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x,y even numbers. Stage 5 + 1, 5 > 0. Let x be the least free number. Let g' = gs\J {(2s + l , x ) } . F o r a l l t G domgs define P 5 +i(£, 2s -fl) = L(gs(t),x),Ps+1(2s + l,t) = L(x,gs(t). For all 2k £ domgs define 25 + 1 has a subset Di with \B{\ different copies of elements having the same (positive and negative) relations as the element a; G 21. Although it is possible b{ = bj for i ^ j but Di and Z}j are disjoint. From Theorem 1.2, £> is an elementary extension of 21. Hence D is a model of T. Define a function F so that it maps all the elements of Di to element bi in B. We can check that F is a homomorphism from £> onto 03. Because the theory T is preserved under onto homomorphisms therefore 03 is a model of T. Theorem 2.5. A Theory T is preserved under onto homomorphisms if and only if T is equivalent to a countable disjunction of countable conjunc tion of positive existential sentences.
< 1),
and i G B «-► [z] n P / 0. As already noticed this implies P = T A and, therefore C = B@LU = T ^4-
4
Open questions
In this paper we have shown that any infinite r. e. set supports any infinite recursive linear ordering, but this is not true for recursive partial orderings: as is shown in Theorem 2.2 there exists a recursive partial ordering
W E A K P R E S E N T A T I O N S OF R E C U R S I V E PARTIAL O R D E R S
45
We may also ask Question 4.2 Do there exist a Turing degree a > 0 such that if a partial ordering C has a weak presentation in a set i G a then C is recursive? As follows from Theorem 3.1 this degree a (if it exists) cannot be recur sively enumerable.
5
Bibliography
[Soare] Soare, R. L, Recursively Enumerable Sets and Degrees, SpringerVerlag, Berlin, 1987. [JSh] Jockusch, C.G. Jr. and Shlapentokh, A., Weak presentations of computable fields, J. Symb. Logic 60, 1995, 199-208.
47
Recursion Theory on Weak Fragments of P e a n o Arithmetic: A S t u d y of Definable Cuts
C T Chong and Yue Yang National University of Singapore
Classical recursion theory (CRT) is built on the axioms of Peano arith metic P. Of these axioms, the induction scheme Ind provides both the source and the power to carry out highly sophisticated constructions (such as priority arguments), on the set of natural numbers, and to verify their correctness. Now Ind may be decomposed into a hierarchy of induction schemes, in increasing order of complexity. Denote by J E n the induction scheme restricted to E n formulas. Then Ind is equal to the union of I E n (n = 0 , 1 , . . . ) . It is clear that each construction in CRT of a definable set on to uses only 7 E n for some n, dictated, roughly speaking, by the number of alternating quantifiers involved therein. It follows that by assuming only fragments of Ind, it is possible to derive substantial number of results in CRT. From the global point of view, one is led to two general problems: (a) Classify the proof-theoretic strength of theorems and results in CRT in terms of fragments of the induction scheme; (b) Study recursion theory in non-standard models of fragments of P. It turns out that these two problems are closely related. In this paper, by a weak fragment of Peano arithmetic, we mean an axiom system that assumes at most the /E2 induction scheme. This is a weak system because it provides just the minimal environment to carry out infinite injury constructions (indeed something even less than Z?E2 is quite sufficient if we work in a special model of / E i (See Section 2)). On the other hand it is strong enough to support a reasonable recursion theory, as we will see in the sequel. This paper gives a survey of recursion theory on weak fragments of P. Our objectives are twofold: (a) To discuss results in CRT which are prov able under a weak system, and to identify their proof-theoretic complexity wherever appropriate, and (b) To summarize the key ideas and techniques introduced into the subject over the past decade. The reader will notice that many of these had originated in ordinal recursion theory developed since the early 1960's. The fact that there is a tight link between recursion theory on ordinals and that on weak fragments of P is a fortuitous coinci dence probably not anticipated by the founders of these two subjects. We believe that there is a two-way interaction between them, and expect to see
48
C H O N G AND Y A N G
more applications going in either direction. We begin with some definitions and terminologies. Denote by P~ the Peano axioms minus the induction scheme, but adjoined with the expo nential function to allow for coding of 'finite' sets. By the E n bounding principle, written P £ n , we mean the following scheme: In a model of P~~, every E n -function total on a proper initial segment (with a last point) has a bounded range. Kirby and Paris [11] showed that, for n > 1, I^n+l
=> P S n + i => J E n ,
and that the reverse arrows fail. This shows that there is a finer hierarchy obtained by the interweaving of P S n . We will see that this finer hierarchy also gives rise to 'finer' recursion theories. The paper is organized as follows: In Section 1 we define the basic notions and notations. Section 2 concentrates on recursion theory under the hypothesis of P E i , and discusses results (such as the Sacks Splitting Theorem) which are provably equivalent to J E i over this base theory. One conclusion to be drawn from results in this section is that finite injury priority arguments may be carried out essentially with only Si-induction. In the absence of this, similar results (such as the Friedberg-Muchnik Theorem) may still be derived without recourse to a priority argument. In Section 3 we move up the ladder of complexity to consider recursion theory under BY,2. In this theory one can prove the Density Theorem. On the other hand, with BT,2 as the base theory, the existence of maximal sets and incomplete high r.e. degrees is equivalent to E 2 -induction. In the final section we discuss some open problems.
1
Preliminaries
In a (non-standard) model M of P ~ , a bounded set is M-finite if it is coded (by an element). A set X in M is recursively enumerable (r.e.) if it is Ei(A / (). X is recursive if both X and M \ X are r.e. We define recursive functions (partial and total) in the obvious way. Assuming only Eo-induction on M, it can be proved that there is a Y,i(M) enumeration {Wc|e G M] of all r.e. sets. An analog of Church's thesis may be formulated in this setting as well. This will be used implicitly throughout the paper. Given subsets A and B of M, we say that A is pointwise recursive in B (written A
eWekKxcB
A K2CiB
x ^ A ^ 3K13K2[(x,l,KuK2)
eWe^KxcB
A #
= %
and
2
n £ = 0],
A STUDY OF DEFINABLE C U T S
49
where K\ and K2 are (codes of) .M-finite sets. We say that A is strongly recursive in B (written A
T,n(M)
A refinement of this result leads to a characterization of the existence of nonlow r.e. sets in BY.2 models (see Section 3).
50
C H O N G AND Y A N G
Finally, there are two important notions which make the subject different from CRT, and show its links with ordinal recursion theory: Definition 1.1 In a model M. of a fragment of Peano arithmetic, a regular set A is one whose intersection with any M-finite set is M-finite. Cuts are examples of nonregular sets. Their existence very often results in the failure of priority arguments, and along with it theorems whose proofs use these arguments. In cases where a theorem continues to hold, different methods, of the non-priority kind, are needed. Definition 1.2 A set A is hyperregular if every function pointwise recur sive in A maps a bounded set to a bounded set. Hyperregular sets and nonhyperregular sets are natural objects to study in connection with the jumps of r.e. degrees (Section 3). In particular, if B is a hyperregular r.e. set, then Ei induction relative to B holds, and this feature is crucial in verifying that certain constructions work.
2
Ei Bounding and Ei Induction: The Role of a Ei-cut
In this section, we fix M to be a model of JE?EI. When M is a Z?Ei model, I is used to denote a Ei cut in M and / : i" —► M a Ei cofinal function. We first discuss Post's Problem in a # E i model. This is contrasted with The Sacks Splitting Theorem, which is equivalent to J E i over Ei bounding. An analysis of the degree of a Ei-cut (and its generalization) leads to the proof of the Friedberg-Muchnik Theorem in E E i . To conclude this section we state some results relating to 7Ei but not BT,2 models, for example the nontransitivity of the weak reducibility relation
51
A STUDY OF DEFINABLE C U T S
T h e o r e m 2.1 In any £?Ei model M, 0 < T / < T 0'Proof. It suffices to enumerate a set A such that A ^ p / , because i" is not recursive. The construction is done in / many stages. At stage i, consider all numbers m less than f(i). We use (m,x) to diagonalize against $ m . If there is some ((ra,x),0,iV) G $ m ,/(i), then we choose an Nmax, which is the maximal negative condition N among them, and for this Nmax, choose the least x such that ({m,x),0,Nmax) £ $m,f(i) a n d x is not in Amj^y Enumerate the least x in the definition of Nmax into Am. This will guar antee that A 7^ $ m ( ^ ) by the following argument. An application of the Pigeon Hole Principle produces an x0 such that (m,xo) is not in A. If ^ m ( ( ^ , ^ o ) ) 7^ 0 then we are done. So let us assume that $ m ((m,a;o)) = 0 and is correctly computed at some stage i. At this stage, Nmax is defined and greater than or equal to the true negative condition used in computing ^ m ( ( ^ 5 ^ o ) ) - It follows that correspondingly the least x in the definition of Nmax is a true witness for $ m ( ( m , x ) ) = 0 / A((m,x)). (Notice that unlike typical injury arguments, witnesses are not found uniformly. They are either of the form x 0 ^ Am or taken to be the least x corresponding to some Nmax.) □ We next sketch the proof that Sacks Splitting Theorem fails in every JBEi model. This was first observed by Mourad [12] using the cut / as an example for the failure of Sacks splitting. We take a somewhat different route here, by first studying the degree (in the strong sense) of a bounded Ei set X. This approach allows one to extract information on the relation between regular, nonregular r.e. sets and Ei cuts in JBEI models. Let a be an upper bound of X. Let Q(y,u) be a Eo formula such that x £ X O (3u)6(x,u). We call the least such u the Ei witness for x. T h e o r e m 2.2 Any bounded Ei set X is either M.-finite or strongly Turing equivalent to I. In particular, all Ei cuts in M. have the same Turing degree. Proof. If there is a uniform upper bound uQ for all Ei witnesses verifying any x to be in X , then X is a bounded Eo set, hence .M-finite. Suppose that there is no uniform upper bound for witnesses of x in X. We show that the degree of X is the same as the degree of i". Consider the set K C J x < a: (e,y) eKoeelAy
f(e))0(y,u).
In other words, (e, y) G K if and only if the Ei witness of y shows up before
He). Now K is coded on Ix < a since it is Ai on Ix < a and so Proposi tion 1.2 (the Coding Lemma) applies. Let K* be an .M-finite set such that
52
C H O N G AND Y A N G
K* fl ( I x < a) = K. Using K* and 7, it is not hard to show that X is strongly Turing reducible to I. Moreover under the assumption that the Ei witnesses of X are unbounded in M, the same code K* can be used to compute I from X as well. □ The above theorem on bounded Ei sets in 7?Ei models may be general ized to bounded E n sets in BT,n models for n > 1. It is now easy to see that I is Turing reducible to any nonregular Ei set, since a bounded nonregular portion is already sufficient to calculate I. Corollary 2.1 Let A be any nonregular Ei set. Then I is strongly Turing reducible to A. Now the failure of Sacks Splitting Theorem follows quite easily (Mourad
[12])Corollary 2.2 Sacks Splitting Theorem fails in every BY,\ model M. Proof. Let A0 and A\ be a splitting of the cut 7. One of them, say A 0 , must be nonregular. By Corollary 2.1, I is strongly recursive in AQ. □ We now give a sketch of Mytilinaios' proof [14] that Sacks Splitting Theorem is provable in P - -I- J E i . Together with Corollary 2.2, this will imply that Sacks Splitting Theorem is indeed equivalent to / E i over the base theory Z?Ei. The proof follows the classical strategy of Sacks preservation. The major difficulty is that E 2 induction is necessary for the original construction to work. Specifically, in the absence of E 2 induction there may be a cut J where no uniform bound exists for the set of restraints {r(e)|e G J } , where r(e) is the restraint imposed by requirement Re. The solution is to use Shore's blocking method [17]. Let M be a model of / E i in which E 2 induction fails (the other case being straightforward). Divide M into blocks of requirements, so that requirements in the same block have the same priority. The size of each block, and the number of blocks, is not determined a priori, but are instead decided in a E 2 manner by the preservation strategy. Thus, for example, the Oth block, written £?(0), consists only of the requirement RQ. As the construction unfolds, the restraint r(0,5) of RQ at stage 5 increases. The size of the block B(l) at stage 5, consisting of all requirements Re, 0 < e < r ( 0 , s ) , will correspondingly increase, finally settling down at r(0), the final position of the restraint for JRO, using Sacks preservation strategy. At each stage 5, all requirements in B(l) have the same priority. Collectively they impose a block restraint which is the supremum of the individual restraints in B(l), and which we denote by r ( l , s ) . Then r ( l , s ) in turn determines the length and size of B{2) at stage 5, and so on. In this way, if B(e — 1) has r(e — 1) as its
53
A S T U D Y OF D E F I N A B L E C U T S
final restraint, then B(e) will consist (finally) of requirements with indices between r(e —2) and r(e—1), and under / E i , the Sacks preservation strategy will ensure that r(e) exists. What one obtains eventually is a £2 cut J so that for each e G J , B(e) and r(e) exist, with the additional property that each requirement belongs to one and only one block. The conclusion is that each requirement is satisfied and so Sacks splitting holds in J £ i . The description above of the construction is an instance of what we call dynamic blocking (Chong and Yang [5]), a technique which was developed in there for infinite injury priority arguments. In the present situation we are concerned with £ 2 dynamics since both the restraints and lengths of blocks stabilize eventually. It turns out that dynamic blocking is a natural tool for priority type constructions in models of weak fragments of Peano arithmetic, as a solution for the absence of a predetermined 'cofinality' (standard objects used in ordinal recursion theory) in the structures under study. The degree of a £ x cut, in addition to providing a solution to Post's problem in models of P~ + £ £ 1 , is also the base of the cone of all nontrivial r.e. degrees. The first result that points in this direction is the theorem of Mourad [13] that / is of minimal r.e. degree. L e m m a 2.1 If A is a regular £1 set, then A is strongly recursive in I. Proof. Let 3uQ(x, u) be a £1 formula defining A. Define K C I x I by (i,j) is in K if and only if there is some element x < f(i) which enters A between stages f(j) and f(j + 1). K is coded on / x 7" by Proposition 1.2, as it is Ai on / x I. Let K* be an .M-finite set such that K* f)(I x I) = K. For any .M-finite set E, we find an i such that E C f(i). Look at all pairs (z, j) G K*. By the regularity of A there is some JQ such that (i,jo) is in K* and for all j > jo in / (ij) is not in K*. That j 0 can be found by using / . By the definition of K, E C A if and only if E C Af{jo) and E fl A = 0 if and only if E fl Af(jo) = 0. □ T h e o r e m 2.3 The degree of the cut I is a minimal £1 degree. Proof. By previous results, it suffices to show that a £1 regular set is either recursive or of the same degree as / . Let A be a regular r.e. set and h : I -+ I be defined by h(i) = k if A [ f(i) settles down before stage f(k). Fix an upper bound e of / . First we will find a coded approximation h(i,j) of h(i). The price to pay is that h(i,j) will overspill to e x e. Consider the set S C I x I x I: S = {(i,j,k)\Af{k)}f(i)
=
AfU)}f(i)}.
By the Coding Lemma, S is coded on / x i" x / , as it it Ai on I x I x I. Define h : e x e —> e by h(i, j) = the least k such that (i,j,k)
G S*.
54
C H O N G AND Y A N G
Then h(i) for i G 7 can be approximated by h(i) = lim/i(z, j). Observe that for each fixed i in 7, the function h(i,j) can be made nondecreasing for j < e. Next with the help of h(i, j ) , we can decide the degree of A based on the following two cases. Case 1 There is a b G e \ 7 such that for all z G 7, h(i,b) = h(i). Then A [ f(i) settles down at stage /(/i(i,&)), therefore A is recursive. Case 2 For all b G e \ 7, there is an i in 7 such that h(i, b) > h(i). We argue that A computes 7 by enumerating 7, the complement of 7, from A as follows. Consider any 6 < e. Enumerate i G I, and calculate /i(i,6). We use A to find the value h{i). Compare h(i) and h(i,b). If /i(z) is less than h(i,b), then b is in 7. □ The origin of the priority method traces back to the solution of Post's problem by Friedberg and Muchnik. In the standard construction of two r.e. sets A and B of incomparable degree (called a Friedberg-Muchnik pair), a finite injury strategy is used, such that each requirement Re is injured at most 2 e times. Simpson had observed some years ago that using Ei induc tion one could derive the same effective bound for each requirement, and hence obtain a Friedberg-Muchnik pair. On the other hand, in the absence of 7 E l 5 an effective bound on the number of injuries for each requirement is no longer assured. With the characterization of the Sacks Splitting The orem as being equivalent to 7 E l 5 the provability of the Friedberg-Muchnik theorem in a weaker system became especially interesting. This was settled by Chong and Mourad [3]: 7?Ei proves the Friedberg-Muchnik Theorem. Since the cut 7 is actually the least nonrecursive r.e. set, any FriedbergMuchnik pair has to consist of nonregular r.e. degrees above that of 7. Nonregular r.e. degrees are typically associated with cuts. Theorem 2.2 states that all Ei cuts have the same degree. Hence one has to go beyond the notion of cuts to look for a solution, and this is found in what is called the union of cuts . For each i in 7, let Li denote the interval [/(z), /(z -h 1)). We say A is a union of cuts if for each i G 7, A D Li is downward closed in L{. We sketch the injury free strategy for making $e(A) ^ B in a JBEi model. The Friedberg-Muchnik pair A and B will each be a union of cuts. In each interval L^, we place two markers a,i and a^. For stages j in 7, marker cti moves up within Li as j increases, whereas a; moves down as j increases. At any stage, ai < di. The idea is that all numbers in Li below a^ enter A, while all those above a; in Li are barred from A (forever). The interval Li has to be 'large' enough (in a precise sense) to allow for the movement of these two markers throughout the construction. Correspondingly, one also
A S T U D Y OF D E F I N A B L E C U T S
55
has, on the B-side, markers 6; < bi in Li (in an actual construction, A would occupy the even intervals while B the odd intervals, in order to take care of certain technical details). The construction is arranged so that at any stage, di — di is large, and bi — bi is large. In addition, this property holds in the limit as well (meaning that 'largeness' is preserved after going through /-many stages). A negative condition N will now consist of an .M-finite set of points, each point belonging to one and only one interval. We describe briefly how a typical construction works with a special case. Suppose $ e j^)((a;,0,iV)) = 0, with N G [a 2 ,a 2 ] and x G [61,61]. There are two possible actions to be taken: Yi N — a>2 is large, we move a 2 down to N and make this computation permanent. Find the least x with these properties (on the A-side). Denote it by XQ. If 61 — XQ is large, move marker 61 up to capture XQ, and we achieve permanent disagreement. If xo — 61 is large, then we move 61 downwards to XQ, and look for a new candidate for diagonalization. On the other hand, if d2 — N is large, then move a 2 up to capture N and look for another computation that allows one to perform diagonalization. To show that the construction succeeds, suppose for the sake of contra diction that $ e (A) = B. In L i , choose an x not in B yet 61 — x is large at every stage of the construction (this needs to be justified). Then there is a true computation (x, 0, JV) G $ e ,/(i) a t some stage i. For simplicity as sume that N G L 2 . Being a true computation, N - a2 must be large. That means that a 2 is moved downwards to capture TV, and either x or some number smaller is captured by 61 as it moves up, ensuring a permanent disagreement. Of course in general the negative condition may have components from different intervals, and so it is more complicated to handle. A careful track ing of where these components reside will be essential. A final point to note is that the requirements are arranged in such a way that for each j , the eth requirement Re (for e < f(j)) is satisfied using less than /-many number of cuts. To do this it is essential that each interval Li is 'large' enough to accommodate the movement of markers, in order to provide space to satisfy such requirements. One can think of the case of the limit cardinal H^, where for finite injury constructions of the Friedberg-Muchnik type, requirements of priority less than an K^ are satisfied by stage H^ + 1 . The fact that each succeeding cardinal has size much larger than the previous cardinal allows sufficient space for markers associated with the relevant requirements to move (there is, of course, a crucial difference here: we are dealing with nonregular sets so that a straightforward cardinality argument, as is done in the case of Nfj will not suffice). As we have seen before, there are theorems in CRT which are equivalent to / E i over the base theory JBEI. In section 3, we will see theorems equiva lent to / E 2 over J 5 E 2 . It is natural to ask whether a weaker base theory, say / E i , is sufficient to establish the equivalence of such theorems with /?E 2 .
56
C H O N G AND Y A N G
No example is known thus far. One of the reasons is that there is a model M of / E i but not / ? S 2 which behaves like the standard model UJ. As such most constructions from CRT carry over with minor modification. The particular model M of P~ + / S i + ->£S 2 has a A2(M) projection g from UJ one-one onto M. Since the projection g is A 2 , it has a recursive approximation. This allows requirements to be indexed by UJ (in a A2 fashion), so that induction can be carried out over UJ via g. For example, in M there is a maximal r.e. set (Chong [1]), and there is an incomplete high r.e. set (Yang [20]). This justifies the need for a base theory stronger than / S i to establish the equivalence of these results with S 2 induction. The construction of such a model M was first done by Groszek and Slaman in their study of the nontransitivity of weak Turing reducibility [8]. We briefly describe the construction. Let AT be a model of P with a nonstandard element p. The model M is built in UJ steps. Initially let M[0] = {p}. Suppose M[k] = {mi,...,mj} C A/". First, throw in all S i witnesses of the first k S i sentences true in A/\ This will make M - ^ A/". Secondly, for each of the first k II1 formulas with one free variable, say ip(v), if there is an x in j\f such that j\f \= ip(x) and x is less than some element of A4 [A;], then we add the AMeast such x to M. This will ensure that M satisfies LIIi, which is equivalent to / S i . Let M = \JM[k],k € UJ. Then M \= P~ + / S i 4- -1BYI2. And the function g : i h-> the z-th element in the construction, is pointwise recursive in 07 in A4, because 0' is absolute in A/" and M.. Therefore, there is a recursive approximation of two variables p(_,s) dig.
3
E2 Bounding and S 2 Induction: The role of a E 2 -cut
We now turn our attention to infinite injury priority constructions, and more generally, constructions which seemingly require more than just S i induction. There are three theorems in CRT which interest us: the Sacks Density Theorem, the Friedberg Maximal Set Theorem, and Shoenfield's theorem on the existence of an incomplete high r.e. degree. The classical proof of each of these theorems comes with a new technique that leads to applications elsewhere. Each construction appears to require at least / S 2 , or even / S 3 , to succeed. Groszek and Slaman [7] have analyzed strategies in infinite priority constructions. Under their classification of complexity of outcomes of strategies, the applicability of all n 2 constructions is equivalent (over the base theory P~ +Exp + IY,o) to / S 2 . However their analysis does not provide much information on individual n 2 constructions, for example Shoenfield's proof on the existence of high sets or Sacks' proof of the Den sity Theorem. As it turns out, the Density Theorem actually holds under P~ + B S 2 , while over this as base theory, the Maximal Set Theorem and
A S T U D Y OF D E F I N A B L E C U T S
57
Shoenfield's Theorem are both equivalent to 7T 2 , showing that essentially strategies of the same level of complexity are required to establish these results. In the discussion that follows, all models satisfy P~ + Z?£ 2 . We begin with the study of the Maximal Set Theorem and Shoenfield's Theorem on high r.e. sets. We will not dwell on the positive aspects of these theorems—that existence of such sets follow from £ 2 -induction. The proofs (Chong and Yang [4]) for maximal sets, and Groszek and Mytilinaios [6] for incomplete high r.e. sets) involve a careful tracking of the complexity of strategies used in the construction, and showing that 7E 2 suffices to do the job. We concentrate instead on the other direction, that if there is either a maximal set or an incomplete high r.e. set in a model M., then indeed £ 2 -induction holds in M. It will be seen that the existence of a £ 2 -cut in a 5 E 2 model disengages the role of many E 2 or II 2 constructions from the theorems (indeed the theorems fail completely in such models). We begin with the Maximal Set Theorem. Definition 3.1 A nontrivial r.e. set M is maximal if its complement is not M-finite, and any r.e. set which contains M is either M or M, save for an M-finite difference. The main result of Chong and Yang [4] states: T h e o r e m 3.1 Let M be a model of P~ + 5 E 2 . Then M has a maximal set if and only if M satisfies 7E 2 . In other words, over this base theory, the existence of a maximal set is equivalent to ^-induction.
Proof. We give a sketch. Suppose that M is a maximal set. There are two cases to consider: Case 1. There is a E 2 -cut I and a E 2 -function / mapping I onto the complement of M . In this case we say that M has a small complement. Fix an upper bound e 0 of / . It is not difficult to define a simultaneous r.e. collection of pairwise disjoint r.e. sets {Hd\d < eo} whose union is M with the additional property that each Hd contains at most one member not in M (the idea of generating such a collection {Hd} originated in Lerman and Simpson [10] where he proved that there is no maximal K^-r.e. set). Now consider the set
K =
{(e,d)\f(e)eHd}
which is a subset of I x I and A 2 on I. By the Coding Lemma (Proposition 1.2) K is coded on / by an .M-finite set K*. Since K is a function on / , we may think of K* as a function as well which we denote by /*. Now U{#/*(2e)|e<e0}
58
C H O N G AND Y A N G
is an unbounded r.e. set which splits the complement of M into two nontrivial pieces, contradicting the maximality of M. Case 2. The complement of M is large, i.e. Case 1 does not hold. The first step is to produce a S2-cut 7, a simultaneous r.e. collection of pairwise disjoint r.e. sets {Hd\d < e 0 } (for some eo an upper bound of 7) such that unboundedly many elements of M\M distributed nicely in U{Hd} (meaning that there are cofinally many d in 7 which contain elements of M). Again let / be a E 2 -function mapping 7 unboundedly into M. Define g(e) to be the first element above / ( e ) not in M (we may arrange / nicely so that there is always a member of M \ M between any two values of / ) . Define K = {(e,d)\g(e)
£ Hd}.
Again K is coded on 7 and so a similar argument proves that M is not maximal. Definition 3.2 Let 0" = {(x,e)\3y\/zy)(x,y,z,e)} where if ranges over all So formulas as e ranges over M. 0" is called a complete E2 set. It is easy to see that every £ 2 - s et is recursive in 0". Definition 3.3 An incomplete r.e. set A is high if its jump A' = {e\{e}A(e) } z*5 Turing equivalent to 0" (i.e. each set is recursive in the other). Shoenfield's high set theorem states that there is a high r.e. set. The proof uses a n 2 -strategy which is a classic example of an infinite injury construction. Groszek and Mytilinaios [6] show that with enough care, this construction actually works in 7E 2 and establishes Shoenfield's theorem in this system. We give here a description of how such a result must fail when only BY.2 is allowed. The corollary is that any construction of a high r.e. set has to have complexity at least that of the standard one. The technique used in the proof of the result also highlights the role played by a £2-cut in recursion theory. T h e o r e m 3.2 (Three Point Theorem) If M is a BT,2 model, then there is no high r.e. set. Indeed there exist three degrees, denoted 0', 0 1 5 and 0", which form the set of jumps of r.e. degrees. Furthermore, the jump of an incomplete r.e. degree is either 0' or 0 1 5 .
The following is a useful observation, which is a generalization of Theo rem 2.2 in Section 2. L e m m a 3.1 In a BT,2 model, all Y,2-cuts are of the same degree.
j
A S T U D Y OF D E F I N A B L E C U T S
59
Our analysis of the jumps of r.e. sets in BY,2 models relies on the no tion of hyperregularity, which originated in ordinal recursion theory. The importance of hyperregularity to the subject of study revolves around the following fact. L e m m a 3.2 In a model M of P~ + BY,2, every incomplete r.e. set A is hyperregular. Furthermore, an r.e. set A is hyperregular if and only if M satisfies /Ex(A). P r o o f of T h e o r e m 3.2. Let 0' be a complete r.e. set, and let / be a £ 2 -cut. By Lemma 3.1, the degree of 7 0 0 ' is well-defined, which we denote by 0 1 5 . The claim is that if A is incomplete r.e., then A1 is recursive in 0' 0 / . By Lemma 3.2, A is hyperregular. Let / be a £ 2 -cofinal map from I into M. Let A'f(e) = {d\{d}f{fele\d) 1}.
Then K=
{(d,e)\A'f{e)U(d)?A'f{e_1)U(d)}
is A 2 on / . Let K* be an .M-finite set which codes K (Proposition 1.2). Since A is hyperregular, for each dm I there is a maximum e in I so that (d, e) e K*. This yields an algorithm to compute A' from 0' 0 i", using K* as the parameter set. The degree 0 1 5 turns out to be strictly between 0' and 0". According to Mourad [13], the degree of 0 1 5 is minimal over 0' (generalizing an argument for the degree of a S i cut in a BY,i model, see Section 2). The argument above shows that the jump of an r.e. degree is either 0', 0 1 5 , or 0". For hyperregular r.e. sets, their jumps must be one of the first two degrees, while the jump of a nonhyperregular r.e. set automatically assumes the third degree. There exist BY,2 models in which only 0' and 0" are the only jumps of r.e. degrees (Mytilinaios and Slaman [15]), and there also exist BT,2 models in which all three degrees, 0', 0 1 , 5 and 0" are jumps of r.e. degrees (Chong and Yang [5]). Indeed a complete characterization is obtained through the notion of tame E2-coding: Definition 3.4 Let M be a BT,2 model. Let I be a S 2 cut. A ^-relation R on I x I is tame if for any a e I, there is a b € I such that if (x, y) G R and x < a, then y < b. In particular, if R is a £ 2 function, then R is bounded on every proper initial segment of I. T h e o r e m 3.3 ([5]) Let M be a BT,2 model Then the following are equiv alent: (a) Every tame T,2-relation on a Y,2-cut I is coded on I; (b) 0 1 - 5 is not the jump of an r.e. degree.
Thus the degree of jumps of r.e. degrees depend on the existence/nonexistence of codes, a phenomenon which hints at a fine structure theory for models
60
C H O N G AND Y A N G
of fragments of Peano arithmetic. We believe that this is an interesting phenomenon that warrants further investigation. The proof that (b) implies (a) in Theorem 3.3 introduces an interesting feature for infinite injury constructions on models of BY.2 (in the absence of S 2 induction) that has also surfaced (though not explicitly) in [16]. As it turns out, its finite injury counterpart for IE]. was present in [14] (discussed in the previous section). This is a technique we refer to as dynamic blocking in [5]. We illustrate with the proof that (b) implies (a) in Theorem 3.3. Suppose for the sake of contradiction that there is a £2 cut / with a tame £2 subset which is not coded. We construct an r.e. set A whose jump is 0 1 5 . The first step to note is that there is an r.e. set B so that for each e G / , B^ = {x\(x,e) e B} is either .M-finite or all of M, and that \Jd<eB^ is recursive. Furthermore, there is no uniform bound for UB^ where d ranges over those in / for which B^ is .M-finite. This is an analog of piecewise recursive r.e. sets in CRT first studied by Shoenfield. The existence of B uses the noncodability of a tame S 2 set in / . Next one constructs a thick subset A of B that is not low. To do this, the technique of blocking is used. Requirements are grouped into blocks of the same priority. One argues that the limit infimum of the restraints on each block exists. Three is, however, an inherent difficulty with using / as the number of blocks. This has to do with the fact that we are dealing with a II2 construction (the limit infimum function on block restraint) over a E 2 cut, so that to argue that each block will impose a limit infimum restraint becomes impossible if we were to specify the number of blocks of requirements a priori. The solution is to use the strategy of dynamic blocking. The number of blocks is not determined before hand. Rather it is decided by the con struction itself. Once the size of a block (say block e) and its limit infimum restraint is known, the size of block e + 1 will be, roughly speaking, equal to the length of this restraint. One then argues that the limit infimum of the restraints for block e + 1 exists, and this determines the size of block e 4- 2, and so on. In the end, there is a sub cut J C J such that for each e G J , the limit infimum of the restraints for block e exists, and that each requirement belongs to one of the blocks. The r.e. set constructed will be incomplete, and its jump will be 0 1 5 since the noncodable tame £2 set we started with is actually recursive in the jump. If A were low, that set would be A2(M) and so coded on / , a contradiction. The Sacks Jump Inversion Theorem states that every degree r.e. in and above 0' is the jump of an r.e. degree. Corollary 3.1 The Sacks Jump Inversion
Theorem is equivalent to IY.2
A STUDY OF DEFINABLE C U T S
61
over the base theory BT>2Proof. The construction of the Friedberg-Muchnik pair in Chong and Mourad [2] generalizes to degrees above 0 1 5 . By Theorem 3.6, these degrees are never the jump of an r.e. degree. Finally we sketch a proof of the Density Theorem in BY,2, which is done by Groszek, Mytilinaios and Slaman [16]. This is an illustration of how, in the absence of E 2 induction, it is still possible to carry out infinite injury constructions successfully, using all the techniques discussed thus far. T h e o r e m 3.4 Suppose that M is a model of P~ + -E?£2. Then the r.e. de grees are dense. Let A and B be r.e. sets such that B
62
C H O N G AND Y A N G
a block {Pe : e < b}. Divide it into two groups. The first one consists of all e < b such that there is some x G C and $e(B;x) = 0. By IY,i(B), it is .M-finite, thus providing only .M-finite computations. The other group of requirements cannot produce unbounded computations at I?-true stages, otherwise T(C) would be total and equal to A, implying that A is recursive in JB, a contradiction. Now we perform a tree construction, similar to the usual one. Firstly each node is associated with a block of requirements. For the preservation strategy, the outcome is, roughly speaking, the collective restraint imposed by the requirements in the block. For the coding strategy, the outcome is the collective computation T performed by the requirements in the block. The point to note is that the size of the next block depends on the outcome of the previous node. Thus, the blocks will settle down in a II2 way. At non-I?-true stages, it can be very large. At B-tvne stages, it will fall back to the true size, a phenomenon which also happens in the proof of Theorem 3.6. We then define the true path to be the leftmost one which is accessed at cofinally many stages, as in CRT. The reason for the existence of a true path is as follows. As in the analysis of block strategies, at sufficiently large B-true stages, we will always see the true length of agreement and the true computation of T. Thus we will visit the same path at these stages. On the other hand, fix any node a on the true path, the nodes to the left of a are abandoned because either we see more i?-true length of agreement in the preservation strategy or we see more computations T correctly calculate A. For a fixed a, the information required from A will be .M-finite. We can use this information as a parameter. Thus the nodes to the left of a can be enumerated from B. IT,i(B) ensures that there are .M-finitely many of them, and hence these nodes can only be visited .M-finitely often. Following a similar argument as in CRT, requirements are satisfied along the true path. The new difficulty is that the true path can be a cut. So we are required to check that every requirement falls into some block associated with a node on the true path. Fix a requirement Rn. Clearly it belongs to some block at stages larger than n. By carefully analyzing the reasons causing Rn to change to a different block, all changes can be traced back to B (again we need an .M-finite amount of A as a parameter). IT,i(B) will guarantee that Rn falls into a block associated with some node on the true path. It remains to argue that C
A STUDY OF DEFINABLE CUTS
63
that is, when the computations of B which could affect 7 have all settled down, and T(C,x) — A(x) with use 7. For a from region (I), we show that such nodes can be enumerated from A. For a fixed node /3, we have argued before that the nodes to the left of (3 can be enumerated from B, provided that we have .M-finite amount of information on A as s parameter. Since we have A as an oracle, and A computes B, all nodes from region (I) can be enumerated from A. Hence we can decide whether 7 associated with a enters C, by using A as an oracle. This completes the proof of the Density Theorem.
4
Some Open Questions
We end this paper with a list of open problems. The first two concern understanding the complexity of higher order priority constructions: (1) Investigate the complexity of 0"'-priority arguments. In particular, are there theorems using such constructions which are equivalent to 7 £ 3 over BT,s or even # £ 2 ? (It is noted in Section 2 that there is a model of / E i which behaves just like the standard model u. It is likely that certain 0"'-priority constructions can be carried out successfully in this model.) (2) Is there a minimal degree below 0' in every (any) £ £ 2 model? Chong and Mourad [2] produced a model in which there is a minimal degree (the degree of a £2 cut). The classical Spector construction was bypassed. This does not seem possible with the problem under discussion. Another ques tion, which appears to be quite difficult at this point, is whether there exists a minimal pair of r.e. degrees in # £ 2 models. 1 Phrased differently, the problem is to decide if the existence of a minimal degree below 0' or the existence of a minimal pair of r.e. degrees is equivalent to £ 2 induction over £ £ 2 . It may be worthwhile to add that the difficulties encountered in attacking these two problems are very similar to those experienced with analogs in the case of H%. It is tempting to conjecture that there is a uni versal, model-theoretic approach to solving these problems. The next two problems deal with understanding the degree-theoretic structure of models of weak fragments of P. (3) In a £ £ 1 model with £x cut / , is / low ? In other words, is I' recursive in 0' ? This question extends to £?£ 2 models M: Is every incom plete r.e. degree in M I0W2 ? As discussed in Section 3, the jump of an incomplete r.e. degree is either 0' or 0 1 , 5 . The latter is in fact the degree of 0' © / , where / is a £2 cut. One is thus asking if the jump of 0' 0 / is recursive in the complete £ 2 set. 2 1 Note added in proof. Chong, Qian, Slaman and Yang have given a negative answer to this problem, i.e., there is no r.e. minimal pair in any BT.2 model. However the minimal degree problem is still open. 2 Note added in proof. Chong, Qian and Yang have obtained some partial results. It
64
C H O N G AND YANG
(4) In a BEi model, / is of minimal r.e. degree, and is not regular. All r.e. sets above I are nonregular. It is known that there is a FriedbergMuchnik pair above / . What is the structure of degrees above / ? Does Sacks splitting or density hold in there ? Working with nonregular sets is challenging in that very often new techniques, very different from those employed in CRT, are required for the solutions. The following problem is admittedly rather vague. However, we feel that there are traces of evidence at hand to warrant a formal statement: (5) Develop a general theory of computation that elucidates the phe nomenon of similarity and inter-applicability of techniques and ideas be tween recursion theory on fragments of Peano arithmetic and ordinal recur sion theory. Finally, we pose a question that is inspired by the Coding Lemma (Proposition 1.2): (6) Is there a fine structure theory, in the spirit of Jensen [9], for models of fragments of Peano arithmetic ?
References [1] C. T. Chong, Maximal sets and fragments of Peano arithmetic, Nagoya J. Math., 115 (1989) 165-183 [2] C. T. Chong and K. J. Mourad, The degree of a E n cut, Ann. and Applied Logic, 48, (1990) 227-235
Pure
[3] C. T. Chong and K. J. Mourad, E n Definable sets without E n induc tion, Trans. Am. Math. Soc. , 334, (1992) 349-363 [4] C. T. Chong and Y. Yang, £2 induction and infinite injury priority arguments, Part I: Maximal sets and the jump operator, J. Symbolic Logic, to appear [5] C. T. Chong and Y. Yang, E2 induction and infinite injury priority arguments, Part II: Tame E 2 coding and the jump operator, it Annals of Pure and Applied Logic, Special Issue on the Recursion Theory Session, ASL 1995 Summer Meeting in Haifa, to appear [6] M. J. Groszek and M. E. Mytilinaios, E2-induction and the construc tion of a high degree, in: Recursion Theory Week, Proc. Oberwolfach 1989 (K. Ambos-Spies et al., eds), Lee. Notes Math. 1432, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1990, 205-223 is now known that in some BEi model, the cut J is not low, and in some BS2 model I' is not recursive in 0". What happens in the general BY.2 model remains open.
65
A S T U D Y OF D E F I N A B L E C U T S
[7] M. J. Groszek and T. A. Slaman, Foundations of the priority method I: Finite and infinite injury, preprint [8] M. J. Groszek and T. A. Slaman, On Turing reducibility, preprint [9] R. B. Jensen, The fine structure of the constructible universe, Annals of Math. Logic, 4 (1972), 229-308 [10] M. Lerman and S. Simpson, Maximal sets in a-recursion theory, Israel J. Math., 14 (1973), 236-247 [11] J. B. Paris and L. A. S. Kir by, E n -collection schemas in arithmetic, in: Logic Colloquium '77, North Holland, Amsterdam, 1978, 199-209 [12] K. J. Mourad, Ph.D thesis, University of Chicago, 1988 [13] K. J. Mourad, Unpublished
manuscript
[14] M. E. Mytilinaios, Finite injury and Ei-induction, J. Sym. Logic 54 (1989), 38-49 [15] M. E. Mytilinaios and T. A. Slaman, E 2 -collection and the infinite injury priority method, J. Sym. Logic, 54 (1989), 38-49 [16] M. J. Groszek, M. E. Mytilinaios and T. A. Slaman, The Sacks density theorem and S 2 -bounding, to appear [17] R. A. Shore, Splitting an a-recursively enumerable set, Trans. Math. Soc, 204 65-78
Amer.
