of it. For example, every automorphism jiking Of is the identity o n the cone above 0 ( 3 )(Nerode and Shore [1980a]). A word is perhaps in order, to explain why there is a factor of three jumps in the results quoted above. The reason is that the methods of proofs use embeddings of partial orderings into the degrees, and three jumps are already needed t o express t h e relation < T , and hence the order relation. To obtain better results, different methods are needed (see Nies, Shore and Slaman [1998]). Improving bounds is only one way to improve results. Another way is t o improve the explicit definitions of degrees or relations, making them more

V. Turing Degrees

552

intelligible or more natural. As a n example, we provide a simple a n d natural definition in 2)' of t h e w-jump operator, which we already know to be definable in 2) by V.7.11. We first extend t h e notion of least upper bound.

Definition V.8.2 (Sacks [1971]) A degree a is a n n-least upper bound for a set of degrees C if at is the least element of

{J") : (Yc E C ) ( c 5 2)). Note t h a t 0-l.u.b.'s a r e the usual l.u.b.'s.

We now consider t h e chain

{ O ( n ) } , E u which, being a n increasing chain, has no 1.u.b. (by V.4.10).

Exercises V.8.3 a) O(") is not a minimal upper bound for {O(")},E,. (Kleene and Post [1954]) (Hint: the proof of Spector's Theorem produces an exact pair for the chain, below 0 6 ).) b) There is a minimal upper bound for (O(")},E, below O("). (Sacks 119631) (Hint: analyze the proof of V.5.21.)

Theorem V.8.4 (Enderton and Putnam [1970], Sacks [1971]) O(") is the ,%least upper bound of {O(n)},c,. Proof. There are two things t o prove: 1. if a is an upper bound of O ( n ) } n E w , then O(") 5 a ( 2 ) T h e hypothesis is t h a t O(" 5 a, for every n. T h e problem is t h a t we

\

d o not have a uniform reduction in general, a n d t h u s we cannot conclude anything about the relationship between O(") a n d a. We however show t h a t O(") 5 a(2)does hold uniformly in n, so t h a t O(") 5 a(2). To do this, we have t o show how t o effectively get a n index of relative t o A ( 2 ) ,for a fixed A E a. This can be done inductively, because @(n+l) = (@(%))', and thus there is a n e such t h a t (~z)(E z @(,+I)

e { e ) ' ( " ) ( z )1)

Since, by induction hypothesis, we can express recursively in A , both sides a r e recursive in A', and t h e quantifier adds one more jump. Thus we can obtain such a n e recursively in A ( 2 ) .

2. there is an upper bound a of {O(")},c, such that O(") = a(') To get an upper bound A of { f ~ ( ~ ) } ~we c ~let, A E T,, where T, is a recursively pointed tree of degree O ( " ) . We use t h e following extension of t h e Totality Lemma V.5.5. Given e and a tree TI there is a tree Q T such that one of the following holds:

nnEw

0

for every A on Q,

is not total

V.8 Degree Theory with Jump * 0

553

for every A on Q, { e } A is total.

This is proved by t h e proof of V.5.5, simply because when t h e relevant trees are not recursive, then we have t o take them into account in our computations. The construction is as follows. Let TObe the identity tree (which is recursively pointed). Given T, recursively pointed of the same degree as @(,I, first get T C T, recursively pointed of the same degree as @(,+'I, by V.5.20, which is possible because ST @(,+'I. Then let Te+l be the Q of t h e Totality Lemma stated above, which is still recursively pointed, and of the same degree as @(e+l). The construction is recursive in @("I and we can, recursively in it, decide whether { e } A is total or not. Then

A(2)ST @(").

0

Corollary V.8.5 The w-jump is definable an Dr.

Proof. By relativization, a(W)is the 2-least upper bound of { a ( " ) } n E w . Clearly, since { a ( " ) } n E w is an increasing chain, a(W)is the 2-1.u.b. of the ideal generated by it, which is t h e smallest ideal containing a and closed under jump. Since the ideals are first-order definable in P' (by Spector's Theorem), so is then the w-jump. 0 Exercisea V.8.6 a) O ( w ) is the least element of the set of double jumps of degrees (Sacks [1971]) (Hint: in the second which we minimal upper bounds of {O(")),E~. part of the previous proof, make A also a minimal upper bound of { O ( " ) } , , E W , using the methods of V.5.21.) b) {O(")},,E~ has no 1-l.u.b. (Sacks [1971]) (Hint: given any upper bound B of it, build another one A such that B' $T A', so that B is not the 1-1.u.b. To make A upper bound, use recursively pointed trees. To satisfy the requirements B' $ {e}A', we have to control twequantifier sentences over A. Start from a tree of sets such that the twequantifier sentences over them are all decided by finite strings: such sets can be obtained by extending the proof of V.2.21, and the tree can be made recursive in 0". Thus all the trees of the construction will be arithmetical. Given one such tree T, to satisfy B' $ {e}A' see if there are two strings on T that would decide, on the same element, {e}A' in two different way. If so, choose the full tree above the one that makes {e}A' different from B'. If not, {e}A' will be recursive in the tree, hence arithmetical and different from B',since B is not arithmetical.)

One trend of next chapters will be to prove a number of results with a similar flavor, producing nice definitions for interesting sets of degrees and operations on them.

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Chapter VI

Many-One and Other Degrees In this chapter we study t h e structure of rn-degrees, with a n approach similar to t h a t used for 3"-degrees in Chapter V. T h e main difference between t h e two cases is t h e fact, proved in Section 1, that the structure of m-degrees is distributive. This is a major regularity, a n d t h e main reason allowing for a nice structure theory. This time t h e material is organized structurally, toward a characterization theorem t h a t will be given in Section 4. We begin in Section 2 by a n observation of Lachlan, a simple grain of sand t h a t inserted in t h e oyster of distributivity will produce the final pearl. Having these two ingredients everything really becomes natural: layer after layer we build all t h e countable initial segments in Section 2, and the uncountable ones in Section 3. We disclose t h e oyster in Section 4, by giving Ershov's characterization of t h e structure of m-degrees up t o isomorphism, as a strongly universal uppersemilattice. From it a number of other global results will follow, after which we will have a fairly complete understanding of t h e algebraic structure of m-degrees, a unique case in Recursion Theory. We conclude with some additional topics, among them a comparison of t h e structures of 1-degrees, tt-degrees, a n d wtt-degrees in Section 5, and t h e structure inside degrees of a given kind with respect to stronger reducibilities in Section 6 .

VI.l

Distributivity

Our subject is t h e structure P, introduced in Section 111.2. Recall t h a t there are t)hree m-degrees containing recursive sets, namely O m , { O } , and {w}.Since 555

556

VI. Many-One and Other Degrees

the last two a r e incomparable and smaller than 0,, b u t all other m-degrees a r e greater than or equal t o Om, by convention we will always consider nontrivial sets, and thus 0, may be considered as t h e least m-degree. We also borrow conventions and notations from Chapter V, e.g. t h e notations for degrees (as boldface letters) and cones.

Proposition V I . l . l A s a partially ordered structure, V,,,is an uppersemilattice of cardinality 2N0 with a least but no maximal element. Moreover, each element has 2N0 successors and at most countably many predecessors. Proof. See V.1.12, and note t h a t 0, is t h e least m-degree, A 5, A’, a n d A @ B is actually t h e least upper bound of A and B w.r.t. m-reducibility. 0 Exercises VI.1.2 The jump operator. a) The jump operator is well-defined o n m-degrees, and preserves the ordering. I n particular, 0, _< a’ for any m-degree a. (Hint: if A
Distributive uppersemilatt ices T h e next concept is going t o be crucial for t h e study of m-degrees.

Definition VI.1.3 ( P ,L,U) is a distributive uppersemilattice i f it is a n uppersemilattice such that if a E b U c there are bo C b and Q c such that a=boU@.

bUc

557

VI.1 Distributivity

The notion of distributive uppersemilattice is consistent with the usual n e tion of distributivity. Recall that a lattice (P,C_, U,n) is distributive if the distributive laws hold in it, i.e.

a n (b U c) = ( a n b) U ( a n c) aU(bnc) = (aUb)n(aUc). By duality, each of the two laws implies the other.

Proposition VI.1.4 A lattice is distributive as a lattice i f and only i f it is distributive as a n uppersemilattice. Proof. If t h e lattice is distributive and a C b U c then a = a n ( b U c) = ( a n b) U ( a n c). Then it is enough t o let bo = a n b and ~0 = a n c t o have distributivity as an uppersemilatt ice. If t h e lattice is distributive as an uppersemilattice, since

( a n b) U ( a n c) L a n ( b u c) always holds, it is enough t o show that the converse holds too. Since

afl(bUc)E b U c holds, for some bo C b and CQ C c is a n ( b u c ) = ~ U Q But . then b 0 , q C_ a, and hence a n ( b U c ) = bo U ~0 C ( a n b ) U ( a n c). 0

Definition VI.1.5 An element of an uppersemilattice is indecomposable, or a n atom, if whenever a = b U c then a = b or a = c. Proposition VI.1.6 I n a distributive uppersemilattice the indecomposable elements below b U c are exactly the indecomposable elements below b or c.

Proof. One direction is trivial. Conversely, if a C b U c then, by distributivity, a = bo U ~0 for some bo C b and ~0 C c. If a is indecomposable then a = bo or a = a,,and hence a C b or a C c. 0 Exercises VI.1.7 a) I n a Boolean algebra the indecomposable elements are exactly the a t o m . b) I n a linear ordering every element is indecomposable. c ) In a finite uppersemilattice every element is the 1.u.b. of the indecomposable elements below it.

Our reason t o introduce distributive uppersemilattices is the following.

VI. Many-One and Other Degrees

558

Proposition VI.1.8 (Lachlan [1970]) D , is a distributive uppersemilattice. Proof. Given A 5, B @ C we want BO 5, B a n d CO 5, C such t h a t A --m BO@ CO.Given f recursive such t h a t z E A w f(z) E B @ C , we let

z E Bo

($

f(.) E B f(z)evenand 2

Then, e.g., BO5, B via

dx) =

{

b

if f(z) not even otherwise,

where b is a fixed element of B . Similarly for COSrn C. Moreover, it is easily checked th at A E, Bo @ CO. 0

Ideals of distributive uppersemilattices T h e special properties of distributive uppersemilattices severely limit t h e possible ideals, a n d thus produce necessary conditions for t h e ideals of V m.

Proposition VI.1.9 A finite distributive uppersemilattice with least element is a distributive lattice. Proof. By VI.1.4 it is enough to show t h a t a finite uppersemilattice with least element is a lattice. But t h e greatest lower bound of two elements a a n d b is t h e least upper bound of t h e set of elements below them, which exists because t h e latter is a nonempty finite set. 0 Corollary VI.1.10 (Lachlan [1970]) Every finite ideal of 27, tive lattice.

is a distribu-

To get conditions for any topped ideal we must develop some of t h e theory of distributive uppersemilattices.

Proposition V I . l . l l (Lachlan [1972a]) Given a distributive uppersemilattice with least element, its ideals ordered b y inclusion form a distributive lattice. Proof. We denote by 0 t h e lea3t element of ( P ,5, U), and by Z t h e set of its ideals. T h e g.1.b. of two ideals is their set-theoretical intersection. T h e 1.u.b. is

VI.1 Distributivity

559

t h e least ideal containing their union. Since P is distributive, given 11,Z2 E their 1.u.b. is actually t h e set

Z

Indeed, Z1,I2 C I1 @ I 2 , because 0 is in every ideal. Every ideal containing ZI a n d I2 must clearly contain I1 @ Z 2 . ZI @ I 2 is closed under 1.u.b. because I 1 a n d 12 are. I t is closed downward because if a C b U c then a = bo U Q, with bo C_ b a n d Q C_ c: if b u c E I 1 @ 12 then we may suppose b E I1 a n d c E 1 2 , a n d thus bo E ZI and ~0 E I2, because 11 and I2 are closed downward. Then a E I1 @ 1 2 . Thus I1 @ 1 2 is a n ideal. Finally, Z is distributive as an uppersemilattice: if I 2 I1 @ I2 then I = I ; @ I ; , where :Z = Z, n I . Indeed, Z;,Z; C_ I and thus Z; @ I; C_ I because @ is t h e 1.u.b. operation. Conversely, if a E I then a E I1 @ Z2, so a = b U c with b E I1 and c E 1 2 : but b , c C_ a and a E Z, hence b E Z; and c E I ; , from which a E I ; @ 1;. 0

Proposition VI.1.12 (Ershov [1975])I n a distributive uppersemilattice P every finite subset closed under l.u.b. is embedded as an uppersemilattice (i.e. with l.u.b. preserved) i n a finite ctistributive sublattice of P.

-

Proof. There is a natural embedding from P t o 2,given by a

iL = t he principal ideal generated by a = { b : b

Given a finite uppersemilattice F

a}.

P , consider F C_ Zso defined:

F = { k :a

E

F}.

Let C be t h e finite distributive lattice generated by F in Z (see V I . l . l l ) . We want to pull back C in P , but the obvious trouble is t hat not every element of C is a principal ideal. Thus we need a one-one homomorphism of upperseniilattices 'p : C + P such t h a t F C 'p(C). T h e obvious approach would be t o take, for I E L, p(Z) = 1.u.b. of I , but not every ideal needs t o have a 1.u.b. (unless it is finite). So we define

GI = a finite subset of I p(Z) = 1.u.b. of G I . In particular this gives 'p(Z) E I , since a n ideal is closed under 1.u.b.

VI. Many-One and Other Degrees

560

To have cp one-one, we pick up one element in each nonempty I1 - 1 2 , for every I,, 1 2 E L. Let H be the set containing these elements, as well as the elements of F . For any a E H , let I , be the smallest element of L containing a. Then I , is the 1.u.b. of indecomposable elements 1; E L, and by V I . l . l l it must be I , = 1: @ ... @ I,". Then there are elements ai E 1: such that a = a l U - . . U a , . Let

Ga = { a ~ , . . . , a n } G = UacHGa G I = U { G n J : J I indecomposable element of L}.

c

We now show that the function defined as ~ ( 1=) the 1.u.b. of G I

has the needed properties:

1.

'p is a homomorphism This follows from the fact that the indecomposable elements contained in I , CBI2 are exactly those contained in I1 or I,. So G I ~ =~G II ,UGI~, ~ and ' p ( z 1 @ 1 2 ) = Cp(I1) u Cp(I2).

2.

is one-one If 11 and I2 are different suppose, e.g., 11-12 # 8, and let a E H n ( I l - Z 2 ) , which exists by definition. Then cp(I1) # 'p(I2), because ( ~ ( 1 2 2 E) 1 2 (al1, because I , is the smallest element ways), but 'p(I1) 12. Indeed, I , of C containing a, and thus 'p(I,) cp(Il),because 'p is a homomorphism. G, Gla by definition, and hence 'p

c

a = ( 1.u.b. of G,) Thus a

( 1.u.b. of G I - ) = ~ ( 1 , ) .

C cp(Il),and if 'p(I1) E 1 2 then a E I2 by downward closure.

VI.2 Countable Initial Segments

561

3. F C c p ( L ) It is enough t o show t h a t cp(a) = a , for a E F . Indeed, I, = a if a E F : ~(1,)= cp(a), by t h e previous part of t h e proof. Conversely, so a cp(ii) C a because p(6) E 6 always holds. 0

Corollary VI.1.13 (Lachlan [1970]) Every topped ideal of P, is the direct limit (in the sense of uppersemilattices) of a n ascending sequence of finite distributive lattices. P r o o f . Consider a n enumeration {w, ul,.. .} of a topped ideal I = V, ( 5a). Start with DO = { O m , u}. Given D, I consider t h e finite uppersemilattice generated by D, U { un}, which has a as greatest element, a n d let Dn+l be t h e finite distributive lattice obtained as in t h e proof above. Since t h e former is embedded in t h e latter as a n uppersemilattice, u is also t h e greatest element of a n d thus 2I. We now have a necessary condition for topped ideals of Dm.I n t h e next section we will show t h a t this condition is also sufficient, thus completely characterizing t h e countable initial segments of Pm.

VI.2

Countable Initial Segments

T h e following observation is going to be crucial for our later development.

Proposition VI.2.1 (Lachlan [1970]) Let A be a set, and a its m-degree. For any r.e. set We and any recursive function f with range We, let:

Then the map @ induces a n onto homomorphism of uppersemilattices between the r.e. sets modulo the finite sets (ordered b y inclusion) and P, ( 5u). P r o o f . We have to check t h e following: 1. the m-degree of @(We)does not depend on f Let g be another recursive function with range W e .If

h(z) = p z [ g ( z )= f(z)] a n d t ( z )= p z [ g ( z )= f (z)] then

VI. Many-One and Other Degrees

562

2. @ ( W e ) I m A By definition.

3. ifB and C are r.e. sets and B C then % ( B )5, @(C) Let B a n d C be t h e ranges o f f a n d g , a n d h ( z )= p [ g ( z ) = f(~)]. Then h is total because B C C , a n d

x E @ ( B )@ f(z)E A @ g h ( z ) E A @ h ( z )E @ ( C ) . 4. if B and C are r.e. sets which differ finitely then @ ( B ), @(C) Let B a n d C be t h e ranges o f f a n d g . Fix a E @(c)a n d b @ iP(c), and

h is recursive because f(x) E B b y definition, a n d B n c is finite (and hence so a r e B n A n and B n2 n T h e n @(B)Snt via h. Symmetrically, @(C)In @ ( E l ) .

c).

@(c)

5. i f C SrnA then there is an r.e. set B such t h a t @ ( B )G~ C Let z E C @ f(z)E A: if B is t h e range of f , @ ( B )= C . 0 T h e result shows t h a t instead of controlling directly t h e m-degrees below

A we can control t h e m-degrees of @(We), for every e. T h i s is t h e technique t h a t we a r e going to use to build initial segments.

Finite initial segments Our goal is VI.3.6, which completely characterizes t h e initial segments of z), . But its proof is quite complicated a n d it involves a number of separate ideas, taking care of different problems. We t h u s actually prove a sequence of results, solving one problem at t h e time. We keep t h e same notations throughout, a n d indicate how to adapt previous proofs to t h e new needs. To illustrate t h e technique of construction of initial segments of m-degrees we s t a r t with t h e simplest case.

Proposition VI.2.2 (Lachlan [1970])E v e r y f i n i t e ordinal i s i s o m o r p h i c to a n initial segment of z), . Proof. We want to build a set A such t h a t t h e m-degrees below its m-degree a r e isomorphic to ( 0 , 1,. . . , n } . W e use t h e homomorphism of V1.2.1 to control t h e m-degrees below t h e m-degree of A , a n d thus we only have to control each @(We).

VI.2 Countable Initial Segments We build n

563

+ 1 infinite recursive sets Pi, for i 5 n, as follows: Pi = { { ( n +l)x+i} : x E w}.

Note that Pi is a set of singletons. In the construction we will build equivalence classes as boxes, and these singletons are our initial equivalence classes.

Consider the r.e. sets R, = Ui<, - Pi, for x we automatically have

I n. Since l& C R1 C . . %,

. . . , Rn} and (0,.. . ,n}, Thus already gives a homomorphism between (h, for any set A. We set up to build A in such a way that the homomorphism is onto and one-one. We simultaneously build, by stages, two disjoint sets A and B , with the intention that B is contained in 3 (i.e. it contains elements we want t o leave out of A). At stage s 1 we have finite approximations A, and B, of A and B , and sets R,,, such that @(R,,,) E , @(R,). As above, Rx,8= Pi,,, and Pi,s is going t o be a strong array: we look a t the elements of the strong array as equivalence classes (boxes), with t h e intention that at following stages we put into A or B not single elements, but equivalence classes. At stage s+l we will do (according t o the order decided by an exhaustive list of requirements) one of the following two types of action, respectively intended t o make @ onto and oneone:

+

uil,

1. to ensure that for each e there is 3: such that @(We) G~ @(R,) Since @(R,,,)+,@(R,),it is enough t o ensure that @(We) E~ @(Rx,s), for some x. We may suppose We infinite, otherwise @(We) is automatically recursive. We look at the greatest x 5 n such that We fl P,,, is infinite and define, for each i 5 x, a new Pi++l with the property that each equivalence class of it intersects We.We do this by moving boxes intersecting We from P,,, t o Pi,,+l, when needed. For z > x we let P,,s+1 = Pi+.