[18] T. A. Slaman and W. H. Woodin, Ex-collection and the finite injury priority method, in: Math. Logic, and Its Applications, Lee. Notes Math. 1388, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1989 [19] R. I. Soare, Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidel berg, New York, Tokyo, 1987 [20] Y. Yang, The thickness lemma from P~ + / E i -f - , ^ E 2 , J. Sym. Logic, 60 (1995)
67
Lattice embedding into d-r.e. degrees preserving 0 and 1 * DING Decheng QIAN Lei Mathematics Department, Nanjing University Mathematics department, National University of Singapore
1
Introduction
We are used to say that a set A is r.e. if A is a domain or range of some partial recursive function. But we can also define r.e. sets by recursive approximations to their characteristic functions. A set A is r.e. iff there is a recursive function / such that for all x: A{x) = l i m / ( x , s ) s
f(x,0) \{8\f(x,8)?f(x,8
= 0 +
l)}\
This is to say, we begin by guessing x is not in A and we may change our mind at most once to put x into A. The natural generalization of this property is to allow the approximation to change more often. In this way we can get the notion of n-r.e. sets. A set A is n-r.e. if there is a recursive function f(x,s) such that for all x:
/(*,0) = 0 \{s\f(x.s)±f{x.s
+
l)}\
So, in particular, the 1-r.e. sets are precisely the r.e. sets. The 2r.e. sets are known as the d-r.e. sets, d-r.e. means "difference recursively enumerable" since a set A is d-r.e. iff there are two r.e. sets B and C such *The work is supported by Chinese National Nature Science Foundation and the Chinese 863 Project
68
D I N G AND Q I A N
that A = B — C. A degree is n-r.e. if it contains an n-r.e. set and if it is (n-fl)-r.e. but not n-r.e., we say it is a properly (n+l)-r.e. degree. The existence of properly d-r.e. degrees shows that the class of d-r.e. degrees is a proper extension of r.e. degrees. The class of d-r.e. degrees is playing a more and more important role in linking the studying of r.e. degrees and Turing degrees. Interest in the d-r.e. degrees stems from the affinity of it with r.e. degrees. We have found that there were some differences and similarity between d-r.e. and r.e. degrees. These differences and similarity let us get well understanding to the structure of degrees. One way to get an understanding to the various degree-theoretic struc tures is to see what lattices can be embedded into them. Lattice Embedding have been to show that such structures have undecidable theory and to show that the theory of such structures is decidable up to certain quantifier level. Studying lattice embedding has a long history. Now it is well known that all countable distributive lattices can be embedded into r.e. degrees and the situation on non-distributive lattices are more complicated: some can be and some not. Based on this results we got well understanding to the de cidability of r.e. structures. Studying lattice embedding into d-r.e. degrees and comparing it with the embedding into r.e. degrees could let us to get a deep understanding to the structure of d-r.e. degrees and the structure of degrees. But, comparing with the studying of r.e. degrees embedding into r.e. degrees, the studying of d-r.e. degrees has a short history. Downey [4] proved that the diamond lattice can be embedded into the structure of d-r.e. r.e. degrees preserving both 0 and 1, which contrasts to the Lachlan's Non-Diamond Theorem of r.e. degrees. In this paper we will continue the study of embedding to the d-r.e. degrees. We improve the Downey's result to show that There are r.e. degree a and d.r.e. degree b such that 0 < a < 0', 0 < b < 0', a U b = 0' and a D b = 0 and prove that M 5 and 7V5 can be embedded into d-r.e. degrees preserving 0 and 1. Our notations and terminologies are standard, which can be found in [1]
and [6].
2
Diamond Lattice
T h e o r e m 2.1 There are r.e. degree a and d.r.e. degree b such that 0 < a < 0 ' , 0 < b < 0 ' , a U b = 0' and a n b = 0.
2.1
Strategy
To prove the above theorem, we shall construct an r.e. set A and a d.r.e. set B which satisfy the following requirements: 5:
T(A@B) = K
(1)
L A T T I C E E M B E D D I N G INTO D . R . E .
P2e :
DEGREES
A / Ve
69
(2)
P2e+l ■ B^^e iVe : $f = $ f = / total =>* / recursive
(3) (4)
Where K G O ' , {
= $ f , r (*)]}•
If l(a, s) > max{/((7, t) : t < 5}, we say 5 is a cr-expansion stage. In the presence of requirement S, we cannot use the minimal pair method without any modification. Our method to meet the requirement Na is as follows: 1. Assign a number ba big enough. Afterwards, if there is a number < ba enters K, we cancel this requirement by canceling all parameters of this requirement e x c e p t ba and restart at 2. 2. Wait for a cr-expansion stage si which satisfies the following condition: there is an (least) x such that x < l(a,so) < l(a,si), $f0(x) ^ $ £ ( # ) and requirement Na was not canceled between stage so and s i , where so is the last cr-expansion stage before s\. In this case, we say Na has an error at x. At stage s\, we put 7(6^) into A and lift 7(7/) for all y > ba to be greater than s 0 (so the requirement S will not injure N^
D I N G AND Q I A N
70
if no element < ba enters K afterwards). Now we say we are ready to get a global win on Na. Set a restraint for B to be s\ and go to the next step. 3. Wait for the next cr-expansion stage s 2 . Now we have that BS2 \(B8l,x,si)
= B8l \(B8l, x, s i ) ,
so $f2(x) = $f2(x) — ^fj(^) 7^ $f0(x). At stage 5 2 , all elements which enter B between stage s 0 and stage Initialize all requirements with lower priority (so B\so is Now we have $A(x) = $f2(x) ^ <&B(x) = 3>f2+1(x) = global win on Na.
we remove s2 from B. preserved). $f0(x) and
Now we consider the interference between the requirements Na's. The main difficulty is as follows: a strategy of Na removes some elements out of B. This action could injure the computation of the £?-side of some require ment Na with higher priority. For instance, at a stage SQ Na has an error and we are ready to get a global win on it. But just before Na takes action to get a global win, Na removed some numbers from B. It could injure the computations of Na so that Na could neither get a global win nor correct its error which appeared at the stage s 0 . However, the tree method can solve the above problem automatically. For any requirement Na, if some requirement Np may hurt it, then (3 D a. This is because if Np has the higher priority than Na, then Na is initialized and the previous action will be abandoned. If (3 >L a, then at any a stage, Np is initialized. So if Np needs to remove some elements from B, these elements must enters B between two consecutive a stages. Hence this will not interfere the action of Na. In the remainder what we should consider is how to make A 0 B >T K. In our construction, we use a mark system j(x). We guarantee that for any re, 7(x) can be redefined at most finitly often and *y(x) can increase at stage 5 only if some 7(2/) for y < x enters A. If x enters K, we put 7(2;) into B. If it is removed at some later stage, then at that stage, we shall put some 7(7/) for y < x into B to replace j(x).
2.2
Construction
A follower x of Pa is said to be realized at stage 5 if ipeiS(x) = 0. A requirement Pa requires attention at stage 5 if it has not been satisfied and one of the following conditions is satisfied: 1. Pa has a realized follower x. 2. Pa has no follower.
L A T T I C E E M B E D D I N G INTO D . R . E .
DEGREES
71
A requirement Na requires attention at stage 5 if stage s is a ^--expansion stage and one of the following conditions is satisfied: 1. ba is undefined. 2. We have not found an error of Na since the last time when Na was canceled. There is an x such that l(a, s) > l(a,t) > x(where t is the last a stage before 5) and the requirement has not been canceled after t. $£j(x) ^ <&£s(x). In this case, we say we have found an error at x and are ready to get a global win on Na. 3. We are ready to get a global win on Na at stage h(a,s) not been canceled since then.
and Na has
"A requirement is initialized" means we cancel all parameters of it and let it be unsatisfied. "A requirement Na is canceled" means we cancel all parameters of it except ba. At any stage s, we define as as follows:
A 't+i
{
(5) cr^O
if s is a
,fiv
a*"l
otherwise
^ (7)
The function h(a, s) is defined to be the last c-stage £, if there is such a stage and JV^ has not been canceled or initialized since the stage t\ otherwise, it is defined to be 0.
Construction: Stage 0: Set A0 — B0 = 0. For all x, define T(x) = x, with use 7(2;) = x.
Stage 5 > 0: Stage s consists of two steps. Step 1: Find the requirement with the highest priority which requires attention at stage 5. If no such requirement, for every a >L &S, initialize requirements Pa and Na, go directly to step 2. Otherwise, adopt the appropriate case below: (1) if this requirement is a positive requirement Pa, do as follows: a) if Pa has a realized follower x, then put it into A or B according to \a\ is even or odd. Initialize all requirements with priorities lower than Pa. b) if Pa has no follower, assign a fresh number to be its follower. Initialize all requirements with priorities lower than Pa. (2) If this requirement is a negative requirement Na then do as follows:
72
D I N G AND Q I A N
a) if ba is undefined, then asign ba as a fresh number. Initialize all requirements with priorities lower than Na. b) We are in case 2. That means we have found an error since /i(cr, s). Assume x is the least such error. We put j(ba) into A and, for all x > &a, redefine 7(2;) to be fresh numbers such that 7(2;) is monotone in x. Initialize all requirements with priorities lower than Na. c) We are in case 3. That means we are ready to get a global win on Na. Remove all elements which enter B between h<7(ha(s)) and 5. let z = min{y : 7(2/) is removed at the stage s}, we put 7(2) into B. Initialize all requirements with priorities lower than Na. Step 2: Put j(ks) into B. For any Na with ba > k8, Cancel Na. End of construction.
2.3
Verification
L e m m a 2.2 There exists an f such that f = liminf 5 as, i.e. the true path exists. Proof: : Obviously. For any n, we denote fn to be
f\n.
L e m m a 2.3 For any n, the following statements hold: (i) There exists a stage s such that after s, there is no requirement Ra (Ra is Pa or Na) with a C fn acts after stage s. (ii) Requirement Pfn is met. (Hi) Requirement Nfn is met. Proof: : We prove this lemma by induction on n. To show (i), by inductive hypothesis and the definition of / , we can find a stage s0 such that after stage SQ, there is no requirement Pa or Na with a C fn or a
To show (ii), assume so &s above. By our assumption of / , there must be infinitely many fn stages. After stage so, if Pfn is assigned a follower x, it will never be canceled by our assumption of SQ. If x is never realized,
L A T T I C E E M B E D D I N G INTO D . R . E .
DEGREES
73
then we have Ve(x) / 0 = A(x) (for n = 2e) or # e ( z ) # 0 = £ ( x ) (for n = 2e + 1). If it is realized at stage s 3 > s 2 , then ^ e ) S 3 (x) = 0 and x enters A or B at the next fn stage, say s 4 . If n = 2e, then the theorem holds obviously. If n = 2e + 1 then at stage S4, every negative requirement with lower priority than Pfn is initialized, so they can not remove x from B. Besides, by assumption, no requirement with higher priority than Pfn could receive attentions after stage 54, so they can not remove x from B too. So (ii) holds. To show (iii), assume 55 > si such that after stage 55, Pfn does not receive attention and bf is defined at stage 55. Let s 6 > 55 such that
K\bfn=KsJbfn. Assume $„ = ®n = 9 ls total, then we must have infinitely many fn expansion stages and / n + i = /nT). In this case, we can recursively compute g as following: for any x, find an fn expansion stage s' after 56 such that Kfn,s') > x-> then we have g(x) = 3>^ 5 ,(x). If the above formula does not hold, then there must be two fn expansion stages 57, s% > SQ such that S7 = h(fn,s$) and there is the least y such that ^n,s7(v) / ^n,s8(2/)Then fn should act at stage sg and 7(6/ n ) is put into A to lift all other 7(771) for m > bfn. At the next fn expansion stage, say 59, we remove all elements enter B between 57 and 59, put 7s 9 (z), where z = min{y : y is removed at stage 59} into B and initialize all requirements with pri orities lower than fn. But at stage s 8 , 7(b/ n ) is lift to be greater than 5g. We have BS7 \s$ = /3 S9 [5 8 . By our assumption, there is no 7(771) such that m < bfn enters B later, so * * , 7 ( y ) = * * , 9 + 1 ( y ) = $ * ( y ) . But
*iJ(y) = <S9(y) = *£,8(y) = <*8(y) * *£*M-
So
*£(*) * *£(*)■
This contradicts to our assumption. Thus Nfn is met. L e m m a 2.4 K
3
7V5 lattice
In this section, we prove the following theorem: T h e o r e m 3.1 There are r.e. degrees a , b > 0 and d.r.e. degree c > 0 such that a < b, a U c = 0' and b Pi c = 0.
74
D I N G AND Q I A N
To prove the theorem, we shall construct d.r.e. set C and r.e. sets A, B which satisfy the following requirements:
A^i>e
P2e
C + i>t
\e+\
Re Ne S
B ± $e(A) $ e (B) = $ e (C) = / total => f recursive 3r(r(,4 ®C) = K) A
Q
(8) (9) (10) (11) (12) (13)
Where K G 0' is an r.e. set and assume {ks : s G UJ) is a recursive enumeration of it, {3>e : e G to} is an effective enumeration of all recursive functionals and {ipe : e G u} is an effective enumeration of all recursive functions. To satisfy the requirement Q, we ensure that all numbers of A are even and B n {2x : x G u } = A. The strategies to satisfy the requirements 5, P and TV are similar to the strategies used in the previous section. The only different is that we should consider the requirement R. The method to satisfy the requirement R is the Friedberg-Muchnic method. Now we describe the construction. We also use the tree method to construct the sets. The tree we use is same as the tree used in the above section. Each node a G T is used both for requirement N, P and R. If a
= $H)5(C5,y))}
We define ml(cr,s) = max{/(cr, t) : t < s}. At any a stage 5, if l((T,s) > m/(cr, 5), then we say 5 is a cr-expansion stage. At stage 5, we define as recursively as follows:
=
A
(14) "0
as
=
if 5 is a al expansion stage. otherwise
a ss
And the function h(a, s) is same as in the previous section.
(15) (16)
L A T T I C E E M B E D D I N G INTO D . R . E .
DEGREES
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A follower x of requirement Pa is said to be realized at stage s \iipejS(x) = 0, where |a| = 2e + i for i < 2. A follower x of requirement Ra is said to be realized at stage 5 if
76
D I N G AND Q I A N
f) We are in case (2). That means we have found an error at the stage h(a,s). Assume x is the least such error. We put j(ba) into A and, for all x > ba, redefine 7(2) to be an even fresh number such that 7(2;) is increasing in x. Initialize all requirements with priorities lower than Na. We say that we are ready to get a global win on Na. g) We are in case (3). That means we are ready to get a global win on Na. In this case, remove all elements which enter C between ha(ha(s)). Let x = mm{y : y is removed at the stage 5}, put x into C and initialize all requirements with priorities lower than Na. We say that we have got a global win on Na. Step 2: Put j(k8)
3.1
into C. For any Np with bp > ks, cancel Np.
Verification
The construction of N$ lattice is similar to that of diamond lattice. So we only discuss the difference between them. In this construction, the requirements P and S are met as the previous section. This is because this construction and the previous construction are same for these requirements. Let / be the true path of the construction and fn stands for
f\n.
For a requirement Rfn, if it is not injured after some stage s0> then it acts at most twice. So its influence to other requirements is finite. Besides, after stage so, it must have a follower x and this follower will never be canceled. If it never receives attention after it gets a follower, then at any fn stage t (note that there are infinitely many fn stages), we have
L A T T I C E E M B E D D I N G INTO D . R . E . D E G R E E S
4
77
M5 lattice
In this section, we prove the following theorem: T h e o r e m 4.1 There are r.e. degree a > 0 and d.r.e. degrees b , c > 0 such that a U b = b U c = c U a = 0' and a n b = b n c = c D a = 0 Note that we can not force b or c to be r.e., because it contradicts to Lachlan's non-diamond theorem. To prove this theorem we shall construct r.e. set A and d.r.e. sets B,C which satisfy the following requirements:
p3e
A^i>e B^i>e
P3e + 1
C^A
P3e+2 iV3e
AWl N3e+2
So Si
s2
*c(B) --- $ e (C) = / total =S> / recursive $e(C) == $e{A) = f total =>• f recursive
*e{A) == $e{B) = f total ==> / recursive 3T0(T0(B ®C)=K) 3 r i ( r i ( c ® A) = K) 3T2{T2{A®B) = K)
(17) (18) (19) (20) (21) (22) (23) (24) (25)
Where K £ 0' is an r.e. set and assume {ks : 5 G UJ} is a recursive enumeration of it. {3>e : e G UJ} is an effective enumeration of all recursive functionals and {ipe : e £ u} is an effective enumeration of all recursive functions. For requirements S, at every stage 5, we put 72(^5) into B and put 70(&5) and 7i(A:5) into C. We also define the length agreement function / as follows(i < 3):
{
maxjx :Vy < x($ c > J ,(£,y) = $ e , 5 (C,2/))} maxja; : Vy < x($e,s(A,y) = $ e , s (C,?/))} max{x : My < x($e,s{A,y) = $ejS(B,y))}
^= 0 z = 1 (26) i=2
the definitions of function ml and expansion stage are similar to that in the previous sections. For requirement A^3e+2, we do as follows: 1.1) Assign a number b big enough. If there is a number < b enters K later, we cancel this requirement by canceling all parameters of this requirement except b. 1.2) Wait for a (3e 4- 2)-expansion stage si such that l(3e + 2, si) > Z(3e + 2,s 0 ) > x, but $e,Sl(A,x) ^
78
D I N G AND Q I A N
(3e + 2)-expansion stage and the requirement was not initialized or canceled between stage s 0 and si. In this case, we say that we have found an error at x. At stage s i , we put 72(6) into A and lift the 72(b) and all other 72(2/) for y > b to be very big values. We say that we are ready to get a global win on -/V3e+21.3) Wait for the next (3e + 2)-expansion stage s 2 - At stage 52 we remove all elements enter B between stage SQ and 52 and, let x be the least number which is removed at the stage 5, put x into B. Now, if the requirement is not injured later, the requirement is satisfied permanently. For a requirement A^ 3e+ i, we do as follows: 2.1) Assign a number b big enough. If there is a number < b enters K later, we cancel this requirement by cancel all parameters of this requirement except b. 2.2) Wait for a (3e + l)-expansion stage S\ such that /(3e + l , s i ) > /(3e + l,5o) > x, but $e,si {A, x) / 3>e>So(A,:r), where so is the previous (3e-hl)-expansion stage and the requirement was not initialized or canceled between stage so and s\. In this case, we say that we have found an error at x. At stage s i , we put 71(6) into A, 70(b) into C and lift the 71(2/) and 70(2/) for all y > b to be fresh values. We say that we are ready to get a global win on iV3e+i2.3) Wait for the next (3e+l)-expansion stage 52. At stage 52 we remove all elements enter C between stage s 0 and 52 and put 71(6) and 70 (b) into B. Now, if the requirement is not injured later, the requirement is satisfied permanently. The strategy to meet a requirement Nse is different to those of the other cases. It has two kinds of possible actions. We call them ^-action and C- act ion. We do as follows: 3.1) Assign a number b big enough. If there is a number < b enters K later, we cancel this requirement by cancel all parameters of this requirement except b. 3.2) Wait for a 3e-expansion stage si such that Z(3e,si) > Z(3e,so) > x, but $ e > a i (i?,a;) 7^ $ e ) 5 o (Z?,x), where s 0 is the previous 3e-expansion stage and the requirement has not been initialized or canceled between stage so and s i . In this case, we say we have found an error at x. At stage s i , if CSl 2 CSQ, then we do C-action, else do ^-action. (We will show later that if we do ^-action, then we must have BSl D BSQ). If we do C-action, then go to 3.2a. If we do ^-action, then go to 3.2b. 3.2a) Put 7i(6) into A, 70(b) into B and lift the 7i(y) and 70(2/) for all y > b to be fresh values. (So C is preserved) Go to 3.3a. 3.3a) Wait for the next 3e-expansion stage 52. At 52, we put 72(6) into A and lift the 72(2/) for all y > b to be fresh values. (So the computation $eiS2(B,x) = $ejSl(B,x) is preserved). Go to 3.4a. 3.4a) Wait for the next 3e-expansion stage 53. At stage 53, we remove all elements enter C between stage SQ and 53 from C and put 70 (b) and
L A T T I C E E M B E D D I N G INTO D . R . E .
DEGREES
79
71(6) into C. Now if this requirement will not be initialized or canceled, then this requirement is satisfied permanently. 3.2b) For ^-action, we put 72(6) into A at stage si and lift 72(2/) for all y > b to be fresh values. (So B is preserved) We also put 70(6) into C and lift 70(2/) for all y > b to be fresh values. (This is because we shall put 70(6) into B at next 3e-expansion stage and this action can not destroy the current calculation.) Go to 3.3b. 3.3b) Wait for the next 3e-expansion stage 52. At stage 52, we put 70(6) into B and 71(6) into A. We lift 70(2/) and 71(2/) for all y > b to be fresh values. (Now, the computation of $CiS2(B,x) is preserved.) Go to 3.4b. 3.4b) Wait for the next 3e-expansion stage 53. At stage 53, we remove all elements enters B between SQ and S3 from B. Now if this requirement is never initialized or canceled, then our requirement is satisfied permanently. Now we consider interactions between requirements. At first, we have the following danger: Let Na have a higher priority than Np and \a\ = 3ei + 2, \(3\ = 3e2 + 2. Assume s 0 and si are two consecutive a expansion stages and we find an error at stage s\. If at some stage t0 such that s 0 < tQ < si> Np put some 7o,*0(fr') m t o B to lift other 70(2/) for y > b'. At a stage t\ such that to < t\ < 5i, y enters K, so we need put 70,^(2/) into C. But assume at some stage s 2 > $1, 7o,t0(b') i s removed from B, so it is possible that 1o,82+i{y) = 7o,t0(2/) < 7o,*!(?/). Now, we should put 7o,S2+i(2/) = 7<M0(2/) into B or C again to meet the requirement So- However, this action will cause B or C is not d.r.e. set, so the construction should be modified. The requirement N3e+2 should act as following: 1.1) Assign a number b big enough. If there is a number < b enters K later, we cancel this requirement by canceling all parameters of this requirement except b. 1.2) Wait for a (3e + 2)-expansion stage si such that l(3e + 2,s\) > l(3e 4- 2, s 0 ) > x, but $ e j S l (^4,x) ^ $e,So(A,x), where s 0 is the previous (3e + 2)-expansion stage and the requirement has not been initialized or canceled between stage so and s\. In this case, we say we have found an error at x. At stage s i , we put 72 jSl {b) into A and lift all 72(2/) for y > b to be fresh values. We also put 7o,So(&) ^nto C- (Note that 7o,*i(&) = 7o,s0(^)This is because if they do not equal, then the requirement N^e+2 must be initialized or canceled between stage s 0 and s\.) Lift 7o,a+i(&) to be a fresh value. 1.3) Wait for the next (3e-f 2)-expansion stage s 2 . At stage s 2 we remove all elements enters B between stage s 0 and s 2 and put 72(6) into B. We also put 7o(&) into B. Now, if the requirement is not injured later, the requirement is satisfied permanently. Note that if there is an element < b enters K between stage SQ and s 2 , then the requirement is canceled, so we can not get 1.3. If y > b enters K at stage t between stage s 0 and si, then we have jo,Sl{b) = lo,s0{b) < 7o,5o(2/)-
80
D I N G AND Q I A N
But even the value of 70(2/) decreases, it will recover to the value at stage so- Actually we have put 7o,si(&) into C instead of 70(2/)- If the element enters K between stage si and 5 2 , the situation is similar. We put 7o,S2(&) into C instead of 70(y). (Because we put 70(b) into C at stage s i , the 70(2/) can only decrease to 70,^+1 (2/) > 7o,*2(&))For a requirement A^ 3e+1 , because we have put jojSl(b) into C at stage s\ and 7o,5 2 W m t o B a t stage 52, the construction needs not change. For requirement TV^, either a .B-action or a C-action need not change the con struction due to the similar reason. Now the last obstacle is that when we remove something from B or C, we may destroy some computations relative to B and C. But if we generate an error later, we can not make a difference by removing some elements from B or C. This obstacle can be overcome by a tree construction. For a requirement Na, because at any ofO-stage we initialize all Np for (3 >L OL, the requirement Np for P >/, afO can not hurt the requirement Na. If a requirement Np can hurt Na, we must have (3 D a. Actually, in our construction, when requirement Np remove something from B or C(say at stage t), we never put anything into A at the same stage t. At the stage t, we initialize all requirements with lower priorities than Np. So the A-side computation of requirement Na can not be injured at stage t. Now at the next iV a -expansion stage, the A-side computation can not generate an error, so we will not find an error at the next 7Va-expansion stage. Next, in our construction, we never remove something from B and at same time also remove something from C. If we remove something from B at stage t, then the .B-side computation of requirement Na may be injured and we can not recover this computation. But at this case, we can not remove any element from C at this stage and all requirements with lower priorities are initialized. So for requirement Na, we can make a disagreement by remove something from C. (This is why we use ^-action and C-action for requirement A^3e). Now the last danger is that a~0 C Pi and a C Pi> Assume Npx remove something from B at first and Np2 remove something from C later, then they may destroy the both sides' computations and both can not be recovered. If Np1 has higher priority than Np2, then when Np1 acts, it initializes Np2. So when Np2 acts next time, we must have pass an Na~Q stage and the computation of B has been recovered. If Npx has a lower priority than iV^2, then at the stage when Np1 act, Np2 can not act (or Npx can not act.). At the next time when Np2 can act, this stage must be an afO stage, so the B side computation also has been recovered. Now we have finished the description of the construction. The verifica tion is similar to the previous sections.
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References [1] S. B. Cooper &; X. Yi, Isolated d.r.e. degrees, Preprint [2] S. B. Cooper, H. Harrington, A. H. Lachlan, S. Lempp & R. I. Soare, The d.r.e. degrees are not dense, Annals of Pure & Applied Logic vol. 55 (2), 1991 pp. 125-152 [3] D. Ding & L. Qian, Isolated d.r.e. degrees are dense in r.e. degree structure , Arch. Math. Logic vol 36, 1996 pp. 1-10 [4] R. G. Downey, D-recursively enumerable degrees and the nondiamond theorem, Bull. London Math. Soc, vol 211989, pp. 43-50 [5] G. E. Sacks, The recursively enumerable degrees are dense, Ann. of Math. vol. 80 (2), 1964, pp.300-312 [6] R.I.Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987
83
On Stationary Reflection Principles Qi Feng
§0
INTRODUCTION
It follows from the works of Godel and Cohen that the Continuum Hy pothesis can neither be proved nor disproved. In the past thirty years or so, many statements in mathematics have been proven to be independent from the axiom of set theory, ZFC. In order to settle these problems one way or the other, people have formulated many new axioms or principles of infinity. Notably, Martin's Axiom, the Diamond Principle, the Box Principles, the Proper Forcing Axioms, the Semiproper Forcing Axioms, Martin's Maxi mum, and the Principles of Stationary Reflection, as well as many large cardinal axioms. These powerful principles do settle many open problems and provide us with more understanding of the infinity. In this talk, I will give a survey on the Principles of Stationary Re flection. We start with a review of stationary sets. Then in section 2, we present some interesting forms of the principles of stationary reflection. In the following sections, we will present some applications of the principles.
§1
STATIONARY, P R O J E C T I V E STATIONARY S E T S O F C O U N T A B L E
MODELS
Primarily, stationary sets are subsets of ordinals. Here we will be con cerned a generalization of the notion of stationary sets to spaces of countable models, due to Jech [8]. Recall that if a is an ordinal and C C a, then C is unbounded in a if a = UC, and C is closed in a if (U(C n (3)) G C for all (3 < a. A set S C a is stationary in a if S C\C is not empty for all sets C closed and unbounded in a. T H E O R E M 1.1 ( F O D O R ' S L E M M A )
Suppose that K is a regular un
countable cardinal. (a) If (Ca | a < K) is a sequence of sets closed and unbounded in AC, then C = {a
84
Qi
FENG
(b) Suppose that S C K, is stationary in K and / : S —> /£ is such that / ( a ) < a for all a G S — {0} (i.e., / is regressive on S). Then there exists a stationary T C S such that / is constant on T. Assume that K is a regular uncountable cardinal and A > K is a cardinal. Let A be a set of cardinality A. We use PK(A) to denote the set of all subsets of A of size < K. A set C C P K (A) is closed if for every C-increasing countable sequence (x a | a < 0) from C of length 6 < K, (xa C x^ for all a < /3 < 0), the union of the sequence U{x a | a < 6} is also in C. C is u n b o u n d e d if for all x G PK(A) there exists some y G C such that x C y. C is a club if C is both closed and unbounded. A set 5 C PK(A) is stationary if for every club C, the intersection S C\ C is not empty. T H E O R E M 1.2
(JECH[8])
(1) All the clubs on PK(A) generate a K-complete normal filter. Namely, if (Ca | a G A) is a sequence of sets closed and unbounded in PK(A) , then
C=
{xePK(A)\\/aexxeCa}
is also closed and unbounded in PK(A) . C is called the diagonal intersection of the sequence. (2) If S is a stationary set in PK(A) , / : S —> A satisfying that f(x)ex for all x G 5, then there exists a stationary T C S such that / is constant on T. Another basic fact is that given two sets A and B of the same un countable cardinality , then there is a natural correspondence between the closed and unbounded sets and stationary sets. Namely if / : A —> B is a bijection, C C PK(A) is a set closed and unbounded in PK(A), letting C* = {f'X | X G C } , then C* is a set closed and unbounded in PK(B) (where f'X denotes the set {/(a) | a G X}). If 5 is a set stationary in PK(A), letting S* = {/"X | X G 5 } , then 5* is a set stationary in PK{B). Assume that B C A. Let C be a set closed and unbounded in PK(A). Then {XnB | X G C} is a set closed and unbounded in PK(B). Conversely, if C is a set closed and unbounded in PK(B), then {X G PK(A) \ XnB G C) is a set closed and unbounded in PK(A). Hence, if 5 is a set stationary in PK(A), then {X n B \ X € S} is stationary in PK(B). If S is stationary in PKIB), then {X G PK(A) \ X n 5 G 5 } is stationary in P«(A). We will be interested mainly the case that K = LUI . For an uncountable set A, we use [A]u to denote the set of all infinite countable subsets of A. This is a set closed and unbounded in PUl(A). If / : [A] A and I C i , then we say that X is closed under / if X is nonempty and for all e G [X]
ON
STATIONARY R E F L E C T I O N P R I N C I P L E S
85
the closure of X under / by clf(X). Notice that given / : [A]
{x
ePK(\)\xnKeK}
is closed and unbounded in PK(X). The following theorem is a folklore. T H E O R E M 1.4 Assume that C is a set closed and unbounded in PK(X) ( K, is regular uncountable and A > K is a cardinal). Then there exists an / : [X]
Qi
86
FENG
A sequence (Na | a < 6) of countable submodels of HK is an increasing continuous C-chain of length 6 if for all a < 0, (a C Na -< HK) and for all a < /3 < (9, (Na C N0), and if a < 0 is a limit ordinal then Na = \Jp
P is cr-closed.
To see this, let (pn \ n < u) be a decreasing sequence of conditions. Let 6n be the largest countable ordinal in the domain of pn. Let 6 = Un
O N STATIONARY R E F L E C T I O N P R I N C I P L E S
87
By the quoted lemma above, let x C T be a closed set of order type a + 1. Let x — {7^ I P < a} be the canonical enumeration. Let Np = f(ip) for (3 < a. Then (Np \ ft < a) G V is the desired increasing continuous G-chain of length a + 1. This proves the theorem. □ At this point, one would like to ask the following question: Which stationary sets of [HK]U contain an increasing continuous G-chain of length LJI? This turns out to be a very interesting and nontrivial question. Suppose S C [H^ is stationary and S contains an increasing contin uous G-chain of length u^. Then necessarily the set {N C\ w\ \ N G S] contains a closed and unbounded set in u)\. This motives the following concept of projective stationary set, intro duced in [4]. D E F I N I T I O N 1.1
The p r o j e c t i o n of a set S
C [HKY
is the
set
Proj(S) = {X H CJI : X G S}. We say that S C [HK]W is projective stationary if for every club C C [#«]", Proj(5 fl C) contains a club in LJ\. Equivalently, S is projective stationary if and only if the set ST = {X G S I X fl u\ G T} is stationary for every stationary subset T of CJI . It follows from the definition that every projective stationary set is stationary. But the converse is not true. For example, let 5 , T be two disjoint stationary subsets of u\. Then both
{Ne[HK]"
INn^eS}
and {Ne[HK]u | 7 V n ^ eT} are stationary sets but none of it is projective stationary. T H E O R E M 1.6 Let A C K be stationary such that all a G A are of cofinality u. Then {N G 1 ^ ] ^ | sup(N H K) e A} is projective stationary. P r o o f Let T C CJI be stationary. Let / : [iJ«]
HK)eA}
88
Qi
FENG
and {N
G [HK]U
| sup{N
DK)<EB}
are disjoint projective stationary sets.
§2
P R I N C I P L E S OF STATIONARY R E F L E C T I O N
We can now ask if every projective stationary subset of [HK]U can con tain an increasing continuous G-chain of length CJI for every regular K > co2The answer is independent of the axiom of set theory. And the positive an swer has many interesting consequences. We now present the positive answer as the following Projective Sta tionary Reflection principle ( P S R P ) [4]: P r o j e c t i v e Stationary Reflection: For every regular cardinal ft > u>2, if S C [HK]" is a projective stationary set, then there exists an increasing continuous G-cnain (Na | a < ui) of countable elementary submodels of HK of length uj\ such that Na G S for all a < UJ\ . We now show that the Projective Stationary Reflection Principle fol lows from Martin's Maximum. Let us recall Martin's Maximum: A forcing P is u)\-stationary preserving if every stationary subset of a>i in the ground model remains stationary in the generic extension. Martin's Maximum is the following statement: IfP is a UJ\ -stationary preserving partially ordered set, if{Da \ a < UJI} is a sequence of dense subsets ofP of size &i, then there exists a filter G C P which meets every Da. It is proved in [6] that if the existence of a supercompact cardinal is consistent with the axioms of set theory, then Martin's Maximum is also consistent with the axioms of set theory. It is shown also in [6] that Martin's Maximum has many interesting consequences. T H E O R E M 2.1 ( F E N G - J E C H [4]) Assume Martin's Maximum. Then the Projective Stationary Reflection principle holds. In fact, if S C [H^ is projective stationary, then there exists a strongly increasing continuous G-chain of length uoi through 5. P r o o f Let S C [H^Y be projective stationary. The idea is to shoot a strongly increasing continuous G-chain of length UJI through 5 . Thus, a condition p = (Na | a < 6) is a strongly increasing continuous G-chain of length 0 + 1 for some 6 < UJ\ such that for all a < 6 {Na € S). The ordering is by extension.
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89
Let P be the set of all conditions. We are going to show that forcing with P preserves stationary subsets of u)\. For a < ui, let Da = {p G P \ a G dom(p)}. For x G HK, let Dx = {p G P | 3 a G dom(p) (x G p(a))}. We claim that each Da and each Dx is dense. To see this, let a < uj\ and x G # K . Let p = (Np | /? < 0) be a condition. Then 5i = {A7" G S \ {p, x} C TV} is stationary. Applying the lemma above, let (M 7 | 7 < a) be a strongly increasing continuous G-chain from S\. We then define q(j3) = Mp for all (3 < a such that j3 > 6 and define q(j3) = p((3) for all (5 < 0. It follows that q < p and q e Da D Dx. We can now show that the forcing preserves stationary subsets of CJI . Let T C CJI be stationary. Let C be a name for a club subset of CJI . We would like to show that Ih f D C / 0. Let p G P be a condition. Let ST = {TV G 5 | N n CJI G T } . Then 5 T is stationary. Let A > (2 2 ) + be a regular cardinal. Consider the structure H = (tfA,G,A,P,C,ST,--->, where A is a well ordering of H\. Let TV ^ H be countable such that TV n HK G 5 T , TV Pi HK -< HK, and {C,S,T,ST,p} Q N. Let S = TV n u>i. Let (,Dn | n < w) be a list of all dense subsets of P which are in TV. Let p0 = p. Inductively, pick pn_i_! G Dn D TV so that p n +i < Pn- Let 0n be the largest countable ordinal in dom(pn). By elementarity and a density argument, we have 8 = \Jn
q=
\JpnumNnHK)} n
is a condition stronger than all pn. Since C G TV, for each /3 < 6, there is a name 7 G TV such that Ih 7 G C & ^ < 7. Each such name corresponds to a dense subset in TV. Hence, q\\- 7 G 6. It follows that glh "C n 5 is unbounded in 6". Thus, glh ^ G C. Therefore, glh- C n f / 0 . It follows then that forcing with P preserves stationary subsets of u)\. Now let G C P be a filter meeting all the dense subsets Da for a < uj\ defined above. Let
(Na\a
Naes.
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This finishes the proof.
□
From the proof above, we have seen that for a given stationary S C [HKY, there is a natural forcing notion Ps associated with S to shoot an increasing continuous G-chain of length uoi through S. COROLLARY 2.1 S C [HK]U is projective stationary if and only if the forcing notion Ps preserves stationary sets of u\. Proof The harder direction has been proved in the above. To see the other direction, assume that S is not projective stationary. Let T C u>i be stationary such that ST = {N G S \ N D uj\ G T} is not stationary. Then T is not stationary in the extension by Pg. □ After the work of Foreman, Magidor and Shelah [6], Todorcevic in a cir culated hand written note [16] in September 1987 formulated the following Strong Reflection Principle (SRP) (See also [1], page 57-60): Strong Reflection Principle: For every K, every S C [K]" and for every regular 6 > K there is an increasing continuous €-chain {Na \ a < coi} of countable elementary models of HQ (with NQ containing a prescribed element of He) such that for all a < u\, NaC\K G S if and only if there exists a countable elementary submodel M of He such that Na C M, M n CJI = Na fl CJI , and M C\K G S. In the following, we show that the Strong Reflection Principle and the Projective Stationary Reflection principle are equivalent. T H E O R E M 2.2 ( F E N G - J E C H [4]) The Strong Reflection Principle holds if and only if the Projective Stationary Reflection principle holds. Proof First we show that the Strong Reflection Principle implies the Projective Reflection principle. Let S C [HK]" be a projective stationary set. Let {Na \ a < ui} be a continuous G-chain of countable elementary submodels of He with 6 > /^, given by SRP for S. We would like to show that {a < u \ Na D HK G S] contains a club in Ml-
Assume not. Let T = {a < ui \ Naf]HK T is stationary. Define D to be the following set
^ S and Naf)uJi
— a}. Then
D = {N G [HeY | HK G N & V/3 e N n CJI N0 G N}. By normality, D contains a club on [HeY■ Since S is projective stationary, there exists an N G D such that N fl CL>I G T, NDHK G S and N is an elementary submodel of He- Let a — NDoui.
O N STATIONARY R E F L E C T I O N P R I N C I P L E S
91
Since if /? < a then Np C N, Na D^ = a = N n^ and Na C TV. Hence Na fl HK G S by the property of JVa. This is a contradiction. This proves that {a < ux \ Naf) HK e S} contains a club in ui. We prove now that the Projective Stationary Reflection principle im plies the Strong Reflection Principle. Let K > uj\ and 0 > K. Assume that S C [K]U and 0 is regular. Define S* to be the following set: for TV G [Ho]", let TV G S* if and only if TV -< (He, G, A) and that there exists a countable M -< (He, G, A) such that NCM,Nnuji=Mnuj1 and M D « G 5 implies that NHKE S. Claim
5* is projective stationary in [He]".
Let g : [i7^]
if r ( a , x i , - - - , x m ) < «, otherwise.
Then h G TV'. Hence h € N' n He C M. Therefore, a = h(ai,- • • , a m ) G Mfl/c. This finishes the proof that 5* is projective stationary in [He]". Applying the Projective Stationary Reflection principle, let (TVa | a < LJI ) be an increasing continuous G-chain of length CJI such that TVa G 5'* for all a < u)\. Certainly, this is what the Strong Reflection Principle needs for the given S. D There are some weaker forms of stationary reflection principles known before Todorcevic's Strong Reflection principle was presented. In the fol lowing, we give some examples. T h e Weak Stationary Reflection principle For all A of cardinality at least H2, for all stationary S C [A]", there exists a subset X C A of cardinality Ni such that S D [X]u is stationary in
[xr.
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T H E O R E M 2.3 ( F E N G - J E C H [3])
Qi F E N G
Let A be a set of cardinality at
least N2- Then the following are equivalent: (1) For all stationary S C [A]u, there exists an X G [A]*1 such that S fl [X]w is stationary in [X]". (2) For all stationary S C [A]", for all / : [A]
Let A be a set of cardinality at
least ^2- Then the following are equivalent: (1) For all stationary S C [A]", for all Y G [A]" 1 , there exists an X G [A]*1 such that Y C X and S n [X] w is stationary in [X] w . (2) For all stationary S C [A] w , for all closed and unbounded C C [A]^ 1 , there exists some X E C such that 5 fl [X]" is stationary in [X] w . (3) There exists a Y G [A]Wl such that for all stationary S C [A]^, there exists some X G [A]Ul such that 7 C I and S fl [X] w is stationary in
[Xf. A subset F C [iJ K ] Hl is Ui-closed if for every C-increasing sequence {Xa I a < to} of length w\ from F , the union \JaXa is in F . F is unbounded if for every X G [H*]**1 there is some Y G F such that I C F . Fleissner [5] introduced the following stationary reflection principle, called A x i o m R. If S C [i?«]w is stationary, T C [i?*]**1 is uJ\-closed and unbounded, then there exists some X G T such that 5 fl [X] w is stationary in [X]". T H E O R E M 2.5 ( F E N G - J E C H [4]) Assume the Projective Stationary Reflection Principle. If S C [#«]'*' is stationary and F C [i7^] Nl is cji-closed and unbounded in [iJ^]^ 1 , then there exists an increasing continuous Gchain (Na | a < u)\) such that {a < LOI \ Na G S} is stationary in UJ\, and UaNa is in F . Hence the Stationary Reflection Principle holds. Notice that the Stationary Reflection Principle and the Axiom R are strictly weaker than the Strong Reflection Principle, because they do not imply that the nonstationary ideal on CJI is saturated. P r o o f Add F to our language as a unary predicate. Let S C [H^. There is another projective stationary set p(S) C [H*]" naturally associated with S.
O N STATIONARY R E F L E C T I O N P R I N C I P L E S
93
Let r c W l . Define ST = {N G 5 | N n LJX G T } . Let F C {T C CJI | T is stationary and ST is not stationary } be a maximal antichain of the smallest cardinality. Define p(S) to be the following set: {N -
[H„]w and (3AeNnF(Nf)uj1
G A) 4=> N # S)}.
Then p(S) is projective stationary in [HK]W. If we assume that (Na \ a < uj\) is an increasing continuous G-chain from p(S), then S is stationary if and only if {a < u\ \ Na G S} is station ary. Let us assume that S is stationary. Toward a contradiction, let us assume that {a < u\ | Na G S} is not stationary. Then the complement of this set contains a club. It follows that for every A G F there is some a < UJ\ such that A G Na. Let / : u)\ - ^ F b e a surjective mapping. For each a < uj\ let Ca C [iJ K ] w be a club to witness that / ( a ) G F . Let C = Aa
r - j i v r i u ; ! iTVGCn^}. Let a G T be such that for all Ae Naf)F there is a /? < a with A = /(/?), for all (3 < a, f(0) G 7Va, and a = 7Va n wi. Then 7Va G 5. This is a contradiction. Hence {a < u>i \ Na G 5 } is stationary in CJI. To see the other direction, assume that S is not stationary. By mini mality, the cardinality of F must be one and the member of F must contain a club. By elementarity, there must be such a club in A^0. Therefore, for all a < c^i, Na n a>i must be in this club. Hence no Na will be in S. We now check that the union of the iV a 's is in E. Let a < u)\. Since 7Va G Na+i, let {Xn \ n < LJ} € Na+i be an enumeration of Na D F . Let X = U n X n . Then X G Na+i. Since it is true that "there is some Y G F such that I C 7 " , A^+i must satisfy this sentence. Let Xa G Na+i be a witness. Then X C X a by elementarity. It follows that the sequence {Xa \ a < U\] is a C-increasing sequence from F . Hence X* = U a X a is in F . If a G iV a , then it is true that "there is some Y G F such that a G T , N a satisfies this sentence. Hence Na C X* for all a < CJI. Since each X a G iV a + i, I a 6 F * = U a A^ a . It is true that | X a | = Ni. Therefore, Y* satisfies this. Let / G F * be a bijection. Since o;i C y * , by elementarity, the range of / must be a subset of y * . Hence XaCY*. It follows that X* = Y*. This shows that the Projective Stationary Reflection principle implies Fleissner's Axiom R as well as Velickovic's form of stationary reflection principle [18]. □ Let us consider another type of stationary reflection.
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Let EK be the following statement: K, is a regular cardinal n> u)\, there exists a stationary subset E C K such that every member of E has cofinality UJ and for every 6 < K,, E fl 6 is nonstationary in S. In [11] Jensen showed that for every K, > UJ, EK+ holds in L. We will see that by the works of Foreman, Magidor and Shelah, and Todorcevic, when Martin's Maximum, or the Projective Stationary Reflec tion principle holds, the negation of EK hold in a very strong sense. This is also related to a question of Friedman. In [7], Friedman asked if every stationary E C K satisfying that if a G E then cf(a) = UJ can contain a closed copy of CJI, for regular K > UJ2. A positive answer follows from the Martin's Axiom, as proved by Foreman, Magidor and Shelah in [6]. Todorcevic showed in [16] that the Strong Re flection principle suffices. T H E O R E M 2.6
(TODORCEVIC
[16])
Assume the Projective Stationary Reflection. Let K > UJ2. Let E be a stationary subset of K such that every a G E has cofinality UJ. Then E contains a closed copy of uj\. Proof Fix a regular cardinal K > UJ2Let E be a stationary subset of K such that every a G E has cofinality UJ.
Consider the following set: S = {N e [HK]U | N -< HK, and sup(7V n/c)G E}. We have seen previously that S is projective stationary. Applying the Strong Reflection Principle, let {Na \ a < Ui} be an increasing continuous G-chain such that 7VQ G S for a G uj\. Then for each a < CL>I, sup(7Va 0 K,) G E. We are done. □ COROLLARY 2.2 Assume the Projective Stationary Reflection prin ciple. Then the principle EK fails.
§3
SATURATION OF
NS(JJI
In this section, let us consider the saturation of the nonstationary ideal on uj\. Let F be a set of stationary sets on UJ\ . F is an antichain if A D B is nonstationary for A ^ B in F. F is a maximal antichain if F is an antichain and for every stationary subset T C u)\ there exists some A € F such that T n A is stationary. The nonstationary ideal on uj\ is saturated if every maximal antichain has size at most H\.
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In [14], Steel and Van Wesep, later improved by Woodin in [20], showed that the nonstationary ideal may be saturated using determinancy. In [6], Foreman, Magidor and Shelah showed that Martin's Maximum implies that the nonstationary ideal on u\ is saturated. In [16], Todorcevic showed that the Strong Reflection Principle is sufficient. T H E O R E M 3.1 (TODORCEVIC [16]) Assume the Projective Station ary Reflection principle. Then the nonstationary ideal on CJI is saturated. P r o o f Let F be a maximal antichain of stationary subsets of ui. Consider the following set:
S = {Ne [H^Y | N ^HUJ2,FeN^nd3AeFnN
(Nf)^
G A)}.
C l a i m S is projective stationary. Let T C uj1 be stationary. Let A G F be such that A fl T is stationary. Let C be a club on [i? W2 ] u '. Then we can find an elementary submodel N of HU2 with the property that N n Ui G A n T and N G C. Hence S is projective stationary. By the Projective Stationary Reflection Principle, let {Na \ a < u>i} be an increasing continuous C-chain of elementary submodels of HUJ2 such that a C Na for every a and there exists a club C C w j such that if a G C then a = Naf) LOI and Na G S. We proceed to check that F C X = \J{Na | a < LOI). Let A G F. Assume that this A $ X. Let Y be the skolem hull of X U {A}. Let Ma be the skolem hull of Na U {A}. Let D C C be a club such that for every a G D we have that Ma n u i = Na fl u\ = a. Let a G D fl A. Then a = Na C\ ui = Ma D LJ\. There must be some B G F fl iVa such that a G -B. This 5 must be different from A. Hence AC\B must be nonstationary. But by elementarity, as both A and B are in Ma, there must be some closed and unbounded subset E G Ma of CJI such that # fl A n -B is empty. We then have a contradiction because E G M a implies that Ma fl CJI G E and M a 0 CJX G i n R Therefore F C X. Hence the size of F is at most Ni. □ We now take a closer look at the saturation of the nonstationary ideal on UJI . It turns out that it itself is a kind of reflection property. To start with, let us define first certain filters. For a regular cardinal ft > ^ 2 , for X C [HK]U, let X be in TK if and only if for every stationary subset A C u>i there exist a stationary subset B C A and a closed and unbounded subset C C [if*]" such that {N e C \ N HUJI e B} C X. T H E O R E M 3.2 ( F E N G - J E C H [4])
The following are equivalent:
(1) The nonstationary ideal i V S ^ on ui is saturated. (2) For every regular cardinal K > u>2, for every stationary set S C [HK]", there exists a stationary set A C CJI such that 5 is A-projective
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stationary (i.e., for every stationary B C A, the set {TV G S \ N fl u\ € B} is stationary). (3) For every regular cardinal K, > UJ2, for every X C [H*]", X G TK if and only if X contains a closed and unbounded subset. Namely, the filter TK is just the club filter on [HKY. (4) For every regular cardinal K > LJ2- for every X G TK, there exists an increasing continuous G-chain (Na \ a < uj\) of countable elementary submodels of HK of length u\ such that Na G X for all a < LUI . Proof (1) =>> (2) Let S C [HK]" be a stationary set. If S is projective stationary, then there is nothing needed to be proved. So we assume that S is not projective stationary. For a stationary T C ui, we let T G F if and only if there exists a club C C [HK]" such that
rn{7vno; 1 |iVGCnS'} = 0. Since S is not projective stationary, F is not empty. Because the nonstationary ideal on ui is saturated, we can take a se quence {Aa | a < CJI} from F so that for every A € F , A - V a < a ; i A a is nonstationary, where Va
= {Ne
[HK]" | Va G T V H ^ TV G Ca}.
Let A = UJ — V a < u ; i A a . We claim that S is A-projective stationary. First notice that A is stationary. This is because Aa
TniNnux | N e CnAa
O N STATIONARY R E F L E C T I O N P R I N C I P L E S
97
(3) implies (4) is trivial. (4) => (1) Let F be a maximal antichain of stationary subsets of CJI . Let SF = {N G [HK]" \3AeFnN(Nncj1e A)}. Then SF G TK. Namely, for a given stationary subset A C ui, let T G F be such that B = A n T is stationary. Let C be the club of all countable elementary submodels of HK which contains T. Then if N £ C and i V f l w i E f i then N e SF. Now by the argument of Example 2.1, assuming (4), F has cardinality at most Hi. This completes the proof. □ The next thing we want to show is that the presaturation of the nonstationary ideal on u)\ is also a kind of reflection property, which in turn is equivalent to the filter TK being countably closed. First let us recall that the nonstationary ideal on u\ is presaturated if for every countable sequence {Fn | n < u} of maximal antichains of the nonstationary ideal on a>i, for every stationary subset T, there exists a stationary subset B C T such that for each n the set {A G Fn | A n B is stationary } has cardinality at most Hi (see [6]). T H E O R E M 3.3 ( F E N G - J E C H [4])
The following are equivalent:
(1) The nonstationary ideal NSUJl on LO\ is presaturated. (2) For every regular K > CJ2, the filter TK is cr-closed. (3) For every regular cardinal K > CJ2, for every countable sequence (Xn | n < u) from TK, for every stationary subset T C CJI, there exists an M ^ i7 K such that wi C M , M has cardinality Ni and {TV G [M] w D PlrKu; ^ n | AT n CJI G T} is stationary in [Af]w. Proof (1) => (2) Let X n G TK for n < u;. Let X = C\n
T = BH f| va
and let n<cu
It follows that if N G C and iV n u i G T then iV G X.
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FENG
Hence the filter TK is cr-closed. (2) => (3) It will be sufficient to show that every l G f K reflects. Let X G TK* Let T be a stationary subset of u i . Let B C T be stationary and let C C [il*]"' be a closed and unbounded subset such that for all N £ C, N HLUI e B implies that N e X. Let (Na | a < cui) be an increasing continuous G-chain from C. Let M = U a ^ ! Na- Then {iVa | a = Na n wi G 5 } is stationary in [M] w . (3) =► (1) Fix a sequence {Fn \ n < u] of maximal antichains of the nonstationary ideal on UJ\. For each n < to, let Sn = {N -<: HK | N is countable and 3 A G F n n JV (N n wi G A)}. Then every Sn £ J7^. Fix a stationary subset T of CJI. Applying (3), let X -< HU2 be such that ui C X , X has cardinality Ni and {TV G [X] w n f | n < u ; Sn \ Nf)ui G T} is stationary in [X]". Write X = \Ja
is a stationary subset of T. We claim that for every n the set { A G F n | A n i? is stationary } has cardinality at most Ni . Assume not. Let n be the least counterexample. Let A G Fn — X be such that A n B is stationary. Let M a be the skolem hull of Na U {A} for each a < CJI . Then C = {a < CJI | a = 7Va D ui = Ma D c^i} is a club. Let a G C fl i fl 5 . Let Z G iVa fl F n be such that a e Z. Since both A and Z are in Ma which is an elementary submodel of i7 K , and Ma Hcoi = a G A fl Z, we conclude that A. n Z is stationary. This is a contradiction. Hence for every n < u, for every A G -Fn, if A fl £? is stationary, then A € X. We are done since X has cardinality Hi. □ To close this section, let us present the proof of Foreman, Magidor and Shelah of the fact that Stationary Reflection principle implies the presaturation of the nonstationary ideal on ui. T H E O R E M 3.4
( F O R E M A N , M A G I D O R AND SHELAH
[6])
O N STATIONARY R E F L E C T I O N P R I N C I P L E S
99
Assume the Stationary Reflection Principle. Then the nonstationary ideal NS^ is presaturated. Proof We prove the theorem by proving two lemmas. Let K be a suffi ciently large regular cardinal. LEMMA 3.1 Let T C ui be stationary. Let {Xp | (3 < 6} be a maximal antichain of stationary sets below T. Then there is a club C C [HK]U such that each iV G C is an elementary submodel and if N G C, iV n CJI e T, then there is some (3 such that iVDcJi G Xp and the skolem hull of NU {(3} contains no new countable ordinals. P r o o f of the lemma. Assume otherwise. Let S denote the set of countable elementary submodels N such that N f) u>i G T and for every j3, either N fl CJI i Xp or N Pi CJI is in the skolem hull of N U {/?}. Then S is stationary in [HK]U. By Reflection, let X be an elementary submodel of cardinality Ki such that coi C X and S Pi [X]^ is stationary in [X]u. Let X be the union of a C-increasing sequence {Na \ a < UJI} of countable models of length CJI. Let 5* = {Na n CJI | Na G S}. Then 5* C T is stationary. Now if (3 G X , then 5* fl X^ is bounded. Hence there is some (3 £ X such that 5* fl X/3 is stationary. Let (3 be the least such. Let Y be the skolem hull of X U {/3} and iV^ be the skolem hull of Na U {/?}. Let a < Ui be such that 7V^ fl CJI = TVQ, Pi CJI G 5* fl X^. Then we get a contradiction. This finishes the proof of the lemma. Let T be stationary in ui. Let {Xn | n < LJ} be a sequences of maximal antichains of stationary sets below T. Each X n = {X^ | /? < 0n}. Let S = {iVG [frK]w | V n < c j 3 / 3 G i V ( i V n u ; i G X £ ) } . L E M M A 3.2
5 is stationary in
[H^.
Proof of the lemma. Let D b e a club in [HK]U. Applying the previous lemma to each n < u, we get a club Cn. Let C be the intersection of all the C n 's. Then C is a club. Using the club C fl D, by induction, we get an
N
eDnS.
To finish the proof of the theorem, we apply the Stationary Reflection principle to the stationary set S. Let X be of cardinality Ni such that X is an elementary submodel, LJ\ C X , and 5 fl [X] w is stationary in [X]^. Let X be the union of a C-increasing sequence {7Va | a < LOI} of countable elementary submodels. Let T* = {Na fl ui \ Na G 5 } . Then T* C T is stationary. This is a desired stationary set. □
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§4
Qi
FENG
O N CARDINAL ARITHMETIC
The first consequence on cardinal arithmetic of the stationary reflection principles was proved by Todorcevic in [15]. Then Velickovic in [18] proved SCH from a slightly stronger version of the Stationary Reflection principle. T H E O R E M 4.1
(TODORCEVIC
[15])
Assume that every stationary set S C [LJ2]U reflects. Then 2W < w2. Proof For a < u>2, let ea : a —» \a\ be a bijection. Let C be the set of all countable A C w 2 such that for all a G A, A is closed under ea and e" 1 , and LJ C A. Then C is a club in [0J2Y • Let us consider the following set S: S={AeC\Va
\OL
is a club in [X]u, disjoint from S. If X fl u\ / CJI , then X n CJI G c^i and the order type of X is LJ\ . In this case, let <5 = X fl uo\. Then the set {e'^S \ a £ X - U!} is a club in [X] w , disjoint from 5*. By the assumption, S is not stationary in [u^]^- Hence C — S contains a club in [002Y■ By a theorem of Baumgartner and Taylor [0], \C — S\ — 2^°. Hence 2*° < cu2. D T H E O R E M 4.2 (VELICKOVIC [18]) Assume that for every stationary S C [HK]U, there is a continuous G-increasing chain of countable models {Na I a < uoi) such that {a < u\ | Na G 5} is stationary in LOI. Then for all regular K, > LO2, KH° = ftAlthough the Stationary Reflection principle implies SCH, it does not solve the Continuum problem. It does not decide which of the two values 2^° must take. Namely, it is consistent that the Stationary Reflection principles plus CH; it is also consistent the Stationary Reflection principles plus 2^° = K2, assuming large cardinals are consistent. However, the Strong Reflection Principle does solve the Continuum problem. Recently, Woodin in [19] has
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O N STATIONARY R E F L E C T I O N P R I N C I P L E S
shown that the Strong Reflection principle implies that 2^° = tt2- In fact. Woodin has proved that it implies the second uniform indicernible is ct^T H E O R E M 4.3 ( W O O D I N [19]) Reflection principle. Then 2K° = tt2T H E O R E M 4.4
(TODORCEVIC
Assume the Projective Stationary
[16])
Assume the Projective Stationary Reflection principle. Then for all regular K > CJ2, K*1 — K- Hence the Singular Cardinal Hypothesis holds. Proof Let E = {a < K, \ cf(a) = to}. Partition E into K many pairwise disjoint stationary subsets, say E — U{Ea \ a < K}. Let {Ta | a < CJI} be a partition of u± so that each Ta is stationary for a < u)\ and for every stationary subset X C ^ i , there is some a < u\ such that X C\Ta is stationary. Let / : uj\ —► K be a strictly increasing function. Then the following set Sf is projective stationary: S = {N G [HK]» \\/a
sup(7V n K) G
Ef{a)}.
To see this, let T C CJI be stationary. Let a be the least countable ordinal such that TC\Ta is stationary. Then the following set is stationary: {N G [H^
| a< NHLJI
eTnTa&sup(NPiK,)
G
Ef{a)}.
Let {Na | a < cvi} be an increasing continuous G-chain from Sf. Let 0/ = sup{sup(Na
C\ K) \ a < LJI}.
It follows that Ea fl #/ is stationary in Of if and only if there is some 0 < uo\ such that a = /(/3). Now the mapping from / to Of is a one-to-one mapping. Hence K?1 — K. By Silver's theorem [13], this in turn implies the Singular Car dinal Hypothesis holds: For every singular cardinal X, if 2 c / ( A ) < A, then A c / ( A ) = A + . □
§5
O N C H A N G ' S C O N J E C T U R E AND THE B O X P R I N C I P L E
The first indication that stationary reflection principles are very strong properties is probably that Chang's conjecture follows from the weak Sta tionary Reflection principle, as proved by Magidor in an unpublished work.
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Qi F E N G
T H E O R E M 5.1
(MAGIDOR)
Assume the weak Stationary Reflection Principle. Then for A suffi ciently large, the following form of strong Chang's conjecture holds. (*) there is (1) (2) Nc\LJi: Proof NeS
For any expansion of H\, A — (H\, E, /i)i< w , of countable language, a club C C [H\]u such that for each TV E C N -< A, for each TV G C 3 a G (u2 - N) such that SkA(N U {a}) n ui =
Assume otherwise. Define S so that for N G [Hx]" {SkA(N
<=> N ^Ak\/aeuj2-N
U {a}) Hun ^ N PILU^.
Then S is stationary. Let X -< Hx be such that \X\ = \X n u2\ = Ni and X -< A, S n [X]u is stationary in [X]u. Case 1. X C\ LJI ^ LJi. We may assume that X f) ui G a>i. Now it is easy to build a normal chain of countable elementary submodels so that X = | J a < U i Na and Na D u>i = X n CJI for each a < u;i. Let iV^ G 5 . Since |X fl LU2\ = Ni, can find a 0 G X fi u2 - Na. But then
SkA(Na\j{p})nui
= Nar\u1.
This is a contradiction. Case 2. ^ C X. Build up an elementary chain X = Ua ^ * Then Ci = {iVa fl UJI : a < c^i} is a club in u\. Let /3 e LJ2 — X. Let y = SKA(X U {£}). Then ^ = SkA(Na U {£}) (a < CJI) form an elementary chain whose union is y . Let C2 = {N^ C\ u\ : a < LJI}. Then C2 is a club. Let Na G 5 be such that
Nanui
= N'af]u;1eC1n
C2.
Then 7V^ = SkA(Na U {/?}). This is a contradiction.
□
Todorcevic in [17] proved that this type of Chang's conjecture implies certain stationary reflection principle. T H E O R E M 5.2
(TODORCEVIC
[17])
Let A be some large regular cardinal such that (*) of the Magidor's theorem above holds. Then every stationary subset of [0^2]w reflects. Proof Assume that S C [u2Y is stationary but STl [a]u is not stationary u in [a\ for all uncountable a < u2.
ON
STATIONARY R E F L E C T I O N P R I N C I P L E S
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For each uncountable a < LJ2, let ea : a>i —> a be a bijection and Ca C UJI be a club in CJI such that for all /3 G C, e'^/3 ^ 5. Since 5 is stationary in [UJ2Y , and by (*), let M and M* be two countable elementary submodels of (H\,e) such that 5 and {ea,Ca \ uo\ < a < UJ2} are in M and M f) UJ2 G S and such that M n ui = M* n CJI, M C M*, and M D u2 ± M* n LO2. Let a be the least ordinal in M* - M. Notice that M* D a = M H CJ2. Then ea, Ca are in M*. Hence M n wi G C a and M fl LJ2 = e'a(M H CJI). This is a contradiction. □ We have seen previously that the D-principle fails. Here we prove something stronger. Let C be a sequence defined on all the limit ordinals < K such all limit ordinals a < K, Ca C a is a club in a and for all limit (3 < a, if 0 e Ca is a limit point of Ca, then Cp = ft n Ca. C sequence if there is no club D C K such that for every limit point
want to that for ordinals is a □* a of D,
Ca = af)D. Let Sc be the set of all countable elementary submodels N of HK such that, letting a = sup(7V fl «), C a fl N is bounded in a. The following theorem is essentially due to Velickovic [18]. T H E O R E M 5.3 The following are equivalent: (1) Sc is projective stationary. (2) Sc is stationary. (3) There is no club D C K such that for all limit points a of JD, Ca = aH D. That is, C is a □* sequence. Proof (2) => (3). Assume that (3) is false. Let D be a witness. Let 6 > ft be a sufficiently large regular cardinal. Let N -< # 0 be a countable elementary submodel containing all the relevant objects. Then N C\ HK $ Sc(3) => (1). This is essentially due to Velickovic [18]. For an / : [HK]
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LEMMA 5.1 (Velickovic) There exists a club Ef C ui such that for every a G Ef, Player One has a winning strategy a? for the game G(a, / ) . Let / : [HK]
game
G(aJ).
Let 6 = (2 K ) + . Let (M a | a < ft) be an increasing continuous G-chain of elementary submodels of He of size < K such that all the relevant objects are in M 0 and Ma n K G ft for all a < ft. Let D — {Ma fl ft | a < ft}. Then D is a club in ft. Let M ^ He be an elementary submodel of size less than ft, containing all the relevant objects, such that M fl ft is an ordinal of cofinality Ui. Let Assume that there is a X G M such that 8 fl X C C 5 and M |= "X is a club in ft ". Let F be the set of all the limit points of X . Then Y G M. Let 7 < /? be in M such that both are in Y. Then 7 and /? are limit points of Cs- Hence C1 = 7 fl C/j. Therefore, for every pair 7 < /? from y , we have (77 = 7 n C/j. This contradicts to our assumption (3). Thus, for every club X C ft, if X G M then there is some ordinal 7 G X fl <5 which is not in Cs- It follows that for every 7 < (5, there is some £ < <5 such that r^ = M^ fl ft is not in C$ and 7^ > 7. Inductively pick £ n so that rj^n 0 C5 and there is some ordinal /3 G Cs with r]U < p < % n + 1 . Let /3 = ( J { ^ n I ™ < <*>}■ Then C,? = p n C*. Consider the following play of the game G(a, / ) . Player One follows his winning strategy a which is in MQ. Player Two plays \xn = m a x ( ^ n fl C$), which is in M$n , in his nth move. Let In and /? n be the nth move of Player One following his winning strategy a. Then /3n G In C (/x n ,n^ n )^for all n < CJ. Let A7" -< M be generated by a U {/3n | n < CJ} which is closed under / . Then N f) u>i G T and supiV n ft = f5 and JV n C/j C J 0 . Hence 7VntfK G S C . This shows that (3) implies (1). □ COROLLARY 5.1 If for every stationary S C [HK]U there is an in creasing continuous G-chain (Na | a < UJ{) such that {a < U\ \ Na G 5 } is stationary, then there is a club C C K such that for every limit point a G C, C a = a fl C. In particular, there is no □* sequence. Proof By the previous theorem, we need only to prove that Sc is not stationary. Toward a contradiction, let us assume that Sc is stationary. Let (Na I a < CJI) be an increasing continuous G-chain be such that T — {a < CJI I Na G S^} is stationary. Let j a = sup(Na fl ft) for a < u\. Let
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6 = sup{7 a | a < ui}. Let a G T be such that j a G Cs and j a is a limit point of 650(7/3 | P < CJI}. We get a contradiction, since a G T implies that Na G 5 c and Cla = 7 a fl Cs whose intersection with Na is not bounded in la□
REFERENCES
[0] J. BAUMGARTNER AND A. TAYLOR Saturation properties of ideals in generic extensions II Trans. A m e r . M a t h . Soc. 271 (1982) 587-609 [1] M. BEKKALI Topics in Set Theory Lecture Notes in Mathemat ics, Vol. 1476, Springer-Verlag, Berlin, New York, 1991 [2] Q. F E N G AND M. M A G I D O R On rejection of stationary sets Fund. M a t h . 140 (1992) 175-181 [3] Q. F E N G AND T . J E C H Local Clubs, Reflection, and Preserving Sta tionary Sets Proc. London M a t h . Soc. (3) 58 (1989) 237-257 [4] Q. F E N G AND T . J E C H Projective Stationary sets and Strong Reflec tion Principle J. London M a t h . Soc. (to appear) [5] W . FLEISSNER Left-separated spaces with point-countable bases [6] M. F O R E M A N , M. M A G I D O R AND S. SHELAH
[7] [8] [9] [10] [11] [12] [13]
Martin's
Maximum,
Saturated Ideals, and Nonregular Ultrafilters. Part I Annals of M a t h e m a t i c s 127 (1988) 1-47 H. FRIEDMAN One Hundred and two problems in mathematical logic J. S y m b . Logic 40 (1975) 113-129 T . J E C H Some combinatorial problems concerning uncountable cardi nals A n n . M a t h . Logic 5 (1973) 165-198 T. JECH Set Theory Academic Press, New York 1978 T. JECH Multiple Forcing Cambridge Tracts in Mathematics, Cambridge University Press, 1986 R.B.JENSEN The fine structure of the constructive hierarchy A n n . M a t h . Logic 4 (1972) 229-308 D. K U E K E R Countable Approximations and Lowenheim-Skolem The orem Annals of M a t h e m a t i c a l Logic 11 (1977) 57-103 J. SILVER On the Singular Cardinals Problem Proc. Internat'l Cong. M a t h . Vancouver, B. C , 1974, Vol. 1, 265-268
[14] J. S T E E L AND R. VAN W E S E P
TWO Consequences of
Determinancy
Consistent with Choice Trans. A . M . S. 272(1) (1982) 67-85 [15] S. TODORCEVIC Reflecting Stationary Sets I Hand Written Notes, 1985
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[16] S. TODORCEVIC Strong Reflections Hand Written Notes, 1987 [17] S. TODORCEVIC Conjectures ofRado and Chang and Cardinal Arith metic Preprint [18] B . VELICKOVIC Forcing Axioms and Stationary Sets Advances in M a t h e m a t i c s 94 (1992) 256-284 [19] H. W O O D I N The axiom of determinacy, forcing axiom and the nonstationary ideal preprint [20] H. W O O D I N Some consistency results in ZFC using AD Cabal Sem inar 7 9 - 8 1 Lecture Notes in Mathematics, Vol. 1019, 172-198 Department of Mathematics, National University of Singapore, Singapore 0511 Email: f eng@math. mis. sg
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Definable Sets of Real Numbers, Infinite Games and Core Model Theory Kai Hauser * Department of Mathematics, University of California, Berkeley, CA 94720, USA Email: hauser@math. berkeley. edu Abstract Applying Steel's £3 correctness theorem for the core model for one Woodin cardinal [St2], it is shown that A^ determinacy follows from the assumptions that every projective set is Lebesgue measurable and has the property of Baire together with the uniformization property for II3 relations.
1. Introduction General questions about arbitrary sets of real numbers - like for instance Cantor's continuum hypothesis CH (asserting that an uncountable set of real numbers is equinumerous with the whole set of real numbers) - have a partial solution within the context of definability. For example a classical theorem of descriptive set theory due to Suslin (cf. [Lul]) states that no analytic set (i.e., continuous image of a Borel set) can give a counterexample to CH. The analytic sets posess further properties indicative of well-behavedness: They are Lebesgue measurable [Lul] and have the property of Baire [LuSi]. (This is the dual notion of measurability in terms of category: A set has the property of Baire if its symmetric difference with an open set is small in the topological sense that it can be covered by a countable union of closed sets with empty interior.) Another regularity property of fundamental interest to descriptive set theorists is the following choice-like principle introduced in [Lu2]: Given a *The author is a Heisenberg Fellow of the Deutsche Forschungsgemeinschaft. Travel support from the Ministry of Foreign Affairs of the Federal Republic of Germany to attend the 6th Asian Logic Conference is gratefully acknowledged.
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subset A in the plane, a partial function F is called a uniformization if it picks for each x in the projection of A onto the real line a real number y putting (x, y) into A. In ordinary set theory the uniformization problem is solved trivially with the help of the axiom of choice, whereas it becomes highly non-trivial when definable uniformizations are sought, even if the given set is of a simple form. Also in this regard definable sets exhibit wellbehavedness. Kondo [Ko] proved that the coanalytic sets (the complements of analytic sets) have the uniformization property: Any coanalytic subset of the real plane can be uniformized by a function with coanalytic graph. Naturally a great deal of work was invested in descriptive set theory to wards extending these regularity results to more complex sets of real num bers. A suitable framework for this investigation is provided by the pro jective sets, i.e., the smallest collection containing all Borel sets and closed under forming complements and (pointwise) continuous images. This can be stratified into a natural hierarchy: E j is the class of all analytic sets; For n > 1, a set is I I * iff its complement is X*, it is S ^ , 1 iff it is the con tinuous image of a II* set and A* iff it is both X)* and I I * . Finally a set is projective iff it is X)* for some n > 1. Thus A\ is precisely the Borel sets [Su] , and any projective set can be generated in finitely many steps by the above two operations. These are the sets occuring in ordinary mathematical practise, and so one would like to have a unified theory guaranteeing the well-behavedness of as many projective sets as possible. There is, however, an insurmountable principal barrier, despite the fact that projective sets somehow seem to be simply definable: Through the work of Godel [Go2] and Cohen [Co] (and many others after them) all the central questions are now known to be unanswerable on the basis of the standard axiomatic system for set theory ZFC (Zermelo-Fraenkel with choice). Historically, the first attempt to overcome this limitation was in the use of large cardinal axioms (postulates implying the existence of sets of large size). In the mid 60s Solovay (cf. 42.11 in [Jc]) showed that the existence of a measurable cardinal (a prominent large cardinal axiom) implies all sets in the second level of the projective hierachy (i.e., all continuous images of coanalytic sets) are Lebesgue measurable and have the property of Baire. This important theorem led to the hope that eventually sufficiently strong large cardinal axioms may decide all open questions in this area. Quite unexpectedly that was achieved with the help of a different class of axioms: determinacy axioms. Let u denote the set of all natural numbers and u^ the set of all infinite sequences of natural numbers. From a topological viewpoint the space UJ^ (with its natural topology) is indistinguishable from the subspace of the Euclidian space, 1R, consisting of the irrationals, and its elements are referred to as reals in descriptive set theory. The advantage of this identification is
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UJU accomodates in a more natural way the notion of a game: With each set A C uu a game GA is associated in which two players cooperatively produce a real by picking alternatively natural numbers in UJ many rounds. Player I wins the game GA if and only if this real belongs to the payoff set A. A strategy for one of the players is an algorithm telling him which number to pick at each of his turns. Such a strategy is winning for a player if it leads to a win for that player whenever he follows it regardless of the other player's moves. The game is determined if one of the two players has a winning strategy. Simple games (for instance the ones with open payoff sets) are deter mined [GaSt]. The same is true for games with a Borel payoff set [Ma], yet additional hypotheses stronger than ZFC are needed for more compli cated games. The Axiom of Determinacy AD [MySt] settles this issue by decree in that it asserts that for all A C a;w, GA is determined. Its con sequences are rather striking (among them Lebesgue measurability [MySw] and the property of Baire (Banach-Mazur, cf. [Ml]) for any set of reals, but unfortunately it contradicts the axiom of choice. The axiom of choice as usual - fails to produce "constructive" non-determined games, and this motivates the introduction of axioms of definable determinacy (as first sug gested in [MySt] and independently by R. Solovay and G. Takeuti) which allow mathematicians to keep the axiom of choice and with it standard tools of mathematics (like the Hahn-Banach theorem for example) while all sets arising "in practice" enjoy the usual regularity properties. A central role in this regard is played by the axiom of projective determinacy PD which states that every game with projective payoff set is determined. It answers all descriptive set theoretic questions about projective sets and extends the structure theory (including a strong form of uniformization) provided by ZFC for the coanalytic sets to all further levels of the projective hierarchy (cf. ch.6 of [Mo]). A natural question therefore arises: How much of PD is "really" needed? In 1981 Woodin [Wol] formulated the following conjecture to the effect that all of PD is needed: (ZFC)
Assume
(1) Every projective set of reals is Lebesgue measurable ("pro jective measurability") and has the property of Baire ("pro jective category"). (2) Every projectiove set can be uniformized by a function projective graph ("projective uniformization").
with
Then PD holds. Subsequently this question has been included as the last Delfino problem in the expanded list of [KeMaSt]. It is to be regarded as a crucial test for
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the naturalness of the hypothesis PD, and rephrased in metamathematical terms, a positive answer would in effect mean that PD is the only way of extending the structure theory of the coanalytic sets provided by ZFC to the entire projective hierarchy. In terms of heuristics, Woodin's conjecture expresses the belief that PD is the underlying structure enabling the con flicting statements (1) (amounting to a frailure of the axiom of choice at the projective level) and (2) (which retains some form of choice in the projective hierarchy) to hold simultaneously. Woodin [Wol] supplies some evidence for this ambitious claim: It is shown there that (1) and (2) imply II* determinacy. The theorem proved below pushes this one level higher in the projective hierarchy (recall that on general grounds, A^ determinacy is (in ZFC) equivalent to Y,\ determinacy by a theorem of Martin cf. [WoMaHa]) T h e o r e m (ZFC)
Assume
(1) Projective measure and category (2) Ilg
uniformization.
Then A^ determinacy
holds.
This may seem a tiny step toward the solution of the Delfino problem. It is, however, worth noting that A2 determinacy represents a definite cut point in the large cardinal hierarchy with far reaching global consequences for the structure of the set theoretical universe. (This is in analogy with the appearance of sharps for reals.) The above theorem falls into the area best described as reverse descrip tive set theory. Here the aim is to show that propostions asserting regularity properties for higher order projective sets which are derivable from strong set theoretic principles (such as axioms of determinacy) conversely imply those principles. The astonishing feature of these theorems is that despite being phrased in purely descriptive set theoretic terms, their only known proofs rely on core model theory. (More examples of problems dealing ex clusively with definable sets of reals that have been solved with core model machinery can be found in [Hj] and [Stl, sec. 7].) Core models occupy a special place in the inner model program where the guiding idea is to exhibit for each large cardinal axiom A a canonical model MA that is in some sense minimal and whose structure theory can be analyzed in detail. This is not a consistency proof for A as one must assume A in order to conclude that A holds in MA- Nevertheless it is to be expected that any hidden inconsistency in A should emerge quickly in the detailed structure theory of MAThe prototype of an inner model is the universe of constructible sets L introduced by Godel [Gol] to establish the consistency of the generalized continuum hypothesis and the axiom of choice with the original axioms of
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set theory. Godel's analysis of L was immensely refined by R. Jensen who pioneered fine structure theory (a detailed study of how individual sets enter into an inner model) in the early 70s (cf. [Je]). Also discovered in Jensen's fine structure laboratory was a new important class of models, core models. Their name refers to the fact that the simplest one, Kpj, introduced in collaboration with A.J. Dodd (cf. [Do]) in the 70s, can be characterized as the "core" of all models with measurable cardinals. Unlike L which comes out the same no matter in which world it is computed in, the shape of KJJJ varies with the large cardinal situation in the surrounding universe, and it mirrors that situation below measurability. The core models considered in connection with the last Delfino problem are of a much higher order: they approximate inner models with Woodin cardinals (this large cardinal notion of far greater strength than measurability is intimately related to projective determinacy, cf. [MaSt] and [WoMaHa]). More specifically, the proof of the above theorem rests on a recent the orem of Steel [St2] about the correctness for II3 truth as computed in the the core model for one Woodin cardinal, K. This model is the contructible closure of the ordinals under a predicate consisting of a sequence of so called extenders. These are directed systems of measures coding up set sized re strictions of an embedding of the universe into a transitive class. Woodinness is defined in terms of the existence of embeddings satisfying additional conditions, and the extenders appearing on the sequence of K are attempts to approximate such embeddings. The construction of K was invented by Steel [Stl] and relies on previous work of Michell and Steel [MiSt] who built a fine structural inner model with a Woodin cardinal (provided some inner model with a Woodin cardinal exists). In order for the K- construction to produce a model with the desired properties (among which is a strong form of the generalized continuum hypothesis, certain fine structural features and the ability to reflect the large cardinal situation in the universe below one Woodin cardinal) one needs background assumptions. The existence of a measurable cardinal suffices (in fact less, cf. [Stl]) if there is no inner class model containing a Woodin cardinal. In the presence of sharps for all sets this is enshured by the failure of A \ determinacy via a theorem of Woodin (cf. [WoMaHa]). Thus the proof of the theorem takes the form of an indi rect argument: Assuming a failure of A2 determinacy one builds a model of ZFC in which K exists and is correct about II3 truth and derives a contra diction. The construction of that model uses the descriptive set theoretic assumptions of the theorem and is essentially a refinement of the arguments in [Wol]. In some sense the proof of the theorem is more important than its state ment: Modulo the appropriate correctness of higher order versions of K and the closure of the universe under the sharp operation for the appropriate higher order analogues of L, the proof given in the next section generalizes to other levels of the projective hierarchy. This issue and related topics are
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discussed in section 3.
2 The Proof The reader who is not familiar with the inner workings of core models can think of K (or rather its relativization Kz to some real z) as a generalization of the constructive universe L. As mentioned in the previous section the proof rests on the fact that (under the appropriate background assumptions) K correctly computes £3 truth about its reals as proved by Steel [St2]. T h e o r e m . (ZFC) Suppose (1) Every projective set is Lebesgue measurable (PM) and has the property of Baire (PB). (2) II3
uniformization.
Then A ^ determinacy
holds.
P r o o f . The argument proceeds in two steps: First a transitive model M of ZFC is constructed which is both £4 correct and £4 absolute in the sense of [Wol]. Then one establishes that A2 determinacy holds inside M from which the theorem follows because A2 determinacy is a II4 sentence. C o n s t r u c t i o n of M. This uses a minor modification of the the construction in [Wol]. Notice that PB of (1) and (2) clearly imply * * II3 uniformization (as defined in [Wol]). This yields as in lemma 2 of [Wol] (3) For any real c Cohen over V, V is £5 correct in V[c]. Next, using (2), fix a II3 definition from a parameter x0 for a function uniformizing the complete II3 set in E x R. Let Unif(n\,xo) be the II^xo) formalization of this situation. Thus by (3), (4) For any real c Cohen over V, Unif(I[\,xo)
holds in V[c}.