VI. Many-One and Other Degrees

564

@(We)=rn @(Rx,s+l) Note t h a t Rn,, U A, U B, = w,and A, U B, is finite. T h e n , by construction and the choice of x, We is contained in R,,,, except for at most finitely many elements. Since @ is invariant under finite differences, we may actually suppose t h a t We C R,,, = R,,,+I. Thus @(We) 5, @(RX,,+1) automatically, by t h e properties of @. Conversely, every box of R,,,+1 contains a n element of W e by construction. If We and R,,,+1 a r e respectively t h e ranges of f a n d g , let

h ( z ) = py[f(y) and g ( z ) are in t h e same box of Rx,s+l]. Then E @(&,,+I)

*

g(z) E A f h ( z )E A

@

h ( z ) E @(We),

(being in t h e same box)

a n d @(R5,,+1) 5 , @(We).Note t h a t it is crucial here t h a t in t h e following we p u t boxes into A, and not only single elements. @(&,,+I)

~

r

a(&,,) n

We only have to prove this for i I x, since for i > z we have Ri,, = Ri,,+l. If i 5 x then Ri,, C Ri,,+l, since we only enlarge boxes, so @(Ri,s) I, @ @ i , S + l ) . Since every new box of Ri,,+l contains an old box of Ri,,, by associating t o every element of Ri,,+l a n element of Ri,, in t h e same box (as above) we get @(Ri,,+l) 5, @(I&,).

2. to ensure that @(R,) & @(R,) if z # y Since x,y 5 n we may suppose, e.g., x < y. Then R, C R,, a n d automatically @(R,) 5, @(R,).To ensure @(R,) $ , @(R,),suppose t h a t for some recursive function h

z E a(%)

* h ( z ) E @(R,).

Let R, and R, be t h e ranges o f f a n d g : then

f(z)E A

H

g h ( z ) E A.

Note t h a t there is a partial recursive function cp with domain R, a n d range contained in R,, and such t h a t w E A @ cp(w) E A for each w on which cp is defined. For example,

p(w) rx g h ( z ) for the smallest z such t h a t f ( z ) = w.

VI.2 Countable Initial Segments

565

To ensure @(RY)g,, (P(R,) is thus enough to look at t h e partial recursive functions 'p with domain containing 4 and range contained in R,. Since R, C Jzy, there is a such that Pa is not contained in R,, but it is contained in &. Consider Pa,,:the construction is such that every box of Pa,, contains a box of from Pa.Choose w E Pafl Pa,,: then ~ ( wE )R,. We ensure that w E A if and only if 'p(w) # A , thus diagonalizing against cp: 0

0

if ~ ( w E) A, then put the box of w into if cp(w) E B, then put the box of w into

B,+1 A,+1

if 'p(w) # A , U B, then, e.g., put t h e box of w into box of cp(w) into B,+,.

A,+1,

and t h e

In this case (P(Ri,,)E, @ ( & , , + l ) is automatic, since we just produce a finite modification, moving the box of w and, possibly, t h e box of 'p(w) too. 0 T h e construction just given is the basis for all the embeddine to be obtained. In t h e following we will limit ourselves to indicate t h e modifications needed to take care of t h e increasingly more complicated situations.

Proposition VI.2.3 (Lachlan [1970])The topped finite initial segments of Dm are exactly the finite distributive lattices.

Proof. T h e condition is necessary by VI.1.10. For t h e converse, let D be a finite distributive lattice. We have t o define R, for x E D in such a way t h a t {@(R,) : 2 E D } is isomorphic t o D . This could appear t o be complicated, since we have to preserve the relationships among t h e members of D , but a small trick makes everything trivial. Let H be the set {ao,. . . ,a,} of t h e indecomposable elements of D (in t h e case of linear orderings we had D = H ) . Since H generates D (VI.l.7.a), we only have t o define: pai = { { ( n + l ) x + i } : x E w } R, = U { P a : a < - x A a E H } .

VI. Many-One and Other Degrees

566

T h e map @ is a homomorphism from D t o D , ( s a), where a is t h e m-degree of t h e set A we build, because D is distributive: if z = y U z then the indecomposable elements below z are exactly t h e indecomposable elements below y or z (V1.1.6), hence R, = R, U R, and @(R,)E, @(R,)@ + ( R z )by , the properties of @. As before we have to ensure that @ is onto and one-one, and this is obtained in t h e same way as in the previous result, with t h e following modifications.

1. ontoness Given We infinite, consider all the i’s such that WenPa,+ is infinite, and make all the classes of Pa,,s+l intersect W e ,for every j such t h a t aj C a; for some such i. If 2 0 , . . . ,i, is a list of all such z’s, we have as before @(We)E m @ ( R x , 8 + iwhere ), z = ai0 U ... U azm. And @(R,)E, @(Rz,s+l) for all 2,since this holds for indecomposable elements as before, and for the other elements because @ is a homomorphism. 2. one-oneness If 5 g y we want @(R,)&, @(Ry). This is obtained as before, because if z g y then there is a n indecomposable element a such t h a t a C z and a g y (otherwise all t h e indecomposable elements below z are also below y, and hence so is their 1.u.b. z). But then we can choose a n element of Pa,, n Pa t o diagonalize. 0

Countable initial segments T h e proofs given above may be easily extended to handle infinite initial segments whose representations can be chosen ahead of time (like recursive distributive lattices), but some new ideas are needed to treat the general case. Again we start with the linear orderings.

Proposition VI.2.4 (Lachlan [1970]) Every countable linear ordering with least element is isomorphic to an initial segment of Vm. Proof. When we consider a countable linear ordering L we cannot define R, ahead of time for all z E L, unless L is recursive. Then we approximate L by finite pieces. We may suppose that L has a greatest element, otherwise we can add one t o it, and we call 0 and 1 the least and greatest elements. Let L= L,, where LO= (0, l}, and each L, is finite and contained in L,+l. Let L, = (a0 C ... C a,}, with = 0 and a, = 1. At stage s 1 we have Pa,+-. If b is a new element of Ln+l, i.e. b is not one of t h e ai’s, we have to create a new row Pb for it. Since 0 and 1 are already in LO,there must be z such that ai C b c ai+l. Then we can simply let Pa,,s+l = Pa,,s if j # i 1,

UnEw

+

+

VI.2 Countable Initial Segments

567

and we create Pb by taking its boxes from Pai+l,s. It is clear that for x E L, it is still R,,,+l = R,,,, and thus @(Rx,,+l)E, @(R,,,). 0

Proposition VI.2.5 (Lachlan [1970])The topped initial segments of Dm are exactly the direct limits of ascending sequences of finite distributive lattices. Proof. T h e condition is necessary by V1.1.10. For t h e converse, let D be the limit of {Dn}nEw,with 0, a finite distributive lattice, and D, C &+I. Suppose also that DOalready contains the least and greatest elements 0 and 1 of D , otherwise start from the first D, that does. As in VI.2.3 we just have t o consider t h e indecomposable elements, but now D is only approximated by D,, and hence a t a given stage we only have

H , = {ao, .. . ,a,} = set of the indecomposable elements of D,. When we step from D, t o Dn+l two new things might happen: 1. a n e w atom b appears As in VI.2.4 we have t o create a new Pb for it, and this time t h e atoms are not linearly ordered. We may simply build Pb as a sequence of boxes, each one being the union of one box from Pa,,, for each i such t h a t b C ai (some such i must exist, because the greatest element of D is already in DO).This allows us t o get @(RZ,,+I) E , @(RX,,) as follows.

If b @ x then there is nothing to prove, since R,,,+l = R X , , .If b c x we may suppose that x is an atom of D,, since we only have t o prove the property for atoms (the rest follows automatically from the fact that @ is a homomorphism). But then: @(Rx,s)I , W %

, S + d

This follows from R,,, now).

R,,,+l (note that Pa may be in Rx,,+l

I m @(RZ,S) As usual it is enough to get effectively from every element of Rx,,+l an element of RX,,in the same box. The only nontrivial case is when we consider elements in boxes of Pb, but each such box contains a box of P,,, by construction.

@(&,,+1)

2. an atom a E H , becomes decomposable in Dn+l In this case we do not want to consider Pa,, anymore, and we simply throw it into B,+1, which then becomes an infinite recursive set (as opposed t o the finite set of the previous propositions). To check

VI. Many-One and Other Degrees

568

we simply note that for a C x is Rx,,+l = Rx,, - Pa, which is still r.e. since Pa is recursive. But then Rz,, = R,,,+1 U Pa,, and, by the properties of @, @(Rx,,)3, @(RX,,+1) @ @(Pa,,). Since Pa,, 3 by construction, because we threw it into Bs+l,@(Pa)has m-degree 0, and thus a(&+) G , @(&++I).

s

Case 2 introduces the need of small adjustments in the proof. 1. We have to suppose that the top element of D is indecomposable in every D,, so that we always have a top line t o use when a new atom appears, and a new line has t o be created. For this it is enough t o add, if necessary, an element on top of the greatest one, and suppose it is already in DO.

2. To show that @ is onto we prove as before that @(We)G , @(R,) for some x, noting that for an appropriate x we have We & A, U B, U R,,,, so that

@(We)--m @(Wen As) @ @(Wen Bs)@ @(Wen &,s). But 0

0 0

@(Wen A,) has m-degree 0, because Wen A , is finite @(Wen B,) has m-degree 0, because B, 2 @(Wen Rx,,) G~ @(RX,,) as usual. 0

Corollary VI.2.6 Every countable distributive lattice i s isomorphic t o a n initial segment of D , . Proof. If D is a countable distributive lattice, let DO= {0,1}. Given D, take a E D - D,, if it exists, and consider the finite distributive sublattice of D generated by D, U {a}. Then D is the direct limit of {D,},Eu. 0 Exercise VI.2.7 D,,, is not a Eattice. (Hint: let D, be the finite distributive lattice consisting of an ascending sequence of n elements, with a diamond on top: the direct limit of the D,’s is an infinite chain with two incomparable elements and their join above it. These two elements have no 1.u.b.)

A different method of proving VI.2.5 has been given by Lachlan [1971], and consists of building the same initial segments for T-degrees using strongly uniform trees (V.6.7). This produces common initial segments of Turing and m-degrees. Exercise VI.2.8 If A is a set of minimal firing degree constructed b y wing strongly uniform trees, then A has minimal m-degree. (Lachlan [1971]) (Hint: it is enough

VI.3 Uncountable Initial Segments

569

to prove that if f ~ , f ~ are , g an e-splitting strongly uniform tree, A is on it, and {e}" is an -reduction, then A I,,(e}". But if B = {e}" is an m-reduction, h(z) E A. To define f recursive such that there is h recursive such that z E B zEA f(z)E B, see if f~ and f~ agree on z. If so, then let f(z)be a fixed element of B if the common value is 1, and a fixed element of B otherwise. If not, z is in the range of h, because the tree is e-splitting. Then let f(z)= h-' (z).)

*

VI.3

Uncountable Initial Segments

The obvious trouble with uncountable initial segments is that we cannot approximate them by finite pieces in countably many steps. We are able to overcome this difficulty by extending given countable pieces.

Strong minimal covers Recall (V.5.16) that a is a strong minimal cover of b when, for every c, if c then c < b.


Proposition VI.3.1 (Lachlan [1972]) Every m-degree has a strong minimal cover. Proof. T h e construction is an extension of the one for minimal m-degrees, which is the special case of VI.2.2 for n = 1. We build two sets Pa = {{2%} : a E w } Pb = ((2% 1) : % E w } ,

+

and let Ra = Pa U Pb and Rb = Pb. Since Rb a(&)5, @(Ra).w e want t o ensure that:

2.

a(&) =, @(We) =,

3.

@ ( R a ) &n

1.

2 R,,

we automatically have

B , for a given set B E b.

@(I?,)

or @(We) 5,

a(&),for every e

a(%).

We indicate the modifications to make in the construction of a minimal degree t o ensure these conditions. 1. Let B E b: we ensure a(&)--m B by putting the x-th box of Pb into A if and only if z E B . This is the main new idea, t o be fully exploited in this subsection. Let Rb = Pb be the range of f . Then a(&,)5, B via g ( z ) = px[f(z) = 22 11, since

+

zE

a(&)@ f(z) E A @ g ( z ) E B ,

VI. Many-One and Other Degrees

570

and B Srna(&) via h(x) = p.z[f(.z) = 2 2 -t 11, since 2

E B H 22

+ 1E A

f h ( z ) E A H h ( z )E

a(&).

This part of the construction makes t h e approximation of A at a given stage infinite. But we can still think of A at stage s as t h e union of a finite part A, on a side (obtained by the other two parts of t h e construction, as in VI.2.2) and of the infinite part of A contained in It is understood that anything falling into t h e x-th box of Pb during t h e construction goes into A , if z E B .

2. This is ensured as before. If Wen Pa,, is infinite then we use appropriate boxes from Pa,, t o force all boxes of Ra,,+1t o intersect We,thus obtaining @(We)E~ @(Ra,,+l). If W e nPa,, is finite then, except for finitely many elements, We Pb,s: then, automatically, @(We)5, a(&,).

c

3. This is also ensured as before, since we can use elements of Pa n Pa,, t o diagonalize against 'p : Ra --+ Rb. 0 Recall (V.5.17) t h a t there is a cone of minimal covers in t h e Turing degrees.

A similar proof shows here that there is a cone of strong minimal covers in D , . This stronger result fails for Turing degrees, and actually (in accord with V.1.16) there is a cone without strong minimal covers in D , since by V.2.26 no degree above 0' can be a strong minimal cover.

Uncountable linear orderings Linearly ordered initial segments are especially simple because of t h e countable predecessor property, which implies t h a t any such linear ordering can be obtained by a (possibly uncountable) sequence of countable extensions. We thus have only to learn how t o do countable extensions.

Proposition VI.3.2 (Ershov [1975])Every countable linearly ordered initial segment Z of D , can be extended to an initial segment L isomorphic to L, for any countable linear ordering L hawing an initial segment I isomorphic t o Z. Proof. The proof combines the ideas of VI.2.4 and VI.3.1. T h e only step t o be added to VI.2.4 is t h e case when a new atom b appearing in L,+l is in I . Then we build the corresponding Pb by using t h e row Pa,,, for t h e smallest a E L, - I (which is automatically above b ) : we may always suppose it exists, by possibly topping L and putting the greatest element in LO. We then order t h e classes of p b , e.g. by ordering in t h e natural way their least elements, and put t h e x-th class into A if and only if z E B , for some fixed B in t h e m-degree b of Z corresponding to b E I .

VI.3 Uncountable Initial Segments 0

571

a(&)fm

B @(Pb)f mB aS in VI.3.1, and Rb = UcCbPC, SO a(&) =m @cgb@(P,). But b E I and I is downward closed, so if c C b is c E I and P, was introduced in the same way as Pb, in particular @(Pc) has m-degree c 5 b. Thus

a(&)has m-degree b.

@ ( R z , s + l ) Gm @(%,s)

For x E L, - I , which is the only interesting case,

R,,,+1

= RZ+.

8

ontoness This is obtained as in VI.3.1, by considering t h e greatest a E L, - I , if it exists, such that Wen Pa,sis infinite.

0

oneoneness This is obtained as in V1.3.1 and VI.2.4, by ensuring a(%) gm a(&) whenever x c y and either (x E I A y @ I ) or (x @ I A y @ I ) . Indeed, for x,y E I this is automatically achieved by the previous strategy. 0

At this point we can already give some final results. Corollary VI.3.3 The order-types of the linearly ordered initial segments of Pmare exactly the linear orderings with least element and countable predecessor property. Proof. T h e conditions are obviously necessary. Given a linear ordering L with least element 0 and countable predecessor property, let I0 = ( 0 ) and 20 = {Om}. For any ordinal cy < w1 let I , and Z , be isomorphic countable initial segments of L and P, . Take a E L - I a , if it exists, and let Ia+l be the downward closure of if I a U { a } in L: is countable because L has countable predecessor property, and I , C Z,+l since I, is closed downward. By VI.3.2 there is a n initial segment Z,+1 of P, isomorphic t o Za+l and extending 1,. At limit stages take unions. This procedure gives an initial segment isomorphic t o L in at most w1 steps, since L has countable predecessor property. 0 In particular, the ordinals of the well-ordered initial segments of D , are exactly the ordinals 5 w1. Abraham and Shore [1986] have shown that the linearly ordered initial segments of P are the same as those of P, .

Uncountable initial segments We are finally a t the last step of our long journey. In the next result all the ingredients so far introduced, and some new ones, will be employed t o produce all possible countable extensions of given initial segments. We will then show

VI. Many-One and Other Degrees

572

how the result can be modified and used to characterize the initial segments of Drn .

Proposition VI.3.4 (Ershov [1975])Every countabZe ideal Z of Drn can be extended to a n initial segment L isomorphic to L , for any countable distributive uppersemilattice L hawing a n ideal I isomorphic to Z. Proof. First note that, by possibly doing countably many successive extensions, we may simply restrict our attention t o topped countable distributive uppersemilattices L . As in VI.1.13 we can see that L is a direct limit of an ascending sequence of finite distributive lattices. The proof then extends the one of VI.2.5. When new atoms appear, we treat them as in VI.2.5 and VI.3.2. Namely, we create Pb as a sequence of boxes, each one being t h e union of one box from Pa,, for each atom a not in I and above b. Two new things might happen here: 1. an atom a E I becomes decomposable As in VI.2.5 we throw Pa,, away, i.e. we put it into B,+1. This now causes a little trouble, since Pa,, contains boxes which were in A and that go now into 2,and A is not built monotonically. We still want t o have @(R%,,+I) -m @(R,+).Suppose a C z: since R,,,+I = R,,, - Pa,, we have, as in VI.2.5, @(I&) z m@(RZ,,+1)@@(Pa,,). It is thus enough t o show that @(Pa,,)5, @(Rx,S+l). We may suppose we already have in R,,,+1 the Pb’s for the atoms that make a decomposable. Now b z for all these b’s, so Pb,s C R,,,+1 and (P(Pb,s)Srn @(RZ,,+1).But b E I, since a E I and I is closed downward: by construction then @(Pb,s) has the mdegree b in Z corresponding t o b in I , and @(Pa,,)has the m-degree a corresponding t o a . But a is the 1.u.b. of these b’s, and so

@(Pa,,)I

m @(&,s+l).

2. oneoneness We want @(It,) y are in I.

&, @(Ex)whenever y

z. This is automatic if z and

If z E I but y 6I then we have no trouble: there is an atom a C_ y such that a 6 I, otherwise y E I because I is an ideal, hence closed under l.u.b., and y is the 1.u.b. of the atoms below it. But a z since z E Z and I is closed downward, so we can use Pa t o diagonalize. Let us thus suppose that z 6I, y E I, and y x. If we simply consider cp : R, -+ R, then we might not be able t o diagonalize, since cp might have range contained in R,, where z 5 z is the 1.u.b. of the atoms of I below z ( z E I because I is a n ideal): thus cp sends rows corresponding t o atoms of I to similar rows, and we do not want t o touch them because

VI.3 Uncountable Initial Segments

573

they code information. But we now show that if @(R,)5, @(R,)then there is a partial recursive function cp such that: cp has domain R, and range contained in R,

w E A if and only cp(w) E A , whenever cp(w) is defined the range of cp intersects Pa, for some Pa

R, and a @ I

Consider indeed t h e usual cp (VI.2.2), and let W be its range. If cp does not have the required properties then W R,, for the z E I considered above. So @ ( W )srn@(R,). Since w E A H cp(w) E A and R, is t h e domain of cp, @(R,)E , @(W)and hence @(R,)5 , @(R,).But y and z are in I , and @ is by definition a n isomorphism on I , hence y C z C x, contradiction. We can then always consider cp with the stated properties, and use a row Pa for some a @ I to diagonalize. 0 T h e result already has a number of important consequences, e.g. every countable ideal of V, has a strong minimal cover, and it is the intersection of two principal ideals. To derive the most important consequence of all we however need a strengthening of it, stated in the exercise. Exercise VI.3.5 For every countable ideal Z of Dm and any countable distributive uppersemilattice L having a n ideal I isomorphic t o Z, there is a continuum of initial segments of D , isomorphic to L, and such that their parts isomorphic to L - I are pairwise disjoint. (Paliutin [ 19751) (Hint: the properties of the construction above are not affected if countably many times we take a box from the line relative to the top element, and put it into A , or B,. We can thus build a tree of sets whose degrees are top degrees of extension of Z as wanted.)

We can now give the promised characterization of t h e initial segments of 'Dm. Note that t h e exercise allows us t o avoid any use of the Continuum Hypothesis, a n d thus t h e result is final.

Theorem VI.3.6 Characterization of the ideals of P, (Ershov [1975], Paliutin [1975])The ideals of z), are mactly, up to isomorphism, the distributive uppersemilattices with least element, countable predecessor property, and power at most that of the continuum. Proof. T h e conditions are clearly necessary. Conversely, given an uppersemilattice L with t h e stated conditions, we want t o define an ideal L of V, isomorphic to L. Let 0 be t h e smallest element of L , and define I0 = {0}, ZO= { O m } , and cpo(0) = 0,. For (Y < 2Nolet I , and Z, be ideals of L and 23, isomorphic via pa, and of power less t h a n the continuum. Take a E L - I,, if it exists: ?L (the set of t h e

VI. Many-One and Other Degrees

574

predecessors of a in L ) is a countable distributive uppersemilattice (because L has countable predecessor property) with u f l Z a as a countable ideal (since both 6 a nd I , are ideals). By VI.3.4 we can extend t he isomorphism p a of domain Z, t o a n isomorphism p of domain a U I,. Let Za+l be t h e ideal generated by a U I,: we want t o show t hat p is extendable t o a n isomorphism pa+l of domain la+]. By VI.l.ll t h e elements of I,+] are of t h e form z U y , for z E ci a n d y E I , . We then let (PQ+l(z

’

y) = p(.)

”

p(y).

To check t h a t pa+l is well-defined, consider z’ U y’ = z U y. Since z’ C z U y , there a r e ~6 & z and y,!,C y such t hat z’ = xh Uyh. But p is a n isomorphism on ii U I , , an d thus p(x6) 5 p(z) and p(y6) 5 p(y). T hen (Pa+l(z’) = (Pa+l(4

Similarly, ~p,+~(y’)

5

pa+&’Uy’)

=

44)u cp(Yh) i cp(z) ” $ 4 9 )= va+l(z U Y ) .

(z U y), and hence

= pa+l(z’) u V,+l(Y’)

5 pa+1(zUy).

T h e converse holds similarly, and then p a i l is well-defined. But if we only use VI.3.4 as stated then there is no reason t o believe t h a t pa+] is also one-one. This is where t he strengthening VI.3.5 comes into play: it provides enough choices t o make t he degrees below p(a) but not in pn(iiLnIa) disjoint from p a ( Z a ) ,and thus pa+] one-one. At limit stages we take unions. T h e procedure gives a n initial segment isomorphic to L in at most 2N0steps, since L has at most t h e power of t h e continuum. 0 Refinements of t he results of this section have been found by Malc’ev [1981],

[ 19841, respectively on localization of initial segments a n d relativization t o t h e uppersemilattice of immune sets.

VI.4

Global Properties

We now tu r n t o th e study of global properties of D , , following t h e pat h set up in Section V.7 for Turing degrees, with two major differences. First of all, we will be completely successful i n characterizing t h e algebraic structure of D , , while a n analogous characterization is not known for Turing degrees, and might even be independent of Z F C (see t he results of Grozsek and Slaman quoted on pp. 467 and 529). Secondly, t h e properties of definability, homogeneity, and automorphisms for m-degrees are exactly t h e opposite of those for Turing degrees: no nontrivial countable set of m-degrees is definable, strong homogeneity

VI.4 Global Properties

575

holds, a n d t h e r e a r e lots of automorphisn~s. However, t h e complexity of t h e theories of Turing a n d m-degrees is t h e same.

Characterization of the structure of many-one degrees W e now characterize P, &s a n algebraic structure. T h i s result is t h e only known example of a n absolute characterization of a degree theory, a n d it provides a n alternative, recursion-theoretical description of t h e continuum. T h e results proved in this chapter, which culminated in VI.3.6, already provide t h e tools needed to characterize P, with a two-line proof. Aguzza qui, lettor, hen li occhi a l vero, chB 'I velo e o r a ben t a n t o sottile, certo che '1 trapassar d e n t r o B leggero.' (Dante, Purgatorio, VIII)

Theorem VI.4.1 Characterization of P, (Ershov [1975],Paliutin [1975])Up to isomorphism, P, is the only structure with the following properties: 1. i t is a distributive uppersemilattice with least element

2. every element has at most countably many predecessors 3. it has the power of the continuum

4.

every ideal Z with power less than the continuum can be extended to a n ideal L: isomorphic to L, for any distributive uppersemilattice L with power less than the continuum hawing an ideal I isomorphic to Z.

Proof. A back-and-forth argument in t h e style of V1.3.6 easily gives a n isomorphism between any two structures with t h e given properties. 0 Note t h a t n o use is made of extra set-theoretical assumptions, like t h e Continuum Hypothesis, a n d t h u s t h e result is absolute.

Definability, homogeneity, and automorphisms The algebraic characterization of P, contains all t h e information a b o u t t h e structure, a n d it is t h u s not surprising t h a t we can easily derive from i t solutions to a number of algebraic problems. T h e y should he contrasted with t h e opposite ones obtained for Turing degrees in Section V.7. 'Sharpen thy sight now, reader, to regard the truth, for so transparent grows the veil, to pass within will surely not be hard.

VI. Many-One and Other Degrees

576

Theorem VI.4.2 Strong homogeneity. A n y two cones of m-degrees are isomorphic. Proof. Given a,it is enough t o show t h a t Vm(2a) satisfies t h e properties of VI.4.1. It then follows t h a t any cone is isomorphic t o P, , a n d hence any two cones are isomorphic. T h e only nontrivial property is t he fourth one. Let Z be a n ideal of P, (2 a) of power less than t h e continuum, a n d L be a distributive uppersemilattice of power less than t h e continuum and with a n ideal I isomorphic to 1. Let 2 be th e downward closure of Z in V,: we can extend I t o a distributive uppersemilattice i isomorphic t o 2. If we show t hat i U L is a n uppersemilattice with I as an ideal, then we can apply t h e methods of VI.3.6 t o extend 2t o a n ideal isomorphic t o I^UL in P, , thus extending Z to a n ideal isomorphic to L in Pm (2a). Since I is a n ideal, so is I by its definition. T o show t h a t I U L is a n uppersemilattice it is enough, by VI.1.11, t o define x U y when x E i a n d y E L. There are two cases: 0 0

if x E L then define x U y as t h e 1.u.b. of x and y in L if 5 $ L then first consider t he 1.u.b. of x a n d a (the element of i corresponding t o a) in i: this is now a n element of L , being above a . Then take t h e 1.u.b. of it and y in L. 0

T h e next proposition allow us t o derive information about definability and automorphisms.

Proposition VI.4.3 Vm( 5 a ) and V, ( 5 b) are isomorphic i f and only i f there is a n automorphism of Pm carryzng a into b. Proof. One direction is trivial, since any automorphism carrying a into b induces an isomorphism of P, ( 5 a) and P, ( 5 b ) . Conversely, given a n isomorphism of P, ( 5a) and 23, ( 5b) we can extend it to a n automorphism of 23, by a back-and-forth argument as in VI.4.1. 0 Corollary VI.4.4 0, is the only m-degree fixed under every automorphism, as well as the only definable m-degree. Proof. Given a > 0, , P, ( 5 a) is isomorphic t o a countable distributive uppersemilattice I . L = I x I is still a countable distributive uppersemilattice, containing two distinct copies of I as ideals. Extend P, ( 5 a) to a n ideal isomorphic to L, by VI.3.4, and let b # a be t h e t o p m-degree corresponding to t h e second copy of I in L. Then P, ( 5a) a nd P, ( 5b ) are isomorphic, and there is a n automorphism of P, carrying a into b. T hen a is not fixed

VI.4 Global Properties

577

under every automorphism of P,, and in particular it cannot be definable. Thus O,, is t h e only m-degree fixed under every automorphism, a n d t h e only definable m-degree. 0

Corollary VI.4.5 Every definable set of m-degrees different from (0, } has power of the continuum. Proof. Given a set S # (Om} of m-degrees of power less than t h e continuum, choose a E S - {Om}. T h e m-degree b obtained as in t h e previous corollary can be taken t o be not i n S, because S has power less than the continuum, while there a r e (by VI.3.5) 2N0possible choices for b. Since there is a n automorphism of P, carrying a into b, S is not closed under automorphisms a n d hence it cannot be definable. 0 It follows t h a t many natural classes of m-degrees a r e not definable, e.g. t h e r.e. a n d t h e arithmetical m-degrees. O n the other hand, there a r e nontrivial definable sets of m-degrees, e.g. t h e minimal m-degrees, and thus t h e result is t h e best possible. Fkom VI.4.3 it easily follows t h a t there are 2N0 automorphisms of P, : given a minimal m-degree a,for any other minimal m-degree b there is a n automorphism carrying a into b (because P, ( 5 a) and P, ( 5 b) a r e isomorphic), and there a r e 2N0 minimal m-degrees. This is not t h e best possible result, since there a r e 22N0possible maps from V, to P, . We now show t h a t this bound is attained.

Proposition VI.4.6 (Shore) There are 22Noautomorphisms of 23,. Proof. Note t h a t VI.4.2 produces a n automorphism of V, , if both cones a r e Dm itself. Moreover, t h e back-and-forth argument of VI.4.1, on which t h e proof of VI.4.2 relies, takes 2N0 steps (we have t o ensure t h a t each nz-degree is in both t h e domain a n d the range). T h e only new step here is to actually build a tree of height 2N0 of automorphisms of P, , by extending every partial automorphism in two different ways (by using VI.3.5) at successor stages, and taking unions at limit stages. Each branch of t h e tree is now a n automorphism of P m , and different branches produce different automorphisms by construction. Thus there are 22Noautomorphisms. 0

The complexity of the theory of many-one degrees We have characterized t h e complexity of t h e first-order theory of P in V.7.3. If we t r y t o adapt t h e proof used there t o V, we run into trouble. T h e main point is t h a t we are unable t o prove t h e analogue of V.7.1, because its proof uses in a n essential way t h e fact that every Turing degree contains a n introreducible

VI. Many-One and Other Degrees

578

A 1L

al;

U

a;

Figure VI.1: Coding by graphs

21

VI.4 Global Properties

579

set (11.6.7), and this is false for m-degrees: a nonrecursive, not immune set is not recursive in each of its infinite subsets (since some of them are recursive), and there are m-degrees containing only nonrecursive, not immune sets (e.g. any m-degree above the m-degree of see III.6.lO.b). T h e next result comes to t h e rescue and provides a different, slightly less direct way of coding arithmetic. We state it in more generality t h a n needed because of its interest.

c,

Theorem VI.4.7 (Nerode and Shore [1980]) Let ( P ,C,U) be an uppersemilattice with least element such that: 1. every countable ideal is the intersection of two principal ideals 2. every countable distributive lattice is isomorphic to an initial segment of

P. Then the theory of Second-Order Arithmetic is 1-reducible to the first-order theory of P. Proof. There are many steps toward t h e result: 1. translate Second-Order Arithmetic into second-order logic on countable sets This is a standard and well-known procedure, based on t h e fact t h a t Peano Axioms actually define w up t o isomorphism, in second-order logic (see p. 22).

2. translate second-order logic on countable sets into the theory o f countable distributive lattices with quantification over ideals This is the crucial step, which we split into two parts. We refer t o the various parts of Figure 1. 0

code relations by graphs (Lavrov [1963], Rabin and Scott) We start with a binary relation R. Recall that a graph is a symmetric, irreflexive, binary relation, which we may picture as a set of points related by lines. First of all we have t o put down t h e elements of the domain. A simple way is t h e following: for each element u add two points a;l and a;, and relate them as in part a) of t h e picture. Thus in t h e graph the elements of the domain of t h e given relation are t h e points in which two lines coming from end points arrive. We now have t o relate points u and u when R(u,u)holds. A simpleminded solution as in part b) of the picture is not enough, since R might be in general not symmetric, while t h e proposal is. So we add two elements, a n d relate them as in part c) to show t h a t R ( u , u ) ,

580

VI. Many-One and Other Degrees as opposed t o R(w,u),holds. This is still not enough since if, e.g., R(u,u)holds then we would have t he situation of part d). But this is ambiguous, because we might then think t h a t R(u,u)holds whenever we see a triangle with a vertex in u,while t h e triangle might come from t he coding of R(u,w) for some u # u. Our final choice is then t h e following: given u and w we add three new elements, a n d relate them as in part e), t o show t h a t R(u,w)holds. If R(u,u) holds then we get t he unambiguous situation of part f). This technique codes binary relations. But n-ary relations are easily reduced t o binary ones, and thus to graphs as above. For example, if R is ternary we can introduce a nonsymmetric binary relation as follows. When R ( z ,y, z ) holds introduce four new elements, a n d relate them as in part g). An arrow u -+ 21 shows t h a t t h e new binary relation holds for (u,u). T h e old elements are those from which no arrow comes out. If there is more than one relation, we simply have t o arrange for their domains to coincide. code graphs by ideals of a distributive lattice Given graphs on a countable domain we build a countable distributive lattice as follows. T h e atoms of t h e lattice correspond to t h e elements of t he domain. Add 1.u.b.k x U y for every pair of atoms z and y, and on t op of x U y add a n element c(x,y) as a code for { x , y } ,see part h) of t h e picture (the reason why we d o not simply take c(z,y) = 2 U y is t hat we want t h e codes to be indecomposable elements, for reason t o be explained shortly). T hen add t h e necessary elements t o get a distributive lattice. A graph on t h e elements is simply a set of unordered pairs, a n d can be translated as a set of codes. There is thus a natural c o m e spondence between ideals of t h e lattice a n d graphs on t h e atoms, as follows: a n ideal I defines a graph R as

a n d a graph R defines t he ideal generated by t h e codes c ( z , y ) ,for every x and y such t hat R ( z , y ) holds. T h e crucial fact is t hat t h e correspondence is one-one: if R1 a n d R2 ar e different graphs, they generate different ideals. Indeed, t h e only obstacle t o this could be t hat , given a graph R, t h e ideal generated by R as above also contains codes c ( x , y ) for z a n d y such t h a t R ( x , y )does not hold, so t hat decoding t h e ideal would not produce

VI.4 Global Properties

581

t h e original graph. That this is impossible follows from VI.1.6 and t h e fact that t h e codes are indecomposable elements.

3. translate the theory of countable distributive lattices with quantification over ideals into the first-order theory of P A formula cp with quantification over elements and ideals can be translated into a formula cp* by replacing the ideals by exact pairs coding them, and quantification over ideals by quantification over exact pairs. By the initial segment assumption on PI cp is satisfiable in the theory of countable distributive lattices if and only if there is an element a E P such t h a t the initial segment determined by a in P is a distributive lattice, and cp* holds in it. 0

Corollary VI.4.8 (Nerode and Shore [1980]) The first-order theory of

P, is recursively isomorphic to the theory of Second-Order Arithmetic.

Proof. We prove that the two theories have the same mdegree by interpreting each in the other, thus providing faithful translations that will preserve theorems. Since t h e translations will actually be one-one, the theories will have the same 1-degree,and hence will be recursively isomorphic by 111.7.13. One direction is clear, since every formula about the ordering of m-degrees can be interpreted] in the natural way, as a formula about sets of integers. Thus t h e theory of m-degrees is interpretable in Second-Order Arithmetic. For the converse, we want to show that Second-Order Arithmetic is interpretable in P, . It is enough t o show that V, satisfies t h e conditions of t h e theorem, which it does: the required initial segments exist by V1.2.6, while the existence of exact pairs follows from VI.3.4, although it is also essentially implied by t h e proof of Spector’s Theorem for Turing degrees (since m-degrees are closed under finite differences). 0 Corollary VI.4.9 (Lachlan [1970) The firsborder theory o f V,,,is undecidable and not axiomatizable. For what concerns the extent of decidability, as for Turing degrees (see p. 490) we have that the two-quantifier theory of P, is decidable (Degtev [1979]), and the three-quantifier theory of V,,,is undecidable (Nies [1996]). Both proofs exploit the distributivity of the many-one degrees, and are thus quite different from t h e proofs of the same results for t h e Turing degrees given in Lerman [1983]. T h e next result can now be proved as in Section V.7 (using the present coding for arithmetic), and has the same consequences as there.

VI. Many-One and Other Degrees

582

Theorem VI.4.10 (Nerode and Shore [198Oa]) If C is an ideal 0123closed under jump, the first-order theory of C has the same degree (and actually the same isomorphism type) as the theory o f Second-Order Arithmetic with set quantifiers restricted to sets with degree in C .

Comparison of Degree Theories 3r

VI.5

In this section we consider other notions of degree introduced in Chapter 111, namely 1-degrees, tt-degrees, a n d wtt-degrees. We will mostly quote results about t h e m , a n d will content ourselves to develop their theories only to t h e point needed to show t h a t they a r e not elementarily equivalent a m o n g thernselves a n d with Turing a n d m-degrees (with t h e only exception of tt-degrees a n d wtt-degrees, for which it is not known whether this holds).

1-degrees We have already seen in Section 111.7 t h a t 231 is L special case a m o n g all t h e degree structures we have introduced, because we cannot talk of a least 1-degree in a n y natural way (111.7.4). T h e next result shows t h a t t h e differences a r e even deeper.

Proposition VI.5.1 (Young [1964]) 23, is neither a n upper nor a lower semilattice, i.e. 1.u.b. and g.l.b. d o not always exist. Proof. By 111.2.14 every nonrecursive r.e. T-degree contains a simple set. We will prove in X.1.7 t h a t there a r e incomparable r.e. T-degrees, a n d t h u s t h e r e a r e two incomparable simple sets A a n d B. Suppose they have a 1.u.b. D w.r.t. 1-reducibility: 1. for some 2 E 0, A 51 D U { z } and B 51D U { z } Since D is a n upper bound for A a n d B , t h e r e a r e recursive one-one functions f a n d g such t h a t zE

A ts f(x) E D

and

zEB

@

g(z) E D.

We show t h a t D fl range of f # 0 a n d D f l range of g # 0. Suppose, e.g., t h a t D n range of f = 0: we show D 51 A a n d t h u s B 5 D 51 A , contradicting t h e fact t h a t A and B a r e incomparable. Simultaneously enumerate D a n d t h e range of f , a n d define h by induction as follows: 0

if z shows u p first in D , let h ( z ) be t h e smallest element of A not yet in {h(O),. . . , h(z - I)}, so t h a t h ( z ) E A .

VI.5 Comparison of Degree Theories -k 0

583

if x shows u p first in t h e range of f , let y be t h e unique element such t h a t f(y) = x. If y $ {h(O),. . . , h(x - 1)) let h(x) = y, so t h a t

x

E

D

@

f ( y ) E D ts y E A e+h(x) E A .

Otherwise, y must have been defined by t h e first clause, so let h(x) be t h e smallest element of A not yet in {h(O),. . . , h(x - 1)). T h u s D n range of f # 8, a n d D n range of g # 8 can be shown similarly. Let t h e n z E -d n range of f a n d z* E D n range of g. A 51 D U { z } via f itself (since z is not in t h e range of f), a n d B 51 D U { z } via 9' so defined: ifg(x) # g'(x) = otherwise.

{ 4:"'

Note that g* is still o n e o n e , because z* is not in t h e range of g. 2. D U { z } < I D , arid hence D is not the 1.u.b. of A and B First note t h a t A , B 51 A @ B , so if D is t h e 1.u.b. of A a n d B t h e n D 51 A @ R . B u t A @ B is simple because so a r e A a n d B , a n d t h e n so is D . Suppose D 51 D U { z } . T h e r e is f recursive such t h a t zED

e f(x) E D U { z } .