Similiarly one obtains (5) For any real c Cohen over V, *Il2 uniformization holds in V[c] where *Il2 uniformization is * * I l 2 uniformization restricted to II2 subsets of the plane whose projection is the whole real line. This restriction seems necessary to arrive at a II5 notion. On the other hand, the restricted version suffices to run the argument in lemma 2 of [Wol] for £4, and with (3) it follows:
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(6) Whenever Ci,c 2 are mutually Cohen generic over V then V[ci] is £4 correct inside V[ci][c 2 ]. Armed with these facts, one can now proceed exactly as in the proof of theorem 1 of [Wol] building a £4 correct and £4 absolute model M of the form LU,1(XQ,F). Essentially, F is a partial function of HC —► HC assigning to a given tuple of terms f for reals (in the forcing language associated to a poset belonging to M) a canonical term for a witness for a £4 formula in those terms using a JI\{XQ) uniformization as in (4), provided such a witness exists. By a theorem of Shelah [SI], PM implies that there is no uncountable projective sequence of distinct reals. (The proof of Shelah's theorem given in [Ra] actually shows that whenever S is an uncountable sequence of distinct reals, there is a non Lebesgue measurable set projective in S.) This implies that the power set axiom holds in M, i.e., M is a model of ZFC. A2 d e t e r m i n a c y in M. Notice that in M there still is a II^xo) definable uniformization of the complete II3 set since II^xo) statements relativize to M by £4 correctness. The rest of the proof takes place inside the model M. Assume toward a contradiction that A^ deteminacy fails. Fix a real z such that Al(z) determinacy fails. Without loss of generality z is Turing above xo. C l a i m 1. There is no proper class inner model with a Woodin cardinal containing z. P r o o f of c l a i m 1. Otherwise, the fact that every set has a sharp (by £3 absoluteness) allows to find a countable ordinal 6 which is Woodin in some inner model TV of the form L(V6N) containing z. By a theorem of Woodin [WoMaHa], A£(z) determinacy holds in JSfColl^^\ Since 6 is countable N generics for Coll(uj,8) exist. But Al(z) determinacy is upward absolute for proper class models , a contradiction. □ Claim 1. Via covering over the Dodd-Jensen core model for two measurables, £4 absoluteness implies the existence of x^ for any set x. Now go to an inner model containing all the reals and two measurable cardinals K < Q and build Kz, the core model for one Woodin cardinal (cf. [Stl]) relativized to the real z up to height 0 . By a theorem of Steel [St2] Kz is £3 correct. Notice that E f)Kz is £3(2) since for any real x x e Kz iff 3M(M
is a countable 1-small iterable
premouse built over z and i G M ) . The displayed formula is indeed £3(2) because iterability for countable premice built over z is a ILj; condition as there is no proper class inner model
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with a Woodin cardinal containing z. The above equivalence holds because for x G Kz one can take the first level of Kz where x appears for M. The other direction comes from comparing such an M with Kz which must win the comparison by universality (cf. ch. 5 of [Stl]). It follows that with i c M x E defined as (x, y) € A iff y is Cohen generic over I K - M : M is a count able 1-small Il^-iterable premouse built over x} A is a II3 relation such that for all c (z, c) G A iff c is Cohen over Kz. C l a i m 2. E D Kz is countable. P r o o f of claim 2. Define 5 c E x E b y (x,y) G S iff x codes some countable ordinal £x such that y is the £arth real in the order of construction of Kz. If the claim fails then S defines an uncountable £3(2) sequence of distinct reals. But this is a n ^ z ) statement which would contradict Shelah's theo rem [SI] in the outside universe where M is £4 correct. □ Claim 2. So there is a real Cohen over Kz, i.e., 3y(z,y) G A. Now let A* be a Ti.\(z) uniformization of A. Fix a real c such that (z, c) G A*. By Shoenfield abso luteness Kz[c] \= (2,c) G A*. Pick a Cohen condition p such that, over i(Tz, p lh (2, c) G A* and such that p C c. Changing a finite amount of information about c, one obtains c' extending p so that JK"z[c;] |= (z,c') G A*. An inspection of Steel's proof [St2] for the £3 correctness of the 1Woodin core model reveals that in actual fact .K^c'] must be £3 correct. (The version of the Martin-Solovay tree for II2 shown by Steel to exist in side the core model survives "small" forcing.) It follows that (z,c') G A* contradicting c / c'. □ theorem
3 Concluding Remarks Once suitable strengthenings of £3 correctness for the core models for more Woodin cardinals are at hand, certain parts of the above proof can be lifted. This generates the hope to prove the following weakened version of the Delfino Problem (which should be compared with the conjecture made in the remarks just after 4.1 of [Hal]). (ZFC)
Assume
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(1) Every projective set is Lebesgue measurable and has the property of Baire (2) n ^ . 2 uniformization
for each n > 0.
Then PD holds. One way of obtaining higher order versions of the one Woodin K proceeds via definable singleton operations coming from inner model theory. [These singletons appear naturally in the course of generalizing the structure theory of the first level of the projective hierarchy to its higher levels (cf. [Kel], [Ke2]), and their connection with inner models for large cardinals related to determinacy was conjectured early on (cf. [KeMaSo]).] In this approach the least failure of an instance of PD (i.e., the least level n at which E l determinacy fails) corresponds to a first occurence of one of these singleton operations being non-total. This can be expressed in terms of a smallness condition about the sequence of the preliminary Kc which in turn yields an iteration strategy for Kc. But now one is in the position to simply repeat the process of extracting K as a certain Skolem hull of Kc just as in the one Woodin setting of [Stl]. (This method is essentially an initial segment of the far more complex procedure of obtaining a core model from a least failure of ADL^ which was developed by Woodin in his solution of the L(E) version of the last Delfino Problem.) In the first step beyond one Woodin cardinal of the above construction the core model is built inside the smallest model of set theory closed under the ?/o operation and containing a rank initial segment of the universe. [For any set (of ordinals) a, y0 (a) denotes the sharp of a fully iterable fine struc ture model with one Woodin cardinal "built above the set a". This is a II3 singleton in any real (generically) coding a] More precisely: The smallest model of set theory closed under the y0 operation and containing a given rank initial segment of the universe can be re-arranged as a fine structure model in the sense of [MiSt]. £5 absoluteness implies the failure of a weak form of the covering lemma over any such model. Woodin (cf. [SchWo]) ob served that this yields indiscernibles for such models, i.e., normal, countably complete ultrafilters that measure all subsets of some ordinal belonging to the model and are weakly amenable to the model. The construction of Kc in [Stl] can be carried out with such an indiscernible playing the role of a measurable cardinal. Under the above mentioned smallness condition, Kc satisfies a II3 iterability condition and one can extract "true" K2 as it will be called here just as in [Stl] from Kc. Otherwise one obtains that the uni verse is actually closed under the next singleton operation (corresponding to the sharp of a fully iterable two-Woodin model). Then - under the smallness condition that the universe fails to be closed under the three-Woodin-sharp operation - one builds a core model inside the smallest model of set theory closed under the two-Woodin-sharp operation above some rank. This core
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model - called K3 in the sequel - will be closed under the two-Woodinsharp operation and satisfy a H\ iterability condition. [In this numerology K1 would denote the one Woodin K and K° corresponds to L] One would now like to work with the core model K3 imitating the proof of the previous section. However, in order to do so one must first get past the core model Zv2, i.e., the closure of the universe under the two-Woodinsharp operation has to be established. This corresponds to closing the reals under sharps (i.e., H\ determinacy) in the proof of the previous section. [Results of Woodin [Wo2] when combined with the transfer theorem in [Ne] show that the closure of the reals under the two-Woodin-sharp operation is equivalent to II3 determinacy.] In that proof already the £3 absoluteness combined with the covering lemma for the core model L yielded the desired closure. Now the fact that the construction of the various core models takes place inside a universe satisfying Eg absoluteness by itself will not suffice to derive the necessary closure as illustrated by the following example: Cut M 2 (the minimal fully iterable fine structure model with two Woodin cardinals) at its bottom Woodin cardinal. This gives a model of ZFC 4- global A2 determinacy with unboundedly many strong cardinals (because M 2 is closed under the yo operation below its bottom Woodin). By a theorem of Woodin [Wo2], collapsing sufficiently many of the strong cardinals of that model to to with finite conditions results in a model N where any prespecified amount of projective forcing absoluteness is available and global A2 determinacy holds, yet II3 determinacy fails. In contrast to this counterexample the universe in the situation where one tries to get past the core model K2 does satisfy a fair amount of uniformization (namely, II5 uniformization). This suggests the following ap proach to getting past K2: First establish the correctness with respect to a lightface fragment of projective truth in V of generic collapses (of a suit able number) of strong cardinals of K2 assuming that K2 exists. [By the appropriate generalization of theorem 3.10 of [Hal] strong cardinals exist in K2 by projective absoluteness.] Then prove a negative uniformization re sult for generic extensions of K2 of the above form. This would contradict that uniformization holds in V and thus K2 did not exist in the first place, i.e., the universe is closed under the next singleton operation. Thus the stage is set for the construction of K3. [Note that in the above mentioned counterexample, N is itself the generic extension of K2 via the collapse of strong cardinals because we cut M2 below its bottom Woodin cardinal (i.e., theorem 2.6 of [Ha2] applies) and by the the forcing absoluteness of K2. Thus presumably II5 uniformization fails in N] Unfortunately, the approach sketched in the previous paragraph would suggest that the solution of the restricted Delfino problem (assuming level by level uniformization) is no easier than that of the full Delfino problem (cf. the remarks made in the end of section 3 of [HaHj]). This is yet another indication that getting past K2 is a serious issue.
D E F I N A B L E S E T S , I N F I N I T E G A M E S AND C O R E M O D E L T H E O R Y
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Provided that K3 is £5 correct and global II3 determinacy can be de rived, the proof of the previous section lifts and one obtains: Assume (1) Projective measure and category (2) II3 uniformization Then A 4 determinacy
and H\
uniformization.
holds.
As a warm-up for showing £5 correctness of K3 one may try and establish £4 correctness for K2 first. This would follow if it could be shown that cofinally in a suitable II3 norm on the complete II3 set witnesses in K2 exist for II3 statements true in V about parameters from K2. [The closure of the universe under the y0 operation implies global A* determinacy. Thus the pointclass II3 is normed. Recall that the initial segments of a II3 norm are uniformly A 3 whence the above cofinality condition transforms £3 correctness into £4 correctness.] The £4 correctness at this level is related to earlier attempts of Woodin to derive the scale property for II3 combinatorially, i.e., from combinato rial properties of the corresponding inner model. Towards this end Woodin [Wo2] has created a scenario requiring a plausible hypothesis about the be havior of K (at the appropriate level) under coarse iterations of of its sur rounding universe. This hypothesis says (under the necessary background assumptions for the construction of the core model): Suppose T is a coarse iteration tree built on a coarse premouse M with last model Me and associated embedding i : M —> Me. If K denotes the core model at the appropriate level, KMe = i(KM) is the final model of a fine-structural iteration tree based onKM. Despite all this the ultimate goal is still to settle the Delfino Problem in its original form. A positive solution will have the added advantage of showing that among all the conceivable patterns leading to projective uniformization a particular one stands out: No matter which universe of set theory we live in, as long as any projective set is Lebesgue measurable and has the property of Baire, projective uniformization can only be achieved by having the uniformization property oscillate between the odd levels on the II side and the even levels on the £ side of the projective hierarchy.
References [Co] P.J. Cohen, The Independence Of The Continuum Hypothesis, I, II, Proc. Natl. Acad. Sci. USA 50 (1963), pp. 1143-1148, 51 (1964) pp. 105-110.
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[Do] A.J. Dodd, The Core Model, London Math. Soc. Lee. Note Series 61 (1984), xxxviii + 229. [GaSt] D. Gale and F.M. Stewart, Infinite Games With Perfect tion, Ann. Math. Studies 28 (1953), pp. 245-266.
Informa
[Gol] K. Godel, The Consistency Of The Axiom Of Choice And The Gen eralized Continuum Hypothesis, Proc. Natl. Acad. Sci. USA 24 (1938), pp. 556-557. [Go2] K. Godel, The Consistency Of The Axiom Of Choice And Of The Generalized Continuum Hypothesis With The Axioms Of Set Theory, Annals of Math. Studies, Study 3 (1940), Princeton University Press, Princeton, New Jersey [Hal] K. Hauser, The Consistency Strength Of Protective Absoluteness, An nals of Pure and Applied Logic 74 (1995), pp. 245-295. [Ha2] K. Hauser, A Minimal Counterexample mitted for publication.
to Universal Baireness, sub
[HaHj] K. Hauser and G. Hjorth, Strong Cardinals in the Core Model, in Annals of Pure and Applied Logic 83 (1997), pp. 165-198. [Hj] G. Hjorth, Two Applications of Inner Model Theory to the Study of £ * sets, Bulletin of Symbolic Logic 2 (1996), pp. 94 - 107. [Jc] T.Jech Set Theory, Academic Press (1978), xi + 621. [Je] R.B. Jensen The Fine Structure Of The Constructible Hierarchy, An nals of Math. Log. 4 (1972), pp. 229-308. [Kel] A.S. Kechris, The Theory of Countable Analytical Sets, Trans. Amer. Math. Soc. 202 (1975), 259-297. [Ke2] A.S. Kechris, Recent Advances in the Theory of Higher Level Protec tive Sets, in The Kleene Symposium, J. Barwise, H.J. Keisler and K. Kunen (eds.), North Holland (1980), 149-166. [KeMaSo] A.S. Kechris, D.A. Martin and R.M. Solovay, Introduction to QTheory, in Cabal Seminar 79-81, Springer Lecture Notes in Math. 1019 (1983), 199-282. [KeMaSt] A.S. Kechris, D.A. Martin and J.R. Steel (eds.), Appendix: Vic toria Delfino Problems II, in Cabal Seminar 81-85, Springer Lecture Notes in Math. 1333 (1988), pp.221-224.
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[Ko] M. Kondo, Sur L'Uniformization Des Complementaires Analytiques Et Les Ensembles Projectifs De La Second Classe, Jap. J. Math 15 (1938), pp. 197-230. [Lul] N. Lusin, Sur La Classification De M. Baire, Comptes Rendus Acad. Sci. Paris 164 (1917), pp. 91-94. [Lu2] N. Lusin, Sur Le Probleme De M. J. Hadamard D'Uniformisation Des Ensembles, Comptes Rendus Acad. Sci. Paris 190 (1930), pp. 95-96. [LuSi] N. Lusin and W. Sierpinski, Sur Un Ensemble Non Measurable B, Journal de Mathematiques, 9 e serie, 2 (1923), pp. 53-72. [Ma] D.A. Martin, Borel Determinacy, 363-371.
Annals of Math. 102 (1975), pp.
[MaSt] D.A. Martin and J.. Steel, Protective Determinacy, Acad. Sci. USA 85 (1988), pp. 6582-6586.
Proc. Natl.
[MiSt] W.J. Mitchell and J.R. Steel, Fine Structure And Iteration Lecture Notes in Logic 3, Springer Verlag (1994), 130 pages.
Trees,
[Ml] R.D. Mauldin (ed.), The Scottish Book, Birkhauser (1981). [Mo] Y.N. Moschovakis, Descriptive Set Theory, North-Holland (1980), xii + 637. [My] J. Mycielski, On The Axiom Of Determinateness 53 (1964), pp. 205-224, 59 (1966), pp. 203-212.
I,II, Fund. Math.
[MySt] J. Mycielski and H. Steinhaus, A Mathematical Axiom Contradict ing The Axiom Of Choice, Bull. Pol. Acad. 10 (1962), pp. 1-3. [MySw] J. Mycielski and S. Swierczkowski, On The Lebesgue Measurability And The Axiom Of Determinateness, Fund. Math. 54 (1964), pp. 6771. [Ne] I. Neeman, Optimal Proofs of Determinacy, (1995), 327-339.
Bull. Symb. Logic 1
[Ra] J. Raisonnier, A Mathematical Proof Of S. Shelah's Theorem On The Measure Problem And Related Results, Isr. J. Math 48 (1) (1984), pp. 48- 56. [SchWo] , The Jensen Covering Property, manuscript (1996). [SI] S. Shelah, Can You Take Solovay's Inaccessible Away?, Isr. J. Math. 48 (1) (1984), pp. 1-47.
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[Stl] J.R. Steel, The Core Model Iterability Problem, Lecture Notes in Logic 8, Springer Verlag (1996), 112 pages. [St2] J.R. Steel, The S3 correctness for the 1-Woodin K, manuscript (1993). [Su] M. Suslin, Sur une definition des ensembles mesurables B sans nombres transfinis, Comptes Rendus Acad. Science, Paris 164 (1917), pp. 88-91. [Wol] W.H. Woodin, On The Consistency Strength Of Protective Uniformization, in Logic Colloquium '81, J. Stern (ed.), North-Holland (1982), pp. 365-383. [Wo2] W.H. Woodin, personal communication. [WoMaHa] W.H. Woodin, A.R.D. Mathias and K. Hauser The Axiom of Determinacy, monograph in preparation.
121
THE DESCRIPTIVE CLASSIFICATION OF S O M E C L A S S E S OF C * - A L G E B R A S
Alexander S. Kechris 1 ) Department of Mathematics California Institute of Technology Pasadena, California 91125
We introduce here a parametrization of separable C*-algebras by a standard Borel space and study the descriptive complexity of various canon ical classes of C*-algebras in this parametrization. This can be viewed as providing an analog of the corresponding classification of classes of von Neu mann algebras (acting on a fixed separable Hilbert space) in the Effros Borel space of von Neumann algebras (see for example Nielsen [13]). However, in contrast with the von Neumann case, where most interesting classes (like: factors, type I, II, III, hyperfinite) turn out to be Borel, in the C*-algebra case many important classes turn out to be co-analytic but not Borel. This makes the situation more interesting from the set-theoretic point of view, and leads to further questions, like the construction of canonical co-analytic norms, which we also address here. Finally, we relate the above to work of Sutherland [17] on parametrization of Polish groups and indicate how one might be able to show that various familiar classes of (second countable) locally compact groups are non-Borel as well. We assume that the reader is familiar with the basics of descriptive set theory (see Moschovakis [12] and Kechris [11]) and C*-algebras (see Dixmier [4] and Fillmore [5]). Acknowledgement. We would like to thank E. Effros for many valuable conversations about this subject. 2
) Research and preparation of this paper were partially supported by NSF Grants DMS-9317509 and 9619880.
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ALEXANDER KECHRIS
§1. Parametrizing separable C*-algebras Let H be a fixed infinite dimensional Hilbert space, say I2, and B(H) the C*-algebra of bounded operators on H. Every separable C*-algebra is isomorphic to a C*-subalgebra of B(H). We can parametrize or "encode" a separable C*-subalgebra A of B(H) by giving a sequence of generators for A. So our parameter space will be r :=
B(H)N.
To each 7 = (7 n ) G T we associate the C*-subalgebra of B(H) given by A{i) := the C* - subalgebra of B(H) generated by {jn : n G N}. We endow B(H) with the Borel structure of the weak topology (which is the same as the Borel structure of the cr-weak, strong and cr-strong topolo gies). This is a standard Borel space. We give T the product Borel structure, so this is again a standard Borel space. D E F I N I T I O N 1.1. Let A be a class of separable C*-algebras. We say that A is Borel, analytic, co-analytic, etc., if A = {7 € r : A( 7 ) G A} is Borel, analytic, co-analytic, etc., in I \ Denote by N the class of separable nuclear C*-algebras, by GCR the class of separable postliminal (or t y p e I) C*-algebras, by CCR the class of separable liminal C*-algebras, by GTC the class of separable C*-algebras with generalized continuous trace, by TC the class of separable C*algebras with continuous trace, by FD the class of separable C*-algebras all of whose irreducible representations are finite-dimensional and by BD the class of separable C*-algebras all of whose irreducible representations have b o u n d e d finite dimension. Finally, let NGCR be the class of antiliminal separable C*-algebras. We then have the following inclusions B D
-
™ I C GTC C CCR C GCR C N.
T H E O R E M 1.2. The classes BD, TC, N and NGCR are Borel. (The result about N is due to Effros). T H E O R E M 1.3. The classes FD, CCR, GCR are co-analytic but not Borel. In fact they are complete co-analytic (i.e., every co-analytic set
D E S C R I P T I V E CLASSIFICATION
123
in a standard Borel space is the inverse image of each one of them by a Borel function). Moreover there is no analytic class of separable C*- algebras A such that F D U C GCR. (So if an analytic property holds for all separable C*-algebras in FD, it must also hold for a non-type I separable C*-algebra). P R O B L E M S . Let H be the class of separable C*-algebras with Hausdorff s p e c t r u m and AF the class of approximately finite sep arable C*-algebras. Then H is co-analytic and AF is analytic. Are they Borel? Finally, is the class GTC co-analytic? (It is not analytic by Theorem 1.3).
§2. Encoding pure states Let A be a separable C*-algebra. Denote by Bi(A*) the unit ball of the dual of A and by B±(A*) the set of positive elements of Bi(A*). Let P(A) C B+(A*) be the set of pure states of A. Thus 0 $ P(A) and P(A) U {0} is the set of extreme points of Bf(A*). Enumerate in a sequence P 1 ? P 2 , • • • all polynomials in non-commuting variables x±, ■ • • , x n , x * , • • ■, x* with complex rational coefficients; say Pi = Pi(xi,--',xni, x j , - - - , x * . ) . If 7 G T, let 7; = Pi (71, ••• , 7 n i , 7 i ,••" »7nJSo again 7 G T and ^ ( 7 ) = ^ ( 7 ) . Moreover, {yn : n G N} is a dense "Q + * Q-*-subalgebra" of A(
124
ALEXANDER
KECHRIS
L E M M A 2.2. Let X be a separable Banach space, B1(X*) the unit ball of its dual with the weak*-topology, Kc(Bi(X*)) the space of compact non-0 convex subsets of Bi(X*) with the Hausdorff metric. Then there is a Borel map F : K^B^X*)) -> £ i ( X * ) N such that for K G JS a Kc(Bi(X*)), F(K) = {x*n K}neN dense sequence in the set of extreme points Ext(K) ofK. The proof of 2.1 is routine so we will omit it. P R O O F OF 2.2. in Dixmier [4], p. 395.
This is based on an argument of Choquet, given
First note that we can find in a Borel way for each K G Kc(Bi(X*)) a point XK G Ext(K). Indeed, fix a dense set xi,X2,- • • in X. Define then inductively K\ = {x* G X : Re (xi,x*)
is maximum},
K2 — {x* G K\ : Re (x2,x*) is maximum}, Kn+i = {x* G Kn : Re (xn+\,x*)
Then f]nKn Borel.
is maximum}
is a singleton, {XK}, and XK G Ext(K).
Easily K 1—> XK is
Fix now an open basis {Un} for Bi(X*) (in the weak*-topology). It will be enough to show how to find for each K G KC{B\(X*)) and each Un with Un fl Ext(K) ^ 0, a point x*nK G Un D Ext(K) such that x*nK G Aj(if, Z), where 2; is a real "encoding" the compact metric space Bi(X*). This will show that the set C = {(n, K) : C/nn Ext(X) / 0} is Borel and will provide a Borel function G : C -> # i ( X * ) such that G ( n , K ) G LTnn Ext(lf).'From this it is easy to define F. Fix K G ^ ( ^ ( X * ) ) . For each real q G R let Um,q = K ^ :
Re (x*,z m ) < g}
^m, 9 = {x* e K : Re (x*,x m ) < q}. C L A I M . If x G Ext(K)
D C/n, there i s m , g G Q with
X G Um^q
^_ ffn.q
b= ^ n -
P R O O F . By Hahn-Banach if y G if \ £/n find m(y),q(y)
G Q with
D E S C R I P T I V E CLASSIFICATION
125
Since K\Un is compact, choose yx , •••, yk e K\Un with Fm^yi)^yi) n- • • n Fm(yk),q(yk) Q Un. Let i ^ = K\Um(yi)tq(yi). As i ^ is convex, compact, so is co(Ki\J---\JKk). Also a; 0co(ifiU- " 1 1 ^ ) , as z is an extreme point of K. So by Hahn-Banach find ra, g G Q with a; G C7m>g and (i^iU- • ■ U i f n ) n F m ^ = 0. Then a; G l/ m , g C F m , g C [ / n . ' H So assuming £/n fl Ext(AT) ^ 8 we can find ra, g G Q with 0 / F m
q
C
We will now find effectively a point in F m ) g D Ext(if). Notice that Fm,q is compact, convex and K\Fm,q is convex. As before we can find effectively a point x in Ext(F m > g ). If this is in Ext(AT) we are done. Else we can find, effectively from x, y / z G K and 0 < A < 1 such that x = Xy + (1 - \)z. Let 6 be the line through y, z. Let also y'', z1 be the end-points of Fm,q fl 6. One of y',z' is in Ext(K). Since ?/', 2/ can be found effectively from y, z we are done. H So we have C O R O L L A R Y 2.3. There is a Borel map D : T -» (A N ) N such that £ ( 7 ) = {c?n(7)}nGN is a dense set in P(A(^y)). Now P(A(7)) is a G§ set in A N , in fact in a uniform way, i.e., we can find Borel maps Fn : T —> i f ( A N ) , where if(A N ) is the compact space of closed subsets of A N with the Hausdorff metric, such that P(A(7)) = A N \ U n ^ ( ^ ) - By standard facts then we can define a complete metric on P(A(7)) which gives the topology of P(A(7)). Since the definition of this metric is effective, given a sequence of closed sets whose union is the complement of P(A(7)), we have C O R O L L A R Y 2.4. There is a family of Borel maps pm,n : T —> R (777,, 77, G N) such that if p1(dm{^(), dn(j)) := pm,n{l), P^ Is the restriction of a complete metric p1 on V{A(^)). Thus the completion of {dn(j)}ne^
under p1 is the space
§3. The class BD We have that A G BD <£=> There is n so that every irreducible representation of A has dimension < 77. In the notation of Dixmier [4], p. 85 this means: 3n(A =
nA).
V(A(^)).
126
ALEXANDER
KECHRIS
Now nA is closed in A, so if $ : P(A) -» A is the canonical open and continuous surjection of P(A) onto A, then ^~1(nA) is closed in P(A). So if {fm} is a dense sequence in P(A), then i E B D ^ 3nVra(/ m G $ _ 1 ( n i ) ) . It follows that 7 G BD o> 3nVra[$(/i 7 (d m (7))) has dimension < n]. So it is enough to show, in order to complete the proof that BD is Borel, that for each n, m the set Dn,m = {7 G T : $(/i 7 (d m (7))) has dimension < n} is Borel. Given 7 G T denote by fm{l) the pure state /i 7 (d m (7)). Then the space of the representation $ ( / m ( 7 ) ) is A(j)/Nmtl, where N m > 7 is the left ideal of x G ^ ( 7 ) with fm(j)(x*x) = 0, and with inner product given by fm{l){y*x). As {7*.} is dense in A^k),{^k/Nmn} is dense in A(j)/Nmn. Thus A(j)/Nm^ has dimension < n iff 3yi • • • 3?/n G ^.(7)Vfc3o:i ■ • • an G ^(7fc _ ^2aiVi € Nm^). This easily gives an analytic definition of DnjTn. To give a co-analytic definition of D n ,m notice that A(j)/Nmy7 has dimen sion < n i f f Vj/i • • -2/n+i € A(7)3(ai • • - a n + i ) 7^ 0 (£c*i2/t G iV m>7 ). Now {(«!,•■• , a n + i ) G C n + 1 : ^ctiVi G Nm,7} forms a subspace of C n + 1 , so given 2/1, • • • ,3/n+i £ -4(7), i f 3(c*i • ■ - a n + i ) / 0 such that YlaiVi € Mm,-?, then such (a 1 ? • • •, a n + 1 ) can be found which is A} (7,^1, • • • , £n+i)> where £1, • • -£ n +i € N N are such that 7$-(i) —>i->oo 2/j- This gives the co-analytic definition that we want.
§4. The class TC By Dixmier [4], p. 106 to say that A is in TC is equivalent to saying that the (self-adjoint two-sided) ideal of all x G A such that IT G A \-> Tr -K(X)-K{XY is finite and continuous on A is dense in A. Let K(A) be the Pedersen ideal of A (see Pedersen [14], p. 175). Then A G TC iff Vz G K(A) (IT i-> Tr 7r(z)7r(:r)* is finite and continuous). Now if C is the class of positive continuous functions on (0, 00) with compact support, then K(A) is the linear span of all x G A+ for which there are xu--- ,xk e A+ and / i , - - - , / f c e C with x < / i ( z i ) + • • • + fk{xk). So Ae
TC
^V(fij)
Ki
V a i - - - a n eViyi-'-ym
TrGii-^
e A+
Tr7r(^ai2/i)7r(^ai2/*)
is finite and continuous).
127
D E S C R I P T I V E CLASSIFICATION
Now 7r G A h-> Tr 7r(rr)7r(x*) is finite and continuous iff / G P(A) »-» Tr $ ( / ) ( x ) $ ( / ) ( x * ) is finite and continuous on P ( A ) . This gives, after a few more messy but routine calculations, that TC is co-analytic. We now want to show that TC is also analytic. This is based on the following facts: (i) the class TC is closed under stable isomorphism; (ii) the stable isomorphism class of A G TC is determined by its so-called Dixmier-Douady invariant and (iii) by a result of Raeburn-Taylor [16], one can construct explicitly a separable continuous trace C*-algebra with a given Dixmier-Douady invariant. We will use these facts to outline a sketch of the proof that TC is analytic. The actual details of the proof involve rather horrible but straightforward calculations so we will omit them. For each Polish locally compact space T, {ATj}iGN locally finite open cover of T and 2-cocycle Xijk : Nijk -> S1 (Nijk = iV* n Nj O Nk), RaeburnTaylor [16] construct explicitly a separable C*-algebra A = A(Ti{Ni},
{\ijk})
whose spectrum is T and its Dixmier-Douady class is represented by ({Ni},
{\ijk}).
Let A be the class of all such A(T,{Ni}, { A ^ } ) . It follows that A is analytic. Now it is not hard to check that the relation A =s B of stable isomorphism between A, B is analytic (i.e., {(j,S) G T 2 : A(*y) — s A(6)} is analytic. Since A e TC <£> 3B[B e A and B ^s A] it follows that TC is analytic.
§5. T h e class N The argument here is due to Effros. For a unital C*-algebra A let Mn(A) be the algebra o f n x n matrices on A and let Fn(A)= Let also Mn(A*)
{M€Mn(A):M>0,||^M«||
be the set o f n x n matrices on A* and let Sn(A)
= {g e M n (A*) : g > 0 , ^ ( 1 ) = % } .
Then the approximation characterization of nuclearity (see Choi-Effros [3]) gives the following, for unital separable C*-algebras, A G N <^V9 G TVVa! • • •, a p G A3n3M \\ai-*(M)oA{g)(a)\\<-l—,
G Fn{A)3g
G Sn{A)V£ = 1, • • • ,p
128
ALEXANDER KECHRIS
where A(g)(a) = [p»j(a)], $(M)([a^]) = Y,aijMij,
$(M)oA(g)(a)
=
so
tnat
'£Mijgij(a).
Now since | | $ ( M ) | | < 1 ( $ ( M ) : Mn -+ A) and ||A(p)||-l(A(^):A-,Mn), it is clear we can restrict a i , • ■ • , a p to any fixed dense subset of A. So if A(pi) — A(j) = A, we have A{j) e N <^VgVm! • • -ra p 3n3M G Fn{A)3g
G 5 n (A)W = 1, • • • ,p
Now (see Takesaki [18], p. 193), Fn(A) consists of all M G Mn(A) of the form [Mij] = [b* bj], with bu--,bn G A and H E ^ I I < 1. Again it is clear that we can restrict the M^ above to be of the form 7^.7^. with E I I 7 ^ 7 A J I < 1- So we have
A(j)
e N oWqVmx • • • mp3n3k1
• ■ • kn3g G Sn(A)
(*)£ 117*^11 <1 (") Il7m, - £ Now M n (A*) = Mn(A)*
7**7^(7™ Jl I < r ^ y -
via the duality ([g{j], [a{j]) = E ^ ' ( a u ) -
So
\9ij] e 5 , n ( ^ ) < ^ p i j ( l ) = <% & V61---6JGA£^(6*6j)>0 ^ P . j ( l ) = «y & Vwi---Wj£^j(7i7i) >0. Let Sn(A) = {[gij] G Sn(A) : | | ^ | | < M). This is clearly a compact set in BM(A*)nxn (where BM(A*) is the ball of radius M in A*), in the (product of the) weak*-topology. Now one can replace "3g G Sn(A)*" above by "3M3g G S*f(A)n and the "p G S™(A)n by a countable dense subset of them, which can be found in a Borel way from 7 as S^f(A) is compact. This shows that N is Borel at least for unital A. If A is not unital and A\ is the canonical unital C*-algebra obtained by adding a unit to A then (see Choi-Effros [3]), A G N <$ A\ G IV, so we are done in the general case as well.
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D E S C R I P T I V E CLASSIFICATION
§6. The class N G C R We have first that A G NGCR <^ VI (I ^ 0 closed ideal of J => J g TC). (This is because any GCR algebra has a non-0 closed TC ideal - Dixmier [4], p. 103). Thus 7 G NGCR <£> V<5 e T(A{6) is a closed ideal of A[n) => 6 # TC). Now A{8) is a closed ideal of ^ ( 7 ) o VnVp3q(\\8p - 7 9 || <
&
VnVp(7 n (5 p ,^7 n G J 4 ( £ ) ) ,
so this is a Borel relation of <5,7, thus NGCR is co-analytic, since TC is Borel. To see that NGCR is also analytic we use the following characterization of Glimm ([6], see also Dixmier [4], 9.5.4), A G NGCR O A is isomorphic to some ^ ( 7 ) such that A"'(7) is a von Neumann algebra of type II. It is easy to see now that the map 7 — i > A" (7) from T into the space of von Neumann algebras on H with the Effros Borel structure is Borel. Since it is known (see Nielsen [13]), that the class of type II von Neumann algebras is a Borel set in this space, it follows that NGCR is analytic.
§7. The classes F D , C C R and GCR: I First we will verify that these are all co-analytic. For F D . We have that Ae
FD & VTT e i3n(dim(7r) < n),
so that in the notation of §3 7 e FD <^ Vx G P(i4(7))3ndim($(/i 7 (x)) < n, so it is enough to show that the set {(7,x) G T x A N : x G P ( A ( 7 ) ) & dim($(/*, 7 (x)) < n}
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ALEXANDER KECHRIS
is Borel. This is done exactly as in §3. For C C R . We have A G CCR
<$ W G A\/x
G A(TT(X) is compact),
so that 7 G CCR <£> VnW G A(j)(7r(jn)
is compact)
<£> VnVx € P(A(7)) ($(/i 7 (x)) (7 n ) is compact). So it is enough to show that {(7,z) G r x A N : x G P ( ^ ( T ) ) & $ ( M X ) ) (in) is compact} is co-analytic. Abbreviate, for x G P(A(7)), hy(x) = fx and $(/a;) = TTX. Then as the space of the representation irx is A(/y)/Nfx, where Nfx — {y G A(j) : fx(y*y) = 0} with inner product induced by fx(y*z), and flxCz/) iz/Nfx) = VzlNfx ■> we have (using the weak- norm continuity charac terization of compact operators): 7ra;(7n) is compact <£> Vxi,x2i
■ • ■ E A{^yiy
G ^ ( 7 ) (fx(x*Xi)
< 1&
f(y*v) < 1 & X
Now Xi/Nfx Mln)
-*w y/Nfx
(?/) iff In^i/Nfx
* / ^ / * ""^ y / ^ / « =* ^ ( T n ) (Si) ~+ 7Ta;(7n) (?/)).
iff Vm(/a.(xJ,2/m) -► fx(y*1m)) ~> 1UVINSX
iff fxiiln^i
~ InVY
and 7r:r(7n) (a;.) -> (in^i
~ InV))
~> 0.
So clearly the property "71^(7^ is compact" is co-analytic and we are done. For G C R . We will use the following characterization of GCR algebras A G GCR o A is of type I O Every representation of A is of type I & VTT : A -* B(H)
So
(TT(A)" is of type I).
_ 7 G GCR <£> V6 G r(A(<5) is a homomorphic image of ^ ( 7 ) =^ A"(8) is of type I).
As 8 >—► ^4"(<5) is Borel from T into the space of von Neumann algebras, and the class of type I von Neumann algebras is Borel, it is enough to show that {(7,8) : A(6) is a homomorphic image of ^ ( 7 ) } is analytic (in T x T). Recall from §2 the definition of 7 for each 7 G T. Then A(8) is a homomorphic image of A(*y) & 3e G T(A(6) — A(e) & 3x G
D E S C R I P T I V E CLASSIFICATION
131
N N such that the map 71^ ( 7 J = € x ( n ) is a norm-decreasing "Q 4- i Qhomomorphism" of {jn : n G N} onto {e n : n G N}) so this is clearly analytic. Next we will show that the above classes are not all Borel, in fact, complete co-analytic and moreover there is no analytic class A such that FD C AC GCR. This will be done by finding a complete co-analytic set P (in some Polish space X) and a Borel map H : X —► V such that x G P =>H(x) G FD x $ P ^H{x)
i GCR.
This will be done by using a certain class of homeomorphisms of 2 N and their transformation group C*-algebras, i.e., crossed products.
§8. Classifying classes of h o m e o m o r p h i s m s of2 N Let H = H(2N) be the Polish space of homeomorphisms with the metric d(g,h) — mdix6(g(x),h(x)) X
+ max<5(^~ 1 (a;),/i~ 1 (x)), X
where S is the metric on 2 N . A homeomorphism / G H(2N) will be called quasi-periodic if every orbit of / is finite. It will be called s m o o t h if the equivalence relation induced by / , i.e., xEy & 3n(fn(x) = y) is smooth, i.e., has a Borel selector. T H E O R E M 8.1. Let QP,S be the classes of quasi-periodic, smooth homeomorphisms of 2 N resp. Then QP,S are complete co-analytic sets in H(2N). In fact there is complete co-analytic set P in some Polish space X and Borel F : X -> tf(2N) such that xeP=> x^P^F(x)
F(x) G QP, <£ S.
(In particular, there is no analytic set R with QPC R C S). To prove 8.1 we will make use of a specific class of homeomorphisms of 2 N , the so-called Lipschitz ones. Denote by 2 n the set of n-tuples of O's and l's. Given permutations it of 2 n and p of 2 m with n < m we write 7T C p if p(xi, ■ ■ •, xm)\n = 7r(xi, • • •, xn).
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ALEXANDER KECHRIS
If 7rn are permutations of 2 n and 7Ti C 7r2 C 7r3 C • • •, then / : 2 N —> 2 N defined by f(x1,x2,--')
=
\j7rn(xu---,xn) n
is a homeomorphism of 2 N . These are the Lipschitz h o m e o m o r p h i s m s if 2 N . Clearly (7rn) is uniquely determined by / . Let L C n n ( 2 n ) ! be the set of all (?rn) with TT± C TT2 C • • -. Clearly, L is a closed set in I l n ( ^ n ) ' ' s o a c o m P a c t metric space. The map (7rn) \-> f is clearly Borel. It will be thus enough to show T H E O R E M 8.2.
(i) For a Lipschitz homeomorphism
/,
/ is smooth & f is quasi — periodic. (ii) The set of quasi-periodic (= smooth) (7rn) G L is a complete set in L.
co-analytic
(Thus in 8.1. we can take X = L, P = {(7rn) : (7rn) is quasi-periodic },F((7r n )) = the corresponding / . ) P R O O F OF 8.2. First we show that if / is smooth, then / is quasiperiodic, the converse being obvious. Call a point x G 2 N s t r i c t l y r e c u r r e n t for / if for each nbhd U of n x £> f ( ) £ ^ \ {x} f° r some n. By Katznelson-Weiss [10] if / is smooth then no point x G 2 N is strictly recurrent. (This is actually a special case of Glimm [7]). We will now show that this implies that every orbit of / is finite. For that we will use a "tree picture" of Lipschitz homeomorphisms, see Jackson-Dougherty-Kechris [9]. We define the orbit tree Tf of / as follows: Given n and an orbit O of 7rn (on 2 n ), exactly one of the following happens: when we look at 7rn+i either O extends to one orbit or two orbits. (In particular \G\ = 2 m for some m). So we can form a binary tree as follows: The nth level nodes of the tree are the orbits of 7rn on 2 n . Every nth level node has one or two (n + l ) t h level nodes extending it according to the above cases. For every x G 2 N there is a unique infinite path ax G [Tf] (= the set of infinite paths through Tf) such that x\n G Oi.x{n). Now if ax has the property that for infinitely many n, ax(n) doubles in the next level ax(n + 1), then x is strictly recurrent, a contradiction. This can be seen as follows: Fix a nbhd U of x. We can assume U = {y : y\n = s}, where 5 is a sequence of length n and n is such that ax(n) doubles in the next level. Say the orbit ax(n) has 2 m elements. Let s~i,s"j be the level n 4- 1 extensions of 5, with s~i — x\(n + 1). Then ^^(s^i) = s"j, so / 2 m + 1 ( x ) = x' where 2m+1 2m+1 x'\{n + 1) = s"j. Thus / ( x ) G U, but f (x) ^ x. So we have shown that for every x G 2 N there is some no such that ax(n) splits in two orbits for all n > n0. Say the orbit ax(n0) has 2 m °
D E S C R I P T I V E CLASSIFICATION
133
elements. Then clearly the orbit of x has 2 m ° elements too and we are done. Finally, the proof that the set of quasi-periodic (7rn) G L is a complete co-analytic set is given in Kechris [11]. H
§9. The clases FD, CCR, GCR: II We will now prove the result stated at the end of §7. T H E O R E M 9.1. There is a complete co-analytic set P, in some Polish space X, and a Borel map G : X —>T such that x G P => G{x) G FD x
G{x) $ GCR.
P R O O F . Take X = L the space of Lipschitz homeomorphisms of 2 N as in §8. Each 7r G L defines an obvious action, also denoted by TT, of Z into C(2 N ). Denote by C(2 N ) \xn Z = C*(?r) the corresponding C*-crossed prod uct of C(2 N ) by Z with respect to n (see Tomiyama [19], p. 68-69). Then by 4.1.9 in Tomiyama [19], 7r in quasi — periodic => C*(ir) G FD. Also by Gootman [8] 7r is smooth <£> C*(TT) G
GCR,
in particular 7r is not quasi — periodic => c*(7r) 0 GCR. We will show that there is a Borel map G : L —> T such that A(H(TT)) C*(7r). It follows by taking P = {TT G L : n is not quasi-periodic } that TT G TT
=
P => G(TT) G FD
$ P =* G(TT) ^ GCR
and our proof is complete. To construct G : First since Z is amenable we can replace C*(K) by the reduced product C(2 N ) i x ^ Z (see Tomiyama [19], 3.2.3). An isomorphic copy of C(2 N ) t
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ALEXANDER KECHRIS
We have that Z, via 7r, acts on C(2 N ) as follows: n •„. / = / o 7rn. Fix a canonical faithful representation A of C(2 N ) into B(H) so that if {/ n } is some fixed dense sequence in C(2 N ) and A(/„) = c n , (so that c = (c n ) G T), then the maps IT \-> (n -^ c m ) n j m , from L into T, where n ■„. A(/) = A(n > / ) is the isomorphic action of Z via 7r in A(C(2 N )) = ^4(c), are Borel. Now let H' = £2(Z,H). on iiT given by
For each n G Z let ixn be the unitary operator
un£(m)=£(m-n),
£e£2(Z,H).
Also consider the representation p^ : A(c) —> B(H') given by ( p „ ( a ) 0 ( n ) = ( - n ) •„ a£(n), £ G P(Z,H),n
GZ
(see Tomiyama [19], p. 71). Then C(2 N ) t x ^ Z is isomorphic to the C*subalgebra of B(H!) generated by {/?7r(a) : a G ^.(c)} U {it n : n G Z}, therefore by {/97r(cm) : m G Z} U {un : n G Z} = d e / {<$£ : n G Z}, where clearly 7r ^ tf71" = (££) is Borel from L into V = T(H'). Fixing an isomor phism of H and H', we finally obtain a Borel map G : L —► T such that A(G(7r)) = C(2 N ) t x ^ Z and the proof is complete. R E M A R K . One can also see that GCR is complete co-analytic using a construction of Behncke, Krauss and Leptin [1], which in turn generalizes a construction of Dixmier [4], 4.7.17: To each countable linear order C one can assign, in a Borel way, a separable C*-algebra A(C) such that C is wellordered <s> A{C) in GCR. Since the set of linear orders on N, which are wellordered is complete coanalytic, the result follows. R E M A R K . Since the crossed products we are using are unital, all the results here hold for unital algebras as well. R E M A R K . The algebras A(C) are in AF, so the result about GCR holds within the class of AF algebras as well.
§10. A co-analytic rank on F D Given a co-analytic set P C X , in some Polish space X , a co-analytic rank on P is a map cp : P —> ui (= the first uncountable ordinal), such that for each countable ordinal a < u>\ the set Pa = {x G P : ip(P) < a} is Borel "uniformly" in a. More precisely, this means that there are analytic and co-analytic, resp., relations R.SCX2 such that xGP=>
[
<^
S{x,y)].
D E S C R I P T I V E CLASSIFICATION
135
One of the basic properties of co-analytic ranks is the b o u n d e d n e s s the orem: If S C P is analytic, there is a < u\ with sup{(p(x) : x G S} < a. We will now define a rank
[/^+i = {f G A : There is open nbhd V of it and positive integer k such that W G V^TT' G ^ or dim(7r/) < A;)}, Ux=
| J ^ , if A is limit.
Let 7^ be the closed ideal of A such that 1$ = U^. Since these notions depend on A we will write U^(A), I^(A) if necessary. Notice that the 7$ can be defined by the following induction: 7O = 0, 1^/11=
h(A/Ii),
Ix= (J/ e ,Alimit. £
Note also that I^A)
= UX{A) = ( J Int {TT G A : dim(Tr) < A:} ifc
- (J Int (fci). k
CLAIM. 4 G F D
<^3^<wi([7e
(<*3£ <<*!&=
=i)
A)).
P R O O F . =>: Since A G FD, A =k A. Since fc^. is closed (Dixmier [4], p. 85) and A is a Baire space (Dixmier [4], p. 79), h(A) / 0, by the Baire Category Theorem. Now notice that A G FD <£> V closed ideal I {A/1 G FD), so if £ is the least (countable) ordinal such that 7^ = 7^ + i we must have 7^ = A, so U^ = A. <=: Say 7T G t/^+i - L^. Then for some fc,dim(7r) < k, so A € FD.
H
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ALEXANDER KECHRIS
Now define for A G FD, tp(A) = least £ such that C/^+i = A (we use this instead of the least £ for which U^ = A for technical reasons). Then ip : FD —> LJ\ is a rank. We want to verify that it is a co-analytic rank. Let $ : P(A) -» A be the canonical open, continuous surjection. Put
Then we can define these inductively by
/ G O^+i <=> 3 open nbhd V of / and fc > 0 such that V/ € V ( / € 0* or d i m ( $ ( / ) ) < fc)), C>A = ( J 0
0
A limit.
£
Now fix any sequence of basic open sets {Vn} in P(A), where some Vn maybe 0, and define the monotone inductive definition ^ A ( ^ 5 S) = ^ ( n , S) on N (n e N, S C N) by *(n,5)^^V/eyn[/G
Denote by ^
| J VmVdim(^)
the £th iterate of ^ , i.e.,
* ° = 0, ^ + i = {n:tf(n,tf*)}, # A = ( J ^ , A limit. £
CLAIM. Oi+1
O^ = ( J { F n : n e * * } .
This is easy to check by induction. From this it is easy to see that = A iff ^ + l = N, so A £ FD <=> 3£ < CJI ( * e + 1 = N)
and
D E S C R I P T I V E CLASSIFICATION
137
We will now verify two things: i) For 7 G FD,
u)?K{n).
ii) If WO C N N is the set of codes of countable ordinals, there is an analytic relation S and a co-analytic relation R such that for w G WO:
<=> S(-y,w),
where \w\ is the ordinal coded by w. From this we get for any 7 G FD:
A}(7)( w G WO &
<^4(7)) < H =* ^(A(«)) < M), which shows that this relation is analytic, and also
Aj( 7 ){w G WO & ip(A(-y)) < \w\ &
Vv G WO(|v| < \w\ =>
(^(7)) < M => vM*)) < M)), which shows that this relation is co-analytic and we are done. P r o o f of i) and half of ii): Fix a sequence of basic open sets {Gn} in A N . Define then the inductive definition \I>*(n, 5; 7), with parameter 7 G T, as follows: Put h7(Gn) = VJ and VJ = P(A) n VJ. So {VJ} is an open basis in P(A(^)). Then let tf*(n, <,;7) *> 3&V/ G V ? [ / G ( J V? V d i m ^ , ) < A:] mes & 3k\/x G A N [z G P(A(7)) n G n
=» x G (J Gm V dim($(/i7(x)) < A;)]. raGS
Thus \P* is a co-analytic monotone inductive definition. Moreover 7
G F D ^ 3 ( < wi({n : (**)* + 1 (n;7)} = N),
where (**)€+1(n;7) ^ *
»
: (**)*(n; 7 )};7)
(#*)A = (J (**)*, A limit,
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ALEXANDER KECHRIS
and
£ = 0: ¥>(A(7)) < 0 o h(A(j))
= i(7).
Now Ji = TJ\ is the union of the interiors of kA. Say Int(fcA) = (Ui)k- So U\ = \Jk(Ui)k- Let (I\)k be the closed ideal such that (h)k = {U\)k- Then (h)i Q (7 1 ) 2 C • • - and /1 = L U A ) * , so i\ = i iff (J^CM* is dense in i . For each y € A denote by I(y) the closed ideal generated by y. Then the above imply the following: CLAIM. tp(A{
7n||
< —^— & m + 1 I* (?/) has only < k — dimensional irreducible representations).
P R O O F . Assume <^(A(7)) < 0. Then \Jk(h)k is dense in A (= A( 7 )). Given 7 n and m there is then k and 7/ G (/i)fc with \\y — 7 n | | < ^ p p Now J*(2/) ^ (A)fc, so I*(y) C (/ijjfe Cfc A, so 7*(y) has only < /c-dimensional irreducible representations. Assume the expression on the right above is true. We have to show that \Jk(h)k is dense in A. Fix 7 n , r a . Then find y,k such that \\y — 7 n | | < ^q-j-, and I*(y) has only < A:—dimensional irreducible representations, i.e., j*&) Qk A. Then f^y) C ( ^ ) f c = I n t ( , i ) , so f{y) (ii)*, so y e (h)k and we are done.
C (7T)fc, i.e., J*fo) C
Now given y G A( 7 ) (represented as a limit of a subsequence of 7 ) we can find in a Borel way from ?/, 7 a 6 G T with A(8) = I*(y), which easily implies, using also §3, that
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D E S C R I P T I V E CLASSIFICATION
L E M M A 10.1. L E M M A 10.2.
l£+1 is the largest closed ideal of A with (p(I) < £. IflCT
are closed ideals and
Granting these lemmas we complete the proof as follows: £—►£+1 : Assume ip(A(j)) < £ is analytic. Then (by the lemmas above) viMl)) < £ + 1 <=> 3/(7 is a closed ideal and
< ^ 3 7 l , 7 2 , . - - , 7 4 , - . - e r ( £ < A) (A(7l)CA(72)C-..CA(7c)C... and each A( 7 ^) is a closed ideal of A( 7 ) and c/?(A(7^)) < £ and ^(A(7)M(«(7l>72,---)))<0), where £(71,72, • ■ •) £ T is an enumeration of ( 7 i ) n , -•, (7f)n> • • • (ft G N,f < A). This shows that cp(A(j)) < A is analytic as well. P R O O F O F 1 0 . 1 . First one can easily show by induction on £ that
ui(i) = uini.
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ALEXANDER KECHRIS
Thus
=I
*>IClz+1. P R O O F OF 10.2. It is enough to assume 7 = 0. So let A be such that
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D E S C R I P T I V E CLASSIFICATION
§11. A co-analytic rank on CCR Fix a separable Hilbert space H. Denote by Fn(H) the class of oper ators in B(H) of rank < n (n = 1,2, • • •). We have the following closure properties of Fn(H): a) Fn(H)
is closed in the weak operator topology.
Proof. Let Sn 6 Fn(H),Sm ->w S. If S £ Fn(H) then for some £i, •••,£„_!_! G H, S(£i ),-"■> £(£n+i) are linearly independent. On the other hand, Sm(£i ),•■•> S m ( £ n + i ) are linearly dependent, so X ) a m , j S m ( f j ) - 0 for some amj, not all 0. By going to subsequences and renumbering £i, • • •, £ n +i, we can assume that for some J C {1, • • •, n + 1}, amj — 0, if j £ J , and amj / 0, if j G J, m a x | a m j | = | a m , i | = 1, and finally jeJ
that amj
- > m <x/. Fix 7? G H. Then £ «m,j(5'm($j)>T?) — 0> s o
77) = 0 .
Since 77 is arbitrary, Yl ajS{(,j)
as
^ ™ —>™
jeJ =
0> t n u s
a
j
=
^' ^ o r
a
^ h
jeJ
contradicting that | a i | = 1. b) Fn(H) is hereditary, i.e., if 0 < S < T, T G F n ( # ) , then 5 G F„(ff). Proof. Let X = TH, so that dim(X)
= 0,
so ( S ^ z ) = (Sll2x,Sl/2x) = 0, i.e, S 1 / 2 ^- 1 - = {0}, so S ^ J f = 1 2 i.e., S / G F n (ff) and thus S = Sll2S1'2 G Fn(H). c) F n ( # ) * = Fn{H) and TFn(H)S
Sl/2X,
C F n (JJ) for any T, 5 G F n (ff).
Now fix a separable C*-algebra A. Consider the following subset R = RA CA defined by x G R O VTT G i3n(7r(x) G F ^ f f * ) ) , where H^ is the space of the representation 7r. Note that this definition makes sense since if 7r, 7r' are equivalent representations and ir(x) G Fn(H^), then TT'(X) G Fn(H^). Denote by if* = J ^ ' x C A the set tf£ = {TT G i : TT(X) G F n ( F „ ) } . LEMMA.
if* is a closed subset of A
P R O O F . This is a uniform version of the preceding argument for Fn(H).
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ALEXANDER KECHRIS
Let / »-* [717] be the canonical surjection of P(A) onto A. We have to show that KZ = {feP(A): 7rf(x) e F^H^)} is closed in P(A). Recall that Hnf is A/Nf, where Nf = {x G A : f(x*x) = 0} with inner product f(y*x). Also 7Tf(x)(y/Nf) = xy/Nf. So let / i e K*Ji ->w* f G P(A). We will show that f € K%. Otherwise there are £1, ■ • • , x n + i G A with 7r/(x)(xi/JV/), • ■ • ,/jrf(x)(xn+i/Nf) linearly in dependent. On the other hand, 717.(x)(xi/Nfi), ■ ■ ,irfi(x)(xn+i/Nfi) are linearly dependent, say
^aijitfiWixj/Nfi)
= 0,
where at least one of the ctij is ^ 0 for each fixed i. Without loss of generality we can assume that there i s J C { l , - - - , n + l } such that a^j = 0 if j 0 J and ctij ^ 0 if j G J , for each i. Also we can assume that m a x | a i j | = |oji,i| = 1 and that a^- —>;_»oo &j for j G J . Since
Y,^i,j(^fi(x)(xj/Nfi),y/Nfi)
=0
for all 2 and y G A, i.e., ^2ai,jfi(y*xxj)
= 0, VWT/ G A,
it follows that
^2ajf(y*xxj)
=0
jeJ
or ^^•(^(x)^/^),
y/Nf)=0,
jeJ
thus OJJ = 0 contradicting that \a\\ = 1.
H
So we have for x G A: x G JR^V7rGi3n(7rG^), where if^ is closed in A. Now define by transfinite induction open sets Ug C i as follows C7"f = {TT G A : 3?i3 open nbhd V 3 TTVTT' G V(7r' G K*)} = | J I n t ( ^ ) , n
E7* = ( J £/*, A limit, £
V TT' G # £ ) } .
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D E S C R I P T I V E CLASSIFICATION
Then it is easy to check, using the Baire Category Theorem in A, that x e R&3£
< UJI(U^+1
= A).
Thus define for x G R, the rank \\x\\R = least au?+1
= A),
and put Rt = {xeR:
= {xeR:
\\X\\R
I7f+1 - i } .
LEMMA. A G CCR «* 3£ < cji[(i^)+ is dense in A + ] . (Here i ? + — the positive elements of B.) P R O O F . <^=: Assume (Rt)+ is dense in A+ for some £ x G (#$)+, then W G A3n(7r G AT*), so in particular W G compact). So there is a dense subset of A, i.e., the linear span which is mapped into compact operators under any -K G A, so A
< ui. If A(7r(x) is of ( i ^ ) + , G CCR.
=>•: Fix a countable dense set S in A+. Let fn : [0,1] —► [0,1], / n (£) < t be nonnegative, continuous, vanishing in a nbhd of 0, such that fn(t) —► * uniformly as n —> oo. So fn (s) —► 5 for 5 G 5 and / n ( s ) G A+ as well. Since W G A(7r(s) is compact), VTIW G A(7r(/ n (s)) = fn(iv(s)) has finite rank), i.e., VnVs G S(fn(s) G # ) . So let f = sup ||/„(s)||fl < u\. As {fn(s) : n G N, s G 5 } is dense in A + , it follows that (Rz)+ is dense in A+. H
We can now define the following rank on CCR: ip(A) — least £ such that (R^)+ is dense in A + . We will show that this is a co-analytic rank on CCR. Suppose A = A(7),7 G I \ It is easy to define a Borel function 7 — t » 7 from T into T such that % G A + and {%, : n G N} is dense in A+. (For instance we can take 7 n = 7 n 7n) Now let fm : [0,1] —► [0,1], / m ( t ) < £ be continuous, vanishing in a nbhd of 0, and fm(t) —> * uniformly as m —> 00. So fm(%) G A + and /m(7n) -> 7n- So if 7 m , n = / m ( 7 n ) G % then ^(A) < 6 We will now prove the converse. LEMMA. Let A = A ( 7 ) , 7 G I \ If ||A|| < £, i.e., ( J ^ ) + is dense in A + , then 7 m , n G ( ^ ) + , V m , n .
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ALEXANDER KECHRIS
P R O O F . We establish first some properties of R^. 1) R^ is hereditary: For that it is enough to show that if 0 < x < y, y e A+ then for all £,U]! C U£. This is proved by induction on £ using the fact that Fn(H) is hereditary. 2) For z e A, zR^z C R^: For that again it is enough to show that for y G A,Uc C U?yz, which is again proved by induction on £ using that for T G B(H), TFn{H)T C F n ( J J ) . We now use an argument as in Pedersen [14], p.175: Fix 7 m ,n = fm(ln)Let g : [0,1] —> [0,1] be equal to 1 on the support of fm and #(0) = 0, 0 < g < 1. Consider x = p(7 m > n ). As / m p = p, we have 7 m , n x = 7 m > n . Also x G A+. As (#f)+ is dense in A+, there is y G (JR$)+ with ||x—T/|| < \. Then _ -1/2 -1/2 _ - 1 / 2 / _ \~l/2 -1/2 -1/2 . i~ -1/2 -1/2 7m,n — 7m,n#7m,n — 7 m , n l ^ 2/J7m,n ~r 7m,n2/7m,n S 2 7m,n ' 7m,n2/7m,,nj
thus 7 m , n < |7m,n2/7m,n, so 7 m , n < z i ^ 2 , where 2: = |7m,n, therefore 7m,n G ^ . H So we have ^ ( ^ ( 7 ) ) = least £ such that (R^)+ is dense in ^ ( 7 ) + = least £ such that 7 m > n G #£ for all ra, n, i.e., ^(^(7)) = s u p | | 7 m , n | | ^ ( 7 ) . ^From this it immediately follows that in order to show that ip is a coanalytic norm on CCR, it is enough to show that for each fixed ra,n the function 0(7) = Il7m,n||i^( 7 ) is a co-analytic norm on the co-analytic set ^m,n(7)^7erA7m,ne^
( 7 )
(Rm,n Q r ) . To see this, first define the monotone inductive definition;
* m ,n(p,5; 7 ) «• 3*v/ e 17 [/ e U V£ v T/ (7 m ,„) e Fk{Hr,)] m(ES «■ BfcVx € A N [x 6 P ( ^ ( T ) ) n Gn
=>
x G | J GmV $(/i 7 (x)(7 m , n )) G - ^ ( ^ ( ^ ( x ) ) ) ] -
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D E S C R I P T I V E CLASSIFICATION
It is easy to check that this is a co-analytic monotone inductive definition. Since now i W ( 7 ) ^ 3 < e ( { p : ^ + 1 ( p ; 7 ) = N}) and \\lm,n\\RM^
= least £({P : ^ + 1 ( p ; 7 ) } - N})
(this is proved as in §10), it follows that for 7 with Rm^n(j) we have 0(7) < cjf ^ ( 7 ) , while for some co-analytic relation P and w G WO. 0(7) < \w\ &
P(j,w).
It remains to find an analytic relation S with e(-Y)<\w\*>S(>y,w),
for w G WO. Let us go back temporarily to an arbitrary separable A. Given x G A and a closed ideal I C. A, define a relativized U?(I) as follows: \J* (I) = {IT el:3n3
open V C 7, V 9 TTVTT' G V(TT' G K * ) } ,
[ 7 ^ ( 7 ) = {TT G 7 : 3 open V C 7, V 3 TTVTT' G V[7r' G C/^J) V TT' G # £ ] } ,
^w - u ^r C L A I M . U*(I) =InU*
(where Ug = U*(A)).
(Easy by induction).
For x G # 7 , let II*, I | | = least aU^i(!)
= !)•
Since Uf is open in A, let 7^ be the closed ideal such that 7 | = U%. C L A I M . J£ +1 is the largest ideal 7 with ||x,7|| < f. PROOF. ||x,7||<^^^+1(7) = 7
^ u*+1 n 7 = 7 ^ 7C ^
+ 1
146
ALEXANDER KECHRIS
We will now show that the relation
IMII<€ is uniformly analytic in x, 7, £, which means that there is analytic T such that for 7 G T,S G T with A{8) = I a closed ideal in ^ ( 7 ) = A, e G N N which represents a subsequence of {7 n } converging to x, and w G WO, with \w\ = £ we have ||x,I||<^T(7,S,e,w). This is done as usual by effective transfinite induction on £:
||*,I||<0<*i7f(J) = I
<*icu? O | J Int [KD D I k
O I C ( J i | , where 7£ = Int(7f£), 7£ closed ideal in A k
<* Vn\/m3y3k(\\y
- Sn\\ < ^ ^
& I(y) C 7£),
where 7(?/) is the closed ideal generated by y. Now "7(2/) C 7£" iff "/(y) C # £ " iff "VTT G 7(T/) TT(X) G F * ( # „ ) " . Since it is enough to restrict "VTT G 7(?/)" to a dense subset of 7(?/), this is clearly a Borel condition, so ||x, 7|| < 0 is analytic.
£ - * + !:
||:r,J||<* + l * l 7 J + 2 ( / ) = .f
For J C 7 closed ideals, put Q(x, J, 7) & VTT G ( 7 / ^ ) 3 ^ open in ( 7 / J ) 371V7T1 6 V V ( x )
€Fn{H„,)).
Then we have l k , 7 | | < £ + 1 ^ ^J{J Q I closed ideal & Q(x,J,7)&||*,J||<0(For this notice that Q(x, J, 7) & J C J ' C 7 => Q(x, J', 7).)
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D E S C R I P T I V E CLASSIFICATION
As the relation Q is also analytic (which can be seen as in the argument for the case £ = 0), this shows that \\x,I\\ < £ + 1 is analytic as well. £ < A —> A : Again we have ||x,/||
2
C - C J
(
C -
&
lk,Jdl<e&0(^LU,/), so we are done as before.
H
§12. T h e problem of co-analytic ranks on G C R The natural way to define a co-analytic rank on GCR is to use one of the canonical composition series for GCR algebras. (For an exposition of this see Dixmier [4] or Pedersen [14]). For example, one can take for each A G GCR the canonical composition series Ja such that Ja+i/Ja = CCR(A/ Ja) = the largest CCR ideal of A/Ja, and define the rank p(A) = least £ such that (J^ = A). Unfortunately, this rank is not a co-analytic rank, since the set of A's with p(A) = 1 is the class CCR, which is not Borel. Another possibility would be to use the composition series Ia+i/Ia — the largest ideal of A/Ia of type IQ (see Pedersen p'(A) = least £ such that (1% = A), then the set of A's with p'(A) set of A's in / 0 , which is easy to see that is an analytic set, but know if it is Borel or not. It is open if p' is a co-analytic rank or
Ia where [14]). If = 0 is the we don't not.
Denoting by Io(A), CCR(A), resp., the largest closed ideals of A of type Jo, in CCR, resp. and assuming we have a uniform way of assigning to each A a closed ideal K{A) with I0(A) C K(A) C CCR(A) we define a composition series Ka with Ka+i/Ka = K(A/Ka). This could be used to define a rank pK(A) — least i such that (K^ = A). It is conceivable that with the appropriate choice of K(A) this would give a co-analytic rank. For example, if K0{A) = L(A), where L(A) = {x £ A : V7T 6 A3n3 open nbhd V 3 ir VTT' G V(*'(X)
G Fn(^))},
then one can easily check that L(A) is an ideal of A, so K0(A) is a closed ideal of A. Also IQ(A) C K0(A) C CCR(^l). If PK0 i s t n e corresponding
ALEXANDER KECHRIS
148
rank, then {A : PK0(A) = 0} = {A : ip(A) — 0}, where I/J is the co-analytic rank on CCR constructed in §11. So {A : PK0(A) = 0} is Borel in this case. We don't know however if this pK0 is a co-analytic rank on GCR. REMARK.
Given a closed ideal K(A) as before, notice that the set {A : K{A) ± 0}
is Borel. This is because K(A) ^$&
A(£ NGCR,
(K{A) / 0 =» CCR(A) / 0 => A g NGCR => J 0 (A) ^ 0 =* K(A) / 0). So it is an interesting question to see if there is a Borel function that picks, given a code for A = A{j) 0 NGCR, an element in K{A). For K(A) = I0{A) this amounts to picking in a Borel way from 7 an abelian element (see Pedersen [14]) of ^ ( 7 ) = A for A 0 NGCR. We do not know the answer here. Similarly, one can ask if one can pick in a Borel way a dense sequence in K(A). This is not possible for K{A) = CCR(A), otherwise CCR would be Borel. Again, we do not know the answer to any of these questions for Jo (A) or K0(A).
§13. Classifying classes of (second countable) locally compact groups We refer here to Sutherland [17] for a Borel parametrization of Polish groups. (For another parametrization, see 12.E, 2) of Kechris [11].) In particular, this gives a Borel parametrization of second countable locally compact groups. Various standard classes of such groups have been verified in Sutherland [17] to be Borel. To these we can add the class of amenable groups (use, e.g., to characterization 18.13 of Pier [15]), Lie groups (use, e.g., the characterization of Lie groups as those containing no small subgroups - a strong version of the Hilbert 5th problem), nilpotent groups, connected groups, etc. (Also note that the operation G —> G on abelian groups is Borel using constructive proofs of the existence of Haar measure.) However, we conjecture that the class of type I (postliminal) groups is not Borel. (It is easily co-analytic.) An approach to such proof would be as follows: First one would find for each countable ordinal £ a group G such that p(C*(G)) > £ (see §12 for the definition of p(A)). If one can then solve the problem (see §12) of constructing a co-analytic rank p* on GCR, which moreover has the property that p* > p (which any reasonable approach for constructing p* would surely satisfy), then by the boundedness theorem for co-analytic ranks we would be done. Similar approaches might work for showing that the liminal (CCR) groups do not form a Borel class (here one would only have to construct groups with arbitrarily large I/J(C*(G))).
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D E S C R I P T I V E CLASSIFICATION
REFERENCES [1] H. Behncke, F. Krauss & H. Leptin, C* -algebren mit geordneten ideal folgen, J. Funct. Anal. 10, 1972, 204-211. [2] L. Brown, Stable isomorphism of hereditary subalgebras of C* -algebras, Pac. J. Math. 71 (2), 1977, 335-348. [3] M-D. Choi & E. Effros, Nuclear C*-algebras and the property, Amer. J. Math. 100 (1), 1978, 61-79.
approximation
[4] J. Dixmier, C*-algebras, North Holland, 1977. [5] P. A. Fillmore, A User's Guide to Operator Algebras, Wiley, 1996. [6] J. Glimm, Type IC*-algebras,
Ann. of Math. 73 (1961), 572-612.
[7] J. Glimm, Locally compact transformation Soc. 101 (1961), 124-138.
groups, Trans. Amer. Math.
[8] E. Gootman, The type of some C*- and W*-algebras associated with transformation groups, Pac. J. Math. 48 (11), 1973, 93-106. [9] S. Jackson, R. Dougherty, A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1), 1994, 193-225. [10] Y. Katznelson & B. Weiss, The construction sures, Israel J. Math. 12 (1972), 1-4.
of quasi-invariant
mea
[11] A. S. Kechris, Classical Descriptive Set Theory, Springer Verlag, 1995. [12] Y. N. Moschovakis, Descriptive Set Theory, North Holland, 1980. [13] O. Nielsen, Direct Integral Theory, Marcel Dekker, 1980. [14] G. Pedersen, C*-algebras and their Automorphism Press, 1979.
Groups, Academic
[15] J-P. Pier, Amenable Locally Compact Groups, Wiley, 1984. [16] I. Raeburn and J. L. Taylor, Continuous trace C*-algebras with given Dixmier-Douady class, J. Austral. Math. Soc. (Series A) 38 (1985), 394-407. [17] C. Sutherland, A Borel parametrization Kyoto Univ., 21 (1985), 1067-1086.
of Polish groups, Publ. RIMS,
[18] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979. [19] J. Tomiyama, Invitation World Scientific, 1987.
to C*-algebras and Topological
Dynamics,
151
DECIDABILITY A N D
UNDECIDABILITY
IN THE E N U M E R A B L E T U R I N G
DEGREES
STEFFEN LEMPP
Department of Mathematics University of Wisconsin Madison, WI 53706-1388, USA ABSTRACT. We survey some recent work on the (recursively) enumerable Turing degrees, with particular emphasis on work relating to decidability and undecidability.
Two fundamental notions of mathematics are those of a computable set and of an enumerable set. A set S is called computable (or recursive) if there is an effective algorithm which for any input x can compute whether x is an element of S. A set S is called (recursively) enumerable if there is an effective algorithm listing all elements of S. Clearly, these notions only make sense for countable sets. However, all "basic" countable sets (such as the set of all fc-tuples of natural numbers for some fixed k, the set of all finite strings of O's and l's, the set of words over some finite alphabet, or the set of first-order formulas over some finite language) are easily seen to be in effective 1-1 correspondence with the set of natural numbers. (The "code number" for an element of such a countable set is often called its Godel number.) Thus, for investigating computability, one can restrict oneself to studying computability over the natural numbers. The above intuitive notions of a computable set and an enumerable set were made precise in various equivalent ways in the mid-1930's and later. For example, Turing [Tu36] defined effective algorithm to mean what is now called a Turing machine, i.e., a very simple-minded computer with no limitations on run time or memory space. This (and its many equivalent def initions) is nowadays generally accepted as the way to define computability (see also Soare [Sota]). 1980 Mathematics Subject Classification (1985 Revision). 03D25. Key words and phrases, (recursively) enumerable Turing degrees, decidability. This research was partially supported by NSF grant DMS-9504474 and a grant of the British Engineering and Physical Sciences Research Council. The author would like to thank the Leeds logicians for their hospitality during his half-year stay in Leeds, and the Chinese logicians for their hospitality during his two-week visit to China. .
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STEFFEN LEMPP
There is a close connection between computability and definability in arithmetic. By Post's Theorem [Po48] (see [K152, p. 293]), a set is com putable iff there are first-order formulas ip and ij) (in the language of arith metic) in which all quantifiers are bounded such that for all x, x e S
iff
3y(p(x,y)
iff
Vyip(x,y);
and a set is enumerable iff there is a first-order formula ip (in the language of arithmetic) in which all quantifiers are bounded such that for all x, x GS
iff
3yip(x,y).
More obviously, and also historically, there is a close connection between computability/enumerability and decidability/axiomatizability. These con nections often led to deep theorems, such as Matiyasevich's solution [Ma70, Ma93] to Hilbert's 10th Problem (based on previous work by Davis, J. Robinson, and others). He showed that a set S is enumerable iff there is a polynomial p(x, J ) G Z such that S = {x | 3yp(x,y)
= 0}.
Given that most sets are noncomputable, the question arises as to how to compare them in terms of their information content, i.e., how to measure noncomputable information. The most general "effective" reducibility was first defined by Turing [Tu39]: A set S is Turing reducible to a set T (denoted by S < T T) if there is an oracle Turing machine computing S (with oracle T, i.e., such that the Turing machine can query membership information about the set T). This reducibility gives a prepartial ordering (i.e., a reflexive and transitive relation) on the power set of N . We can now define two sets S and T to be Turing equivalent (denoted by S = T T) if they are Turing reducible to each other. This gives an equivalence relation on the power set of N . The equivalence class of a set S is called its Turing degree (denoted by deg T S or simply deg S) and intuitively denotes the "information content" of the set S while stripping away all the facts about S inessential from a computational point of view, such as whether a particular number is an element of S. These Turing degrees then form a quotient structure of the power set of N , partially ordered by the relation induced on it by Turing reducibility. We denote this structure by D , and the substructure of the Turing degrees of the enumerable sets by E. Note that the latter structure can also be defined as the set of Turing degrees of solution sets of diophantine equations (by the above-mentioned result of Matiyasevich [Ma70, Ma93]) or as the set of Turing degrees of word problems of finitely presented groups (by Boone [Bo66], Clapham [C164], and Fridman [Fr67]). There is one more operation on the Turing degrees that will be used later, namely, the so-called Turing jump. Given a set 5 , its jump is defined as the
D E C I D A B I L I T Y AND UNDECIDABILITY
153
set S' = {e | eth oracle Turing machine with oracle 5 halts on input e}. This operation can be iterated: 5^ 0) = S, and 5 ( n + 1 ) = ( 5 ( n ) ) ' , and we have S
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STEFFEN LEMPP
be extended to an embedding ofV into E. Since this conjecture would have allowed back-and-forth constructions, it would have implied a number of "nice" results about E, such as the saturatedness of E and the No-categoricity and decidability of its first-order theory. However, Shoenfield's Conjecture was almost immediately shown to fail quite dramatically. We mention here three Refutations of Shoenfield's Conjecture. (1) (Lachlan [La66] and Yates [Ya66]) There is a minimal pair in E, i.e., there are nonzero degrees with infimum 0. (This precludes that any embedding of the diamond lattice as an upper semilattice can be extended to an embedding of the five-element lattice obtained by adding a new element below the two atoms of the diamond lattice.) (2) (Yates (unpublished) and Cooper [Co74]) There is a noncuppable degree in E, i.e., there is a nonzero degree a such that no incomplete degree b joins a to 0'. (This precludes that any embedding of the three-element linear order can be extended to an embedding of the diamond lattice.) (3) (Cooper, Sui, Yi [CSYta]) There is a superminimal pair in E, i.e., there is a minimal pair a 0 and a x such that for any i < 1, every degree x £ (0,a;] joins ai_i to ao U a i . (This precludes that any embedding of the diamond lattice can be extended to an embedding of the six-element lattice obtained by inserting two new elements, one below one old atom, and the other as the join of the first new element and the other old atom.) 3. Logical aspects. Shoenfield's Conjecture, even though it failed, was crucial in that it stimulated the next generation of algebraic investigations which revealed the "not so nice" structure of E and culminated nearly two decades later in the proof of the undecidability of its theory by Harrington and Shelah [HS82]. Soon afterwards, Harrington and Slaman (unpublished, see [SWta]) succeeded in showing that in fact the first-order theory of E is as complicated as possible, namely, as complicated as first-order arithmetic. Since the proof of this result is not readily available in the literature, we will briefly sketch it here, with some later simplifications due to Slaman and Woodin [SWta] as well as Nies, Shore, and Slaman [NSSta]. Clearly, the first-order theory of the enumerable degrees can be interpreted in first-order arithmetic in the usual way. The proof in the other direction proceeds in several steps: Step 1: We code the natural numbers with addition and multiplication by a computable partial ordering (P, <) such that the natural numbers n are coded by the minimal elements pn of P and such that the operations
D E C I D A B I L I T Y AND UNDECIDABILITY
155
are coded, e.g., by m 4- n = k iff 3p, q € P (p is minimal over p m and p n , and q is maximal in P and minimal over p and over p&), and similarly for multiplication. Step 2: Code this partial order (P, <) into E with parameters (i.e., de grees) a, b , c, and d such that the elements p of P are coded by the minimal degrees x = x p < a with the property that c < x U b , and such that p < q in P iff x p U d < xq U d. We will call such a quadruple of parameters a, b , c, and d a coded standard model of arithmetic. The remaining steps are now needed to sort out the coded standard models of arithmetic from other coded models of (a finite fragment of) Peano arithmetic (PA). Step 3: For any nonzero degree a € E, there are low parameters below a coding a standard model of arithmetic. (A degree a is low if a' = 0'.) Step 4: Given two coded models M 0 and M\ of (a finite fragment of) Peano arithmetic, we want to code an embedding (i.e., an order-preserving injection) / from Mo into Mi in a similar fashion. If Mo is a standard model of arithmetic coded by low parameters, then, for any coded model M\ of (a finite fragment of) Peano arithmetic, such a coded map / always exists. Step 5: We can interpret first-order arithmetic in the first-order theory of E as follows: Fix a sentence cp in the language of arithmetic. Then cp is true in N iff in E, the following sentence holds: 3a > 0 3M 0 l= (finite fragment of) PA coded below a ((VMi l= (finite fragment of) PA coded below a 3 embedding / : M 0 —► M x coded below a) and Mo t= ip). Note that the clause ( V M i 3 / . . . ) here ensures that M 0 is a standard model of arithmetic since by Step 3, some standard model must be coded below a by low parameters, and by Step 4, Mo is "more standard" than any other model of (a finite fragment of) Peano arithmetic coded below a. Once undecidability of a theory has been established, the immediate next question is at what quantifier-level undecidability first occurs since mathe maticians are usually only interested in statements with a small number of alternations of quantifiers. By an old observation of Sacks [Sa63], the IIi-theory of the enumerable degrees is decidable since any finite partial order can be embedded into E. By a recent result of Lempp, Nies, and Slaman [LNSta], the II3-theory is undecidable. This result is shown by coding finite bipartite graphs (in a language without equality) into E using only Ei-formulas with parameters,
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STEFFEN LEMPP
and then applying Nies's Transfer Lemma [Ni96] to transfer the hereditary undecidability of the II3-theory of bipartite graphs without equality to that of E. The gap remaining in this line of research thus is at the n^-theory, which we will discuss in more detail in the last section. 4. Model-theoretic aspects. The above research focusing on undecid ability also led to some results concerning the type structure and questions of definability in E. Lerman, Shore, and Soare [LSS84] exhibited infinitely many 3-types real ized in E by embedding an infinite number of lattices into E, all generated under meet and join by three elements. This showed the non-Ho-categoricity of the first-order theory of E, thus disproving another consequence of Shoenfield's Conjecture. Later, Ambos-Spies and Soare [AS89] found infinitely many 1-types realized in E (namely, degrees bounding n but not n + 1 many degrees forming pairwise minimal pairs). And Ambos-Spies and Shore [AS93] showed that continuum many 1-types are consistent with the firstorder theory of E (namely, that given any subset S C u with at least three elements, there is a degree coding a partial ordering with maximal chain of length k 4- 1 (incomparable to all other elements of the partial ordering) iff keS). Clearly, the types of the least and the greatest element of E are isolated. But it is open whether there are any other isolated 1-types, and whether in fact all 1-types are isolated, i.e., whether E is a prime model of its theory. This naturally leads to questions of definability. A fair number of results were shown in this respect over the years. The most exciting is probably the following recent T h e o r e m (Nies, Shore, Slaman [NSSta]). If S C E is definable in firstorder arithmetic, and closed under double jump (i.e., for any degrees a and b , a" = b " and a G S implies b G S), then S is definable in E (in the language of partial ordering). This result has a number of interesting consequences; e.g., it shows the definability of the classes of the high n and low n enumerable degrees (for n > 2) in terms of the partial ordering only. By a small trick, they also obtained the definability of the class of the high (i.e., highi) enumerable degrees. (They show that a is high iff for any b there is c < a with b " = c". Here, an enumerable degree a is lown if a^n) = ()(n), and highn if a^n) = 0( n + 1 ).) Obviously, the least and the greatest element of E are definable. It is open whether any other enumerable degrees are definable in E (in the language of partial ordering without parameters). The following question addresses an interesting partial result in this direction which seems more accessible to current methods: Question (Li Angsheng (see [SI])). Are there degrees a < b < c in E and a formula ip(x) in the language of partial ordering without parameters such
D E C I D A B I L I T Y AND UNDECIDABILITY
157
that for all enumerable degrees x G (a, c), (E, <) |=
x = b?