_ _

T h e n (since z E D ) D has a n infinite r.e. subset { z , f ( ~ ) , f ( ~ ) ( z. .).}, , contradiction. T h u s DU { z } < 1 D , since DU { z } 51D clearly holds. By a symmetrical argument (using z E D a n d D a n d B have n o g.1.b. 0

-

{ z } ) one can show t h a t A

Even if t h e r e is 110 least 1-degree, one can consider segments above O 1 . Lachlan [ 19691 proves t h a t every distributive uppersemilattice which is the direct

limit of an ascending sequence of finite distributive lattices is isomorphic to a segment of Dl above 01. T h e proof consists in forcing all t h e m-degrees of t h e initial segment of D, built in VI.2.5 to contain only cylinders (see VI.S.l), so t h a t they a r e actually 1-degrees. Exercise VI.6.2 I f A is a set of minimal niring degwe coilstructed b y u s i n g strongly uniform trees which is neither immune nor wimmune, then A hay minimal 1-degree. (Hint: as in VI.2.8, using recursive subsets of A and 2 t o make the mreductions one-one.)

A complete characterization of t h e segments of 1-degrees above 01 is not known, even for t h e finite ones, a n d t h e following results of Lachlan [1969] show t h a t it might be complicated:

584

VI. Many-One and Other Degrees

1. every finite segment of ZY1 is a lattice (this is not as trivial as VI.l.lO, since PI is not a n uppersemilattice) 2. some finite segment of ZY1 is nondistributiue

3. not all finite lattices are isomorphic to a finite segment of ZY1. W h at is known is however enough for t he analogue of Simpson’s Theorem, proved by Nerode and Shore [1980]: the first-order theory of ZY1 is recursively isomorphic to the theory of Second-Order Arithmetic. T h e proof uses VI.4.7, once some problems are solved. T h e first problem is t h a t ZY1 is not a n uppersemilattice, a n d thus VI.4.7 h a s t o be rephrased for directed sets, in which every pair of elements has a n upper bound. T h e second problem is t hat t he segments we have for ZY1 are only above O1. This does not introduce complications, because O1 is definable in PI: (8) and {w} are th e only minimal l-degrees (in t h e sense of being degrees with no smaller degree), and O1 is t he srnallest degree above bot h of them. T hus we can work only above 01. T h e final problem is t h e existence of exact pairs. T h e same proof of Spector’s Theorem V.4.3 shows t hat for any countable set of l-degrees in which every pair of elements is bounded there is a pair such t h a t every set l-reducible t o it is also l-reducible t o t h e disjoint union of finitely many finite modifications of representatives of t he given l-degrees. By the restriction above we only work with l-degrees above 01,which are closed under finite modifications because a set A whose l-degree is above 01 is neither immune nor coimmune, and thus infinite recursive subsets of A and 2 can be used to patch u p finite modifications. Closure under disjoint union is needed only for t h e ideals generated by subsets of t h e codes of t h e distributive lattices used for VI.4.7, and it is proved in Nerode and Shore (19801.

Truth-table degrees and weak truth-table degrees We will treat the two structures P t t a n d Dwtt simultaneously, in t h e sense t h a t we will state our results f o r tt-degrees only, but note here that they all hold for wtt-degrees as well, either by t h e same proofs or by minor changes t h a t we will indicate when needed. To get a n elementary difference between ZY an d ZYtt we must develop some theory of t h e latter. We use for it the usual notation for t h e j u m p operator, which is well-defined on tt-degrees by V.1.6. T h e next result provides t he analogue of th e J u m p Inversion Theorem V.2.24.

Theorem VI.5.3 Jump Inversion Theorem for Dtt (Mohrherr [1984]) The range of the jump operator on ZYtt is the cone ZYtt(>O:,).

VI.5 Comparison of Degree Theories *

585

Proof. By V.1.6, for any tt-degree a we have a ' 2 O t t . To get t h e converse, let C be a set such t hat K < t t C : we want t o get A such t h a t A' - t t C . Consider th e construction of V.2.24, with t h e understanding t hat 'the least string u 2 us such t hat {e}"(e) 1' means 'the least string u 2 us such t h a t t h e search of a pair (u,t) (in a n exhaustive recursive list of them) for which {e}:(e)l succeeds'. Then t o be such a u is a n r.e. predicate. 1. A' < t t C By induction on e we want t o determine a truth-table which is satisfied by C if an d only if e E A'. Recall t h a t e E A' is decided at stage 2e 1 of th e construction, since

+

e E A' e {e}"ze+l(e)l

@ (37

2 ~z,)((e}"(e)

I),

where us+l is inductively defined as follows:

us+l

=

{ 1:

pu(u

ifs=2i A i#A' A i E A' if s = 2 i + 1 .

2 usA { i } " ( i ) J )if s = 22

* (C(i))

Since we only need to determine uze, we only have t o use i < e. We can thus fix two initial segments TO and 7 1 of A' and C of length e. Recalling t h e initial observation, t he following is a n r.e. predicate: there is a string 132, that satisfies t he above inductive definition with r O a nd T~ in place of A' a n d C , and a string u 2 u2, such t h a t {e}.(e) 1.

It can thus be reduced t o a question on K and hence, using t h e fact C, t o a tt-condition which is satisfied by C if a n d only if t h e that K itt predicate is true. We still have t o express t h e fact t hat O r and TI really are, respectively, initial segments of A' and C. By induction hypothesis, for each i < e we already have a tt-condition which is satisfied by C if and only if i E A'. Th u s there is a tt-condition t hat is satisfied by C if and only if TO of length e is a n initial segment of A'. And it is trivial t o find a tt-condition satisfied by C if and only if 7 1 of length e is a n initial segment of C itself. Th u s we find a tt-condition, depending on TO and T ] of length e, which is satisfied by C if and only if e E A'. We still have t o eliminate t h e reference t o TO a n d T ~which , is easily done by considering all possible pairs of strings TO a n d T ] of length e, and the disjunction of t he tt-conditions relative to them.

VI. Many-One and Other Degrees

586

2. C A' By induction on e we want to determine a truth-table which is satisfied by A' if a n d only if e E C. Recall t h a t e E C is decided at stage 2e 2 of th e construction, since

+

eEC

*

We thus have t o find

g ~ e + 2 ( 1 ~ ~ 2 e += l ( )1

1uZe+l

1,

102e+1(

E A.

which can be explicitly defined as

where

use (2, A ) =

i E A' other wise.

\ p ( a 2 u2, A {2}"(i)1)1 if

+

Indeed, at stage 22 1 we see if we can make { 2 } A ( i ) converge, a n d if so we take a string t hat does it, otherwise we leave 0 2 i + l = u 2 i . T hus use (2, A ) determines t he length increase due t o forcing t h e j um p, while coding C always produces a one-point extension, and this accounts for th e factor e in t h e expression for J u 2 e + l ) . Unraveling th e definition of 1u2e+ll (and using t h e first e values of C , which we suppose to know by induction hypothesis) we can write down 1 0 2 ~ + l ) , and thus a tt-condition on A' whose truth-value is equivalent to e E C. T h e trouble is t h a t we have used A' explicitly, while tt-reducibility allows only recursive procedures. T h e first step is t o consider, as above, approximations TO a n d TI of A' an d C , respectively of length e 1 and e. This however might cause a problem when computing use ( i ,A ) , since we might look for a string u such t h a t {i}"(i)1 because t he approximation to A' tells us t h a t 2 E A', and we thus believe t hat such a string exists, while this might not be t h e case. But we only need t o look for a string u of length bounded by t he true u s e (2, A ) , since we may ask whether (30 A ) ( { i } " ( i1): ) this is a question r.e. in A , which can be translated into a tt-condition on A'.

+

T h e fact th a t TO and 7 1 are, respectively, initial segments of A' and C can be dealt with as above, this time using t h e induction hypothesis on C. And reference to TO and 7 1 can also be eliminated as above. 0

The Jump Inversion Theorem actually holds for any reducibility 5, between < T , because if K 5, C then K S t t K @ C E, C. By t h e theorem there is A such t h a t A' -tt K C, and hence A' z,. C.

S t t and

VI.5 Comparison of Degree Theories *

587

Kallibekov [1973] showed that O:, is not a minimal cover in the r.e. tt-degrees, a n d a simple modification of his proof actually shows that Oit is not a minimal cover in the tt-degrees. This result relativizes, a n d shows t h a t no tt-degree which contains a jump is a minimal cover. By the Jump Inversion Theorem we then get t h e next result, which provides a n elementary difference between P a n d Dtt. T h e proof uses methods and notations typical of t h e study of r .e. degrees (priority, coding, and Sacks agreement strategy), and will be best understood with some knowledge of Chapter X.

Theorem VI.5.4 (Mohrherr [1984])P t t ( > O i , ) is dense. Proof. By t h e J u m p Inversion Theorem for Ptt, it is enough to show t h a t given sets A and C such t h a t A
(0,z)~e B ~ E A . This codes A into B , and produces A Stt B . The other columns of B will be globally built as a set r.e. in C, hence tt-reducible t o C’. Being B t h e join of its columns, B S t t C’ follows from A &t C’. T h e requirements for t h e construction of B are:

Pe : B Ne : C‘

$tt

gtt

A via (pe B via (p,.

Fix a n enumeration {Cl}9cwof C’ recursive in C (since C’ is r.e. in C ) ,and of A recursive in C and correct in the limit (by t h e a n approximation Limit Lemma IV.1.17). Let l p ( e , s ) = max{z : (YY < z ) ( V e , s ( Y )I A Y E Bs e As L ( e , s ) = max{z : (VY < z)(pe,s(Y)I A Y E CL e Bs T h e construction is the following. At stage s

I=ap,(y)>} k cpe(y))}-

+ 1, for each e 5 8,

+

( e 1, x) E Bs+l if z E Ci A z < l p ( e ,s ) ( i + l , z ) E B , + l if e < i < s A z < s A x S l , ( e , s ) . Thus P, takes action only in the e-th column, while N , takes action on t h e i 1 column, for every i > e . We show t h a t P, and N , are satisfied, by induction on e . Suppose P, and N, are satisfied for every i < e . Then, for i < e,

+

lim l p ( i ,s ) 9-00

< 03

and

lim Zn(i,s) < 00. S + M

Thus Pi puts only finitely many elements on the ( i and so does N , for i 5 e . Then:

+ 1)-th column for i < e,

VI. Many-One and Other Degrees

588

P, is satisfied Suppose B 5tt A via (P,. Then lim8+m Zp(e,s) = 00 and, by construction, (e+l,X)E B

e XEC'

except for at most finitely many elements, and C' I t t B Stt A , contradicting the hypothesis A
+

N, is satisfied Suppose C' S t t B via (p,. Then lims+m ln(e, s) = 00, and each (i 1)-th column with i > e is contained in B. But then B E t t A , since t h e @th column codes A , the (i 1)-th column for i 5 e is finite, and the other columns are contained in B. Then C' S t t B I t t A , again contradicting the hypothesis A
+

+

The proof actually shows that f o r any reducibility caught in between 5%and no degree containing a jump is a minimal cover. T h e crucial property is that if a computation converges at infinitely many stages then it converges. The property and the result both fail for Turing reducibility (every Turing degree has a minimal cover, and every degree above 0' is a jump). For reducibilities (like 5 t t and L w t t ) for which the Jump Inversion Theorem holds it follows that no degree above 0' is a minimal cover. This fails for m-degrees, and we thus have a different proof of the fact that the Jump Inversion Theorem fails for m-degrees (VI.1.2.d). The next result is a typical example of a transfer method t h a t allows us t o carry embedding results from Turing degrees t o tt-degrees. Swtt

Proposition VI.5.5 (Martin) Any hyperimmune-free minimal Turing degree is also a minimal tt-degree. I n particular, there is a minimal tt-degree. Proof. Let a be a hyperimmunefree Turing degree, and A E a. We show that if B ST A then B S t t A. If B N { e } A then the function

f (4= PS<{e>8"(41) is recursive in A and, being a hyperimmunefree, it is majorized by a recursive function g. Then B
{e}c

_N

{ ie}c(x)

if it converges in less than g ( z ) steps otherwise

then { e} c is total for every C, and

21

B.

VI.5 Comparison of Degree Theories *

If also B E a,by symmetry B minimal Turing degree, B ST A

E~~A

if B

* B recursive or B G T A

Thus A has minimal tt-degree as well.

589 E-TA.

It follows t h a t , if A has

B recursive or B

E~~A.

0

For future reference, note that the method just used is not useful below 0’, because no nonzero Turing degree comparable with 0’ is hyperimmune free (V.5.3.d). T h e method however gives the stronger result t h a t the Turing degrees below a hyperimmune-free Tiring degree are all tt-degrees (recall t h a t t h e hyperimmunefree degrees are downward closed). T h e proof of VI.5.5 can be relativized, and shows t h a t for any set A there is a set B which is a minimal cover of it with respect t o tt-reductions via functions recursive in A . This falls short of being a minimal cover with respect t o ttreductions via recursive functions. In particular, it is not even true in general for some function f S T A . t h a t A S t t B , although z E A a B Since every hyperimmune-free Turing degree has a hyperimmune-free minimal cover, many tt-degrees have minimal covers. But we cannot improve much on this, by VI.5.4.

Corollary VI.5.6 Failure of homogeneity. Dtt and D t t ( 2 qt) are not elementarily equivalent.

Proof. T h e sentence asserting the existence of a minimal degree is true in Dtt, but false in Ptt(LOit). 0 Dtt is obviously an uppersemilattice with least element O t t , and t h e proof of Spector’s theorem V.4.3 shows that any countable ideal of D t t is t h e intersection of two principal ideals (because tt-degrees are closed under finite modifications). Thus VI.4.7 allows one to prove that the first-order theories of Dtt and Dwtt are recursively isomorphic to the theory of Second-Order Arithmetic (Nerode and Shore [1980]), provided one has enough initial segments. T h e proofs of t h e initial segment results for Turing degrees can easily be modified t o get t h e same results for hyperimmune-free degrees (for topped initial segments it is enough t o ensure this for t h e top degree, since t h e hyperimmunefree Turing degrees are downward closed), and hence for tt-degrees, as in VI.5.5. Then any uppersemilattice with a least element, countable predecessor property, and power at most N1 is isomorphic to an initial segment of P t t (Abraham and Shore [1986]). In particular, this holds for countable distributive lattices, and thus VI.4.7 applies. Other global results about Dtt and Dwt t have been obtained by Nerode a n d Shore [1980a] and Mohrherr [1984], although the full analogues of t h e results on

VI. Many-One and Other Degrees

590

absolute definability a n d automorphisms proved for Turing degrees in Section V.7 a r e not known to hold.

Elementary inequivalences W h a t we have proved until now allows us to state t h e main result of t h i s section, on t h e comparison of degree theories.

Theorem VI.5.7 (Young [1964], Lachlan [1970], Shore [1982a]) T h e theories of PI, P,,,, P t t , V w t t and , P are pairwise not elementarily equivalent, with the only possible exception of Vtt and V,tt. Proof. D ldiffers from all t h e remaining structures because it is not a n uppersemilattice, by VI.5.1, while all t h e other ones are. P, differs from all t h e remaining structures because every m-degree has a strong minimal cover, by V.5.16, while this fails in t h e other cases, by VI.5.4 a n d V.2.26 ( t h e latter implies t h a t no Turing degree above 0’ h a s a s t r o n g minimal cover). Ptt a n d Pwtt differ from P because not every tt-degree or wtt-degree has a minimal cover, by VI.5.4, while every Turing degree does, by relativization of V.5.11. 0 We do not know whether P t t a n d P W tatr e elementarily equivalent. In any case, t h e following result shows t h a t their relationship is a special one.

Proposition VI.5.8 (Shore [1982a]) Ptt and Dwtt have isomorphic cones. Precisely, Dtt(>O:,) and D,tt(>O’,,,) are isomorphic. Proof. Consider t h e natural m a p @ : Dtt --+ P,tt defined as follows, for any tt-degree a: @(a) = t h e wtt-degree of any A E a. In particular, @ ( O i t ) = O L t t . Since It-reducibility is stronger t h a n wtt-reducibility, we have a

5

b in Vtt

3

@(a)<@(a) in D,,,tt.

In particular, Oi, 5 b 3 0Ltt < @ ( b ) , a n d hence @ is a homoniorphism from Dtt(>O:,) to P w t t ( > o : t t ) . To show t h a t @ is o n t o let @(a) 5 b, a n d choose A in t h e tt-degree a,a n d B in t h e wtt-degree b. T h e n A F w t tB a n d A I l t A @ B - w t t B , a n d t h u s b is the image of t h e tt-degree of A 63 B under @. To show t h a t is one-one we prove t h a t if b is a tt-degree above 0 i t , then @(a) 5 @ ( a )in Dwtt

3

a

5

b i n Dtt.

VI.6 Structure Inside Degrees

*

59 1

It is enough t o show that if K < t t B and A SwttB then A Stt B . Let A N p,", with recursive bound f . Given z there are 2f(Z)+1 sets X (0,. . . ,f(z)}, each of them recursive. Then recursively in K we may know if p$(rc) 1. Since K stt B , we can then build a truth-table reduction of A t o B . 0 Note that no other pair of structures among Pm,P t t , P w t t ,and P admits isomorphic cones, because the properties used in VI.5.7 to show elementary inequivalence actually hold on a cone. Thus, even if Dtt and Dwtt were elementarily inequivalent, they would resemble each other more than any other of t,hese pairs. Bulitko [1980] and Selivanov [1982]have shown that the only possible truthtable-like reducibilities are: S m , & t ( l ) , 51, _ < d , and Stt (see pp. 268 and 331 for definitions). Among them, the structures induced by S c and arc isomorphic, since

sc,sp,

A Sc B e

2 < d B,

arid so are the structures induced 5 , and & ( I ) (Malc'ev [1985]). The remaining ones are pairwise elementarily inequivalent (Degtev [1979], [1985]), with the only possible exception of the structures induced by Spand Stt.

VI.6

Structure Inside Degrees *

In the last section we have compared various notions of degree from the point of view of the structures they induce. Despite the elementary differences proved i n V1.5.7, we have noted a resemblance of methods of proofs in t h e study of l-dcgreej and m-degrees on one side, and tt-degrees, wtt-degrees, and Turing degrees on the other. In the present section we take a different perspective, and analyze the possible structure of degrees of one type inside dcgrecs of another. We will again discover that l-degrees and m-degrees are close, i n the sense that thcy coincide in a large number of cases, and the same will hold for tt-degrees and Turing degrees (and hence for wtt-degrees). On the other hand, m-degrees and tt-tlcgrees never coincide, and this shows where t h e real deniarcation among degree notions lies

Cylinders Rccall that the jump operator provides a honiomorphism from

2)

to Dl , since

A ST B e A' 51B'. We now define canonical Iiomomorphisnis from Dm and Ptt to D ]

VI. Many-One and Other Degrees

592

Definition VI.6.1 (Myhill [1959], Rogers [1967]) T h e cylindrification o f a set A is the set A .N = { ( ~ , n : 2) E A } .

A is a cylinder if A

G

A . N , i.e. A and A . N are recursively isomorphic.

Proposition VI.6.2 (Rogers [1967]) A

*

A<,B

G,

A . N . Moreover,

A.Ns1B.N.

Proof. A 51 A . N via f(x) = (x,x),a n d A . N 5, A via g(x) = ( x ) ~T.h u s A_,A.N. Let A 5, B : since A . N SwLA , t h e n A . N B via s o m e recursive function f , a n d A . N < I B . N via g ( x ) = (f(x),x), which is o n e o n e . Conversely, let A.N 51 B . N : sinceA A - N and B.N B , then A B. 0

srn

<,

<,

<,

We now give some conditions for a set to be a cylinder.

Proposition VI.6.3 (Young [1966a], Rogers [1967]) The following conditions are equivalent: 1. A is a cylinder 2. for every set B , B I , A

* B 51A

3. there is a recursive function f such that, f o r every x, 0

0

W,,,) is infinite 2 E A * W,(,) 2 A x E ?I =+ W,(,) c_ A.

Proof. 1 implies 3 because if A G A . N then there is a one-one recursive function g reducing A . N t o A , a n d t h u s it is enough to let

W,(,) = { g ( ( x 4 , ): 71. E w}3 implies 2 because we can use W,(,. t o t u r n a many-one reduction g of B to A into a o n e o n e reduction, as follows. Let h(0)= g(0). Given h(z),let h ( x + l ) be t h e first element generated in W f ( 9 ( s + al )n )d not in {h(O),.. . , h ( z ) } . 2 implies 1 because A < 1 A . N a n d A . N 5 , A always hold, a n d by 2 t h e latter implies A . N < I A. T h u s A G I A . N , a n d A G A . N by 111.7.13. 0 Exercises VI.6.4 a) A set A is a cylinder if and only iJ f o r some recursive function g and every D, # 0,

D, C A

* g(z) E A - D,

and D, C 2

* g(z) E 2 - D,.

VI.6 Structure Inside Degrees

*

593

(Rogers [1967]) (Hint: the condition is equivalent to VI.6.3.) b) Logical theories are cylinders. (Hint: show that condition VI.6.3.3 is satisfied, using the fact that cp and ~ i c are p equivalent.) c) Every cy2inder is a splinter. (Myhill [1959]) (Hint: see the proof of III.7.lO.c.) d) A cylinder is either recursive or pseudocreative. (Hint: by c) and III.7.10.a.) e) The only recursive sets which are cylinders are 0,w , and the infinite winfinite sets. (Hint: by VI.6.3.3.) It follows that not every splinter is a cylinder. Young [1966] has shown that there are infinite coinfinite splinters which are not cylinders. f ) Every creative set is 0 cylinder. (Myhill [1959]) g) Not every pseudocreative set is a Cylinder. (Young [1964a]) (Hint: let B be a simple set. Then B . B is pseudocreative, since if z E then {z} * B is an infinite Choose B such that B . B gmB, see 111.8.5. Then B . B is not r.e. subset of a cylinder, otherwise there is f like in VI.6.3.3, and since W,(,, is infinite and B is simple there is (a,b) E Wf(,) such that one of a and b is in B. Let g(z) be the other. Then z E B . B ej g(z) E B, contradiction.)

B

m.

We now turn to t h e analogue of the notion of cylinder for tt-reducibility.

Definition VI.6.5 (Rogers [1967]) The tt-cylind$cation set

Att = {x : A A i s a tt-cylinder i f A

of a set A is the

k oz}.

Att.

Proposition VI.6.6 (Rogers [1967]) A

~

t

Att. t Moreover,

A 5tt B e At" 51 Btt. Att via f such that CT~(~. is the tt-condition 'z E X ' , and Proof. A Att Stt A via the identity function, since x E Att e A uz by definition. Let A Stt B : since Att 5tt A then Att _
+

2

E Att

*Bk

f(.)

E Btt.

Thus Att 5, Btt. But f can be made one-one by induction, by substituting tt-conditions with equivalent ones if necessary (obtained by adding redundant clauses, like n E X V n @ X ) . Thus Att 51 B t t . Conversely, if Att 51 Btt then A < t t B , since A 5tt Att and Btt Stt B . 0

Proposition VI.6.7 (Rogers [1967]) The following conditions are equivalent: 1.

A is a tt-cylinder

2. for every set B , B

I t t

A + B 51A

VI. Many-One and Other Degrees

594

Proof. 1 implies 2 because if B Stt A then B SrnAtt by definition, and hence B 51 Att as in the proof of VI.6.6. If A is a tt-cylinder then Att 51A , and hence B 51A . 2 implies 1 because A
Inside many-one degrees Since 51 is stronger than Srn,an m-degree can be thought of as consisting of 1-degrees.

Definition VI.6.8 An m-degree is irreducible i f it consists of only o n e 1-degree. Obviously, a n m-degree i s irreducible if and only if it contains o n l y cylinG~ A . N always holds, Trivial examples of irreducible m-degrees are (0) and {w}. The first example of a nontrivial irreducible m-degree was given by Myhill [1955], who showed that the m-degree of K: is such a degree (see 111.6.6 and 111.7.5). By relativization, t h e m-degree of a jump set A' is irreducible, and thus every Turing degree above 0' contains an irreducible m-degree.

ders, since A

Proposition VI.6.9 (Kobzev [1975]) If A is r.e. and nonrecursiwe, t h e m-degree of Att is irreducible. Proof. Consider B Z E Att. ~ Since Att is a cylinder, from B 5 , Att we have B 51 Att. For the converse, let Att 5, B via f : we want t o show t h a t we have infinitely many equivalent choices for each value o f f , so that f can be turned into a one-one reduction, and Att 51 B. Consider

for a given y: 0

if z E Att then A

0

if

5$ !

Att then A

b o,,and thus

k loz, and thus

VI.6 Structure Inside Degrees *

595

These are still only equivalences, but if we choose y @ Att we explicitly have: 0

if x E Att then z E Att and f(z) E B

0

if x @ Att then z @ Att and f(z) @ B.

Fix then a n r.e. subset C of

f(z) E

F . There is a recursive function g such t h a t

Wg(,) e+ (3Y E C)[az= (0,A

v (10,A ay)].

Moreover, Wg(,)is an r.e. subset of B if x E Att, and of otherwise. It remains t o choose C in such a way that W,,,. is always infinite. Note t h a t it is enough t o have C and Att recursively inseparable, because then if W,(,, were finite t h e set

(with a, as above) would be a recursive set separating C and At t . Indeed, C C R by definition. Moreover, once x is fixed the only thing t h a t matters for z , a n d hence for f(z), is whether y is in Att or not. Since for y 6 C we have y E R, a n d C p , it cannot be t h a t y E Att n R, a n d hence R p. -To find C as wanted, note that A 51Att and Att 51 F (the latter because Att < t t Att, a n d Att is a tt-cylinder). Thus A 51F . Let C be t h e image of A under this 1-reduction: it is a n r.e. set because A is, and it cannot be separable frorn by a recursive set, otherwise A would be the inverse image of this set,

c

c

a n d it would then be recursive.

0

Corollary VI.6.10 Every nonrecursive r. e. tt-degree contains an irreducible m-degree.

Proof. A

=tt

Att.

0

While not every nonrecursive tt-degree contains a n irreducible m-degree (Degtev [1979]), every nonrecursive Turing degree contains such a n m-degree (Downey [1993]). We only prove a weaker result, namely t h a t almost every nonrecursive Turing degree contains a n irreducible m-degree.

Proposition VI.6.11 (Degtev [1979])Every n r i n g degree not below 0' contains a n irreducible m-degree.

Proof. We modify t h e proof of the previous result to show t h a t if A is nonrecursive and the m-degree of Att is not irreducible, then A E A;. T h e result follows from t h e Limit Lemma IV.1.17, and the fact t h a t A =T Att. Since Att 51 p (because Stt A t t , and Att is a tt-cylinder), it is enough to show t h a t Att E E!. Since At' is a cylinder, if its m-degree is not irreducible

VI. Many-One and Other Degrees

596

t h e n it must contain a set B which is not a cylinder, a n d t h u s such t h a t Att B but Att B (by V1.6.3.2). Let f be a recursive function such t h a t z E Att ts f ( z ) E B . Given n, let

srn

f(.)

E Wg(n,,)

@

(*)[a2 = 0, A ( on v (3Y)[.2

= 0,

4 1v

v (-on Aol,)I.

We show t h a t

n E A t t e ( ~ E ) ( W ~ finite). (~,,) T h e n Att E E!, becaiise finiteness of an r.e. set is a E! condition, since it can be expressed as saying t h a t there is a number such t h a t all greater ones a r e not in t h e set.

1. If

TI E

At' then A

A

bg,

A

b u?,, and

hence

(onV uy) ts A

0,

V (yonA al,)ts A

b a,.

If Wg(,,,, is infinite for every z t h e n we can easily get ALt51 B , contradicting t h e hypothesis.

2. If n @ Att then A zE z

Att

# Att

--oTL, a n d hence 3

* *

A uz A ( on V oy)iff A I= al, {f(z) : oz = a, A ( on Va,)} is infinite A u, V ( 1A oV) ~ iff ~A ay {f(z) : uz = a, v (-on A oU)}is infinite,

otherwise A is recursivc (because A 5, A t t Srn B ) .

0

Exercises VI.6.12 A set A is perfect if it is q-closed, for some nontrivial r.e. equivalence relation q whose only recursive yclosed sets are 0 and w . Note that every equivalence class of q is infinite and r.e. a) If A is perfect then its m-degree is irreducible. (Ershov [ 19711)(Hint: if B <_m A via f, define g one-one reducing B t o A by letting f(z) and g(z) be in the same B via f, consider the equivalence class: this works because A is ?closed. If A new equivalence zq'y ts zqy V f ( z ) = f(y). A is still closed and perfect w.r.t. q * . f m-reduces [zItl-to f([zIq.),and thus the latter is still infinite. Define g one-one reducing B to A by letting g(z) E f([z],.).) b) Not every irreduczble m-degree contains a perfect s e t . (Denisov [1974]) (Hint: modify III.6.23.c to prove that if B 511 A and B is perfect then A is not hypersimple. Then use V1.6.10.)

<,,

We have now examples of irreducible m-degrees, as well as of m-degrees containing infinitely many 1-degrees (like Om). These a r e t h e only possible cases.

VI.6 Structure Inside Degrees

*

597

Proposition VI.6.13 (Young [1966a]) An m-degree contains either only one or infinitely many 1-degrees. Proof. If there is more than one 1-degree in a given m-degree, there is a set A which is not a cylinder. Note that

are all in t h e same nr-degree, since @ induces the 1.u.b. for m-degrees. Since B < I B @ B always holds, the two following facts produce, by induction, an infinite ascending chain of 1-degrees in t h e given m-degree:

1. B @ B 51 B 3 B @ B cylinder Given f one-one such that

Z E B @ B@ f ( z ) E B e 2 f ( z ) E B @ B ,

2. B @ B cylinder 3 B cylinder If g satisfies VI.6.3.3 then

It is thus enough t o let

Wh(,) = ( 2 : 22 E W&) v 22+ 1 E Wg(2,)}. 0 T h e proof shows that if a n m-degree contains infinitely many 1-degree then it contains a n infinite chain. Young [1966a] shows that actually every countable linear ordering is embeddable in the 1-degrees of such a n m-degree. It is not known whether there must always be an infinite antichain as well. Exercises VI.6.14 a) Every m-degree contains a grentest 1-degree (consisting exactly of the cylinders in the given m- degree). (Rogers [ 19671) (Hint: given A consider A . N.) b) There are m-degrees without least 1-degree. (Dekker and Myhill [1960]) (Hint: if A is simple and z E 2 then A - { z } =m A but A - { z } <1 A, by the proof of

VI.5.1.)

VI. Many-One and Other Degrees

538

Inside truth-table degrees We now look at m-degrees inside tt-degrees.

Proposition VI.6.15 (Jockusch [1969]) Every nonrecursive tt-degree contains infinitely many m-degrees. Proof. Let C be a nonrecursive set. Consider t h e t r e e of binary sequence numbers, a n d t h e branch A defined by C : zE

A

* Seq(z)A

(~~)l<"g"(z)l(~ =) C " ( n-

111.

By definition A ~ t Cl (since there a r e only finitely many binary sequence numbers of a given length). Moreover, A is retraceable via

f ( b I , . .. ,

zni

&+1))

= (51

1 . .

. ,G I ) .

Being nonrecursive a n d retraceable, A is immune (by 11.6.5). I t is also not hyperimmune, because t h e strong array whose elements a r e t h e sets of binary sequence numbers of a given length intersects A . For n 2 1, let (Z)% be t h e recursive product of with itself n times:

(xi)"+'

We now prove $, (2)71, Clearly, (7)"-tt A , a n d (7)"2, (7)n+1. so t h a t t h e tt-degree of A (and hence t h a t of B ) contains infinitely many m-degrees. T h e plan of t h e proof is t h e following. Suppose (Z)"+' 5, (A)n.We want to find a n r.e. subset B of 7 so big t h a t at most 2n elements at a n y level of t h e tree a r e in B. We t h e n get a contradiction as follows. From t h e fact t h a t A is not hyperimmune we have a recursive function g such t h a t g(s)majorizes t h e z-th element (i.e. t h e o n e of level z) a, of A . T h e n we can define S, such t h a t IS,I 5 2n as follows: take all elements less t h a n g ( ~ which ) are o n level z of t h e tree, a n d eliminate those t h a t a r e generated in B , until at mast 2n remain. Clearly, a, is not eliminated, because B 5 We t h u s get a strong array of bounded cardinality intersecting A , contradicting t h e fact t h a t A is immune (11.6.10.b). I t remains to find B . Note t h a t

x.

(7)"-, {y : ID,I I n so t h a t t h e hypothesis {y : )D,I

A D,

nA # 8),

(x)n+l 5, (A)"can be reformulated as:

5 ? L + 1 A D , nA # 8) I ,

{y : ID,]

In

A

D, nA # 8).

VI.6 Structure Inside Degrees

*

599

Thus there is a recursive function h such that

Let

where zID, means that z and any element of D, are not sent by iterations off on a same element, and x I+ D, that z is sent by iterations of f on a n element of D,. Clearly, B1 U B2 is r.e. Moreover:

1.l3cz

Suppose z E B1 n A. For some y, z E D, n A and zlDh(,), so that D, n A # 0 and hence n A # 8. But the latter contradicts x/Dh(,), because the elements of A are all sent by iteration o f f on a same element (e.g.1 ao).

Suppose z E B2nA. For somey, "ID, and z H Dh(v),so that D,nA = 0 (because x E A and xlD,) and hence Dh(,) r l A = 0. But the latter contradicts z t-)Dqy),because z E A and it is thus sent only on elements of A by iterations of f .

2.

has at most 2n elements at any level of the tree For t h e sake of contradiction, suppose there is { X I , . . . ,~ 2 ~ + 1 C} B, whose elements we may suppose to be ordered from the left on the tree. Consider

D, = {z1,23,z5r. .. 1 2 2 n + 1 } 1

+

1 elements. By hypothesis D, E B,and in particular Since ID,[5 n 1 and zpi+l E D,, then it cannot be z2i+llDh(,). Since IDh(,)I 5 n and ID,I = n + 1, there must be i < j such that z2i+l and z z j + l are sent on a same element of Dh(,). Since zzi+l and zzj+l are on the same level of the tree, also z2i+2 (which is between them and on the same level) must be sent on the same element, by the definition of f . But this is impossible, since then 22i+z(D, but z2i+2 t-+ Dh(,), and hence z2i+2 E B2, contradicting the hypothesis t h a t which has n

D, G

xpi+p

B1.

EB.

+

0

T h e proof shows that every nonrecursive tt-degree contains an infinite chain of m-degrees, and Stephan [199?] has proved the same for infinite antichains.

600

VI. Many-One and Other Degrees

Exercises M.6.16 a) Every tt-degree contains a greatest m-degree and a greatest f-degree. (Rogers [1967]) (Hint: consider Att.) b) No nonrecursive tt-degree contains a least m-degree. (Jockusch) (Hint: let A be an immune retraceable set in the given tt-degree, by 11.6.13. Then A and 2 form a minimal pair of m-degrees. Indeed, let z E C f(z) E A g(z) E 2. If f(z) and g(z) are sent by the retracing function into the same element, this is in A. The set of such elements is finite, being r.e. Let a E A be greater than its maximum. If C is not recursive the ranges of f and g are unbounded, and A - (0,. . . ,a - 1) is r.e.,

*

*

contradicting immunity: if z 2 a choose z such that f(z),g(z) > z . See if one is sent into z by the retracing function. At most one of them can be, by the choice of a, and if z E A one does.) c) Every nonrecursive tt-degree contains incomparable m-degrees. (Jockusch [l968a]) (Hint: let A be a semirecursive set in the given tt-degree, by 111.5.5. Then A and 2 are m-incomparable.)

Inside Turing degrees We now look at tt-degrees inside Turing degrees.

Definition VI.6.17 A Turing degree is irreducible if it consists of only one tt-d egree. The irreducible Turing degrees are old friends.

Proposition VI.6.18 (Jockusch [1969], Martin) A &ring degree is imducible if and only if it is hyperimmune-Fee.

Proof. We have already proved in VI.5.5 that if A has hyperimmune-free degree and B ET A then B ~ t A.t Thus a hyperimmune-free degree contains only one tt-degree. Suppose now that the Turing degree of A contains only one tt-degree. We may suppose that A has the greatest m-degree in it (otherwise we can consider Att). Suppose f ST A is not recursively majorized. With no loss of generality we may suppose f increasing. Let e E B e { e } ( e )converges in less than f ( e ) steps, and {.}(el $ A. Then B ST A , and A ET A@B. We now show that B grnA: thus A < m ABB, contradicting the fact that A has the greatest m-degree. Suppose B SrnA via g, Let eo, e l , . . . be indices of g , with e, > x. Then

e, E B e g(e,) E A

@

{e,}(e,) E A.

Thus {e,}(e,) must converge in more than f(e2) > f(x) steps (recall that f is increasing), and the number of steps needed t o compute {e,}(e,) is thus a

VI.6 Structure Inside Degrees recursive function majorizing

*

601

f, contradiction.

0

As in t h e case of m-degrees, there are only t w o possibilities. Corollary VI.6.19 A Tiiring degree contains either only one or infinitely many tt-degrees.

Proof. T h e proof given above works also in the case t h a t a Turing degree has a greatest tt-degree, by choosing A in t h e greatest m-degree of t h e greatest tt-degree. Since A @ B G-TA , A @ B I t t A because A is in t h e greatest tt-degree, a n d hence actually A ~ t At @ B . Thus A <m A @ B produces a contradiction, because A has greatest m-degree in its tt-degree. But if there are finitely many tt-degrees, there is a greatest one (their 1.u.b.). T h u s t h e Turing degrees consisting of finitely many tt-degrees are still t h e hyperimmune-free ones, i.e. the irreducible ones. 0 Exercises VI.6.20 a) A Turing degree has a greatest tt-degree i f and only i f it is irreducible. (Hint: by the proof of the corollary.) b) Non-irreducible Turing degrees contain an infinite chain of tt-degrees. (Hint: by part a.) c) Non-irreducible Turing degrees contain a n infinite antichain of tt-degrees. (Deg= (a0 < a1 < - - - } ,A tev [1972]) (Hint: this uses the priority method. Let hyperimmune. We build A , in the Turing degree of A so that, for m # n, A , &t A,. Code A into A , by letting z E A 2z+1 E A,: thus A
*

*

+

This Page Intentionally Left Blank

Bibliography We include only references quoted in the book. A mmplete bibliography of Recursion Theory has been edited by Hinman as Volume IV of t h e R-Bzbliogruphy of Mathematical Logic, Springer Verlag, 1987. On-line bibliographies can b e found at t h e following Web addresses: 0

http://vvv-1ogic.uni-kl.de/bibl/index.html http://vvv.nd.edu:80/-cholak/computability/bib/bib.html

We indicate publications by abbreviating their original names. T h e following is a list of full names of Russian publications and of their official translations.

1 . Algebra i Logika (Algebra and Logic) 2 . Doklady Akademii Nauk S.S.S.R. (Soviet Mathematics, Dokladi)

3. Izvestiya Vysshikh Uchebnykh Zavedenij Matemataka (Soviet Mathematics, Izvestiya) 4. Kibernetika (Cybernetics)

5. Matematicheskiz Sbornzk (Mathematics of the U.S.S.R.) 6. Matematicheskie Zametki (Mathematical Notes of t h e Academy of Science U.S.S.R.)