The strongest possible definability result would be the following Biinterpretability Conjecture (Slaman, Woodin [S191]). There is a map f from E into a standard model coded in E such that for all a e E, deg Wf{a) = a. This conjecture would have implied in particular the rigidity of E (i.e., that the only automorphism of E is the identity). This consequence, and thus the Biinterpretability Conjecture, was refuted by Cooper as described in the next section. 5. Second-order a s p e c t s . The earliest results on automorphisms of E concerned automorphism bases (i.e., sets S C E such that any automorphism which is the identity on S must be the identity on E). A number of nontrivial automorphism bases were found, mainly in the 1980's. An interesting recent result here is due to Ambos-Spies [Amta] that any nontrivial initial segment of E forms an automorphism base. However, the question of whether there are any automorphisms of E other than the identity remained open until Cooper's recent results [Cota] about the existence of such automorphisms. He also showed that there is an automorphism mapping a low to a nonlow enumerable degree. Thus the low enumerable degrees are not definable in E from the partial ordering alone, in contrast to all other jump classes as mentioned above. Once a full proof of Cooper's results is available, a whole number of questions will arise: How many automorphisms are there (e.g., are there continuum many)? Are all automorphisms arithmetical? Is there a finite automorphism base? 6. T h e n 2 - t h e o r y . The main open question about the enumerable Turing degrees at this time, accessible to currently available methods, is in our opinion the decidability of the ^ - t h e o r y of the enumerable degrees. It is not hard to see that the n 2 -theory can be rephrased in purely algebraic terms as follows: Equivalent formulation of t h e decidability of t h e n 2 - t h e o r y . Decide if given any finite partial orders P C Q0,... ,Qn (for some n > 0), any embedding of P into E can be extended to an embedding of Qi into E for some i < n. (Note here that i may depend on the embedding of P into E.) A natural subproblem of the above is obtained by setting i = 0, i.e., deciding whether any embedding of a finite partial order P into E can be extended to an embedding of a finite partial order Q D P into E. (This is usually called the extension of embeddings problem.) A solution to this problem was given by Slaman and Soare, which we state in a modified version due to Lempp and Lerman:
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STEFFEN LEMPP
E x t e n s i o n of E m b e d d i n g s T h e o r e m (Slaman, Soare [SS95], rephrased). | Fix a finite lattice P and a finite upper semilattice Q extending P as an upper semilattice and respecting the lattice structure of P. Then any embedding of P into E (as a lattice) can be extended to an embedding ofQ into E (as an upper semilattice) iff Vo, b e P Vx e Q - P(a = min{c G P | O x } & x ^ - ^ x V 6 = a V 6 ) . Lerman calls the above the "Saturation Axiom". It is a generalization of the phenomenon encountered in the superminimal pair mentioned earlier. Lerman [Le96] then suggests the following general approach to deciding the n 2 -theory: Expand the language of partial ordering to include • the language of bounded upper semilattices (i.e., <, V, 0, and 1); • (n + 2)-ary meet predicates M(a, b0,..., 6 n ) (for all n > 1): This denotes that all x < 6 0 , • • •, bn are also < a and takes into account that the meet of two degrees need not always exist; • saturation predicates generalizing the phenomenon of the Satura tion Axiom mentioned above; and • a unary predicate for the so-called promptly simple degrees. (The class of the promptly simple degrees was shown by Ambos-Spies, Jockusch, Shore, and Soare [AJSS84] to coincide with a number of other interesting classes (such as the degrees which do not form one half of a minimal pair). Prompt simplicity interacts nontrivially with saturation, e.g., Cooper, Slaman, and Yi (unpublished) observed that saturation cannot occur between a promptly simple degree a and a non-promptly simple degree b , namely, given any such degrees a and b , there is an enumearable degree x < a with x £ b and x U b < a U b . ) Lerman's approach then consists in deciding the III-theory in this ex panded language in order to give a decision procedure for the Il2-theory in the language of partial ordering. Lerman's approach thus highlights another natural subproblem, the socalled Lattice Embeddings Problem, namely, to characterize the finite lattices which are embeddable into E. This is actually a very old open problem going back to the early investigations in the 1960's of the algebraic structure of E and, in particular, of its finite substructures. Of course, the minimal pair theorem of Lachlan and Yates mentioned earlier yields an embedding of the diamond lattice into E. This was soon generalized to showing that all finite distributive lattices are embeddable into E by Lerman (unpublished) and Thomason [Th71]. Lachlan [La72] proved that the two non-distributive five-element lattices M5 and N& are embeddable into E. However, Lachlan and Soare [LS80] exhibited a finite lattice, iSs? which is not embeddable into E. The known embedding and non-embedding techniques up to the late 1980's were then distilled into two
D E C I D A B I L I T Y AND UNDECIDABILITY
159
conditions (the so-called Embeddability Condition (EC) and Nonembeddability Condition (NEC)) by Ambos-Spies and Lerman [AL86, AL89]. EC turned out to be a rather unwieldy condition and received little at tention. NEC, on the other hand, was a nice, algebraic condition and was conjectured by many to be the correct condition characterizing exactly the non-embeddable finite lattices. In particular, NEC requires the existence of a so-called critical triple in the lattice. Definition. A triple (a, b, c) of elements of a finite lattice L is called a critical triple if a, 6, and c are pair wise-incomparable, a V b = a V c, and b A c < a. Ambos-Spies and Lerman [AL86] observed that in a finite lattice, the absence of critical triples is equivalent to another property which is easier to verify. P r o p o s i t i o n . A Unite lattice L with least element 0 fails to have critical triples if for alia < d in L such that the interval (a, d) is empty, the difference of the intervals [0, d] — [0, a] has a (unique) least element. The above proposition allows one to organize the enumeration of elements for an embeddings proof for a lattice without critical triple much more easily, so the absence of critical triples was conjectured by many to ensure the embeddability of a finite lattice. This was recently refuted, however, by Lempp and Lerman [LLta], who exhibited a finite lattice, L 2 o, which is not embeddable into the enumerable degrees but does also not contain a critical triple. The search for a characterization of the finite lattices embeddable into the enumerable degrees continues and is likely to be hard; it involves ana lyzing the obstructions to embeddability in a typical pinball machine proof and using them to produce a nonembeddability proof in an effective fash ion if possible. Only then, a decision procedure for the I^-theory of the enumerable degrees can reasonably be attempted. REFERENCES
[Amta]
K. Ambos-Spies, Automorphism bases for the recursively enumerable degrees (to appear). [AJSS84] K. Ambos-Spies, C. G. Jockusch, Jr., R. A. Shore, and R. I. Soare, An alge braic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Trans. AMS 281 (1984), 109-128. [AL86] K. Ambos-Spies and M. Lerman, Lattice embeddings into the recursively enu merable degrees, J. Symbolic Logic 51 (1986), 257-272. [AL89] , Lattice embeddings into the recursively enumerable degrees, II, J. Sym bolic Logic 54 (1989), 735-760. [AS93] K. Ambos-Spies and R. A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Ann. Pure Appl. Logic 63 (1993), 3-37.
160
[AS89]
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K. Ambos-Spies and R. I. Soare, The recursively enumerable degrees have in finitely many one types, Ann. P u r e Appl. Logic 4 4 (1989), 1-23. [Bo54-57] W . W . Boone, Certain simple unsolvable problems of group theory I, II, Indag. M a t h . 1 6 (1954), 231-237, 492-497; / / / , IV, Indag. M a t h . 1 7 (1955), 252-256, 571-577; V, VI, Indag. M a t h . 1 9 (1957), 22-27, 227-232. [Bo66] , Word problems and recursively enumerable degrees of unsolvability, a sequel on finitely presented groups, Ann. of M a t h . (2) 8 4 (1966), 49-84. [CD93] P. A. Cholak and R. G. Downey, Lattice nonembeddings and intervals in the recursively enumerable degrees, Ann. P u r e Appl. Logic 6 1 (1993), 195-222. [Co74] S. B. Cooper, On a theorem of C. E. M. Yates, h a n d w r i t t e n notes. [Cota] , Beyond GodeVs Theorem - the failure to capture information con tent, Logic Colloquium '95 (V. Harnik and J. A. Makowsky E u r o p e a n S u m m e r Meeting of the Association for Symbolic Logic, eds.), Springer Lecture Notes in Logic, Springer-Verlag, Heidelberg (to a p p e a r ) . [Cota] , The Turing degrees are not rigid (to a p p e a r ) . [C164] C. R. J. C l a p h a m , Finitely presented groups with word problems of arbitrary degrees of insolubility, P r o c . London M a t h . Soc. (3) 1 4 (1964), 633-676. [ELTT65] Yu. Ershov, L. Lavrov, A. Taimanov, M. Taitslin, Elementary theories, Russian M a t h . Surveys 2 0 (1965), 35-105. [ET63] Yu. L. Ershov and M, A. Taitslin, Undecidability of certain theories, Alg. i Log. 2 no. 5 (1963), 37-41. (Russian) [Fr67] A. A. F r i d m a n , Degrees of unsolvability of the word problem for finitely pre sented groups, Nauka, Moscow, 1967. [Fr57] R.M. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, P r o c . Natl. Acad. Sci. USA 4 3 (1957), 236-238. [HS82] L. Harrington and S. Shelah, The undecidability of the recursively enumerable degrees, Bull. AMS (N. S.) 6 no. 1, 79-80. [K136] S. C. Kleene, General recursive functions of natural numbers, M a t h . A n n . 1 1 2 (1936), 727-742. [K152] , Introduction to metamathematics, North-Holland, A m s t e r d a m , 1952. [La66] A. H. Lachlan, Lower bounds for pairs of r.e. degrees, Proc. London M a t h . Soc. 1 6 (1966), 537-569. [La72] , Embedding nondistributive lattices in the recursively enumerable de grees, Conference in M a t h e m a t i c a l Logic, London, 1970 ( W . Hodges, ed.), Lec t u r e Notes in M a t h e m a t i c s No. 255, Springer-Verlag, Berlin, 1972, p p . 149-177. [LS80] A. H. Lachlan and R. I. Soare, Not every finite lattice is embeddable in the recursively enumerable degrees, Adv. M a t h . 3 7 (1980), 78-82. [La63] I. Lavrov, Effective inseparability of the set of identically true formulas and finitely refutable formulas for certain elementary theories, Alg. i Log. 1 (1963), 5-18. (Russian) [LLta] S. L e m p p and M. L e r m a n , The decidability of the existential theory of the poset of recursively enumerable degrees with jump relations, Adv. M a t h , (to appear). [LNSta] S. L e m p p , A. Nies, and T. A. Slaman, The Tls-theory of the enumerable Turing degrees is undecidable, Trans. AMS (to a p p e a r ) . [Le96] M. L e r m a n , Embeddings into the recursively enumerable degrees, C o m p u t a b i l ity, enumerability, unsolvability (S. B . Cooper, T . A. Slaman, and S. S. Wainer, eds.), London M a t h e m a t i c a l Society Lecture Notes No. 224, Cambridge Uni versity Press, Cambridge, England, 1996, p p . 185-204. [LSS84] M. L e r m a n , R. A. Shore, and R. I. Soare, The elementary theory of the recur sively enumerable degrees is not N 0 -categorical, Adv. M a t h . 5 3 (1984), 301-320.
DECIDABILITY AND UNDECIDABILITY
[Ma70] [Ma93] [Mu56] [Ni96] [NSSta] [No55] [Od89] [Odta] [Po44] [Po48] [Ro67] [Sa63] [Sa64] [Sh65]
[S191]
[SI]
[SS95] [SWta] [So87]
[Sota] [Th71] [Tr53] [Tu36]
[Tu39] [Ya66]
161
Yu. V. Matijasevic, Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 1 9 1 (1970), 279-282. (Russian) Yu. V. Matijasevic, Desyataya problema GiVberta, Nauka, Moscow, 1993; H i l b e r t ' s | Tenth P r o b l e m , M I T Press, Cambridge, MA. A. A. Mucnik, On the unsolvability of the problem of reducibility in the theory of algorithms, Dokl. Akad. Nauk SSSR, N.S. 1 0 8 (1956), 194-197. (Russian) A. Nies, Undecidable fragments of elementary theories, Alg. Universalis 3 5 (1996), 8-33. A. Nies, R. A. Shore, and T . A. Slaman, Standard models of arithmetic and definability in the enumerable degrees (to a p p e a r ) . P. S. Novikov, On the algorithmic unsolvability of the word problem in group theory, T r u d y M a t . Inst. Steklov 4 4 (1955), 1-143. (Russian) P. Odifreddi, Classical Recursion Theory, vol. 1, North-Holland, A m s t e r d a m , 1989. , Classical Recursion Theory, vol. 2 (to a p p e a r ) . E. L. Post, Recursively enumerable sets of positive integers and their decision problems, Bull. AMS 5 0 (1944), 284-316. , Degrees of recursive unsolvability, Bull. AMS 5 4 (1948), 641-642. H. Rogers, Jr., Theory of recursive functions and effective computability, Mc Graw-Hill, New York, 1967. G. E. Sacks, Degrees of Unsolvability, Ann. of M a t h . Studies No. 55, Princeton University Press, Princeton, N . J . , 1963. , The recursively enumerable degrees are dense, Annals of M a t h . (2) 8 0 (1964), 300-312. J. R. Shoenfield, Application of model theory to degrees of unsolvability, Sym posium on t h e T h e o r y of Models (J. Addison, L. Henkin, A. Tarski, eds.), North-Holland, A m s t e r d a m , 1965, p p . 359-363. T . A. Slaman, Degree structures, Proceedings of the International Congress of M a t h e m a t i c i a n s , August 21-29, 1990, Kyoto, J a p a n , vol. 1, Springer-Verlag, Tokyo, 1991, p p . 303-316. , Open Questions in Recursion Theory, list of open questions on t h e world-wide web at http://www.nd.edu/~pcholak/computability/computability.html. T . A. Slaman and R. I. Soare, Algebraic Aspects of the Computably Enumerable Degrees, P r o c . Natl. Acad. Sci. USA 9 2 (1995), 617-621. T.A. Slaman and H. Woodin, Definability in degree structures (to a p p e a r ) . R. I. Soare, Recursively Enumerable Sets and Degrees: The Study of Com putable Functions and Computably Generated Sets, Perspectives in M a t h e matical Logic, Q-Series, Springer-Verlag, Berlin, 1987. , Computability and recursion (to a p p e a r ) . S. K. T h o m a s o n , Sublattices of the recursively enumerable degrees, Z. m a t h . Logik G r u n d l a g . M a t h . 1 7 (1971), 273-280. B . A. T r a k h t e n b r o t , Recursive separability, Dokl. Akad. Nauk SSSR 8 8 (1953), 953-956. A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, P r o c . London M a t h . Soc. (2) 4 2 (1936), 230-265 4 3 (1943), 544-546. , Systems of logic based on ordinals, P r o c . London M a t h . Soc. (2) 4 5 (1939), 161-228. C. E. M. Yates, A minimal pair of recursively enumerable degrees, J. Symbolic Logic 3 1 (1966), 159-168.
163
THE THEORY OF FINITE MODELS * LUO Libo Department of Mathematics Beijing Normal University, Beijing 100875
(Abstract) We study the model theory of finite structures. By releasing the requirement of a predefined equal sign and using formulas of infinite lengths we obtain new theorems for preservation of model extensions, submodels, model homomorphisms and inverse homomorphisms. This kind of theorems were discussed in Chang and Keisler's Model Theory systematically for general models but Gurevich obtained some different theorems at this direction for finite models. In our paper the old theorems managed to survive in finite model theory. There are some differences between into homomorphisms and onto homomorphisms in preservation theorems too. We also study reduced models and minimum models. The characterization sentence of a model is given which derives a general result for any theory T to be equivalent to a set of existential - universal sentences. Some results about completeness and model completeness are also given. SECTION 0. INTRODUCTION The general model theory is studied for a long time. Chang and Keisler in [1] give a detailed introduction. It seems that the model theory is mainly studying infinite models. For the case of finite model theory some people would think that it was easy to deal with and not as important as the infinite case. The development of computer science gives us a different thought. Database is finite, operating system is finite, even computer itself is considered as a finite machine. The finite model theory becomes more important. Another reason for studying finite model theory separately is that many of the theorems in general models theory cannot be used in the finite cases. For example the Godel's Incompleteness Theorem, Craig's * Project supported by the National Natural Science Foundation of China
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Interpolation Theorem and Compactness Theorem. Without these theorems the finite model theory looks quite different from the general model theory. The research also finds out that it is not always easy to prove a theorem in the finite model theory in comparing with the same theorem in general model theory. In some cases the proofs are even more difficult. A partial reason for that is the lacking of the theorems mentioned above. Gurevich in [2] and [3] gave a systematic discussion for the finite model theory in connection with the theory of computer science. Then in [4] he investigated the preservation theorems for finite model theory. He gave counter examples in finite model theory to some of the theorems for infinite models and some of the others are still valid in finite model theory. In our paper we prove some different kinds of preservation theorems for finite model theory. In general model theory the equal relation is supposed to be built into the system but we treat all relation equally. Our notations are as follows: 21, = < A, ci, . . . , Cp, # i , . . . , Rq > is a model with the universe A, p constants ci, . . . , cp and q relations Ri, . . . , Rq. The universe A is always finite and the language L = < A, ci, . . . , cp, Ri, . . . , Rq > is also finite. The logical formulas in our paper are usually first order formulas. We also use infinite conjunctions and disjunctions of first order formulas. In the latter case we will use capital Greeks to denote a formula with infinite length. We adopt the idea in [4] to say that two formulas are equivalent if they are equivalent in all finite models. They are logically equivalent if they are equivalent in all finite and infinite models. The notation 21 |= T means that all first order sentences in T are true in 21. T\ and T 2 are theories. The notation T\ |= T 2 means that for any model 21, 21 J= T\ implies 21 |= T 2 . Ti is equivalent to T 2 if and only if T\ |= T2 and T 2 \= T\. A theory T is consistent if and only if T has a (finite) model. SECTION 1. ELEMENTARY PROPERTIES As in the general model theory we can define elementary equivalence between models, Elementary submodels and elementary extensions of mod els with almost the same definitions. Most of the theorems about these concepts in general model theory are still true in finite model theory. The others are similar. We provide some of them here and skip the proofs. Theorem 1.1. Two models 21 and 93 are elementary equivalent to each other if and only if the following two conditions are true: (1) There is a constant keeping function F from A to B such that for any relation R in the language L with arity k and any A:-tuple of elements ai, ..., ak in A, 21 [= R[au . . . , ak] if and only if
T H E O R Y OF F I N I T E M O D E L S
165
(2) There is a constant keeping function G from B to A such that for any relation R in the language L with arity k and any fc-tuple of elements 61, ..., bk in B, 05 |= R[bu . . . , bk] if and only if 21 ^ R[Gbu . . . , G&fc]. Proof. (Outline) By the functions F and G we can find 21 = 2lo 2 2li D 2t2 2 ■-., OS = 050 2 ®i D 052 2 ..., such that 21; = 05 i + i and 05; = 2t i + 1 . Since they are finite models the two sequences will be the same models in finitely many steps. Definition 1.1. Model 21 is an elementary submodel of model 05 if A C B and for any first order formula ip(xi, . . . , xk) and for any A>tuple (21, . . . , zk) from A |J {ci, . . . , cp} the sentence ^(21, . . . , zk} is true in 21 if and only if the same sentence is true and in 05. Theorem 1.2. Model 21 is an elementary submodel of 05 (or model 05 is an elementary extension of 21) if and only if there is a constant keeping function F from B onto A such that for any relation R in the language L with arity k and any A:-tuple of elements 61, ..., bk in B, 05 |= R[bu . . . , bk]ii and only if 21 \= R[Fbu . . . , Fbk]. We also give the definitions of reduced models and minimum models. Definition 1.2. Two elements a and b are said to satisfy the same re lation in a model 21 of language L — < A, c i ? . . . , c p , R\, . . . , Rq > with respect to elements a i , . . . , a n if for all relation R(xi, . . . , xk) all A;tuples (zi, . . . , Zfc) from {ai, . . . , a n } (J {ci, . . . , c p } whenever the relation R(zi, . . . , 2i_i, a, zi+i, . . . , ^ } is true the relation R(zi, . . . , ^ _ i , 6, ^ + i , . . . , 2?jt} is also true and vise versa. In a model 21 if two elements a and b satisfy the same relations with respect to all elements of the model 21, then we can reduce one of them to form a new model 05. The new model is an elementary submodel of the original model 21. Definition 1.3. A model in which any two elements do not satisfy the same relation with respect to all elements of the model is call a reduced model. Definition 1.4. A model 21 of a theory T is called a minimum model of T if 21 itself is a reduced model and it does not contain any proper submodel of T. Theorem 1.3. For any model 21 there is a reduced model 05 such that 05 ^ 21. The reduced model 05 is unique up to isomorphism. Theorem 1.4. A theory T is complete if and only if T has only one reduced model up to isomorphism. We can define model completeness like in the general model theory, but here the theorem is different. Theorem 1.5. If a theory T is complete then T is model complete. Proof. If 21 C 05 are T models, then both of them can be reduced to the same reduced model.
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Theorem 1.6. For any model 21 of language L = < A, ci, . . . , cp, Ri, . . . , Rq > there is a characterization formula ip such that for any model 03, 03 \= ip if and only if 03 is elementary equivalent to 21. The characterization formula is an existential universal sentence. Proof. Suppose that model 21 has n elements {ai, . . . , an}. We define a conjunction formula X V! = ail%
. . . ,y
(2/i» •■•» 3/m) m
=
airn
with free variables (T/I, . . . , y m ) where T/I represents a^, ..., ym represents a; m . The formula x is a conjunction of all positive and negative relations eR(zi, . . . , Zk) where the fc-tuple (z\, . . . , Zk) runs over all possible com binations (including repetitions) of symbols in Z = {yu
. . . , ym}\J{au
..., an}|J{ci, ...,
cp}.
The symbol e is chosen according to the truth of the relation R(zi, in 21. We give the following definition: X
(3/1, •••> 2/m)
A H(Z1, {*1.
.... ••.
( z
zfc) € L fc> C ^
.A.
. . . , zu)
A
2l|=il(zi,
/\
* ( * » ■••>**) ...,
Zfc)
^ ( z l 9 . . . , z fc )).
* £ * ( « ! , ..., Zfc)
Using formula x we can define the conjunctive formula of the model 21. Xa V3*1' *■'' *Z'n/
=
X I
l =
a
V^IJ •••? ^nj I
l
n = a
n
Formula x a ( x i , • • • > ^n) is true if and only if elements xi, . . . , xn have the same positive and negative relations as elements a\, . . . , an in model 21. Next we define a formula ipi for an element yi to have the same relations as the element a* with respect to all a's. il>i(Vu xi, ...,
xn)
=
X
(2/t, » i , •■-, xn).
Putting together ^(z/ij x i 5 • • •, xn), for i = 1, . . . , n and replacing ?/ for 2/i we obtain a disjunction. n i=l
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THEORY OF FINITE MODELS
The above formula is true if and only if y satisfies the same relations as one of the elements in a i , . . . , a n . with respect to all a's. If 2/ix, • ■ •, Vik have the same relations as elements aix, . . . , aik with respect to all elements of 21 we want the list (y^, . . . , yih) to satisfy the same relations as (aix, . . . , aik). In the following the arity k is considered as the maximum of the arity of all relations in the language L and the fc-tuple (Vh, • • • 5 Vik) runs through all possible strings of length k in (yi, . . . , yn). The formula is as follows: k
( A ^M^i' Xl> •••' x^)
A {y»i> •••> y i f c } c { y i , ..., i / n }
-"
X
.7=1
(2/*i, •■■> 2/*J)-
Now we are ready to define our main sentence (p. The meaning of (p is that there exist n elements x\, . . . , xn such that they satisfy the same relations as a i , . . . , an and for any y, y satisfies the same relations as one of the x's and for any A:-tuple (y^, . . . , yik) chosen from {2/1, . . . , yn} if each y^ satisfies the same relations as a^, then the fc-tuple (y^, . . . , yik) satisfies the same relations as (a; 15 . . . , a;fc). (f =
3xU
. . . , Xn ( X a ( » l , • • • ,
Xn)
n
• A.\A/(\/ i/>i{y, xu
...,
xn))
i=l k
. A.V3/1, . . - , 2/n (
A {Vii» •••> I / i f c } C { y i , ..., 2/n}
(f\^ij(yii>
X
l> '■■»
X
J
.7 = 1
We will prove that for any model 03, 53 (= c/9 if and only if 05 = 21. Let 05 be a model of ip and B = {61, . . . , 6 m } be the universe of 05. We can set up a many to many and onto mapping from B to A satisfying F(bi) = a,j *
if b{ has the same relations as aj with respect to a\, . . . , an.
For any relation R in the language L with arity k and for any &-tuple (6 i l 5 . . . , 6 i f c ) i n B , ® ^ U ! ^ , . . . , b i f c ] i f a n d o n l y i f 2 l h # [ ^ 1 , . . . , ^ i j With this property we can prove that the two models 21 and 05 are elemen tary equivalent to each other. Observe that there is not a quantifier in formulas ?/> and %• Therefore ip is an existential universal sentence.
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The following two theorems are of special interest for finite model theory. We only give a simple proof for Theorem 1.7 and skip the prove of Theorem 1.8. Theorem 1.7. Any theory T is equivalent to a countable disjunction of first order existential universal sentences. Proof. For every model 21 with n elements we can write the character ization sentence
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T H E O R Y OF F I N I T E M O D E L S
Proof. The right to left implication is trivial so we only have to prove the left to right implication. Assume T is preserved under extensions. For every finite model 21 in language L = < cu . . . , c p , # 1 , . . . , Rq > with n elements we can write an existential sentence cp which describes all relations and negative relations among all elements. Suppose that the n elements are A — {ai, . . . , an}. We first define a conjunctive formula Xxn=an
0 l > •••>
x
n)
with free variables xi, . . . , xn where x\ represents a\, ..., and xn represents an. The formula x is a conjunction of all positive and negative atomic formulas for all relations e R(zi, . . . , Zk) where the fc-tuple (zi, ..., zk) runs over all possible combinations (including repetitions) of Z = {yi, . . . , i/m}(J{ai, •••, a n } ( J { c i , . . . ,
cp}.
The symbol e is chosen according to the truth of relation R(zi, . . . , z^) with the corresponding fc-tuple in model 21. To specify the details we give the following definition: x
l
=
a
l t
.
. . , x
an
=
n
A
(
R ( 2 1 , . . . . zk) { z i , • • • , zk}
€ L C Z
A
*(*!> •••' z^-A-
A ...,
Zk)
-.#(*!, ..-, Zk)).
m^R(Zl, ..., zk)
The maximum existential conjunctive sentence is defined as follows: 99(2l) = 3xi, •-., xn
X
(si, -.., sn).
Let 5^ be the disjunction of the above sentence
E = V *(*)• Every model 21 of the theory T is a model of the formula J2 because 21 satisfies at lease a disjunct of £). Conversely, every model 0$ of the formula ^2 satisfies a disjunct of ^ which is written according to a model 21. Model 0$ is an extension of the reduced model of 21. 21 is a model of T. Therefore 05 is a model of the theory T. Theorem 2.2. A Theory T is preserved under submodels if and only if T is equivalent to a set of universal sentences.
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Proof. The theory T is considered as a conjunction fj of (possibly infinitely many) sentences. We consider the theory
T
—n= A *■ l
The negation of it is
£= V ^1<2<00
It is easy to see that ^ is preserved under extension of models. In the proof of Theorem 2.1 the theory T is very general. We do not require the T be a set of first order sentences or in any other special form. So we can use it for ^ . Hence ^ is equivalent to a countable disjunction
£ ' = V *■ l
where each ipi is a first order existential sentence. Therefore the original theory T is equivalent to
-£'= A ^l
The left hand side is equivalent to a set of first order universal sentences. A homomorphism is considered as a mapping h from a model 21 into (including onto) a model
f\ R(zly
...,
( zk)
G L
{*1. ■••» zfc> c z
f\ <&\=R(Zl,
R(ZU . . . , Zk)) ...,
Zfc)
where x\ represents a i , ..., xn represents an and Z = {xi, . . . , x n } | J { a i , . . . , a n } | J { c i , . . . ,
cp}.
171
THEORY OF FINITE MODELS
Let Y be the disjunction of the above sentences over all models of the theory T. Every model 03 of the theory T is a model of the formula Y because 03 satisfies at least a disjunct of Y which is written according to the model 03 itself. Conversely if model 03 satisfies a disjunct of Y which is written according to the model 21. Model 21 can be reduced to a model <£. 03 also satisfies the positive existential sentence written according to model <£. Hence 03 is a homomorphism image of €. £ is a model of T. Therefore 03 is a model of the theory T because T is preserved under into homomorphisms. Theorem 2.4. A Theory T is preserved under inverse into homomor phisms if and only if T is equivalent to a set of negative universal sentences. Proof. Similar to the proof of Theorem 2.2. For the case of onto homomorphisms our result is weaker. We will use countable disjunction of countable conjunctions of first order positive sen tences to be equivalent to the original theory. We first give a lemma. Lemma 2.1. A theory T in the language L = < A, c\, . . . , c p , R\, . . . , Rq > is preserved under onto homomorphisms. Let A be the set of all positive sentences in L. If 21 is a model of T and every sentence S G A holds in 21 holds in model 03, then 03 is a model of T. Proof. Suppose that model 21 has n elements {ai, . . . , a n } . We define a conjunctive formula
y\
=
a
X i1
0/1, •••> Vm)
• ■ • • ■ Vn =
°i
m
with free variables (2/1, . . . , ym) where 2/1 represents a^, ..., ym represents airri. The formula x 1S a conjunction of all positive relations R(zi, . . . , zu) where A:-tuple (zi, . . . , Zk) runs over all possible symbols in Z = {2/1, . . . , ym}[J{a>i,
•••, a n } | J { c i , ■••, c p } .
We give the following definition: X
A
(2/1 > •••> Vm)
(
H ( Z 1 , . . . . z fc ) e L {«!, . . . . z f c }C2T
A
fl(*i,
.■•. **))■
2 1 1 = ^ ( 2 1 , ..., Zfc)
Using formula x we can define the positive conjunctive formula of the model 21. X a (si, ..., zn) =
X
_
0*1. •••>
x
n)-
Formula x a (»i, • • • > ^n) is true if and only if the n-tuple (xi, . . . , x n ) has at least the same positive relations as (ai, . . . , an) in model 21.
172
LIBO
Luo
Let 03 = < B,L > where the universe is B = {&i, . . . , bm} with m < n. Choose a set of new variables Z — {zi, . . . , z{\ where l = m — n. Let Q be the set of all functions from Z to the universe of 21. Q = {UJ\UJ is a function
of
Z to
A}.
We define a formula I/J^ with free variables {zi, . . . , zi} saying that the /tuple (21, . . . , zi) has at least the same positive relations as (LJZI, . . . , UJZ\) with respect to all elements of 21. V ^ (*1, . . . , *j) =
X z
Putting together i/jn(zi,
i =
w r
i . • • • - zi
(El, ••-, »n) =
"zi-
• • • 5 ^/) f° r w G f i we obtain a formula:
Formula \P is true if and only if (zi, . . . , 2/) satisfies at least the same positive relations as one of the /-tuples of elements (UJZI, . . . , uzi) with respect to all elements of 21. Now we define our main formula (p.
. . . , 2//).
THEORY OF FINITE MODELS
173
Proof. For every T model 21 with n elements we can write a countable conjunction of positive sentence
$= A ^ 0
as in the Proof of Lemma 2.1 such that $ is true in a model 03 if and only if 03 is an image of an onto homomorphism of an elementary extension of 21. Let Yl be the disjunction of the above formula $ over all models of T. Every model of T satisfies at least a disjunct of ^ which is written according to itself. Conversely, every model 03 of the sentence 53 satisfies a disjunct which is written according to a model 21. 03 is an image of an onto homomorphism of an elementary extension of 21. Therefore 03 is a model of T. Theorem 2.6. A Theory T is preserved under inverse onto homomorphisms if and only if T is equivalent to a set of countable disjunction of negative universal sentences. Proof. (Similar to the proof Theorem 2.2.) In Section 3 an example is given which shows that a sentence ip is pre served under extensions of models but it can not be equivalent to an existen tial sentence. Of course from our Theorem 2.1 the sentence (p is equivalent to a countable disjunction of existential sentences. SECTION 3. EXAMPLES We give an example to show how we work differently in the finite model theory and the general model theory. Example 3.1. The language L has two relations x < y and x S y expressing the relations 'x is smaller than y' and 'x is succeeded by y' respectively and two constants m and M representing the minimum and maximum elements of the linear order. The symbol 'x ~ y' is defined as the simplification of the following: x ~ y : Vz((x < z —► y < z). A .(z < x —> z < y)). Then we define the following sentences: (1) \/xyz{x < y Ay < z :—* x < z), (2) Vx-i(x < x), (3) Vxy(x
174
LIBO
Luo
We need strong axioms here because there is not an equal sign in our language. It is not necessary to discuss the independence of the axioms. The sentences (1) - (6) give a linear order with two end elements. (7) (10) give a successor relation which is consistent with the linear order. The sentence ip is now given as the conjunction of the above all 10 sentences. 10
v = Awi=i
The theory T consists of only one sentence {y>}. It is not difficult to see that the theory T is model complete but not complete. If we change our sentence to the following: 9
i& = A w - ( i o ) , i=i
then the sentence ip is preserved by extensions of models but it can not be equivalent to a first order existential sentence. This was the original example given by Gurevich in [4].
REFERENCES [1] Chang, C. C , and Keisler, H. J., Model theory, North-Holland, Am sterdam, 3rd ed., 1989. [2] Gurevich, Y., Toward logic tailored for computational complexity, Computation and proof theory (Ed. M. Richter et al.), Springer Lecture Notes in Math., v.1104, 1984, pp. 175 - 216. [3] Gurevich, Y., Logic and the challenge of computer science, Current Trends in Theoretical Computer Science (Ed. E. Borger), Computer Science Press, 1988, pp. 1 - 57. [4] Gurevich, Y., On finite model theory, Feasible Mathematics, (Ed. S. R. Buss et al.), A mathematical sciences institute workshop, Ithaca, New York, 1989, pp. 211 - 219. [5] Lyndon, R. C , An interpolation theorem in the predicate calculus, Pacific J. math., v. 9, 1959, 155 - 164.
A N o t e on Weak Segments of PFA
175
Tadatoshi MIYAMOTO A b s t r a c t . We consider a family of large cardinals below supercompact and weak segments of PFA. This generalizes Si-reflecting cardinals and B P F A i n [Go]. Introduction In [Go], the £1-reflecting cardinals and the Bounded Proper Forcing Axiom (BPFA) are devised and the equiconsistency between them is estab lished. Since Si-reflecting cardinals are weaker than Mahlo cardinals, the corresponding forcing axiom BFPA sits near the "bottom" of the Proper Forcing Axiom (PFA) whose consistency is gotten by a supercompact car dinal. In this note, we first define the localized reflecting cardinals which generalize the Si-reflecting cardinals. We then mention a couple of rela tions between these cardinals and supercompact cardinals and provide a list of equivalent formulations for some cases of these cardinals involving elementary embeddings. It turns out that we can force Laver type diamond sequences to some cases of the localized reflecting cardinals. We move on to consider the forcing axiom S(A) for the infinite cardinals A. The S(A) are weak segments of the PFA. The BPFA is our E(CJI). The large cardinals we introduce are sufficient for obtaining levelwise consistency of some of these weak segments of PFA. We actually have an equiconsistency result for S(cj2) along the line of [Go]. We also make use of known results to get very rough estimate on the consistency strength at some of higher levels. This note is a shortened version of [Mi]. Preliminaries For a set P / 0 and a binary relation < on P , the pair (P, <) is said a preorder iff (1) Vp G P p < p (reflexive) ; (2) Vp,g,r G P if q < p and r < q, then r < p (transitive). For a preorder (P, < ) , p G P , and A C P , A is predense below p (in P ) iff Vx G P x < p3a G A such that x and a are compatible in P , i.e., 3y G P s.t. y < x,a. If A is predense below every p G P , then A is simply said predense (in P ) . A is a centered subset of P iff for any nonempty finite subset S of A, there is x G P (may or may not x G A) such that for all y G S, x < y. For a set x, \x\ denotes its cardinality and TC(x) denotes the transitive closure of x. For an infinite cardinal A, H\ denotes the set {x : | TC(x)| < A}. We consider G-structures (H\,e) and their elementary substructures
176
TADATOSHI M I Y A M O T O
(Y, G) (written as Y -< H\). A formula (p is a formula in the language of set theory. So the only non-logical symbol is the binary relation symbol G. We may assume cp is in H^ as an object. An Easton support iteration (Pa \a < K) is constructed by taking the direct limit at every strongly inaccessible cardinal a < K, and inverse limit at any other limit a < K. For other notions not found here, see [Ku] and [Je]. §1. Localized Reflecting C a r d i n a l s We consider a hierarchy of large cardinals which generalize the £ 1 reflecting cardinals in [Go]. 1.1 Definition. Let A be a cardinal and K be a regular cardinal with A > K > uo\. We say K is Hx-reflecting iff for any a G Hx and any formula tp, if Hx \= > ( a ) " holds for some cardinal x, then {X G [HX]
K, K is
then
Hx-reflecting.
(4) Let [i be a strongly inaccessible cardinal with fi > A > K. If K, is Hxreflecting, then V^ \= "K is Hx-reflecting". And so, (5) If A is a strongly inaccessible cardinal and if K, is Hx-reflecting A > hi, then Vx |= "AC is supercompact".
with □
1.3 N o t e . Therefore, if we are interested in an Hx-reflecting cardinal K, whose consistency is weaker than supercompact, then we may restrict A to A < K*, where K,* denotes the least strongly inaccessible cardinal above hi, if any. In particular, going into VK*, if necessary, we may assume that
W E A K S E G M E N T S OF
K is H\-reflecting universe.
PFA
177
and K is the greatest strongly inaccessible cardinal in the
We next give a list of equivalent formulations for some of the localized reflecting cadinals with associated diamond sequences. 1.4 T h e o r e m . For any cardinal X and any strongly inaccessible car dinal K with X
G iV and Va < ftVs G a N s"a G JV.
there is (j, M) such that • M is a transitive set with j,N,Hp G M and -PM C M. • j : N —> M is an elementary embedding with crit(j) p<j(hz).
• J(f)M
— K and
= <*>■
Proof. We just give a proof for (1) implies (3). The rests are left to the readers or can be found in [Mi]. For each (3 < K, we define f(/3) = x, if h(/3) = (xi,x2,x3,x4,x) for some xi, #2, xs, and x 4 . Otherwise, we just set f(/3) — 0. We show this / : K —> HK works. Notice that / G HK+ C Hx+. Fix (p,a,N) as in the hypothesis of (3). We prove by contradiction. So suppose to the contrary that there were no (j, M) as such. Let us abbreviate this by writting -i3(j,M) i/;(j,M,p,N,K,f,a). We would like to derive a contradiction.
178
TADATOSHI M I Y A M O T O
To this end, we choose a regular cardinal 6 so that for all /x < 0, jjbpX < 6. Then He satisfies the following : • There is a cardinal p s.t. • For all cardinals /z, ppX exist. • K < \N\ = A, H\+ exists, and a G H\+. • -n3(j, M) ip(j, M, p, AT,«, / , a). So by (1), we may choose (X, 0',//) so that • N,K,\,f,a a < K.
G X -< i^A+j 1-^1 < ^J
an(
l -X" H ft = RW
= a for some
• p' < 6' < K, p' is a cardinal, 0' is a regular cardinal, and for all p < #', ^
A
< 0' hold. W
• a < \N \
= \(X)
< 6' and a<*> G ff
(x)+
.
• H9, |= " - 3 ( i , M ) ^ , M , p ' , A T ( x ) , a , / x ) , a W ) " . .fc(a) = ( J V w > a , A w > r , > a W ) . And so, . / ( a ) = fiW. Claim. We may construct (Y,M,j)
so that
(1) (N n X ) U A(X) U Hp, U fT x ( x ) + U 7 V W U { # ^ 7 V ( X V , / } C F ^ 7V? < P ' - A ( x ) F c y ; and | y | (2) Let
_(y)
M, Vx,y J(a){Y) f^ (4)
<6'.
: Y —► M denote the transitive collapse. Then -p'x e M {x,y}
M C
(X)
G M, V/3 < p' • A Vs G^ M s"P e M, and
= a<*> fto/ds.
i — { ( ^ * \ ^ y ) ) | x G Af f l X } zs an elementary embedding from N into M and j(f )(a) = a ^ G M holds. 4>ti,M,f/,Nlx\a,?x\aW).
But since j,M eHe,, is a contradiction.
so #*/ |= > ( j , M , ^ 7 V ( X \ a , / X ) , a W ) " . This
Proof oi claim. For (1) : We first remark, since for all x,y G AT, {x,y} G N and since for all (3 < K and all 5 G^ A/", s"/3 G AT, we have <«j\T C JV and # „ C N. We also necessarily have that p / ,A ( X ) < 6' < K and that i f y , # _ ( x ) + , AT(X) G ify e HK C N. So (AT n X ) U A W U ifp* U H-xix)+ U iV ( x ) U {Hp>, iV (x) , / / , / } C AT holds.
W E A K S E G M E N T S OF
179
PFA
But for any B C N with \B\ < Q', since 6' is a regular cardinal s.t. for all fi < 0', pp'~x
< 6', we have Y s.t. B C Y -< N, ^ ' ^ V
C y , and
|y|<0'. For (2) : It is clear that _
Therefore j is well-defined and is an elementary embedding from TV into W X) M. Since X f= "TV |= " / is a function with domain «"", so TV (= " / is a function with domain a" and so M f= "j(f ) is a function with domain j(a)". But a < X(X) < *
)(a) makes sense and j ( /
)(a) = /
(a) = f
(oj( y )) =
{Y)
J(a)
= a<*> G H-{x)+ C M . For (4) : It remains to observe that j G M , crit(j) = a, and that —(X)
p' < j(a). ^ )
W 6
Let e : A
—(X)
—► TV n X be a bijection. Then for each i < X
TV ( X ) G M and ^ )
5 = {(z, (e(z)
, e(i)
( y )
)) | z G A
G M , so ( i ^ i f * * , ^ 1 0 ) }. Then 5 GA
G M.