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Notation Index Introduction set of natural numbers 13 power set of w (set of all subsets of w ) 13 set of total functions kom w to w 13 set of partial functions from w to w 13 cardinality of A 14 disjoint union of A and B 14 recursive product of A and B 14 characteristic function of A 14 bounded existential quantifier 15 bounded universal quantifier 15

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Qi 3i

Ii

successor function 19 least number operator 21 constant zero function 22 i-th projection of n arguments Robinson Arithmetic 23 Peano Arithmetic 24 integer difference 25 bounded sum 25

22

bounded product 25 x divides y 25 xis aprime 25 the 2-th prime number 26 bounded poperator 26 26 exponent of k in the decomposition of pairing function 27 right and left inverses of 9 27 G a e l ’ s &function 28 numeral for n 32 system of equations 32 Tarski, Mostowski and Robinson arithmetic 44 state of a Turing machine 48 symbol of a Turing machine 48 instruction of a Turing machine 48

643

Notation Index assignment 62 abstraction operator 76 fixed-point paradoxical combinator 79 combinators 83 sequence number mding 5 1 , . . . ,x n 88 n-th component of x 88 length of x 88 x is a sequence number 88 concatenation of x and y 89 subsequence predicates 89 history function (course-of-values) o f f 89 normal form predicate 90 normal form function 90

Chapter I1

IWT

PR

equality for partial functions 127 'p is defined (converges) a t x 127 is undefined (diverges) at x 127 p extends a as a partial function 127 n-ary partial recursive function of index e 130 n-ary partial recursive function of index e 130 parametrization function 131 n-ary r.e. relation of index e 134 set of indices of total recursive functions 146 diagonal r.e. set 147 master r.e. set 150 index set of A 150 Kolmogorov complexity of x 151 Turing reducibility 176 structure of Turing degrees 176 functional 177 normal form predicate for functionals 179, 180 partial recursive function with oracle A of index e partial recursive function with oracle A of index e set of partial functions from w to w 186 basic open set determined by u 186 set of continuous functions fmm D to D 194 order relation on c.p.0.'~ 195 inverse limit of a chain of c.p.0.'~ 195 enumeration reducibility 197 weak Turing reducibility 198 set of partial recursive functions 205 set of total recursive functions 208 system of indices 215 set of characteristic indices of recursive sets 226 set of r.e. indices of recursive sets 226 finite set of canonical index e 226 set of r.e. indices of finite sets 228 r.e. class of r.e. sets 228 reducibility for enumerations 236 equivalence for enumerations 236

181 181

Notation Index Lo(A)

L(”4

645 structure of equivalence classes of r.e. enumerations of A 236 structure of equivalence classes of enumerations of A 238

Chapter I11 0 0’ I m , Em

vrn Om

o:, Ie I d

IP cn I t t , -tt

I1

Dtt

T-degree of recursive sets 252 T-degree of K 252 m-reducibility 257 structure of m-degrees 257 m-degree of nontrivial recursive sets 257 m-degree of K 257 conjunctive reducibility 268 disjunctive reducibility 268 positive reducibility 268 truth-table condition 268 truth-table reducibility 268 linear reducibility 269 structure of tt-degrees 270 tt-degree of recursive sets 270 tt-degree of K 270 Q-reducibility 281 e-state of x at stage s 291 positive equivalence relations 300 7)-closure of A 300 *reducibility for functions 308 I-reducibility 320 structure of 1-degrees 320 1-degree of infinite coinfinite recursive sets 320 1-degree of K 320 recursive isomorphism 324 recursive equivalence 328 set of isols 328 extensions to isols o f f and R 329 bounded truth-table equivalence 331 structure of btt-degrees 331 btt-degree of recursive sets 331 btt-degree of K 331 weak truth-table equivalence 337 structure of wtt-degrees 337 wtt-degree of recursive sets 337 wtt-degree of K 337 s-reducibility 340 bounded search reducibility 340

Chapter IV e, t’ A, A* Ab’p

x:, n:,

AS1 A0 C&, , n&, , A&,

languages for first-order arithmetic 364 structures for first-order arithmetic 364 ‘p is true in A 364 arithmetical hierarchy classes 367 arithmetical relations 367 bounded arithmetical hierarchy classes 368

Notation Index

646

Boolean hierarchy classes 373 relativized arithmetical hierarchy classes 374 A:-reducibility 375 arithmetical reducibility 375 structure of arithmetical degrees 375 languages for second-order arithmetic 376 structures for second-order arithmetic 376 'p is true in A2 376 analytical hierarchy classes 379 analytical relations 379 set of characteristic indices of recursive well-founded trees 383 ordinal of the node u on the tree T 383 ordinal of the tree T 383 Church-Kleene ordinal 384 tree associated to the ordering 4 385 ordering associated to the tree T 386 set of characteristic indices of recursive well-orderings o f w 386 relativized analytical hierarchy classes 394 A:-reducibility 394 structure of A;-degrees 394 ZermeleFkaenkel set theory 399 generalized Kripke-Platek set theory 399 Z F minus the power set axiom 400 cumulative hierarchy classes 400 universe of set theory 400 structure for set theory 400 'p is true in A 401 set-theoretical language hierarchy classes 406 settheoretical theory hierarchy classes 406 set-theoretical structure hierarchy classes 406 transitive closure of x 409 hereditarily finite sets 414 constructible hierarchy classes 423 constructible sets 423 Von Neumann-Gael-Bernays set theory 423 axiom of constructibility 423 well-ordering of L 430 constructible sets of natural numbers 432 hereditarily countable sets 441 relative constructibility 444 constructibility reducibility 444 structure of constructibility degrees 444

Chapter V alb aUb

an76 @ncIAn a'

Q' ~ ( n )a(n) ,

incomparable degrees a and b 448 least upper bound (join) of a and b 449 greatest lower of bound of a and b 449 infinite join 449 jumpof a 450 structure of degrees with jump 450 n-thjump 451

Notation Index

647 w-jump 451 wneabove a 452 axiom of determinacy 453 a string 456 length of u 456 concatenation of u and r 457 coinfhite condition 484 principal ideal below a 487 a tree 493 nodesofT 493 left and right functions of an admissible triple 516

Chapter VI cylinder of A 592 %cylinder of A 593

This Page Intentionally Left Blank

Index n,'

0' 252, 459, 461, 462, 464, 468, 469, 470,

374 368, 369 no classes 505-512 367,369-372, 514 II' 38Ck387, 419, 420 II!, 374-381 H E K p 411-413 c,' 373,374 E:,, 368, 369 Co 368 C i 367,369-372, 514 C' 361,420, 437-440, 442 E? 379-381 E z K P 411-413 C f F - 426, 430 w-BRFT 222-224 w-consistency 160, 163, 167 w-jump 451, 543, 552, 553 wyk 384, 385, 387, 443 wf 432

I:,

488, 495, 498, 500, 501, 503, 508, 513, 570, 589, 594, 595 0" 464, 497, 498, 500,501, 523 0(3)551 O ( w ) 543, 544, 548, 552, 553 1-generic 468, 484 1-1.u.b. 553 1-tree 520 2-1.u.b. 552, 553 a-rule 77 @-rule 77 A;' 374 A:,, 368, 369 A' 285, 341,373, 507,514, 515 A, 367, 369,372-375, 514, 515

a

A: 367 A' 387-392, 395 A! 396, 437 A: 5, 379-381, 391, 395, 438, 540, 543 A; 379 AFKP 40G410, 419 A$KP 4&411. AfF- 426, 430, 431,443

Abel 7 Aberth 9, 214 Abian 189 Abraham 529, 571, 589 absoluteness 418-420, 426, 442, 537, 575 acceptable system of indices 215221,236, 271, 272, 292, 308, 346 Ackermann 165, 415, 417 A D 45S456 Addison 10, 373, 391, 393, 395, 440 adjacent strings 520 Adler 9 admissible ordinal 9, 443, 444 set 421 triple 516, 520 Agenvala 74 algebra 7, 8 combinatmy 22S225

€-induction 399 7)-

x-

closed 300 finite 300 (hyper)hypersimple 300 infinite 300 maximal 280, 301, 302 rule 82 simple 300

abstraction 76 algebra 224 calculus 76-87, 194-196, 22S225 definability 84, 105, 132 model 224, 225 p-recursion 21, 128, 129

649

650 algebraic c.p.0. 192, 201 ALGOL 70, 186 Ambos-Spies x analogue machine 109-113, 117 analysis 2, 8, 9, 213, 214, 376, 422 see also second-order arithmetic analytic set 392, 393 analytical hierarchy 375-397, 418, 419, 420, 438, 441, 442 engine 134 set 3, 361, 376, 377, 540, 543 Anderson 115 antichain 464, 466, 530, 597, 599, 601 Appel 242, 243, 295 Arbib 50, 53, 116, 117, 133, 172, 173 Aristotle 113, 127, 363 arithmetic 11 first-order 24, 363-365 intuitionistic 10 Peano 23, 28, 46, 164, 169,417, 418, 442, 510-515 primitive recursive 23 R 44, 99, 101, 159, 162, 165-168, 307, 353, 355, 356, 359, 368, 513 Robinson 23, 44, 165, 169, 359 second-order 24, 376, 430, 442, 5 3 6 539, 541-543, 545, 546, 550, 551, 579, 581, 582, 584, 589 arithmetical (Q) degree 375, 541-545, 547 hierarchy 362-375, 381, 392, 393, 415-417, 438 reducibility 375 set 3,5,361, 364, 371,375,390,415, 419, 451, 514, 541-545, 548, 550, 577 arithmetization 11, 87-90 array 228 Arslanov ix, 255, 256, 277, 308, 338 Asanuma 117 Ashby 115 assembly programs 60 associate 200 atomism 51, 113 automaton cellular 109, 170, 173, 174 finite 52, 53, 116 Life 173, 174 automorphisms 546551, 576, 577, 590 basis 548, 550

Subject Index autoreducible 501, 502, 523, 528 axiom choice 399, 430-432, 442, 444, 453455 collection 398, 421, 427, 442 comprehension 397, 398, 442 constructibility 393, 423, 429-432, 434, 44&442, 544 determinacy 393, 453-456, 544,546 extensionality 397, 4 W 4 0 5 foundation 399, 422 infinity 398, 400, 415, 417, 418, 426, 442 Martin 475 pairing 398 power set 398, 400, 442 replacement 398 separation 398, 421, 427 union 398 axiomatic Recursion Theory 221 axiomatizability 350, 352, 357-360, 427, 510, 537, 581 finite 357, 510 independent 357-360 Babbage 113, 134 Backus 64, 68 Baire vii, 4, 392, 448, 471, 473-475, 477, 478, 484,495 category 471-484, 495 theorem 475, 477, 495 property 477 space 471, 473 Baker 22 1 Banach 193, 475, 478 fixed-point theorem 193, 194 -Mazur game 475, 476, 478 Bardzin 53 Barendregt 83, 196, 223, 225 Bar-Hillel 397, 423, 427 Barna 193 bar-recursion 382 Barwise 421 basis 507-512 automorphism 548, 550 Kreisel 507 low 508 Scott 510 Beeson 11, 196, 210, 214, 223-225 Beige1 298 Bell 112 Benioff 5 2

Subject Index Bennett 51, 52 Berlekamp 173 Bernays 1, 72, 90, 169, 352, 423 Bernstein 324 Berry (paradox) 262 Bessel 111 Beth 118 Bezboruah 169 Bienestock 117 bi-immune 266, 267, 498 biology 172 Birkoff 112 Bishop 214 Blackwell 393 Blum 220, 314 Bohm 70, 81 Boole 113 Boolean hierarchy 373, 374 Boolos 436, 437 Boone 8 Bore1 8, 213, 392, 456 Born 107 bounded p-operator 26 arithmetical hierarchy 368, 369 product 25 quantifier 15, 25, 365, 369, 405, 411 reducibility 340 sum 25 truth-table (btt-) completeness 331, 333-335, 337, 341-343,347, 349 degree 331 reducibility 331, 332, 373, 591 Bourbaki 113 brain 115118, 122 branch 381,494, 505-510 Bremermann 51 BRFT 222 Brouwer 119, 122, 127 Brown 189 Bulitko 269, 591 Burks 172-174 Bush 117 Byerly 223, 225 canonical index 226 system 105, 143, 144, 253, 254 Cantor 1, 23, 26, 146, 324, 398, 402, 471, 473, 474,494 space 471, 473

651 theorem 146, 402 cardinal 9 measurable 393, 456 Cardone x Carnap 165 Carpentier 319 Cartesian closed category 196, 218 Casalegno 199 category 196, 218, 223 Baire 471-484, 495 Cauchy 211, 213 Cayley 193 Ceitin 209, 211, 213 cellular automaton 109, 170, 173, 174 chain of degrees 466, 489, 505, 523 Chaitin 151, 263 characteristic function 14, 15 index 225227 Chinese remainder theorem 29 choice axiom 399, 43@432, 442, 444, 453455 functions 137, 138, 229, 230, 232 Chomsky 144 Chong 421, 443 Church 1.37, 78,82-84, 98, 1 W 1 2 3 , 147, 148, 152, 162, 164-166, 254, 385 -Kleene ordinal 384, 385, 387, 443 rule 121 theorem 164, 165 thesis 101-123, 254 class 400, 423, 427 Cleave 319 Cobham 44 Codd 173 coding 26, 27, 31, 88 procedure 279 Cohen P.J. 2, 164, 392, 397, 432, 437, 468 Cohen P.F. 266, 273, 294, 316 cohesive 288-290, 498 coinfinite condition 484 extension 484, 493, 500, 52Cb523 collapsing lemma 402-405, 421, 434, 435, 439, 442 collection axiom 398, 421, 427, 442 Colmerauer 39 combinator 81, 83, 87, 223 cornbinatorial (set) function 329 combinatory

652 algebra 22S225 logic 223, 224 comeager 473-477, 479, 483, 489, 495, 503, 527, 546, 549-550 compactness 181, 190, 192, 199, 206, 209, 211, 473 compiler 61 complementary index 225 complete IIi set 383,386 y-l l , n-1 , a -, , set 374 EE, n: set 370, 372, 451 EA, IIk set 381 chain- 189, 190 enumeration 237 extension 51G515 formal system 350 partial ordering 189, 192 complete r.e. set 306, 341-349 1- 320, 321, 348, 349 bs- 340, 341 htt- 331,333-335,337,341-343,347, 349 c- 275, 306, 311, 313, 341-343, 348, 349 d- 306, 311, 312, 335, 341-344, 347349 rn 258, 259,305,307,308,319,341, 343, 347-349 p 295, 341, 343, 344, 347, 349 Q- 282, 286, 287, 294, 295, 297, 299, 301, 306, 314, 316, 34-343, 349 T- 253, 255, 264, 265, 277, 293-295, 304, 309, 339, 341, 342, 344, 349,353 tt- 27G272, 274, 334,341,344, 346, 347,349 wtt- 338-342,344, 347,349 completely autoreducible 502, 523, 528 productive 304, 322 component 89 comprehension axiom 397, 398, 442 computability 6, 17, 101, 102, 104, 125126, 197, 202-205, 361, 386, 39 1 flowchart 61-70, 99, 101, 132, 197198 Herbrand-Gael 36-38,98, 101, 105, 132, 198 prime 203-205, 222, 396

Subject Index search 204, 205, 396, 421 Turing 53-54, 99, 101, 132, 197 computation deterministic 18, 50, 107-109, 197 (ir)reversible 51, 52, 174 nondeterministic 18, 50, 74, 198 theory 223 tree 91 computers 6, 64, 104, 110, 112, 115-117, 133 concatenation 89 cone 452, 454, 503, 526, 544, 545, 548, 550, 551, 576, 584, 590, 591 configuration of a Turing machine 49 conjunctive (c-) completeness 275, 306, 311, 313, 341-343, 348, 349 reducibility 268, 282, 591 consistent extension 491, 492, 51G515 formal system 114, 115, 168-170 with Z F C 431, 432, 434, 435, 438, 440, 445, 453, 454, 529 constructibility axiom 393, 423, 429-432, 434, 44G442, 544 constructible 4, 5, 361, 422-445, 543 constructive mathematics 7-10 constructively immune 267 contiguous degree 279 continuous function(a1) 187-192, 205, 206, 211, 214, 393 effectively 188, 189, 191, 207-210, 212, 213 continuum hypothesis 164, 392, 422, 430, 434, 459, 468 contraction of quantifiers 365, 377, 405 contraproductive 310, 311, 313,319, 322 conversion rule 77, 82 Conway 173 Cook 197 Cooper 285, 498, 501, 523 countable functional 200, 201 course-of-value 89, 410 Cowan 116 Craig 357 creative 30&312, 314, 318, 319, 321, 322, 323, 327,348, 356,593 p s e u d e 323, 324, 593 quasi- 311-313,336, 348 semi- 313 s u b 314, 315, 349 Crick 117

Subject Index Crossley 8, 330 cumulative hierarchy 400 Curry H.B. 79, 83 Curry J. 193 cut 295 cylinder 583, 592-594, 597 tt- 593, 594 2, 176, 449, 451-455, 457, 459, 462, 463,

483, 488, 490-492, 493, 516, 526, 528-530, 534, 536-550, 587, 591 D' 450, 467, 469, 541, 55Ck-553 D1 320, 582-584, 591 D, 257, 555,556,558, 561,562, 565568, 5 7 6 5 7 7 , 581, 582, 591 Vtt 584-591 D,tt 584, 584-591 Dalil 68 Dante 100, 152, 162, 170, 304, 575 Darwin 152 Davis53, 70, 106, 133, 135, 179-181, 197, 200, 368, 391 De Broglie 107 De Leeuw 50 De Witt 107 decidability 253, 350, 462, 492, 581 Dedekind 1, 7, 19-24, 213, 328, 490 deficiency set 263, 265, 266, 287, 349 stage 246, 277 definability absolute 540-543 arithmetical first-order 361, 364, 415 second-order 361, 376, 436, 437, 542, 543, 545, 551 invariant 44, 45 of functions 45, 46 over D 449, 450, 530, 534, 54C543 over D' 550, 551, 553 over D1 584 over D, 576, 577 over Dtt 590 over 'HC 442 over '?dF4 15 over L 43S440, 443 set-theoretical 361, 401, 423 degrees (notions of) 2, 4 1- 320, 325, 582-584, 590, 591, 594, 597, 600 A:- 394, 440, 543

653 a- 375, 541-545, 547

b t t 331 e- 340 hyper- 395 L 444, 445, 543 m- 238, 257, 352,502, 555-582, 590, 591, 594-598, 600 partial 197, 199 Q- 282 T- 175, 176, 263, 276, 278, 287, 297, 299, 304, 310, 313, 316, 323, 328, 351, 447-553, 568, 570, 574, 575, 583, 589-591, 595 tt- 270, 276,296, 317,338, 352, 584591, 595, 5 9 S 6 0 1 wtt- 337,338, 584-591 degrees (types o f ) below 0' 459, 461, 462, 464, 487, 488, 498, 500, 501, 589 completely autoreducible 502, 523, 528 hyperimmune-free 495, 496, 497, 500, 501, 505, 509, 520, 523, 588, 589, 600 low 508, 513 minimal 465,479,483,484,498-502, 513-523, 527, 528, 549, 577,

583, 588 r.e. 252, 257, 263, 270, 276, 278, 287, 297, 299, 304, 313, 316, 317, 320, 323, 328, 331, 337, 338, 351, 352, 508, 509, 513, 577, 595 Degtev x, 244, 248, 298, 302, 328, 340, 581, 591, 595, 601 Dehn 8 Dekker 150, 205, 228, 234, 238, 239, 241, 242, 245, 247-249, 259, 263, 275-277, 289, 304, 306, 308311, 313, 322, 323, 327, 328, 330, 498, 597 Democritus 113 Demongeot 174 Deni.sov ix, 317, 596 Dennett 115 dense

immune 273 (open) set 474, 475, 495 simple 273, 289 Descartes 113, 115, 170 descriptive set theory 10, 392-394 determinacy axiom 393,453456,544,546

654 projective 393, 544 deterministic computation 18, 50, 107109, 197 Detlovs 145 Devlin , 423, 424 diagonalization 11, 145, 146, 152-154, 162, 468, 496, 500 lemma 496, 517 diamond 527, 528 Dijksterhuis 106 Dijkstra 68 Di Pmla 218, 223 directed 192 Dirichelet 75 disjoint array 228 disjunctive ( d - ) completeness 306, 311, 312, 335, 341-344,347-349 reducibility 268, 339, 591 distributive 529, 555561, 565, 567, 568, 572, 579-581, 583, 584 DNA 172 domain of a partial recursive function 16, 192 domination 273, 274 dovetailing 26 Downey 360,595 Drake 398 Dreben 165 Du Bois Reymond 146 Duff 174 e-splitting 463, 498 e-state 291, 292 Eccles 115, 116 effective algebraic c.p.0. 192, 210 definitional schemata 203 metric space 211 operation 20.5213 on partial recursive functions 205 -208 on total recursive functions 208210 weak 213 topological space 21 1 topos 214 effectively bi-immune 267 extensible 356 hyperhypersimple 288 hyperimmune 277

Subject Index hypersimple 277, 280 immune 267 inseparable 318-319, 327, 356 maximal 293 non-recursive 280, 304-306, 318 not m-reducible 305 simple 26S266, 271, 280, 294, 338, 349 effectivity principle 280 Ehrenfeucht 159 Eilenberg 53, 218, 223 Elgot 64, 197, 218, 223 Einstein 107, 112 elementary (in)equivalence 540, 544-546, 551, 587, 590, 591 map 547 Ellentuck 329, 330 embeddings 459, 461, 462, 49@493, 526, 528, 529, 537 Enderton 552 energy (mnsumption) 51-52 Engels 152 Entscheidungsproblem 165 enumeration of Ez or relations 370 CA or n: relations 380 C z K P o r n g K prelations 411 classes of r.e. sets 228-236, 248 higher type recursive objects 200 partial recursive functions 130, 215, 216 operators 197 primitive recursive functions 96 r.e. set 138-140 recursive functions 146 operators 197 set 139-140 enumeration (types of) complete 237 minimal 236 o n e o n e 140, 229 principal 236 reducibility (e-) 197 enumerations (theory of) 23G238 Epimenides 166 Epstein ix, 5, 468,500, 501, 516, 529, 538, 55 1 equations 31-38 equivalent 1- 320

Subject Index a- 375

btt- 331 712- 257 L 444 T- 176 tt- 268 w t t 337 Ershov A.P. 65, 202 Ersh0vY.L. ix, 8, 192, 201, 205, 211, 237, 238, 300, 319, 341, 353, 373, 555, 559, 570, 572, 573, 575, 596 topobgy 205, 211 essential undecidability 353-357, 513 Euclid 2, 26 Euler 111 exact pair 485-489, 584 expansionary 190, 193 extension coinfinite 484, 493, 500, 52Ck523 complete 51&515 consistent 491, 492, 51&515 finite 456, 457, 477-479, 487-490, 493, 496, 500 extension of embedding 49Ck493 formal system 354, 510 Peano arithmetic 51CL515 string 456, 494 extensional A-calculus 82, 194-196, 224-225 combinatory algebra 224-225 operator 155, 205 extensionality axiom 397, 4 W 4 0 5 fo-space 192 fap 202, 204 Farber 145 Farmer 172, 174 Feferman x, 6, 120, 159, 201, 351 Feiner 551 Fenstad 200, 223, 421 Feynman 52, 106, 112 Feys 83 filter space 201 finite 6 autoniaton 52-53 Dedekind- 9, 328 definability 33-36, 97, 101 extension 456, 457, 477-479, 487490, 493, 496, 500 finitely

655 axiomatizable 357, 510 branching tree 509, 510 strongly hypersimple 285, 286, 298 finitism 23, 119 firsborder arithmetic 24, 363-365 formal system 35&352, 593 theory of degrees 5 3 0 , 5 3 6 5 3 9 , 546, 579-582, 584, 589 Fischer 332 fixed-point 157, 182, 189, 192-194, 255, 256, 308, 338 least 182-185, 189-191 operator 79 theorem 79, 152-158, 165, 184, 185, 217, 235, 237, 255, 413 Banach 193 double 155 with parameters 155 Florence 235 flowcharts 58, 61-70, 99, 101, 132, 197, 198 Fogelman Souli6 117 folbwer 230 ‘for’ 70, 71 forcing method 468 formal systems 11, 39, 45, 46, 98, 99, 114, 159-170, 254, 349-360 fornialisni 119, 158-165 FORTRAN 64 foundation axiom 399, 422 fractals 193 Fraenkel 2, 397, 399, 423, 427 Redkin 52 Rege 119,397 Freivald 213 Freud 152 Freyd 225 Friedberg 213, 217, 230, 232, 277, 288, 290, 306,311, 337, 338, 468 Friedman 11. 164, 202-204, 222, 223 Friedman S. 421 full subtree 494 function choice 137, 138, 229, 230, 232 coding 27, 31, 89 combinatorial 329 continuous 187-192, 214, 393 expansionary 190, 193 history 8 9 monotone 181, 189, 190

656

Subject Index

partial recursive 125132, 157, 158, 222, 223 primitive recursive 4, 20, 22, 24-28, 7&74, 88-90, 96, 147, 407 recursive 3, 22, 28, 34, 37, 43, 54, 65, 84, 97-102 functional 177 countable 200, 201 partial recursive 3, 178-181, 188, 196, 197, 269 restricted 177, 179

grammar 144 graph 135-137, 538, 579, 580 model 194 Grassniann 20, 22 Greenspan 109 Greibach 144 GriUiott 200, 204 Griswold 145 Graszek 467, 529, 574 Grzegorczyk 33, 72, 213, 214 Guy 173

Gale 453, 454 Galilei 106 Galois 7 game 453, 456, 475, 476 Gandy 107, 108, 110, 200, 201, 442 gap 436, 437 garden-of-Eden 174 Garnett 193 Gasarch x, 298 Gauss 7 genetics 172 Georgieva 74 Gill 286, 298, 308, 314, 316, 341 G K P 399, 405-421, 426, 441 Gladstone 73, 74 g.1.b. of degrees 449, 465, 488, 489, 582 global results 458, 53Cb549, 574-582 ‘go to’ 64 God 165, 175 G a e l 1, 2, 5, 9, 18, 22, 27, 28, 31, 34, 3G39, 43, 44, 88, 98, 102, 105, 106, 11S115, 120, 132, 147, 159, 160, 162, 164-167, 169, 170, 198, 254, 350, 356, 365, 392, 393, 397, 398, 401, 402, 418, 422-427, 43C-432, 440, 443 theorem first 114, 115, 162-164, 167, 254 second 16S170, 401, 427 theory 356 Gold 226, 373 Goldfarb 165 Goldstiue 62 Goles 174 Goodstein 214 Gordon C.E. 203, 204 Gordon G. 63, 421 GPSS 63 Graham 107

Hack 74 Haddon 116 Hajnal 444 halting problem 150, 263 Hamilton 1 0 8 , 109 Hanf 351, 510 Hauson 8, 353 Hardy 146 Harrington x, 164,421,438,542,544,545, 547,548 Harrow 368 Hartmanis x, 50, 53, 221 Hausdorff 186, 393 HC 441-443 head of a Turing machine 47, 50 Hebb 116 Heisenberg 107 Heller 218, 223 Helm 212, 213 Herbrand 18, 33, 34, 36-39, 132, 198, 350 Herbrand-Gael computability 3G-38, 98, 101, 105, 132, 198 hereditarily countable 441-443 finite 414-418, 424, 438, 441 hereditary set 298 Hermea 53, 133 hidden-variable 112 hierarchy analytical 3 7 5 3 9 7 , 418, 419, 420, 438, 44 1, 442 arithmetical 362-375, 381, 392, 393, 415417,438 Boolean 373, 374 bounded arithmetical 368, 369 constructible 4 2 S 4 4 5 cumulative 400 Grzegorczyck 72 hyperarithmetical 362, 391 Jensen 423

Subject Index Levy 397-421, 438 projective 377, 392, 393 ramified analytical 436, 437 set-theoretical 397-421, 438 theorems 371! 381,413, 425 higher type 199-2,01, 422 Higman 8 Hilbert 1, 7, 23, 88, 90, 105, 119, 135, 169, 422 f tenth pmblem 135 Hindley 83, 196, 225 Hinman 5, 200, 396 history function 89 HF 414-418, 424, 438, 441 Hoare 68 Hobbes 113 Hodges 134, 164 Hofstadter 18, 115, 170 homogeneity 452, 54S546.551, 576, 589 Hopcroft 50, 53, 133, 144 Hopfield 117 Horn 39, 330 Horowitz 322 Howard 232, 236 Hugill 527, 528 Hyland 201, 214 hyperarithmetical 5, 33, 361, 391, 510 degree 395, 510 hyperimmune 284, 285, 498 hypersimple 282-288, 294, 297, 316, 349 immune 272,273, 276, 277,495, 498 effectively 277 -free 495, 496, 497, 500, 501, 505, 509, 520, 523, 588, 589,600 simple 272-280, 285, 287, 295, 297, 317, 338, 339, 349, 357, 358, 514 strongly 285, 286 lanov 198, 202 ideal 487, 488, 530, 538-540, 546, 558, 561, 572, 573, 575, 579, 580, 582 immune 141,239, 259,263, 266, 267, 273, 321, 328, 498, 579, 583 hyper- 272, 273, 276, 277, 495, 498 hyperhyper- 284, 285, 498 incomparable degrees 457, 459, 463, 464, 468, 479, 483, 600 incompatible strings 456

657 inmnipletenes 11, 114, 115, 162-164, 167, 254, 262 indecomposable 557 independent d e g e e s 462, 464, 46G468 of ZFC 392, 467, 468, 475, 540,574 sets 460 iudependently axiomatizable 357-360 index of finite sets 226, 227 partial recursive functions 129, 150 r.e. sets 134 recursive functions 90,91 recursive sets 225, 226 index set 150, 370 index system acceptable 215221, 236, 271, 272, 292, 308, 346 standard 215, 220, 221 index (types of) canonical 226 characteristic 225, 226 complementary 225 r.e. 225-228 induction 1&%21, 46, 399, 402, 510 infinity axiom 398, 400, 415, 417, 418, 426, 442 initial segment of V 487, 492, 516, 52G529, 537,538 V1 583, 584 V m 561-574 Vtt 589 string 456, 457 injury 261, 292 inseparable A:- 372 effectively 318, 319, 327, 356 recursively 148, 31G319, 334, 353355 interpretable 353 interpreter 61 introreducible 241, 502, 534, 579 intuitionism 10, 22, 119-122, 210, 214 invariant choice function 137 definability 44,45 irreducible 594-596, 600, 601 irreversible computation 51 isols 328-330 isomorphism of mnes 545, 551, 576, 590, 591 theorem 325

658 type 324, 325, 536, 538 iteration 72, 73, 192-194 theorem 132

Jampini 70 Janiczak 350 Jech 397, 475 Jensen 423, 436, 438 Jeroslow 169 Jockusch ix, 241, 256, 266, 268, 271, 273, 276, 279, 280, 294-297, 316, 337, 338, 340, 341, 469, 489, 498, 501-503, 50G510, 51% 515, 523, 528, 537, 540-542, 544-551, 598, 600 John 363 join 448, 449, 452, 523 Jones 44, 135 Joyce 16 Julia 193 j u m p 450-452, 467-471, 501, 509, 546, 550-553, 556, 584, 586, 588, 594 inversion theorem 468, 556, 584, 586, 588

K 147, 148, 164, 252, 254, 258, 259, 265, 266, 268, 280, 298, 299, 305, 307, 310, 317, 332, 337, 349, 353, 373, 374, 450, 579, 594 K O 150, 254, 307 Kallibekov 587 Kalniar 33, 103 Kandel 116 Kanoui 39 Kanovich 277, 338, 339 Karp 310, 410, 425, 430, 443 Kechris 200 Kelley 186, 201, 471 Kfoury 202, 204 Khutorezkii 232, 237 Kino 438, 442, 443 Kleene vii, ix, 1, 5, 10, 21, 22, 28, 34, 37, 79, 83, 84, 86, 90,98, 100, 118, 119, 121, 122, 128-132, 134, 137, 138, 140, 142, 146148, 152, 156-158, 178-182, 184, 198-201, 318, 328, 350, 354, 364, 367, 364-372, 376, 37& 382, 385387, 391, 393, 395, 436, 44S452, 457, 460, 462, 463, 485, 488, 490-492, 506, 552

Subject Index Kleene-Brouwer ordering 386 Klein 327 Klop 224 Knaster 189, 190 Knorr 2 Kobzev x, 334, 335, 337, 594 Kolmogorov 10, 107, 151, 236, 261, 263 complexity 151, 152, 261-263 Konig 214, 270, 473, 505-507 lemma 505-507 Kowalski 39 Koymans 196 Kozmnikh 74 K P 421, 443 Kreiselix, x, 6, 10,33,38, 45,97, 102-104, 107, 109-112, 1 1 s 1 2 3 , 179, 200, 202, 208, 211, 213, 224, 357, 359, 379, 397, 430, 442, 459, 507, 508 basis lemma 507 Kreitz 214 Kripke 122, 356, 399, 411, 421, 443 Kripke-Platek set theory 399, 421 Krivine 397 Kronecker 7, 22 KuEera 256, 515 Kunen 397 Kuratowski 142, 366, 378, 393, 406, 476, 48 1 Kuratowski-Ulam theorem 476, 477 Kurtz 484 K n h e r 214 Kuznekov 272

L 423-440,444, 445 La Budde 109 La Mettrie 113, 116, 118 Lachlan 213, 221, 232, 234, 235, 241, 255, 265, 272, 293, 309, 311, 313, 319, 335, 341, 344, 346, 347, 520, 522, 523, 528, 529, 537, 555, 558, 561, 562, 565, 566569, 583, 590 Lacombe 9, 204, 208, 211, 213, 214, 237, 238, 485, 500 Ladner 339 Lagrange 7 lambdaabstraction 76 algebra 224 calculus 7 6 8 7 , 194-196, 223-225 definability 84, 105, 132

Subject Index model 224, 225 Lanibek 64 Laniola 116 Landauer 51, 52 Landin 87 Langton 173 lattices 15, 488, 489, 528, 529, 537, 538, 557, 558, 561, 565, 567, 568, 57S-581, 583, 584 Laventrieff 149 Lavrov 8 , 74, 298, 353, 579 leaf 381 least fixed-point 182-1 85, 189- 191 number principle 21, 399 possible j u m p 469, 479, 501 upper bound of degrees 449,479,489,490,552, 582 principle 214 Lebesgue 22, 381, 392, 484 Leheuf 529 Lee 165 Leeds 437 left-narrow 402 Leibniz 87, 88, 140, 164, 397 Leighton I06 length of sequence number 89 string 456 Lerman x, 5, 256,462, 490,492, 501, 516, 529, 538, 543, 581 Leucippus 113 Levy 397, 406, 411, 421, 423, 426, 427, 442-444, 475 absoluteness 442, 443 hierarchy 397-42 1 , 438 Lewis 165 Li Xiang 267 liar paradox 166 Life 173, 174 limit lemma 373 sets 4, 361, 373 limitation of size 398 linear (1-) ordering 385, 527 reducibifity 269, 591 Lipschitz 110 Lipton x LISP 7, 87, 186 Ltib 169, 170, 213

659 theorem 169, 170 local results 458 logic 11, 165 Lolli x Longo 201 low basis theorem 508 degree 508, 513 Liiwenheim 432, 433, 435, 439, 440, 442, 537 Liiwenheim-Skolem theorem 432, 433, 435, 439, 442, 537 Luckham 202 Lucretius 113 Lullus 87 Lusin 148, 376, 37%382, 385, 386, 392, 393 Lynch 308 Mac Lane 218 machine analogue 10CL113, 117 nondeterministic 50, 51 probabilistic 50, 51 random access 64 SECD 87 SK 87 Turing 4 6 6 1 , 116, 132, 203, 204, 254 Machover 443 Machtey 217, 221 Malc’ev A.A. 574, 591 Malc’ev A.I. 237, 238, 300, 319 Manaster 330 Mandelbrot 193 Manna 186 Mausfield 240 rnany-one (m-) completeness 258, 259, 305, 307, 308, 319, 341, 343, 347-349 degree 238, 257, 352, 502, 5 5 5 5 8 2 , 590, 591, 594-598, 600 reducibility 251, 257, 331, 591 Mapertuis 109 Marchenkov 232, 248, 295, 301 Marek 436 Margolus 174 Markov 8, 10, 145, 210, 213, 214 algorithms 145 chains 112, 122 Marques 314 Martin D.A. ix, 255, 264, 266, 273, 286, 287, 289, 296, 297, 360, 393,

660 454-456, 475, 481, 484, 4 9 5 497, 500, 501, 503, 505, 517, 523, 528, 544, 546, 588, 600 axiom 475 Martin R.L. 166 M a n 152 materialism 118 Matijasevich 7, 135, 164, 368 maximal 279, 2 8 s 2 9 4 , 316 7- 280, 301,302 maximum degree principle 280 Mazur 475, 478 McCarthy 87, 186, 203 McCarty 122, 328 McClelland 117 McCulloch 52, 116 McLaughlin 242, 243, 249, 266, 267, 287, 295, 296, 310, 311, 319, 328, 330 McNaughton 206, 229, 427 meager 474-477, 479, 481, 495, 502, 507, 549 measurable cardinal 393, 456 measure 455, 456, 484 mechanics lot-113 mechanism 11S-115 mechanisms 15CL-152 Medvedev 272 Melzak 64 memory 47, 52, 53, 64, 117 Mermin 112 Metakides 7 method Baire category 471-484, 495 coinfinite extension 484, 493, 800, 52G523 e-state 291, 292 finite extension 456, 457, 477-479, 487-490, 493, 496, 500 forcing 468 permitting 277-280, 338 priority 232, 261, 292 splitting 463, 464 tree 493-495, 500 Meyer 70, 71, 194, 224 mezaic 322,323 Miller 495497, 500, 501, 505,517, 523 mind 113-118 minimal cover 502, 503, 523, 527, 569, b70, 573, 587, 588, 589 degree 465, 479, 483, 484, 498-802,

Subject Index 5 1 S 5 2 3 , 527, 528, 549, 577, 583, 588 enumeration 236 pair 464-466, 479, 490, 499 predecessor 483, 528 upper bound 502-505, 528, 552,553 minimality lemma 499, 504, 518, 520, 526 Minsky 27, 50, 53, 133, 144, 253 Mirimanoff 399 Mitchell 309, 311 model of X-~lCUluS194-196, 223-225 graph model 194, 225 D , 195, 225 arithmetic 536-539, 541-543 set theory 401 modulus of continuity 188, 209 Moggi 201 Mohrherr 584, 587, 590 Moldestad 204 Monk 397 monotonicity 181, 189, 190, 206 Montague 24, 426, 427 Moore 50, 174 Morley x, 164 Morris 286, 308, 314, 316, 341 Moschovakis ix,5,200,203,204,211, 213, 222, 223,393, 394, 396 Moses 165 Mostowski 8, 33, 39, 40, 42-45, 134, 140, 142, 159, 162, 352-354, 364, 367, 36S371, 391, 393, 402, 419, 421, 424, 437, 440, 461 Muchnik 155,213,233,258,277,310,318, 319, 327 MuLy 214 Mundici 51 Mycielski 453 Myhill 11, 174, 196, 198, 206, 207, 211, 213, 229, 234, 239, 241, 242, 244, 245, 249, 276, 288, 289, 305, 307-310, 321, 323, 325, 329, 330, 459, 477, 498, 592, 593, 594,597 Nerode x, 7, 8, 186188, 191, 269, 330, 538, 540, 547, 548, 551, 579, 581, 582, 584, 589, 590 neuronic net 52, 116, 117 Newton 106, 108, 192, 193 NGB 423, 427 Nies 551,581

Subject Index n-1.u.b. 552 node 381 Nogina 211 nondeficiency 246, 248 deterministic computations 18, 50, 74, 198 standard integer 330 norm 331 normal canonical system 144 object 200 normal form of A-terms 78 sets380 analytical relations 378 arithmetical relations 366, 367 higher type recursive objects 200 partial recursive functions 129 functionals 179, 180 r.e. relations 134 recursive functions 90 set-theoretical relations 406 Normann 201 Novikov 8, 148 number theory 7, 10, 29 numeral 32, 39, 84