,
Let
M and so we have seen
(X)
that j = s"\ G M. Since X n K, = a < A W < A W + 1 C 7 , so for every (3 < a, j(/3) = j ( £ ( x ) ) = £ m = /J. Also j(a) = j(*<*>) = *( y ) > p' ■ A(X) + 1 > p', a. D
□ We may force associated diamond sequences to some of localized re flecting cardinals in the following situation. 1.5 T h e o r e m . Let K be an Hx+-reflecting cardinal for some cardianl A with \
K-C.C
(2) In VPK , \
180
TADATOSHI M I Y A M O T O
Proof. Let us consider Q = {IN, OUT, STOP, l g } partially ordered by
but STOPG
Ea+i.
Namely, a + la is the coordinate > a where STOP appears first. It is clear that la is well-defined by genericity. Then we define • g(a) = {i < la | ING
Ea+i}.
So g(a) corresponds to the set of coordinates in the interval [a, a + la) where IN gets appear. It can be shown that this g : K —► [K]
§2. Weak S e g m e n t s of t h e Proper Forcing A x i o m 2.1 Definition. For any infinite cardinal A, let E(A) denote the fol lowing : For any preorder P with \P\ = A and any sequence (Bi \ i < OJ\) of subsets of P, if there is a proper preorder Q such that \\-Q "there is a filter G in P s.t. for all z < cou G n B{ ^ 0", then there is a filter G in P s.t. for a l i i
W E A K SEGMENTS OF PFA
181
Notice that in this definition the i V s a r e just subsets of P and Q may have any size at all. The following is not hard to observe : 2.2 Proposition. (1) £ ( u ) is equivalent to the PFA restricted to the Cohen p.o. set Fn(u),2). (2) Y,(ui) is equivalent to the BPFA of [Go]. (3) Con(ZFC) implies ConfEW
+
^fa)).
(4) For infinite cardinals Ai and X2 with X2 > \\, ^{X2) implies E(Ai). (5) PFA iff for all infinite cardinals A, E(A). (6) In particular, E(A) implies the PFA restricted to the proper preorders P with \P\ < X. □ 2.3 Proposition. For any cardinal X > uj\, the following hold : (1) The PFA restricted to the proper preorders P with \P\ < Aw -A+ implies (2) And so, E(AW • A + ) implies - o A . So for example, we have • T,(UJ2) + 2W = u2 implies - Q ^ . • E(CJ 3 ) + 2^ = LO2 implies - c ^ 2 .
• E(cj4) + 2U = LO2 implies - Q ^ . Proof. By pages 198,199 in [Sc], a p.o. set R concerned is an iteration of: (1) Forcing a club subset C of A + of order type w\ by countable conditions. (2) Specializing a relevant tree on the C by finite conditions. So by considering an appropriate dense subset of R, we may assume \R\ < (A+) w -(wi xu)
2.4 Proposition. For any cardinal X > u\, «/E(A) holds, then there is an uj\-closed and A + -dense p.o. set R such that \\-R "E(A) + n\ +XU < A + and so -iE(A + )". So for example, we have • Con(E(u;i)) implies Con(E(cJi) + -iE(w 2 )). • Con(£(o>2)) implies Con(E(cj2) + ^ E ^ ) ) . • Con(E(c<j3)) implies Con(E(u3) + -iE(o;4)). Proof Suppose E(A) holds. We force a n^-sequence via an cji-closed and A + -dense p.o. set ([Bu]). Since A > CJI, it is easy to see E(A) remains in
182
TADATOSHI M I Y A M O T O
the generic extensions. By genericity, [X]w can be coded in the cu-sequence. So we have Aw < A+. We are done. D
We give a list of equivalent formulations for the axioms. 2.5 T h e o r e m . For any cardinal A > CJI, the following are equivalent : (1) For any proper preorder P and any sequence (Ai | i < UJ\) of predense subsets of P s.t. for all i < UJI \Ai\ < X, there is a centered subset C of P s.t. for alii <cu1,CnAi^ 0. (2) For any a G H\+ and any T,Q-formula (p, if \\-p "3ytp(y,a) holds in some transitive G-structure" for some proper preorder P, then {N G [Hx+]UJl | (1) a G N -< Hx+ ; (2) 3y(p(y,a,^) holds in some transitive G-structure of size c^i} is stationary in [H\+]Ul, where ~(N^ denotes the transitive collapse of N. (3) S(A). Proof. (1) implies (2) : Fix a G H\+, a proper preorder P , and a Eoformula (p s.t. \\-p"3yip(y,a) holds in some transitive G-structure M " . We may assume that | ( - p " | M | = |A| = u\" by further forcing with a count ably closed forcing. We may also assume that P is a complete Boolean algebra (sans its zero). The following is Lemma 4.2 and 4.7 in [Go] with minor changes. C l a i m 1. / / \\-p"E\ is a well-founded binary relation on u\", then there are at most u)\ many predense subsets of size at most CJI s.£. ifGCP is a filter which meets all of these predense subsets and if we define E\ = {(a,/3) G U! : 3p G G p \\-P "aErf"), then (1) (u)\, E\) is well-founded. (2) For any formula ijj and any a G
{= «^(3)»". D
Wl
We recall that every club C C [A] has a function h : [X] —> A s.t. C is induced by h. Namely, {X G [A]^1 | (1) ux C X ; (2) h"[X]
Ul
C/"wi.
W E A K S E G M E N T S OF (3)
183
PFA
^"[/"CJII^C/"^.
(4) Furthermore,
if \\-P "f is one-to-one",
then so is f. a
We make use of the following easy fact on the Mostowski collapses. Claim 3. Let E be a binary relation on a set X (^ 9) s.t. (X,E) is transitive collapsed to (M,e) by TT. If Y C X is an E-initial segment of X (i.e., for any x G X and any y €Y, if xEy, then x G Y), then ir"Y is a transitive set and so (Y,E\Y) is uniquely transitive collapsed to (7r"Y, G) via ir\Y. □ Now we proceed in four steps. S t e p 1. Since we want to keep track of a, while taking a copy (ui, E\) of (M, G) in Vp, we first code (TC(a) U {a}, G) in V as follows : We split A into A C A and A - A so that 0 G A, \A\ = |TC(a) U {a}\ < A, and |A — A\ = A. Take an isomorphic copy (A, E0) of (TC(a) U {a}, G) via some 7T0 with 7r0(0) = a. So for £,77 G A, ££077 iff 7r0(£) G 7r0(?7). S t e p 2. In F p , we may assume M \= u3y(p(y,a)n with \M - (TC(a) U {a})| = |A| = CJI by a simple manipulation. So we may take a copy (A, 2£) of (M, G) via some -k s.t. 7r[A = 7r0. And so E\A — E0. Since TC(a) U {a} is an G-initial segment of M, A is an .E-initial segment of A. S t e p 3 . In Vp, we now take a copy (UJI,EI) of (A,£7) via some / : ( w i , ^ ) —> (A,£7) with /(0) = 0. So {u)UE{) is well-founded and (a;i,.Ei) |= " The Axiom of Extensionality and 3yip(y,0)n. S t e p 4. Let C C f ^ + J ^ 1 be an arbitrary club. We may assume C C {N e [Hx+]Ul I 7T0 G JV -< F A + } by taking the intersection of C and the latter. Notice that for N £ C, since 7r0 G iV, a G iV. Since {AT fl A G [A]W1 I iV G C} contains a club in [A]"1, we have /i : [\}
is well-founded.
(b) (LJI,EI)
\= "The Axiom of Extensionality and
3y(p(y,0)n.
(c) / : UJ\ —> N n A is one-to-one and onto with ui C iV D A for some Now take a copy (TV fl A,E*) of (LJI,EI) (TV D A, £*) to (M, G) via, say, 7r.
via / and transitive collapse
184
TADATOSHI M I Y A M O T O
It is easy to check the following : Claim 4. (1) A D N is an E*-initial segment of N D A. (2)E0\{AnN)
=
E*\{AnN). D
Since 7r0 G N, 7r0 \(A f) N) is an isomorphism between (A D iV, P 0 [(A D N)) and ( ( T C ( a ) U { a } ) n i V , e ) . Since ((TC(a) U {a})OiV, G) is an G-initial segment of (A^,G)," ( i V ) r((TC(a)U{a})niV) collapses ((TC(a)U{a})nA^,G) onto a transitive set. Hence the composition of functions ~(N) o -K0 \(A fi A7") is the unique collapse ir\(A ON). In particular, 7r(0) = a W and so M \= u 3y
f:K—>HK.
(2) For any a G Hx+, any cardinal p, and any N s.t.
W E A K S E G M E N T S OF
185
PFA
• N is a transitive set of size A and / , ft G N. G N andVa < KVS £a N s"a G N.
• Vx,y e N {x,y} there is (j, M) s.t.
• M is a transitive set with j,N,Hp • j : N —► M is an elementary
G M and -PM C M. embedding with crit(j)
= K and
P<J(K).
• i(/)0) = a. Then there is a countable support iteration (Pa \ a < K) of proper preordes such that PK has the K-C.C. and \\-K "£(A) + 2" = (jj2 = K". For example by 1.3 note and 1.5 theorem, we have • Con(HK+ -reflecting K exists) implies Con(£(u;2) + 2^ = co2)• Con(HK++-reflecting
K exists) implies Con(£(u;3) + 2U = cu2).
• Con(iJ (2 -)+-reflecting K, exists) implies Con(E(2 u;2 ) + 2^ = LJ2). Proof. The basic idea of the proof can be found on page 102 in [Ma]. We construct a countable support iteration (Pa | a < K) together with (Qa I OL < K) by recursion on a so that (1) For all a
and \\-a"Qa
6 H^{
al
is a proper preorder".
PKeHK+. PK has the AC-C.C.
It suffices to describe what Qa names in
VPa.
C a s e 1. In V, a is a strongly inaccessible cadinal and f(a) = ((\a,<),(Bi
\i
for some ordinal A a , some P a -name <, and some sequence of P a -names (Bi I i
=
The second case is intended to collapse cardinals between ui and K. This completes the construction of the iteration. We claim that this iteration works. To see this, let GK be an arbitrary P K -generic filter over V and suppose in V[GK] that (P, < p ) is a preorder
186
TADATOSHI M I Y A M O T O
with P = A, that (Bi \ i < u)\) is a sequence of subsets of A, and that there is a proper preorder Q (of any size) with \\-VQ[G"] "there is a filter G in P s.t. for a l i i < CJI, G n 5 i ^ 0". We want to find G as such in V[GK\. To this end, we fix a P K -name
Then
(1) In both N[GK] and M[GK), (A,
KJ
and is a proper preorder with
"there is a filter G in P which hits the Bi".
Proof. We consider (2). Since we assume Hp G M for some cardinal p which is sufficiently large, we may have Q € Hp^ ^ = HP[GK] = Hp ■ • A In particular, we may assume that the power set of Q in V[GK] and M[GK] are the same, that [2'^']^ is absolute between V[GK] and M[GK], and that V[GK] and M[GK] share the exactly same stationary subsets of [2^1]^. Since the preservation of the stationary subsets of [2^1 ] w is enough to imply Q is proper (see pages 76,77, and 83 in [Sh]), we may conclude (2).
Now we define (P™ | a < J(K)) = j((Pa J(K))=K(Q* I <*<*)).
\ a < K,)) and define (Q% \ a <
W E A K SEGMENTS OF PFA
187
C l a i m 3. We may assume that (1) In M, (P™ | a < J(K)) and (Q% \ a < j{K)) are defined using j(f) in the corresponding manner as (Pa \ a < K) and (Qa | a < K) are defined using f in V ( and N). (2) For all a < K, Pa = P™ and PK = P™ hold. Proof. For (1) : Since N is gotten by collapsing some elementary substructure of Hp which reflects V well, (Pa \ a < K) and (Qa \ a < K) are defined using / in N in the same manner as in V. So by the elementarity of j , we have (1). For (2) : Since Pa <E HK and j\HK = id, so Pa = P™ for all a < K. Since we may assume K is a strongly inaccessible cardinal in M, so P^f is the direct limit of {Pa | a < K). But since M is fat, we may assume p
TDM
u Claim 4. We next force over V[GK] via P^rK\- So let G™jtK\ be a P^.ygeneric filter over V[GK] and denote the P¥Kygeneric filter GK * G
KJ(K)
0Ver
M
^
G
¥(K)'
TheTl
We
hCLVe
the
f°llowin9
in
V G
[ K\[GK3(K)\
:
(1) j : N —> M is extended to an elementary embedding j \ : N[GK] —► M[Gf{K)] such that h(GK) = Gf{K). (2) In M[G^fK\]j there is a directed subset F of ji(P)
s.t. for all i < u)\,
FnhiBi)^®. And so by the elementarity of j \ , we are done. Proof. For (1) : For any p E PK, since the support of p is bounded below ft, we have juGK C GMKy Therefore it is standard that ji — {(TGKJ(T)GMJ
I r G Np«}
works.
For (2) : We have arranged so that in M , K is a strongly inaccessible cardinal, that j(/)(/c) = ((A, < P ) , {B{ \ i < ui)), that in M[GK], (A,
□
188
TADATOSHI M I Y A M O T O
§4. T h e Strength of E(A) We give some account concerning the consistency strength of E(A). 4.1 Theorem([Go]). The following are equiconsistent : (1) There is an H^-reflecting
cardinal K.
(2) Y,(u)i) holds. Proof. A Ei-reflecting cardinal K in [Go] is called i7K-reflecting in our hierarchy of localized reflecting cardinals. BPFA in [Go] is equivalent to E(CJI) as we have seen. So this theorem is a restatement of a theorem in [Go].
□ 4.2 T h e o r e m . The followng are equiconsistent : (1) There is an H'K+ -reflecting cardinal K. (2) E(w 2 ) holds. The proof is given below. 4.3 Proposition. (1) E(a;3)+2 u; = CJ2 implies - c ^ 4- ->3u2-Aronszajn tree. (2) -eh., 4- ->^U2-Aronszajn tree implies 0#. Proof. For (1) : Since E(u#) implies - o ^ (see pages 198, 199 in [Sc]) and since E ^ ) implies -iEL;2-Aronszajn tree (see page 110 in [Ma]), we are done. For (2) : Let K = co2- Since nK holds in L, we have (AC + ) L < LJ3. The rest is the same as page 216 in [Ab].
□ 4.4 N o t e . We may consider a stronger version E(A, semiproper) of E(A) by expanding the class of preorders from proper to semiproper (more precisely, {a;i}-semiproper) preorders. By paying attention to the sizes of relevant p.o. sets, we may note from pages 57-60 in [Be] : (1) E(2 W2 , semiproper) implies 2W = cu2In particular, E(2 W2 , semiproper) iff £(2 2 W 1 , semiproper). Actually, on page 63 in [Go], it is claimed that £(u;i, semiproper) implies 2U = cu2(2) E(2 2 *, semiproper) implies NS Wl is ^ - s a t u r a t e d . And so there are inner models of set theory with many measurable cardinals (see page 430 in [Je]). (3) E(2 A , semiproper) implies AWl = Aw = A for regular cardinals A > u2([Be] and [Fo])
W E A K S E G M E N T S OF
189
PFA
We now give a proof to 4.2 theorem which is a reworking of [Go] ad justments made to fit into our context. Proof of 4.2 theorem. We need to show (2) implies (1). Let us put AC = uj2- We first mention that the following weak form of E(«) is sufficient to conclude that K is an L(K+^L-reflecting cardinal in L. • For any A0, • • •, An C K, any a G H^, and any E 0 -formula ijj, if \\-Pu3yip(y, A 0 ,- • •, An,a) holds in some transitive G-structure" for some finite iteration of c.c.c. and wi-closed preorders P , then {a < K | 3yil>(y, A0 n a, • • •, An n a, a) holds in some transitive G-structure of size o;i} is stationary in K. If 0 # exists, then every Silver indiscernible \i is an L^+)L-reflecting cardinal in L. In particular, the uncountable cardinal K, is L(K+)i. -reflecting in L. So we may assume 0# does not exist. In V, let 0 = 2U'1. By applying the covering lemma, we have A C 9 such that for any X G [ON]- W l , X G L[A]. In particular, we have • For any X G [ON]" 0 , X G L[A] and so LOX =
LJ[[A].
To show K is I/(K+)L-reflecting in L, it suffices to show the following : • For any subset X G L of K and any formula ip, if L x |= "
190
TADATOSHI M I Y A M O T O
We next force with < C J l x over V
Now going down to L[5], we set T in L[B] as follows : t G T iff
• t=
((0l\i
• 0 '
• p
with i < j < p\ /*, = /?..
Then (T, < y ) is a tree of height oj\ such that each node is contained in a cofinal branch through T and that there are at least c^i-many cofinal branches through T. Notice this remains in the bigger universe V "ie* U1 x. So by § 3 in [Go], we have a finite iteration of c.c.c. and u;i-closed preorders Q in V "le* "lx which forces the "sealing devices" (Ba | a < u\) and g : T —> UJ to this tree T. Namely, • The Ba are pairwise disjoint end-segments of cofinal branches through TwithT = [j{Ba I a
6 * < u ; i x * Q by P. This
(( Objects and Properties in Vp )) Now we choose a sufficiently large regular cardinal 6 > x
m
^P
s
° that
• T G # 5 L [ i ? ] and for any p < x, ( [ / ? H L ^ G F 5 L [ B ] . • (fl a | a < o ; 1 ) , p G i 7 ] / P . Then since A C 6 < 6 and since B,C arguments (see page 105 in [De]), we have
C ui < 8, via condensation
191
W E A K SEGMENTS OF PFA
AeL6[A]=H^[A].
.
B,TeL6[B]=H^[B].
•
• For a l H < uu B n i € L«[B n i] =
fl^[Bni].
• C € L « [ C ] = ff4L[c]. C ff* M c itf'*' C i ^ [ c > C l f j " \ so
Since H ^ . For all i <
Wl,
L«[B f1:]C L«[A] C L«[B] C L«[C] C
Hj".
• For a l i i < wi, B n i € L«[A], 4 € LS[B], and J5 e L6[C\. Notice that for every i < uiu LS[B Hi] = (L[B n i]) L «M, L ^ ] (L[A])L'W,
LS[B] = {L[B])L^C\
L e t M = if]/
=
and LS[C] = ( L [ C ] ) < " .
and we list the following rather ad hoc properties in Vp
(1) M is a transitive set and u)\, AC, 0, x, A, B,C,X
G M.
(2) X C «, X G L x , and L x f= ' V ( X ) " . (3) A C 0, J3 C wi, C C u)U ux < K < 6 < x < S = MOON, and 6 is a limit ordinal. Once again, (4) A G L6[A], B G LS[B], C G LS[C\, Bni I*[C] c M ( i < w i ) . Since CJI = c j j L so w e h a v e /^
, Ls[A) _
J
L6[B] _
= cjf
, so CJI = c;1 L
L 6 [C] _
G LS[BM\ J
C LS[A] C LS[B] C
= CJ1 L J . Since CJ 1 L
J
= CJ 1
l
J
,
M _ ,,
Since L[B] \= "|0| = wi and x remains a cardinal", L[C] |= "|x| = wi", 0 <x<S,L6[B] = Hs[B\ and L6[C] = tff[C], we have (6) L6[B] \= "|0| = «ji and x is a cardinal" and L6[C] |= "|xl = " i " . Since 5 C w b we have L[B] \= " for any cardinal /x with cf(/x) = CJ, |[/x]^| = /x + " via condensation. But since 6 has chosen so that L[B] \= " for every cardinal \i < x, [M]W £ W a n d t n a t £<s[^] = H$ > so we nave (7) LS[B] \= "V/x < x if M is a cardinal with cf(/x) = CJ, then |[/x]w| = /x+". By the assumption on y4 and the any limit ordinal (3 < x, as |x| = u\ or cj!, so either there is h G L[A] s.t. h G L[C] s.t. h : wi —> (5 increasing, Therefore, we certainly have
fact that
(13) \fp<xmu)Li[B]
([^] a ; ) L [ B ] = ([PY)H^B\
and
([/3]W)L[C1 -
= mu)Li[C]-
And finally, since {Ba \ a < u\), g G M = Hj
, we have
(14) There is T in L6[B] and {Ba \ a < CJI), #, and {{{Of \ i < UJ{), (/?• | i < j < CJI)) | o: < LJI) in M such that • T = {{{0i \i
< to, ({Of | i < /?), (/?■ | t < j < /?)) G T.
• For all a < UJU there is p < UJX s.t. Ba = {({Of \ i < 0), (/g | i < 3 < P)) \ P < P < u\} (i.e., £ a is an end-segment of the cofinal branch {((Of \i
\a<(j1}.
• g : T —► CJ and for any t,s € T with t < 5 in T, if g(t) = g(s), then there is a < ui with t,s € Ba. Since every cofinal branch through T in V p has the direct limit which is a well-order,
W E A K S E G M E N T S OF
193
PFA
• Va < CJI37 < 6 s.t. 7 is the direct limit of ((Of | i < ui), (/£•
\i<j<
S t e p 2. Let $(a;i,/c,X,
r\a,tp) says : 7
For each a G 5 , we have M ' , 6', A , £ ' , C", 0', x' G HK such that (1)' M ' i s a t r a n s i t i v e s e t j C J i j a ^ ' j x ' j A ' j S ' j C A T l a
G M ' , and so | M ' | =
CJi.
(2)' XnaCa,Xna<E
Lx>, Lx> |= V ( X n o ) " .
(3)' A' C 0', B' C CJI, C" C cjx, CJI < a < 0' < X' < S' = M ' n O N , 6' < K is a limit ordinal. (4)' A' G M A ' ] , £ ' G L6r[B'l C G L t f /[C], and for a l i i < uu L 5/ [£' fl t] C Ltf/ [A7] C Ls. [B'} C L*< [C] C M ' . (5)' o ^ ' ' = wf"[*'] = wf " ^
= < ' =
B' n i G
Wl.
(6)' M # ' ] \= "\6'\ = LJ! and x' is a cardinal" and L6,[C] \= "|x'l = wi". (7)' Lgi[B'] \= " For any \i < x', if M i s
a
cardinal with cf(/x) = a;, then
(8)' For any limit ordinal ft < x', either (a) there is /i G L^/ [A'] s.t. /i : w —> (3 is increasing and cofinal, or (b) there is h G Ls'[C] s.t. h : CJI —► (3 is increasing, cofinal, and for all i < u\, h \i G L^/[A ; ]. (9)' M C ] (= "V 7 GON3y C 0 N 3 ^ = L 7 [T/]". (10)'
(11)' (12)'
M # ' ] t= "VA G ON V£ GON V«0< I i < fi),(hn_ \ t < V < P)) Vh: P + l —> fjL + 1 increasing 3y y = ((0h{i) \i < P), (fh(i)h(j) I * < 3
(14)'
There are T' G M ^ ' ] , K h < ^i> € M 1 , ( ( ( C | t < ^ ) , ( / ^ | i < j < CJI)) I 7/ < LJi) G M ' , ^ G M ' such that • V = {((Oi I i < (3),(fij \i < 3 < P)) G L ^ [ 5 ' ] I (1) 0 < ^ ; (2) Vz < P,6i < x1 ; (3) Vi, j with i < 3 < P, fij : ^ —-> ^ is an order-preserving map ; (4) commutative }.
194
TADATOSHI M I Y A M O T O
• Viy < CJXV/3 < wi
{(O?\i<0),(f%\i<3
• V77 < u^P < co, s.t. B'v = { ( ( C Ui}.
\i
• For any 77,77' < <JI, if 77 ^ 77', then B'^ n £^, = 0.
• r = U{^U
g'(s),
• V77 < u;i37 < 6' s.t. 7 is the direct limit of ((6'*1 | z < u>i), (/■? | z < J<wi». S t e p 3 . It suffices to show, by (2)', that x! 1S a cardinal in L[B']. We establish this by comparing L^[B') and L[-B'] as follows : L e m m a 1. (1) For any P < X', (cfP = u)L[B']
iff (cfP = LJ)LS'[B'] iff
cfp = u. (2) For any p < x ' , ([P]U)L[B'] = {W)LAB'](3) For any \i < x', if cf'6'^ ^(/x) = u and fi is a cardinal in L[B'], (Mu\=H+)LlB'] And so,
then
and(\W\=V+)I"'[B']-
(4) For any fi < x'> if cf^M — w
an
d ^ is a cardinal in L[B'],
then
Proof. For (1) : This follows from (4)' and (8)'. For (2) : It suffices to show ([P]")LW C ( [ / ? H L ^ ' l Fix y G ([0\U)L^ arbitrarily. Since p < x' < & < «, B' C UJU and Lx> G LK[B'] = HK[B'\ there is 7 < K s.t. y G Ly[B'] and Lx> C L 7 [ £ ' ] . So in V, we may fix a continuously increasing sequence (Xi \ i < uj\) of countable elementary substructures of L^B1] s.t. y U {y} C XQ. Since Lx> C Ly[B'], so (XiC\Lx> \ i < LJ\) continously increases to Lx>. So by (12)', we may find i < ui s.t. X{ fl Lx> = Y{ and X ; fl CJI = z. And so ?/ C Y{. Let 7r : Xi —► L^[B] be the transitive collapse. Then since Lx< is transitive, ir\Yi = TT*. By (4)', £ = TT"(X; n £ ' ) = 7r"(£' fl z) = B' n z G Lffi[JB']. Since |X<| = w, so 7 < LJX < 8'. By (4)' again, LS,[B'] C L ^ [ C ] and so B G £*<[[C] and so ir(y) G Ltfi[C"]. By (12)', y = {( G Y{ \ T T ^ ) G ic(y)} G L^[C']. Since |z/| = LJ in L[£'], so in V and so L ^ C ' l N "M = <*>" by (11)'. But by (13)', we conclude
yemu)L''lB']. For (3) : Suppose \i < x ' , ciLi'^B \fi) — u) and fi is a cardinal in L[S']. By (7)', we have (\\n\u\ = fi+)L^B'l Since cf L I B '](^) = w, L[B'] f= "IH W | > M+"- Since B ' C W l = w^[B'] by (5)', we have L[B'] \= "\[n]w\ = fi+" by a condensation argument.
195
W E A K SEGMENTS O F P F A
For (4) : By (1), c f L * ' ^ ( / x ) = <j. So by (3), (|[p] w | = p+)LlB'l and (Mu\ = ^)LAB>]But by (2), ([^)Ls'{B'} = ([nY)L[B']. T h u s i n L[Bi^ we have /x+ = \[fi)"\ = |(/x+) L <'^']| and so ( M + ) J W < ( M +) L «'[ fl 'l. But {^)L[B')
> ( M +)L,i[B']
h o l d s
trivialiy>
□ L e m m a 2. (%) For any cardinal xx < x' in L[B'], we have H^B
*C
(2) For any (3 < X', cffi = W l iff 3h G L6.[C] s.t. h : ux —► 0 is increasing, cofinal, and for each i < uj\, h\i G L$f[B'\. (3) For any cofinal branch B through T', there is n < uj\ and 7 < 8' s.t. B'^CB and 7 is the direct limit of B. (4) For any cardinal \i < x' i>n L[B'\ with cf \i — u)\ and any 7 with M < 7 < (ft+)L^B h there is a cofinal branch B through T' s.t. 7 is the direct limit ofB, so 7 < 8'. And so (fi+)L^ < 8'. Proof. For (1) : Fix a cardinal /x < x' H^
= L^B']
LJI = ool
G Ls'[B'].
m
L[B'].
So we may assume LJ1 *
If /x = u, then < /x. But since
and B' C uj\, we may consider in two cases. r rD'l
C a s e 1. n = cui : For any x G H^L , since B' C CJI, we have z,j < CJI s.t. x G Lj[B' fl z] via a condensation argument. And so x G L Wl [ 5 ' Hi] C
M # ' ] by (3y and (4)'. C a s e 2. LJI < fi : For any x G -fl^ , since /x is a cardinal in L[B'}, we may find j < /x s.t. x G £j[-B/] via a condensation argument in L[B'). Since /x < x' < ^', we have x G £ M [#'] C L x #[5'] C L$/[Z?']. Thus in this case, we actually have H^} J = L^[B'\. For (2) : This is immediate from (4)' and (8)'. For (3) : Suppose B is a cofinal branch through V. By (14)', we have g' : V —► u. Since \B\ = ui, there are n < u and £ ' G [£] W l s.t. for all t G £ ' #'(£) = n. Therefore for all t,s G B' with £ < 5 in T', there is 77 < u)\ s.t. t , s G B'. But since (£?^ | rj < u\) is disjoint, there is rj < u)\ s.t. B' C JB^. Since B is a cofinal branch and B^ is an end-segment of a cofinal branch, B'^ C B holds. Now by (14)' again, we may get 7 < 8' such that 7 is the direct limit of the cofinal branch B. For (4) : Fix a cardinal /x < x' m £ [ # ' ] with cfy/x = CJI and fix 7 with /x < 7 < (/x + ) L t B/ l. Since I7I = /x in £ [ # ' ] , we have a one-to-one and onto map / G L[B'] s.t. / : /x —► 7. For each £ < fi, let 0$ be the order type of / " £ and let /^ : 0£ —► / " £ be the isomorphism. For each f < 77 < /x, let / ^ = Z " 1 o £ . Then 0C < xx (< x')> / ^ : ^ —> ^ is an order-preserving map, and ( / ^ | £ < 77 < /x) are commutative. And 7 is the
196
TADATOSHI M I Y A M O T O
direct limit of this system. By (2), there is h G Ls'[C] s.t. h : u\ —► \x is increasing, cofinal and for each i < a>i, h\i G L$>[B']. For each i < j < CJI, define 0; = 0/^) and fij = fh(i)h(j)- We claim that for each 0 < UJ\, (0* I * < /?}, (/;; I t < j < /?) € M # ' ] and s o 5 = { ( $ \ i < 0), (/^ | t < j < /3)) | /3 < u;i} is a cofinal branch through T". Since /i is cofinal in //, 7 is the direct limit of B. And so by (3), 7 < 6' holds. Now to see (Oi \i < 0), (fij \i < j < 0) e LS.[B'] for each 0 < LJU let £ = h{0) < < X' < S1- Since (0* U < A), p, + 1 is in L6>[£'], so by (10)', we have ((0* | i <
M
P), (fij \i<3<
P)) = ((0h(i) \i<0),
(fh(i)hU) \*<J<0))£
L
AB'Y □
L e m m a 3 . (1) Ls>[B'] \= "x' ™
a
cardinal",
l
(2) For any cardinal fi < x' in L[B ], (/i + ) L t B 'l < 6' holds. And so, (3) For any cardinal \x < x' in L[B'], cardinal in L[B'] and so in L.
(/i + ) L [ B ' ] < x' holds. So x' is &
Proof. For (1) : This is immediate from (6)'. For (2) : Fix a cardinal \i < x' m L[B']. Since \i < x' < & < « = ^2 , we have either cfy/x = u or civ(i = u\. But by lemma 1 and 2, we have (fi+)LW < 6'. For (3) : If fi = u, then (fi+)L^ = ux < x'■ So assume u[[B'] = UJX < + L B \x. For any 7 with fi < 7 < (fi ) ^ J, there is a one-to-one and onto map / : ji —► 7 with / £ L[B']. But by a condensation argument in L[B'], since B' Cuu we may find j < (fi+)L^ such that / G Lj[B']. So / G Z ^ [ £ ' ] by (2) and so 7 is not a cardinal in L6>[B']. Thus by (1), {ix+)L^B'^ < x' must hold. D D
References [Ab] U. Abraham, Aronszajn trees on H2 and K3, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 213-230. [Be] M. Bekkali, Topics in Set Theory, Lecture Notes in Mathematics, Vol. 1476, Springer-Verlag, 1991. [Bu] D. Burke, Generic Embeddings and the failure of Box, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2867-2871. [De] K. Devlin, Constructibility, Springer-Verlag, 1984.
Perspectives in Mathematical Logic,
W E A K S E G M E N T S OF
PFA
197
[Fo] M. Foreman, M. Magidor, and S. Shelah, Martin's Maximum, sat urated ideals, and non-regular ultrafilters. Part I, Annals of Mathe matics, vol. 127 (1988), pp. 1-47. [Go] M. Goldstern and S. Shelah, The Bounded Proper Forcing Axiom, Journal of Symbolic Logic, vol. 60 (1995), pp. 58-73. [Je] T. Jech, Set Theory, New York, Academic Press, 1978. [Ku] K. Kunen, Set Theory, An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, North-Holland, 1980. [Ma] A. Mathias, Surveys in Set Theory, London Mathematical Society Lecture Note Series 87, Cambridge University Press, 1983. [Mi] T. Miyamoto, Localized Reflecting Cardinals and Weak Segments of PFA, a typed note, 1996. [Sc] A. Schimmerling, Combinatorial Principles in the Core Model for One Woodin Cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153-201. [Sh] S. Shelah, Proper Forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, 1982.
Department of Mathematics Nanzan University 18, Yamazato-cho, Showa-ku Nagoya, 466, Japan e-mail : [email protected]
199
ON STRUCTURAL INFERENCE RULES FOR GENTZEN-STYLE NATURAL DEDUCTION, PART I Koji Nakatogawa Takeshi Ueno Department of Philosophy, Department of Mathematics, Hokkaido University, Sapporo, Japan e-mail: [email protected] [email protected] Abstract For a system of Gentzen-style natural deduction, we present characteri zations of two structural rules, namely, weakening and contraction, in terms of the numbers of assumptions discharged at an application of introduction rule for logical implication. It is shown that the number is zero for the weak ening rule, while the number is reduced to exactly two for the contraction rule.
1
Introduction
The purpose of the present paper is to characterize two structural inference rules, namely, weakening (Theorem 2) and contraction (Theorem 3), for a system D (defined below in section 3) of Gentzen-style natural deduction. Our characterizations of these two rules are given in terms of the number of assumptions discharged at an application of the introduction rule for implication. For the weakening rule, the number is zero, while it is reduced to exactly two for the contraction rule. We start with introducing a propositional language equipped with the logical connectives, implication D, conjunction A, disjunction V, and negation ->. Then, the system D is defined as a system without weaken ing, contraction, and exchange rules. (0ide[9] introduced sequent calculus systems where the number of formulas allowed to occur on each sides of a
200
K O J I NAKATOGAWA
sequent is restricted to zero or one. This system provided us of motivations and stimulations leading to the definition of D.) In this base system D, the elimination rule for implication is the ordinary modus ponens. For the in troduction rule of implication , it is customary not to put any restriction on the number of discharged assumptions. In our £), however, the number of assumptions discharged at each application of the introduction rule D I1 for implication is restricted to exactly one. It will turn out that if an introduction rule D 1° for implication, where the number of discharged assumptions is restricted to zero, is added to D as a new rule, it works as the weakening rule. (See Theorem 2 below) If the introduction rules D In(n > 2), where the number of discharged assumptions is any positive integer strictly greater that or equal to two, are added to D, they work as the contraction rule.(See Theorem 3 below) For the contraction rule, the number of discharged assumptions can be reduced to two (Lemma 4). Inference rules for conjunction, disjunction, and negation are also formu lated by using the standard symbols A , V , -> for those logical connectives . However, due to the restrictions on introduction rules, the inference rules for these logical connectives have to be formulated in such a way that the "con texts" in a proof can be taken into consideration. Thus, the substitutions of D with —o, A with multiplicative conjunction
2
Language L
The propositional connectives of our basic language L are the implication symbol D, the negation symbol -■, the conjunction symbol A, and the disjunction symbol V. We also use the constant symbols 1 (one) to denote the unit element for the conjunction, _L to denote the unit element for the disjunction. The formulas of L are constructed by combining the constant symbols 1 , J_ and the propositional constant symbols, A,J9,C, •••
STRUCTURAL INFERENCE RULES
201
together with the connectives D, ->, A, V, ( (left parenthesis), ) (right parenthesis). Some lower case Greek alphabets 0, I/J , 0, • • • denote a formula. We use capital letters such as T, A, \&, $, • • • to represent a multi-sets of formulas. In what follows we usually omit "multi-" when it is clear from the context. The sequent of L have the following form
r-> A . The left hand side of a sequent may be empty. The right hand side of a sequent is either empty or consists of a single formula. To specify the element of T and A we write 7 o r - - > 7 m - i -> 6. When both sides of a sequent are empty, we write
and call it the empty sequent.
3
Natural Deduction System D
We now introduce a formal system D of natural deductions which is obtained by restricting and modifying Gentzen's natural deduction system NK. Our system D has the following inference rules. The non-negative integer n used as the superscript of I, E indicates that the number of assumptions cancelled at the application of an inference rule is restricted to exactly n.
The introduction of D with the number of cancelled assumptions being restricted to exactly one
[A]
ADB The elimination of O A
ADB
„
202
K O J I NAKATOGAWA
The introduction of -> with the number of cancelled assumptions being restricted to one [A]
i A * The elimination of A
^A _L
The introduction of A B
AAB
M
The elimination of A with the number of the cancelled assumptions being restricted to one for A, and one for B respectively [A] [B] AAB
C C
AE1
The 1 on the left side of the superscript of E indicates that the A used as an assumption is cancelled just once at the application of A.E 1,1 . The 1 on the right side of the superscript of E indicates the similar restriction for the assumption B. The first introduction of V with the number of cancelled assumptions being restricted to one
B VI\ Ay B The second introduction of V with the the number of cancelled assumptions being restricted to one
hB]
STRUCTURAL INFERENCE RULES
203
The above two rules are modifications of the introduction rules of the multiplicative disjunction to suite to the present circumstances . Instead of trying to give the meanings to these rules within an already available framework, their meanings are to be sought in the actual use of these rules themselves. The first elimination of V Ay B
-IA
B
VEi
The second elimination of V -.£ A V B — 4 — v *
„ 2
The above two rules are adopted from the elimination rules of the multiplicative disjunction to suite to the restrictions under which we have been working. They may be regarded as expressing a sense of logical incompatibility. The introduction of D for the propositional constant 1 where the number of cancelled assumptions is restricted to zero
A
IDA
DI°(1)
The superscript 0 of / indicates that the number of cancelled assumptions at the application of D 1° is restricted to zero. The symbol 1, regarded as a formula, is the axiom of D. There are other inference rules not belonging to D.
The introduction of D with the number of cancelled assumptions being restricted to zero
ADB The superscript 0 of I indicates that the number of cancelled assumptions at the application of D 1° is restricted to zero.
204
K O J I NAKATOGAWA
The introduction of D with the number of cancelled assumptions being restricted to n(n > 2) n times [A]--[A] B
D/n
A D B
The superscript n of I indicates that the number of cancelled assumptions at the application of D In is restricted to two or more than two.
4
Sequent Calculus S
Next, we introduce the sequent calculus S as follows. We say that (Ti —» Xi, T 2 —> X 2 , • • •, T n —> Xn I r —> X ) is an instance of a certain inference rule if it has the form indicated by the corresponding figure. If (Ti —► X i , T 2 —> X 2 , ■ • • , T n —> X n / T —> X ) is an instance of an inference rule a , we call I \ —► X; the i-th upper sequent of a , and T —► X the lower sequent of a.
The sequent ^4 —► ^4 is called an axiom. The sequent —>1 is called the unit axiom of A. The sequent _L —► is called the unit axiom of V. The left rule of D
Dleft
r,ADB,±,-+c The right rule of D A,T-^B T-+ AD
B D right
The left rule of
.A,r-
-left
205
STRUCTURAL INFERENCE RULES
The right rule of -• aright
T->^A
The right hand side upper sequent of aright
have to be empty.
The left rule of A A left
AAB,T^C The right rule of A
T^A A-> B A right I\A-> AAB The first left rule of V A,r->
B,A->C
Av5,r,A.c
7
,
vleftl
The right hand side of first upper sequent of V/e/ti has to be empty. The second left rule of V A,r-+C £,A i V 5 T , A ^ C
V left2
The right hand side of second upper sequent of Mleft2 empty.
has to be
The first right rule of V This and the next rules are adapted from the introduction rule of multiplicative disjunction. Our main concern here is neither the subformula property nor the nomarization of proof procedures. We regard these rules as expressing a certain meaning of logical compatibility in its own forms. ->A,T^B —
— V riqhU
The second right rule of V ->B,T-*A . ^ — — V riqnto y r - > AM B
206
KOJI
NAKATOGAWA
The left rule of 1
r^4 i,r->4
Heft
The right rule of _L
r -► j _
■Lr%9ht
The right hand side of upper sequent of ±right
have to be empty.
The left exchange rule
ex.left
T,B,A,A^C The cut rule T->A
A,A-+C -cut
r,A-+c
There are other structural inference rules not belong to S.