";

Odifreddi 341 Omanadze 311, 340 one-one (I-) completeness 320, 321, 348, 349 degree 320, 325, 582-584, 590, 591, 594, 597, 600 enumeration 140, 229 reducibility 320 open set 186, 205, 472, 474 dense 474, 475 effectively 187, 207 oracle 175, 251 ordinal of node 383 structure 402 tree 383 ordinal (types o f ) admissible 9, 443, 444 Church-Kleene 384, 385, 387, 443 countable 384 gap 436, 437 recursive 384-386 stable 443

661 Ord-Smith 109 Ostrowski Owings 154, 298 Oxtoby 476 padding lemma 131, 218, 237 pairing axiom 398 function 27 Paliutin 573, 575 paradoxes 23 Berry 262 liar 166 Russell 76, 81, 148, 159, 397 paradoxical combinator 81, 155, 185 parallelism 74, 197 parametrization theorem 132, 215, 221 Paris 164, 484 Park 194, 202 partial degree 197, 199 function 127 ordering 15, 385, 461 algebraic 192, 210 chain-complete 189, 190 complete 192 effective 192, 210 linear 385, 527 reflexive 195 recursive function 12S132, 157, 158, 222, 223 functional 178-181, 201 operator 196198 tree 494 uniformly 178, 199 Pascal 113 PASCAL 4, 7, 70 Pasen, 39 Paterson 202, 502, 523, 528 Paul 166 Peano 23, 24, 28, 39, 45, 84, 442, 510 arithmetic 23, 28, 46, 164, 169, 417, 418, 442, 51cb515 Pearson I 15 Peitgen 193 perfect set 494, 596 permitting method 277-280, 338 permutation of quantifiers 365, 377, 405 recursive 324, 327 Peter 1, 64, 65, 74, 89, 96, 99

662 Peterson 75 Petri (net) 74, 75 Picard 9 Pilate 363 Pitta 52, 116 Plank 1 0 6 Platek 203, 399, 411, 421, 430, 443 Plato 113, 175 platonism 119 Plotkin 194 Podolski 112 Poincare 22, 399, 422 pointed tree 503, 504 Poliakov 74 Polonsky 145 Popper 115 positive (p-) completeness 295, 341, 343, 344, 347, 349 equivalence relation 300 information topology 186 reducibility 268, 306, 340, 591 Posner 465, 468, 500, 501, 549, 550 Post 1 , 8, 18, 46, 48, 53,63, 64, 103, 105, 106, 114, 118, 130, 132, 134, 140, 142-144, 147, 162, 176, 178, 251-254, 256-259, 267, 268, 27&272, 274, 275, 279283, 288, 294, 296, 304-307, 320, 324, 327, 331, 333, 339, 341, 350, 353, 372, 395, 396, 421, 444, 449-452, 457, 460, 462, 463, 485, 488, 49Ck492, 506, 552 hypenimple set 271 problem 251-254,256,258,259,263, 267, 268, 270, 282, 288, 293, 294, 304, 305, 321, 333, 349, 353, 421, 446 simple set 259, 265, 266, 271, 272, 275, 278, 279, 294, 339, 346, 349, 506 theorem 140, 372, 373, 3 9 6 3 9 7 Pound v Poundstone 174 Pour El 9, 110, 111, 211, 213, 214, 232, 234-237, 356, 357, 359, 360 power set axiom 398, 400, 442 precomputation theory 222 predicate calculus 165 predicativity 6, 422 prenex normal form 366, 367, 378, 406

Subject Index Preston 174 Previale x Priese 50, 133 prime computability 203-205, 222, 396 primitive recursive arithmetic 23 function 4, 20, 22, 24-28, 7&74,8890, 96, 147, 407 principal enumeration 236 ideal of degrees 487, 488 Principia Mathematica 253, 254 priority method 232, 261, 292 probabilistic machine 50, 51 probability 51, 111, 112 productions 143, 144 productive 306-311, 322 completely 304, 322 contra- 310, 311, 313, 319, 322 program schemata 202 programming languages 4, 6, 59-61, 63, 64.68-70,86,87,144, 145, 186 ALGOL 70, 186 FORTRAN 64 GPSS 6 3 LISP 7, 87, 186 PASCAL 4, 7, 70 PROLOG 7, 38, 39, 186 SNOBOL 7, 145 projective determinacy 393, 544 hierarchy 377, 392 set 393 PROLOG 7, 38, 39, 186 provability 16G170 pseude creative 323, 324, 593 simple 323, 334 Putnam 232, 234, 235,352,354,355,373, 435437,552

Qcompleteness 282, 286, 287, 295, 297, 299, 301, 306, 316, 341-343, 349 degree 282 reducibility 281, 282 Q 23, 24, 44, 165, 169, 359 quantifier 15, 25, 365, 369, 377-381, 406, 411, 462, 490, 492, 538, 581 contraction 365, 377, 405

294, 314,

405, 530,

Subject Index one- 462 permutation 365, 377, 405 tw- 490, 492, 581 quasicreative 311-313, 336, 348 Quiue 31

‘R44.99, 101,159,162, 165-168,307,353, 355, 356, 359, 368, 513 Rabin 7, 579 Rakic 117 ramified analytical hierarchy 436, 437 Ranlsey theorem 164 random access machine 64 object 116, 152, 261-263, 265 range of (partial) recursive functions 138140 realizability 10 recursion theorem first 181-186, 189, 192 second 156, 157, 165, 184, 185 Recursion Theory axiomatic 221-223 basic 222 classical 1 generalized 1, 421 Recursion Theory o n abstract domains 158, 202-205, 222, 396 admissible sets 421 continuum 3 9 5 3 9 7 higher type objects 158, 1 9 S 2 0 1 ordinals 423, 443, 444 recursive admissible triple 517 analysis 213, 214 branch 506 completely 150 enumeration 236 equivalence 328 function 3, 22, 28, 34, 37, 43, 54, 65, 84, 97--102 graph 136, 137 in E 391 in K 373 isomorphism type 324, 325 linear ordering 527 operator 197, 198 ordinal 9, 384-386 partial function 12.5132, 157, 158, 222, 223

663 functional 3, 17%181, 188, 196, 197, 269 permutation 324, 327 potentially 147 predicate 22 product 14 program 185, 186 real 8, 213 schemata 203, 204 set 3, 139-142, 159, 233, 23S241, 257, 258, 270, 298, 308, 320, 323, 327, 331, 353, 355, 367, 368, 507, 514, 593 topos 2 14 tree 382, 494, 5&510 uniformly 178, 199, 233 recursively bounded tree 509 independent 460 inseparable 148, 31C319, 334, 353355 invariant 327, 328 isomorphic 324 pointed tree 503, 504 recursively enumerable class of partial recursive functions 150, 205, 206, 228, 232 r.e. sets 228-236 completely 205, 228, 229, 236 degree 252, 257, 263, 270, 276, 278, 287, 297, 299, 304, 313, 316, 317, 320, 323, 328, 331, 337, 338, 351, 352, 508, 509, 513, 577,595 index 225-228 set 3, 134-143, 147-149, 159, 242249, 252, 254, 255, 257, 268, 281, 288, 304, 331-333, 355, 368, 376, 515 without repetitions 229-236 reducibilities 1- 320 AX- 394 n- 375 bounded 340 btt- 331, 332, 373, 591 c- 268, 282, 591 d- 268,339, 591 e- 197, 340 1- 269, 591 L 444

664

Subject Index r n 251, 257, 331, 591 p 268, 306,340, 591 partial 341 Q- 281,282 3- 340 T- 175, 176,254, 280, 294,306,338,

340

tt- 251, 268, 306, 337, 591 wT- 198 wtt- 337 reduction property 142, 372 rule 77, 82 reflection 413, 426, 427 reflexive c.p.0. 195 regressive 238, 242-245, 249, 276, 284, 285, 297, 327, 328 regular predicate 21 relative recursion for functionals 199 partial functions 176 total functions 175-177, 251 relativity 51 relativization 177, 374, 375, 394, 444 Remmel8 replacement axiom 398 representability 39-44, 98, 99, 101, 15% 166, 198, 307, 352, 353, 355, 5 13-5 15 strong 40, 42-44, 99 weak 40, 41, 43, 98, 99, 15%163, 165, 166, 307, -352, 353, 513515 representation theorems for rI: sets 382, 385 requirements 232, 261, 292, 477, 478 restricted functional 177, 179 retraceable 239-241, 2 G 2 4 9 , 276, 284286, 297, 327, 328 reversible computation 51, 52, 174 Rice 150, 151, 205, 213, 228, 229, 272 theorem 150, 151,237 Richards 9, 111, 214 Richardson 9 Richter L. 551 Richter P.H. 193 Riemann 111 rigid structure 547 Ritchie 70, 71 r-maximal 285 RNA 172 Roberts 1 1 7

Robinson A. 64, 197 Robinson J. 31, 74, 135 Robinson R.M.8, 23, 39, 40, 42-44, 7274, 159, 162, 165, 213,352-354 arithmetic 23, 44, 165, 169, 359 Robinson R.W. 285, 286 Rogers ix, 5, 133, 156, 184, 196, 197, 210, 215, 216, 218, 219, 221, 227, 254, 305, 306, 308, 311, 322, 324, 327, 337, 338, 544, 592, 593, 597, 600 root 381 Rose 288 Rosen 112 Roseubloom 79 Rosolini 214 Rosser 78, 83, 84, 134, 140, 142, 148, 161, 162, 167, 355, 356 theory 356 trick 161 Rouse139 Rowbottom 432, 435 Rubel 110, 111, 117 rudimentary predicates 368 Rumelhart 117 Russell 76, 81, 119, 148, 159, 262, 397399, 422 paradox 76, 81, 148, 159, 397 Rutledge 197 Ryll-Nardmewski 24, 33 S,"-theorem 131, 132 Sacks vii, 264, 421, 436, 444, 461, 462, 466, 467, 477, 484, 492, 493, 50S505, 527, 552, 553, 587 Sammett 63, 64, 70, 87, 145 Sands 106 S a w 178, 179, 188, 198, 199, 501 SEedrov 164, 210, 214, 225 Schrnerl 492 Schnorr 221 Schiinfinkel 83 Schroder 324 Schrodinger 107 Schwartz 116 Scott 192, 194, 196, 211, 224, 510, 511, 513, 579 basis theorem 510 topology 192, 211 search computability 204, 205, 396, 421 SECD machine 87 second-order

Subject Index arithmetic 24, 376, 430, 442, 5 3 G

539, 541-543, 545, 546, 550, 551, 579, 581, 582,584,589 definability 361, 376, 436, 437, 542, 543, 545, 551 Seldin 83, 196, 225 selection theorem 200 selfapplication 76 membership 76 reference 146, 153, 157, 162, 165170 reproduction 170-174 Selivanov ix, 269,591 Selverston 11 7 semicreative 313 intuitionists 22 recursive 140,294-299,302,337,349 separable 354,355,506 separated space 473 separation axiom 398,421,427 sequence number 89, 381 Set Theory 2, 9 descriptive 10,392-394 G K P 399, 40.5421, 426, 441 K P 421, 443 N G B 423,427 Z F - 400, 419, 425427, 430, 431, 459 Z F ( C ) 2, 399, 401, 417, 420, 427, 430-432, 434-438, 440-445, 453, 467, 475, 529, 537, 540, 574 Shanker Shannon 50, 52, 110, 133 Shapiro 50, 206, 229 Shepherd 117 Shepherdson 44, 64, 159, 169, 197, 202, 206, 207,211, 213, 229,426 Shoenfield 208, 211, 213, 311-313, 316, 318, 334, 351, 353, 373, 374, 394, 395, 420, 430, 437, 464, 471,493,498, 503,508,529 Shore x, 156,490,492,529,538,540-545, 547, 548, 551, 571, 577, 579, 581, 582,584,589, 590 Sierpinski 149, 376, 379, 381, 382, 385, 386, 392, 393 simple hyper- 272-280, 285, 287, 295, 297, 317, 338, 339, 349, 357, 358,

665 514 hyperhyper- 282-288, 294, 297, 316,

349 effectively 263-266, 271, 280, 294,

338, 349 pseude 323,334 set 259-267,270,276, 278,280, 284,

295, 301, 308, 313, 317, 321, 323, 333,334, 338,339, 349 Simpson M . x Simpson S. 164,421, 444, 498, 527, 5 3 6 538, 546,551 theorem 536-538, 546 simultaneous iteration 72 recursion 89 Singer 111 SK machine 87 Skolem 1, 7,22-24,89,432,433,435,439, 440, 442,537 Slaman vii, 467, 529, 530, 534, 538, 547, 551,574 Smith 174 Smorynski 170 Smullyan 31, 155,258,263,319,327,354356, 360,368 SNOBOL 7,145 Soare x, 5, 256, 271, 279, 280, 328, 469, 50&510,51%515, 550 Socrates 113 Solomonov 151 Solovay 256,393, 438,456,475,511, 551 Soloviev 277, 281, 286, 311,339, 340 space Baire 471,473 Cantor 471,473 countable To- 211 fo- 192 filter 201 metric 211 separated 473 topological 21 1 Specker 9,213, 214 Spector 120,385,386,467,470,471, 485, 487-489, 491, 49a500, 503, 517-519, 530, 537, 581, 584, 589 theorem 485-487,530,537,584,589 splinter 244,323,324,593 splitting method 463,464 Spreen 211 Srebrny 436

Subject Index stable ordinal 443

standard class 235 model of arithmetic 53G539, 541543 part of a model 44 structure 401-405, 4 1 g 4 2 1 system of indices 215, 220, 221 state of a Turing machine 47, 50 Statman Stearns 50, 53 Steel ix Steinhaus 453 Steinitz 7, 351 Stephan 599 Stephenson 109 Sterling 329 Stewart 453, 454 Stillwell 479, 484 Stoltenberg-Hansen 204 Stoy 196 string 456 Strong 185, 222 strong array 228, 229, 235 homogeneity 544, 545, 576 minimal cover 503, 527, 569, $70, 573 normalization 78 strongly effectively immune 267 simple 266, 273, 294, 316, 339, 349 hyperhyperimmune 284, 285 hypersimple 285, 286 representable 40, 42-44, 99 uniform tree 520-523, 568, 583 Sturgis 64 subcomputation 217, 223 creative 314, 315, 349 sequence 89 tree 494 substitution property 178, 179 Sudan 147 Sullivan 193 superthesis 103, 115-118 Suslin 380, 397, 392 Suslin-Kleene theorem 387, 391 Suzuki 225, 233 system of indices 215-221

acceptable 215-221, 236, 271, 272, 292, 308,346 standard 215, 220, 221 t a g 253 Taimanov 8, 353 Taislin 8, 353 Tait 33, 38, 97, 201 Takeuti vii, 425, 438, 442, 443 tape of a Turing machine 46, 50 Tarski 8, 24, 39, 40, 42-44, 88, 114, 159, 160, 162, 166, 189, 190, 352354, 363, 364, 366, 371, 376, 378,400,406 theorem 114, 166, 376, 413 Tchuente 174 Tennenbaum 24, 238, 241, 274, 281, 289, 318, 513 terse 298, 299 Thatcher 165 Thomason 529, 537 threeelement chain 523-528 quantifier theory 492, 581 Thue 8, 144 Titgemeyer 524-526 Toffoli 52, 109, 174 topology 18C192, 20!+208, 472, 473, 494, 495, 527 Ershov 205, 2 11 positive information 186 Scott 192, 211 total functional 177 recursive function 129 totality lemma 496, 517, 520, 552 Trakhtenbrot 53, 148, 269, 502 transitive closure 409 structure 401-405 tree 381, 493 1- 520 binary 505 e-splitting 498, 500 finitely branching 509, 510 identity 494 method 493-495, 500 of trees 497 pointed 503, 504 recursive 494, 5W510 recursively bounded 509 strongly uniform 52CL523, 568, 583

Subject Index total 494 uniform 516, 527,583 Troelstra 10, 121, 122 true stage 246 t r u t h 114, 166, 363, 364, 370, 376, 400,

401, 468 truth-table (tt)268,473 completeness 270-272, 274, 334,

341,344,346,347,349 cylinder 593, 594 degree 270, 276, 296, 317, 338, 352,

584-591, 595,598401 reducibility 251, 268, 306, 337,591 weak 337-342, 344, 347, 349, 584-

591 Tucker 204 Turiiig vii, 1, 6, 46, 48, 53,54, 64,79, 99,

102, 106, 115, 117, 118, 130, 132-134, 146, 150, 164, 165, 175, 176, 197, 213, 254, 337, 538, 581, 582, 589, 590, 595, 600,601 completeness 253, 255, 264, 265, 277, 293-295, 304, 309, 339, 341, 342, 344, 349,353 computability 53-54, 99, 101, 132, 197 degree 175, 176, 263, 276, 278, 287, 297, 299, 304, 310, 313, 316, 323, 328, 351, 447-553, 568, 570, 574, 575, 583, 589-591, 595 machine 46-61, 116, 132, 203, 204, 254 reducibility 175, 176, 254, 280, 294, 306,338, 340 Turner 87 twequantifier theory 490, 492,581 type

isomorphism 324, 325 equivalence 328

Ulam 476,481 Ullian 244,267, 288,323 Ullman 50,53, 133, 144 undecidability 11, 103, 114, 147,148, 150,

151, 162-166, 253, 254, 263, 350, 352-357, 492, 513, 537, 550,581 essential 353-357, 513 uniform tree 516, 527,583 strongly 520-523,568,583

667 uniformization 137 uniformly recursive 178, 199, 233 union axiom 398 universal canonical system 254 constructor 172, 173 differential algebraic function 1 1 1 id330 partial function 132, 217 Turing machine 132, 133, 172 unsolvability 103, 148, 150, 151, 254 uppersemilattice 15, 451, 529, 556, 558,

559, 561, 572, 573, 575, 579, 583, 589 urelement 399, 400 Uspenskii 107, 137, 186188, 191, 207, 215, 234,236, 272, 318, 323 Van Der Mey 81 Van Emden 39 Van Heijenoort Vaught 44, 352 verbose 298, 299 vertex of a tree 381 Vichniac 109 vicious circle principle 399, 422 Von Neumann 6, 62, 109, 112, 115, 116,

151, 171, 173, 407, 410, 423 Von Neumann-Wel-Bernays set theory

423, 427 VuEkovich 319 Wagner 222 Wang H. 50,64, 65,99, 118, 120, 133 Wang P.9 Watson 172 weak array 228, 229 effective operation 213 partial ordering 14 Turing (wT-)198 truth-table ( w t t ) completeness 338-342, 344, 347,

349 degree 337,338, 584-591 reducibility 337 weakly representable 40, 41, 43, 98, 99,

15Ck163, 165, 166, 307, 352, 353, 513-515 Webb 18 wellfounded

668 relation 407, 421 structure 402 tree 381-384, 386 ordering 395, 386 of L 430, 431 principle 399 Weierstrass 8 Weihrauch 214 Weisbuch 117 Wexelblat 63, 64, 70, 87, 145 Weyl 422 ‘while’ 68-70, 185, 186 Wiener 117 Winklmann 217, 221 winning strategy 453 Winograd 116 Wirth 70 Wittgeustein 8 4 Wolfram 174 Woodin 530, 534, 538, 547 word problem 8 Xiang Li 267

y 81, 155, 185 Yates 212, 233, 247, 248, 277, 278, 283285, 287, 292, 293, 313, 453, 478, 495, 528, 544, 551 Young J.Z. 116 Young, P.R. 211-213, 217, 221, 235, 244, 285, 324, 341, 582, 590, 592, 593, 597 Zakharov 340 Zbierski 442 Zernielo 2, 399 Zermel~Fraenkelset theory 399 Z F - 400, 419,425-427,430, 431,459 Z F ( C ) 2, 399, 401, 417, 420, 427, 43& 432, 434-438, 44Ck445, 453, 467, 475, 529, 537, 540, 574 Zorn 464

Subject Index