The left contraction rule A,A,T^C c. left A,T->C The left weakening rule
A,T-+C
5
w. left
Provability
Our notations about provability in D and S ( and in systems resulting from them by certain enlargements) are fairly standard. Thorough definitions will be found in Gentzen [1], Prawitz [2], and Troelstra [4]. A deduction E in D is either a single formula or a ("tree-like") figure constructed by applying inference rules of D finitely many times, ("in D" is often deleted if it is clear from the context.) When (j> is the formula located at the bottom ("root") of a deduction E , we say that E is a deduction of c/> in D.
STRUCTURAL INFERENCE RULES
207
We distinguish proofs from deductions. A proof of 0 in D is a deduction where all the assumptions of E are cancelled. (/> is provable in D if we can construct a proof of (j) in D. When E is a deduction of 0 in D, and r is the set of uncancelled assumptions in E , we say E is a deduction of 4> from T in D. In this case, following the common practice, we often say that E is a " proof of (j) from T in Z), even if some assumptions T of 0 are remaining uncancelled in the deduction E . If the context clearly tells which E is meant, we use the notation, T \- (f) in D. Since the formula 1 is an axiom of D, the formula 1 appearing in E is regarded as being cancelled. Thus, one might feel inclined to think that 1 is not listed in the assumptions T of E. But we explicitly display 1 among the assumptions r whenever 1 appears in E. A formula <j> is provable from V in D if we can write a deduction E of (f) from r in D. Provability for a sequent is also defined in a usual way. A proof II in S is a list of sequents defined inductively as follows. 1. A sequent axioms A —> A, —> 1, J_ —> are a proof in S. 2. If II is a proof in 5, then the figure obtained from II by applying an inference rule of S to the last sequent of II is also a proof in S. 3. The proofs in S are only those obtained by applying of (1) and (2) finitely many times. Let a be an inference rule or axiom. Let X be a formal system. X U {a} means a new enlarged system resulting from the addition of an inference rule or axiom to a system X . For example, D U {D I0} is a system obtained from D by adding a new inference rule D 1°. The notions and phrases about provability carry over to those enlarged systems.
6
Characterization of Structural Inference Rules
In Gentzen [1], structural inference rules are formulated in classical and intuitionistic sequent calculuses called LK and LJ, respectively . In the corresponding Gentzen-style systems of natural deductions, namely NK and NJ in Gentzen [1], there is no structural inference rules. The powers of the structural inference rules in sequent calculuses are distributed unevenly over the logical inference rules of natural deduction systems, and can not easily be detected. Our strategy here is then, first, to introduce a system S of sequent calculus which is small enough not to include any structural inference rules and any effect resulting from their presence, but, large enough to capture basic characteristics of logical connectives . (The idea of removing and
208
K O J I NAKATOGAWA
restricting structural inference rules in sequent calculuses is found in Oide [9].) Second, we carefully construct a natural deduction system, here called D, in such a way that the provability in D is the same as £, so that D is free of the effect of structural inference rules. Third, the effect of structural rules such as weakening and contraction will be captured in the corresponding axioms, and then the equivalence of the systems enlarged by these axioms are proved. In this way, the effects stemming from structural rules in sequent calculuses are isolated and added to natural deduction systems in such a way that these two systems posses the equal proving capabilities. Equivalence proof of S and D is obtained by adapting to our situation, the standard equivalence proof of LK and NK , which is carried out in detail in Gentzen [1]. Since the lack of structural inference rules require some care, we present the inductive proof of the equivalence of S and D. Lemma 1 Let (j) be a formula of the language L. Let V be a set of formulas of L. Then, is provable from T in D if and only if the sequent T —► <j> is provable in S. Proof First, we assume that 0 is provable from V in D. Then, we can construct a deduction of 0 from r in D Let A, Band C be formulas of L. Let E be a deduction of from r in D. " We use induction on the number |jE of the application of inference rules in E . If the number of the application of inference rules in E is zero, then E consisted of only one formula (j). So we make a proof in S as follows .
We assume the result for the case where jjE is less than n, and prove it for n. C a s e 1 The bottom inference rule in E is D I1 AD B.
and the form of (j) is
By induction hypothesis, the sequent A, V —► B can be proved in S. So we deduce the sequent V —> A D B by D right rule. C a s e 2 The bottom inference rule in E is D E and (j) is B. By induction hypothesis, the sequent Ti —> A can be proved in S (Ti U T 2 = T) .
and T2 —> A D B
We deduce the sequent i D 5 , i - > 5 from A —► A and B —► B by D left rule. Then we deduce T 2 , A —> B from A D B,A Ti,T2^B
—> B and T2 —> A D B by cut rule. So we deduce from I \ -> A and Y2,A-^B by cut rule.
209
STRUCTURAL INFERENCE RULES
C a s e 3 The cases where the bottom inference rules in E are -"7 1 , -*E , Al are omitted. C a s e 4 The bottom inference rule in E is AE11
and 0 is C.
By induction hypothesis, the sequent Ti —> AAB can be proved in S ( I \ U 1^ = Y) .
and A , B , T 2 —> C
We deduce the sequent A AB,T2 -> C from A, B,T2 —> C by A/e/t rule. Then we deduce 1^, T 2 —> C from I \ - > i A 5 and i A 5 , r 2 - ^ C by CM* rule. C a s e 5 The bottom inference rule in E is V/J- and 0 is C. By induction hypothesis, the sequent -*A, T —> B can be proved in S . Then we deduce the sequent Wrighti rule.
r -> A V 5
from
->A,T -+ B
by
C a s e 6 The cases where the bottom inference rules in E are Wl2, V£?i, \/E2 are omitted. C a s e 7 The bottom inference rule in E is D /°(1) and <j> is 1 D A. By induction hypothesis, the sequent T —> A can be proved in 5 . We deduce the sequent 1,T-> A by 1/e/t rule. Then we deduce the sequent T -^ 1 D A from 1,T —> A by D right rule. Next we assume that we can prove a sequent T —> 0 in 5. r —> in 5 is defined to correspond to T h ± in D. Let II be proof of the sequent T —> 0 in 5. We use induction on the number of the applications of inference rules in S. This number is denoted by the notation jjll . If JII is zero, then II consists of one of the axioms, A —> A, —> 1, J_ —>. These proofs in £ correspond to deductions or proof; A, 1, J_ in D respectively. We assume the result for the cases where (tn is less than n , and prove it for n. C a s e 1 the bottom inference rule in II is D right rule and the form of T ^ 0 is T -^ AD B. By induction hypothesis, the formula B is provable from r u { i } in D. So we deduce the formula A D B from B by applying the inference rule D I1. Then A D B is provable from T in Z>.
210
K O J I NAKATOGAWA
C a s e 2 The bottom inference rule in II is D left rule and the form of r - > 0 is TUAD B,T2 -*C. By induction hypothesis, the formula A is provable from Ti in D and the formula C is provable from T2 U {B} in D. So we deduce B from A and A D B by D £ rule. Then C is provable from r i U r 2 U { A D B} in £>. C a s e 3 The cases where the bottom inference rules in II are -
aright,
C a s e 4 The bottom inference rule in II is V/e/ti rule and the form of
r-> is AvB,r1,r2
->c .
By induction hypothesis, the formula J_ is provable from Ti U {^4} in D and the formula C is provable from Y2 U {B} in D. We deduce -yA from JL by applying ->IX rule. So we deduce B from ->A and A\/ B by applying VJ5?i rule. Then C is provable from Ti U T 2 U {A V £ } in D. C a s e 5 The bottom inference rule in II is lie ft rule and the form of T —► (j) is l , T i -► A . By induction hypothesis, the formula A is provable from I \ in D. We deduce IDA from A by applying D 7°(1) rule. So we deduce yl from 1 and 1 D A by applying D E rule. Then A is provable from Ti U {1} in D. C a s e 6 The cases where the bottom inference rules in II are V/e/t 2 , .Lright, ex./e/t, cut are omitted. This completes the proof of Lemma 1. To prove the following two theorems, we do not have to introduce no tions such as labeling for the formulas appearing in a proof and the antilexicographic ordering on the labels . However, for an inductive argument to provide a concrete guide in re-writing a given proof to obtain another proof possessing an intended properties, we need labels and anti-lexicographic or dering. (Our definitions of these notions are not given for the general cases, but, limited to the cases relevant to our proofs.) Let R be the set of all finite sequences consisting of zero's and one's. This R is a countably infinite set and is called the set of labels. If two sequences 5 and t in R are distinct form each other, there must be the first place, namely the z-th terms s; and t{, where the two sequences take different values for the first time, so that si / t{. If Si < ti ( in the order of the usual magnitude of natural numbers), we put s >R t. If si > t{, we set 5
STRUCTURAL INFERENCE RULES
211
if and only if there is some i in N (the set of natural numbers including zero) such that S{ < t{ and that Sj = tj for any j < i, where < is the usual order of magnitude among the natural numbers. It is not difficult to see that this
212
K O J I NAKATOGAWA
The set P of all formulas appearing in £ are ordered by the antilexicographic ordering
Now, we consider a deduction E of qi inside £ , which has q\ as its last formula. By the leastness of q\ , E contains no application of D 1° except at the very bottom. Then E can be re-written to E' in the following way, so that E' does not contain any application of D 1° at all.
B D J° QI=ADB AD B The above deduction E can be re-written as follows.
BD(ADB)
-DE B Let £ ' the deduction resulting from £ by replacing E with E'. We then notice that the number of applications of D 1° in E' is strictly less than n (in fact n — 1). By our induction hypothesis, we obtain the proof Eo of 0 having no application of D 1°. AD
(2) =» (1) Let E be a deduction of <> / in D U {axiom X D (Y D X)}. We use induction on the number of the axioms of X D (Y D X) in E. If E contain no axiom of X D (Y D X) in E , then we are done. We assume the result for the case where the number of the axioms of X D (Y D X) in E is less than n, and we prove it for n. The set P of all formulas appearing in E are ordered by the antilexicographic ordering
A2i AD(BDA)
3/0
D/x
STRUCTURAL INFERENCE RULES
213
where q\ is A D (B D A) . Let E' be the deduction resulting from E by replacing the axiom qi by the above proof. We then notice that the number of the axioms X D (Y D X) in E' is strictly less than n (in fact n-1 ). By our induction hypothesis, we obtain the deduction E 0 of cj> having no axioms of X D (Y D X) . (2) « . (3) This is an immediate consequence of Lemma 1, together with the fact that we can prove the axiom of X D (Y D X) if and only if we can prove the sequent axiom of —> X D (Y D X). (3) =* (4) Let II be a proof of V —> 4> in S U {sequent axiom —> X D (Y D l ) } . We use induction on the number of the sequent axioms of -> I D ( F D *
)
■
If II contain no sequent axiom of —► X D (Y D X ) , then we are done. We assume the result for the case where the number of the sequent axioms of —> X D (Y D X ) in II is less than n , and we prove it for n. The set T of all sequents in II are ordered by the anti-lexicographic ordering < # , which we define earlier. Let U be the set of sequent axioms of —► X D (Y D X) in II. Then, U , as a subset of T, is ordered by
The axiom follows.
u\
can be proved in S U { the left weakening rule } as
A
.
A
A-+ A
B,A^ A A-^BD A 'ADIB'DA)
w.left D right D
right
Let IT be the proof resulting from II by replacing ui by its above proof. We then notice that the number of the sequent axioms —> X D (Y D X) in II is strictly less than n (in fact n — 1). By our induction hypothesis, we obtain the proof n 0 of T —► (f) having no sequent axiom —> X D (Y D X). (4) => (3)
214
K O J I NAKATOGAWA
Let II be a proof of T —* <j> in S U {the left weakening rule}. We use induction on the number of the applications of w.left in II. If II contain no application of w.left, then we are done. We assume the result for the case where the number of the applications of w.left rule in II is less than n , and we prove it for n. The set T of all sequents in II are ordered by the anti-lexicographic ordering < # , which we define earlier. Let U be the set of sequents which is the lower sequent of w.left rule in II. U , as a subset of T, is ordered by
un.
Now, we consider a proof 0 of u\ inside II , which has u\ as its last sequent. By the leastness of wi , 0 contains no application of w.left rule except at the very bottom. Then 0 can be re-written to 0 ' in the following way so that 0 ' does not contain any application of w.left rule at all. 0 i4,ri,-c w.left B,A,Tl -+C The above proof 0
ui = B, A,Yi -> A
can be re-written as follows. ■B
A^A
A
BDA,B^..
AD(BDA),A,B^A
[
0
cut
A,B^A
A,fl,ri - > c B1A,Tl
Dleft J eJt
^C
~A,T~i-*C cut
ex.left
where \ is a formula A D (B D A). When the set {A} U Ti of formulas is empty, we use 1 as the cut formula instead. This proof is as follows.
lD(BDl)
B^B 1->1 1 BD1,B^1 1D(5D1),5^1 — B->1 B^C
D left
-".%: Dr left cut
Ul
0 ^C 1 -> C
cut
We then notice that the number of applications of w.left in II' is strictly less than n (in fact n — 1). By our induction hypothesis, we obtain the proof n 0 of T —> (j) having no application of w.left. This completes the proof of Theorem 2.
STRUCTURAL INFERENCE RULES
215
T h e o r e m 3 ( C h a r a c t e r i z a t i o n of c o n t r a c t i o n ) Let (f), X and Y be formulas of L. Let Y be a set of formula of L. Then the following four statements are logically equivalent. (1) We can prove (j) from
T in D U {D In(n
> 2)}.
(2) We can prove (/> from T in D U {axiom (X D {X D Y)) D (X D Y)}. (3) We can prove a sequent V —► (j) in S U {sequent axiom -+ (X D (X D Y)) D {X D Y)}. (4) We can prove a sequent T —>
if and only if we can prove (f)
Proof of Theorem 3 (1) =» (2) We assume Lemma 4. By Lemma 4, we can prove (j> from T in D U {D I2} if and only if we can prove (j) from T in D U {D In(n > 2)}. So, it is sufficient to show that (f> from T in D U {D I2} can be proved if and only if (p can be proved from r in D U {axiom(,4 D (A D B)) D (A D B)} . Let E be a deduction of 0 in D U {D I2}. We use induction on the number of the applications of D I2 in E. If E contain no application of D I2 , then we are done. We assume the result for the case where the number of the applications of D I2 in E is less than n, and we prove it for n. The set P of all formulas appearing in E are ordered by the antilexicographic ordering
Now, we consider a deduction E of q\ inside E , which has q\ as its last formula. By the leastness of qi , S contains no application of D I2 except at the very bottom. Then S can be re-written to E' in the following way, so that E' does not contain any application of D I2 at all.
216
K O J I NAKATOGAWA
[A]JA]
innrDl2
^ = ADB
The above proof E can be re-writ ten as follows.
\A][A] n ADB
Dl Dl1 1 U
AD{ADB)
(AD(ADB))
D(ADB) D E
ADB
Let E' be the poof resulting from E by replacing E with E'. We then notice that the number of applications of D I2 in E' is strictly less than n (in fact n — 1). By our induction hypothesis, we obtain the deduction E 0 of (f) having no application of D I 2 . (2) => (1) Let A and B be formulas of L. Let E be a deduction of 0 in DU{axiom (X D (X D Y)) D (X D Y)}. We use introduction on the number of the axioms of (X D (X D F ) ) D ( I D 7 ) in E. If E contain no axiom of (X D (X D y ) ) D ( I D 7 ) in E , then we are done. We assume the result for the case where the number of the axioms of (X D (X D y ) ) D (X D y ) in E is less than n, and we prove it for n. The set P of all formulas appearing in E are ordered by the antilexicographic ordering
A
^
->
&
D I2 A D B
(AD
(ADB))
D(ADB)
STRUCTURAL INFERENCE RULES
217
Let E' be the deduction resulting from E by re-writing the axiom q\. We then notice that the number of the axioms of (X D (X D Y)) D (X D Y) in E' is strictly less than n (in fact n — 1 ). By our induction hypothesis, we obtain the deduction E 0 of <j> having no axioms of (X D (X
D F))
D(XDY)
.
(2) <* (3) This is an immediate consequence of Lemma 1, together with the fact that we can prove the axiom of (X D (X D Y)) D (X D Y) in D U {axiom (X D (X D Y)) D {X D Y)} if and only if we can prove the sequent axiom of -> (X D (X D Y)) D (X D Y) in S U {sequent axiom - + ( I D (X D Y)) D(XD Y)}. (3) =» (4) Let II be a proof of r —> in S U {sequent axiom —► (X D (X D
y))D(XDy)}. We use induction on the number of the sequent axioms of
—> (X Z>
{x D y)) D (x D y). If II contain no sequent axiom of —> (X D (X D Y)) D (X D Y) , then we are done. We assume the result for the case where the number of the sequent axioms of —> (X D (X D Y)) D (X D F ) in II is less than n , and we prove it for n. The set T of all sequents in II are ordered by the anti-lexicographic ordering < # , which we define earlier. Let U be the set of sequent axioms of -* (X D (X D F ) ) D (X D F ) in II. U , as a subset of T, is ordered by
un.
The axiom of Ui can be re-written as follows
A^
A
A^A B-^B AD B,A-^ B AD (AD B),A,A-+ B A, A, AD (AD g ) -» B A,AD(ADB)^B AD
(AD
g ) -► AD
-^AD(ADB)D(ADB)
B
Dleft Dleft ex.left n times cleft D right D right
Let IT be the proof resulting from II by re-writing the sequent axiom
218
KOJI NAKATOGAWA
We then notice that the number of the sequent axioms of —► (X D (X D Y)) D (X D Y) in II7 is strictly less than n (in fact n - 1). By our induction hypothesis, we obtain the proof n 0 of V —► (j) having no sequent axiom of -> (X D (X D Y)) D (X D Y). (4) => (3) Let II be a proof of r —> (j) in SU { the left contraction rule }. We use induction on the number of the applications of cleft in II. If II contain no application of cleft, then we are done. We assume the result for the case where the number of the applications of cleft rule in II is less than n , and we prove it for n. The set T of all sequents in II are ordered by the anti-lexicographic ordering < # , which we define earlier. Let U be the set of sequents which is the lower sequent of cleft rule in II. U , as a subset of T, is ordered by
un.
Now, we consider a proof 0 of u\ inside II , which has u\ as its last sequent. By the leastness of wi , 0 contains no application of cleft rule except at the very bottom. Then 0 can be re-written to 0 ; in the following way so that & does not contain any application of cleft rule at all. 0
A
AJ\ScC'left
b*=A>T^C)
The above proof 0 can be re-writ ten as follows. 0 D right
A,ri ^ADC -i rr\ r i -> A D (A 5~CT D mght -> X
rlt(AD(ADC))D(ADC),A^C (AD {AD C)) D{A D C),TUA-* TuA-^C T-p -p
A->A C^C ADCA^C
D
~ C
ex ie/t
'
n
D
. ,. c £ ^
Umes
CUt
, ft ex.left
n
times
where x is a formula (A D (AD C)) D {AD C). When the formula of C is not present, we use J_ as the cut formula instead. The proof in this case is as follows.
219
STRUCTURAL I N F E R E N C E R U L E S
0 A,A,TX -► A,A,Ti -> _L A,Ti - ^ A D l ->X
_Lrzp/i£ D ri^/it
r1,(Ap(Ap_L))3(Ap_L),A-> (ADiADl^DiADL)^-^^ f1! A — ► -T-U A., 1 1
A -+ A
_L ->
~~ f/*
" "™e5
ex
n
ex.left
D
LeJt
n times
>
where x is a formula (A D (A D J_)) D We then notice that the number of applications strictly less than n (in fact n — 1). By our induction the proof IIo of T —► 0 having no application of the proof of Theorem 3.
(iDl). of cleft in II7 is hypothesis, we obtain cleft. This completes
Proof of Lemma 4 First, we assume that (j) is provable from T in DU{D I2}. Then it is obvious that we can prove from T in D\J {D I2(n > 2}. Next, we shall assume that we prove 0 from T in D U {D I2(n > 2)}. Let E a deduction of 0 in D U {D J 2 ( n > 2)}. We use induction on the number of the applications of U{D In(n > 2) in E . If E contain no application of D In(n > 3) , then we are done. We assume the result for the case where the number of the applications of D In(n > 3) in E is less than n, and we prove it for n. The set P of all formulas appearing in E are ordered by the antilexicographic ordering < # , which we defined earlier. Let Q be the set of formulas which is the consequence of application of D In(n > 3) in E. Q, as a subset of P , is ordered by
[A]-[A] ABB
=>
7
> * 3)
The above deduction of E can be re-written as follows.
as of to of
220
K O J I NAKATOGAWA
[A]--[A] B
A
2
ADB
B ADB
D
*
D I
After these re-writing, the number of the application of D I2 becomes n — 1, and the number of the application of D E becomes n — 2. Let £ ' be the deduction resulting from E by replacing E with H'. We then notice that the number of applications of D In(n > 3) in £ ' is strictly less than n (in fact n — 1). By our induction hypothesis, we obtain the deduction E 0 of (p having no application of D In(n > 3). So we can prove (f) from T i n D U { D / 2 } .
7
Summary
In the present paper, we characterized two structural inference rules, weak ening and contraction, in Gentzen-style natural deduction. We plan to investigate the exchange rule by using the anti-lexicographic ordering on the set of labels for a proof. We also would like to investigate the natural deduction systems which include additive conjunction and disjunction in a sequel to the present paper.
References [1] Gerhard Gentzen, The collected papers of Gerhard Gentzen, ed. by M. E. Szabo, Noth-Holland, 1969. [2] D. Prawitz, Natural Deduction, A Proof-Theoretical Study, Almqvist &; Wiksell, 1965. [3] Gunnar Stalmarck, Normalization theorems for full First-order Clas sical natural Deduction, The Journal of Symbolic Logic, Volume 56, Number 1, March 1991. [4] A. S. Troelstra, Lectures on Linear Logic, CSLI Lecture Note Num ber 29, Center for the Study of Language and Information, Stanford University, 1992.
STRUCTURAL INFERENCE RULES
221
[5] A. S. Troelstra, Natural deduction for intuitionistic linear logic, Annals of Pure and Applied Logic, Volume 73, 1995. [6] S. Maehara, Suri Ronri Gaku Jyosetu (Introduction to Mathematical Logic), Kyoritu Syuppan, 1966. [7] S. Maehara, Suri Ronri Gaku (Mathematical Logic), Baihu-kan, 1974. [8] G. Takeuti, Senkei Ronri Nyumon (Linear Logic), Nihon Hyoron-sya, 1995. [9] A. Oide, Quantum Logic and Related Systems, Annals of Japan for Philosophy of Science, Volume 6, 1983. [10] H. Ono, Semantics for Substructural Logics in: Substructural logics, K.Dosen and P. Schroeder-Heister eds., Oxford University Press, 1993. [11] H. Ono, Joho Kagaku ni okeru Ronri (Logic in Information Science ), Nihon Hyoron-sya, 1994.
223
Linear set theory with strict comprehension Masaru Shirahata*
Abstract In this paper, we study the extensionality axiom in the set theory with the unrestricted comprehension based on linear logic. We first review Grishin's result which shows the imcompatibility of the ex tensionality axiom and the unrestricted comprehension in linear set theory As one way to remedy this situation, we introduce the notion of "strict comprehension" and formulate a system of linear set the ory which contains the extensionality and the strict comprehension. The consistency of such a system is then proved by a cut-elimination argument. Finally, we sketch the development of arithmetic in our system.
1
Introduction
The set theory based on contraction-free logics [2, 8] has been studied for the reason that the use of unrestricted comprehension does not necessarily yield the contradiction in such logics [3, 4, 7, 9, 10, 13]. However, it has been also noticed that the standard extensionality axiom cannot be added to such a set theory without causing inconsistency [5, 14]. In this paper, we show one way to remedy this situation, using the notion of "strict comprehension." The basic idea is as follows. Contraction of a formula in a sequent calculus corresponds to copying resources, or proofs of that formula (terms of that type). In linear and affine logics [2], the contraction rule is not available and, as a result, we can discharge (abstract) at most one occurrence of the assumption A (variable x : A) in forming the implication A —o B ((Xx : A) M). We apply the similar restriction to the formation of set terms {x : A} so that we have the logical equivalence between the formulas A[s/x] and s e {x : A} only when the variable x has at most one occurrence in the formula A. Such a comprehension rule will be called strict (or linear) comprehension. A similar idea to this was already found in Fitch's work on combinatory logic [1]. *This work was partially supported by JSPS research fellowship for young scientists.
224
M
SHIRAHATA
We first review Grishin's result which shows the imcompatibility of the extensionality axiom and the unrestricted comprehension in linear set the ory. We then introduce the notion of strict comprehension and formulate a system of linear set theory which contains the extensionality and the strict comprehension. The consistency of such a system is then proved by a cutelimination argument. Finally, we sketch the development of arithmetic in our system.
2
The extensionality and linear set theory
The extensionality principle says that two sets are identical if they have the same members. This is a very natural principle, given our intuitive notion of sets. However, it turned out that the set theory with the unrestricted comprehension based on affine (BCK) logic becomes inconsistent with the extensionality added. The proof is due to Grishin [5]. We assume the standard axioms for the equality: \-s = s
s = t,A\-
A[t/s\
where the formula A[t/s] is obtained from the formula A by replacing the term t for some of the occurrences of the term s. We work in the set theory with the unrestricted comprehension r,A[s/x]\T,se{x:A}\-
A A
rh Th
A[s/x],A se{x:A},A
which is based on affine logic, i.e. the logic without contraction but with weakening. First, let's recall that the left contraction rule for the formula A is equiv alent to the axiom A \- A <8> A, which may be called duplication or copying. Then, note that we can easily recover the contraction rule for the equations 5 = t as follows: \- s = s
\- s = s s = t\-s = t<8>s = t
Secondly, we define the terms s = {x : x = q} and t = {x : x = q <8> A} for the formula A and show that A is logically equivalent to the equation s = t. For this, we use the extensionality written as the rule
r,x e s\- x et,A
r,x ett- x e s,A
r\- s = t,A
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225
where the variable x has no free occurrence in the multi-sets T and A. We state the proof of the logical equivalence in the informal fashion, from which one can easily recover the derivation in the sequent calculus. Let's assume x G t. We unwind the term and obtain x = q ® A. Dis carding (left weakening) A, we have x — q and hence, x e s. This amounts to the derivation of the sequent x G t h i G s . On the other hand, let's assume A and x e s. We can immediately obtain x = q ® A. Winding this up, we have x G t. Hence, we have the derivation of A, x G 5 h x G t. Then, combining the two directions by the weakening and extensionality, we have A\- s = t. For the other direction, assume s = t. Since q = q always holds, we have q e s. Hence, q G t. Unwinding this, we obtain q = q
3
The system STCOM of linear set theory
Our guiding idea is to restrict substitution -A[£/s] under the equation s = t so that at most one occurrence of 5 is replaced by t. To achieve this effect, however, we need to modify the comprehension principle as well. To see the point, let's consider the formula A = qes(^sep. Under the assumption 5 = t, we can perform the substitution to obtain q G t<3>s G p, but we should not be able to obtain q G t (8> t G p. However, A is logically equivalent to 5 G {x : q G x
t,se{x:qex®xep}\-te{x:q€x®xep}
to derive s =
t,q€s®sep\-qet®tep.
226
We therefore restrict the comprehension rule so that we can form the formula s € {x : A} from the formula A[s/x] if and only if the variable x occurs at most once in A. This type of comprehension will be called strict comprehension. This condtion can be relaxed when additives are involved. The formulas A&B and A © B in linear logic represent "choices" between A and B in the computational interpretation. Therefore, even when a term occurs in both A and B, only one of the occurrence of the term is used in computation (i.e. cut-elimination). Hence, two occurrences of the variable x in A(x) and B(x) can be identified in A(x)&B(x) and A(x) © B(x), and we allow the strict comprehension to create the term {x : A(x)&B(x)} and {x : A(x) ©£?(#)}. Note that the strict comprehension does not prevent the set-theoretical paradoxes in classical logic, since one can easily write a term such as {x : 3y(x = y
induction
are terms. They are free and strict
2. If s and t are terms, then s G t and s = t are (atomic) formulas. They inherit the free variables from the terms. The free variables x are strict in them when x is strict in s or t, but not in both. 3. If A is a formula with a strict variable x or without any occurrence of x, then {x : A} is a term. This term inherits the free and strict variables of A except for x. 4- If A and B are formulas, then i(8> B and A*$B are formulas. They inherit the free variables from A and B. The variables x are strict in them when x is strict in A or B, but not in both. 5. If A and B are formulas, then A&B and A® B are formulas. They inherit the free variables from A and B. The variable x is strict in them when x is strict in A or B, and possibly in both. 6. If A is a formula and x is a variable, then A1-, VxA and 3xA are formulas. They inhrerit the free and strict variables from A except for x. We write s £ t and s ^ t for (s 6 t)
and (s = t)L,
respectively.
227
LINEAR SET THEORY
In the formulation of the deductive system, we use the Gentzen style two-sided sequent calculus. We write e[t/x] for the expression obtained from the expression e by replacing the term t for a strict variable x in e. If e does not contain x at all, e[t/x] is e itself. Axioms: s€thset s = t,(pe q)[s/x] \~(pe q)[t/x]
s = t,{Pe q)[t/x] h (p G q)[s/x]
R u l e s of inference: r,Ah A rh A-\A r,A,BhA r,A05hA
rhi,A r.A^A rhi,A 6h£,£ r , e i - A®£,A,E Th A,B,A ThA^B.A
r,AhA 0,5hE r,e,A^hA,E r,Af-A r,A&5hA
r,BhA I\A&£hA
r,Ah-A r , B h A rAeshA r,A[g/a]h A r,V^A(-A
ThA,A T\-B,A r h A&B, A
rhAA r hBA n-Ae£,A ri-A©5,A Th A,A rhVxA,A
where the variable x has no free occurrence in T or A in the righthand rule. T,Ah A r,3xAhA
T\-
A[s/x],A T\-3xA,A
where the variable x has no free occurrence inT or A in the lefthand rule. r,A[s/x]h A r, s e {x : A} h A
T h A[s/x],A r h s G {x : A}, A
where A is a formula with the strict variable x or without any occurrence of x. r,x e s\- x et,A r,x et\- x e s,A r h s = t,A where the variable x has no free occurrence inT or A.
rhA,A e,Ahs
rhA
r,0hA,E
r,AhA
ri-A ThA,A
228
M
SHIRAHATA
P r o p o s i t i o n 2 The sequents 1. h s = S 2. s = t\-t
= s
3. s = t,(p = q)[s/x][-(p
= q)[t/x}
4. s = t,(p = q)[t/x] \-{p = q)[s/x] are derivable. Proof We have the derivations: x£shx €s x Gshx Gs \- s = s
s = t,x G t \- x G s s — t,x G s \~ x G t s — t\- t — s
For 3 and 4, suppose x is strict in p but not in q. We have the derivations: s = t,y G p[t/x] \~ y e p[s/x] p[s/x] =q,y G p[s/x] \-y e q s = t,p[s/x] =q,y e p[t/x] \~y eq and p[s/x] =g,y €q\- y G p[s/x] s = t,y G p[s/x] H y G p[t/x] s = t, p[s/x] =q,y eqt- y e p[t/x] ^From them, the sequent 5 = t,p[s/x] = q \- p[t/x] = q is obtained by the extensionality. Other cases are similarly proved. I In order to prove the properties of the system, we need to introduce the size of proofs. ; Definition 3 The size o"(7r) of a proof TT is defined
inductively:
1. If 7r is an axiom, tften its size is 1; 2. If 7r is obtained from r by one of the one-premise rules, then o"(7r) is
LINEAR S E T THEORY
229
Proof We only consider the case when the equation is s = t, since the other case is entirely similar. The proof is by induction on the size of w. When 7r is obtained from ir± (and 7r2) by an application of one of the inference rules, we call 7Ti (and 7r2) the immediatel subproof(s) of TT. If A is not the main formula of the last inference of 7r, we apply the induct ve hypothesis to the immediate subproof in which A occurs. For the right V and left 3 rules, we assume the appropriate renaming of the eigenvariable. If' A[s/x] is obtained by one of the weakenings, so is A[t/x]. Otherwise, we have five cases. C a s e 1 A[s/x] has the form (B®C)[s/x] or (B>9C)[s/x\. Then, it suffices to perform the substitution in exactly one of the formulas B[s/x] and C[s/x] since the strict variable x appears in at most one of them. Hence, we only need to apply the inductive hypothesis to the appro priate immediate subproof. C a s e 2 A[s/x] has the form (B&C)[s/x] or (B@C)[s/x]. For the left & rule and the right © rule, it suffices to perform the substitution in one of the formulas B[s/x] or C[s/x\. Hence, we use the inductive hypothesis to the immediate subproof. For the right & rule and the left 0 rule, the strict variable x may appear in both B and C. However, we can use the inductive hypothesis to both of the immediate subproofs as well without creating additional occurrence of the formula s = t in the end sequent. By the definition of size, this increases the size of a proof at most by one. C a s e 3 A[s/x] has the form B[s/x]±, \/yB[s/x] or 3yB[s/x}. We then apply the inductive hypothesis to the immediate subproof. C a s e 4 A[s/x] has the form (p G q)[s/x]. Then, we will use the axiom s = t,(pe q)[t/x] h (p G q)[s/x] or s = t, (p G q)[s/x] h (p G q)[t/x] to perform a cut. This increases the size of a proof by one. C a s e 5 A[s/x] has the form (p = q)[s/x\. If s = t is obtained as the main formula of an appplication of extensionality, then apply the inductive hypothesis to both of the immediate subproofs of 7r. This increases the size of the proof at most by one. If the formula (p — q)[s/x] is on the lefthand side, then it is from one of the axioms: (p = q)[s/x], (a G b)[p[s/x]/y] h (a G b)[q[s/x]/y] or (p = q)[s/x], (a G b)[q[s/x]/y] h (a G b)[p[s/x]/y] where x in fact occurs in at most one of p or q. Without the loss of generality, we can concentrate on the first case with x occurs in p but not in q. We then have the following.
230
M
s = t,(ae
SHIRAHATA
b)[p[s/x]/y] h (a G b)[p[t/x]/y] p[t/x] = g, (a £ b)[p[t/x]/y) s = t,p[t/x] = q,(a£ b)[p[s/x]/y] h (a <E b)[q/y]
h ( a E b)[q/y]
Other cases are entirely similar. In any on those cases, we only add one application of cut with an axiom. Hence, the size of a proof increases by one. I
4
The cut-elimination theorem
We use three types of reduction rules, i.e. 1) permutative reductions, 2) symmetric reductions and 3) axiom reductions. For the full exposition of those rules which do not involve the equality, we refer the reader to [9, 10]. P e r m u t a t i v e r e d u c t i o n s This type of reductions are applied when one of the cut formulas is not the main formula of the immediate subproof. They are mostly standard and we only give the rules for the extensionality: ThA,A
A,Q,x 6 s h x e t,E A,Q,x eth x e s,E A,e\-s = t,Z r , e i - s = t,A,E
is tranformed to ( ^ ) r h A , A A , 0 , y 6 s h y G t , E Th A,A A,0,y € t h y G s,S r, 0, y G s \- y e t, S r,9,y6thyes,S I \ 0 h s = t,A,E where y is a fresh new variable. The case when the cut-formula is on the righthand side in the extensionality is similarly handled. Since (max(m, n) + 1)4-/ = max(m-\-l,n-t-l)-\-l, this transformation does not change the size of a proof. In general, one can establish by observation that the permutative reductions preserve the size of a proof. In particular, we can freely permute two consecutive applications of cut without changing the size. S y m m e t r i c r e d u c t i o n s This type of reductions are applied when both of the cut-formulas are the main formulas of the respective immediate subproofs, except for the weakening reduction which is applied when one of the cut-formulas is the main formula. They are all standard. Hence, we only state two exemplary cases. rihA.Ai r2t-B,A2 e,A,£hs r i , T 2 h A®B,Al7A2 Q,A®B\-X ri,r2,0HAi,A2,E
"
rihA,Ai e,A,Bhs T2 h B,A2 Tlt 0, B h Ai,S ri,r2,ei-Ai,A2,i:
231
LINEAR S E T THEORY
T\- A[s/x],A
6 , A[s/x] h E
r\- s e {x: A},A
e, s e {x -. A] \- E
r,0hA,E
rn^/x],A "*
r,0h
e,A[g/a;] \- E A,E
We note that the size of a proof is strictly decreased by an application of any of the symmetric reductions. A x i o m reductions This type of reductions are used where one of the cut-formulas is in an axiom. Similar reductions were defined for the nor malization of a natural deduction system of set theory [6]. For this type of cut, there are cases where no reduction is defined. When we substitute the term t for the term 5 in p G s to obtain p G t, we call such a substitution critical. Otherwise, the substitution is non-crtical. When there are two reduction rules which only differ in the position (i.e. the lefthand side or right hand side of the symbol h) of cut formulas or the order of terms in equation (i.e. s = t and t = s), we only state one of them explicitly.
s G t,rh A The size of a proof is strictly decreased by this reduction. For non-critical cuts, we have the reduction: A\p/x][t/y],T\-A s = t,(pe{x:
A})[s/y] \-(pe{x: A})[t/y] (p G {x : A})[t/y],T s= t,(p£{x:A})[s/y},T\-A
h A
is tranformed to (~») S
s=
=
tiA\p/x][8/y]iT\-A ti(pe{x:A})[s/y]iT\-A
where the new proof is obtained using IT for the subproof IT of the sequent A\p/x][t/y],T h A. Since
r,x
Gshx G
t, A r,x e t \- x G S,A r i - s = t,A s = t,pes\-pet ■
r,pe
—
^
T - ^ U ^ A r.pGsrpGt.A
s\-pet,A
where the new proof is obtained by replacing the term p for the variable x in the subproof of the sequent T,x e s \- x e t,A. By this transformation, the size of a proof is strictly decreased.
232
M SHIRAHATA
Each reduction rule gives a one-step reduction. We say that a proof IT reduces to a proof r if there is a finite sequence of one-step reductions which transforms it into r . In order to prove the (partial) cut-elimination theorem, we need to de fine what the results of reductions are. For this, we first describe a certain configuration of cuts in a given proof 7r. The subproof r of 7r will be called a chain if r is a consecutive application of cuts with the axioms for substi tution, i.e. it has the form sn = tn,pn e qn\- pn+i € qn+i so — t 0 ,po E go I- pi € qi Si = t i , . . . , sn = tn,pi G q\ h pn+i so = to,. . . , sn = tn,p0 e qo h pn+i € qn+i
€
where p ^ + 1 G qi+i is obtained from pi £ qi by the substitution of Si (or ti) for U (s^. The equations involved in the chain are called the gates. They are either critical or non-critical according to whether the substitutions with them are critical or not. We call a chain non-critical if it consisits entirely of noncritical substitutions. The formulas p0 G qQ and pn+i € g n +i are the endformulas of the chain r . They are called critical as well if the substitutions for them are critical. Now suppose that the chain in a proof IT is followed by a consecutive application of cuts such that 1. the cut formulas for each cut are (a) one of the non-critical gates and the main formula of a subproof ended with the extensionality, or (b) one of the end-formulas, which is critical, and the main formula of a subproof ended with one of the comprehension rules, 2. none of the critical gates and non-critical end-formulas are used as cut-formulas in -K. Such a configuration of cuts will be called a cluster in the proof 7r. To be precise, a configuration consists of a chain and the cut-formulas involved in the cuts following the chain. A cluster a is closed if one of the end-formulas of the chain is used as a cut-formula in a. If both of the end-formulas are used as cut-formulas in a, it is closed at both ends. If a cluster is not closed, then it is open. L e m m a 5 Suppose that the proof ir is obtained from 7Ti and TT2 by a cut, where the last inference rule of TTI is the right (left) comprehension to create
LINEAR SET THEORY
233
the cut formula p G q (pf G q') and TT2 is a non-critical chain with the endformulas p G q and p' G q''. Then, TT reduces to a proof r such that the last inference rule of r is the right (left) comprehension to create p' G q' (p G q) and O~(T) < O~(TT).
Proof We only consider the case where the cut-formula is created by the right comprehension since the other case is entirely similar. The proof is by induction on the size of the non-ctitical chain. Note that the cut-formula p G q originated from the substitution axiom 5 = t,p G q h p" G q". We first use the permutation to create the cut with TT\ and the substitution axiom, followed by the cut with the rest of the chain. We then apply the non-critical axiom reduction to the upper cut. The result r'again has the configuration described in the condition of the lemma, with p" G q" replaced for p G q and the size of the chain decreased, so that we can apply the inductive hypothesis. Furthermore, the size of the entire proof does not change by the non-critical axiom reduction so that a(r)
< CT(T') < O~(TT).
I
Definition 6 The proof TT is called normal if the cuts in TT appears only as part of clusters in TT. T h e o r e m 7 For any proof TT, there exists a normal proof r such that TT reduces to r and a{r) < O~(TT).
Proof The proof is by induction on the size of a proof. If TT does not end with a cut, we apply the inductive hypothesis to the immediate subproof(s) of TT. If TT ends with a cut, we can use the inductive hypothesis to asstfme that each of the immediate subproof is already normal. Hence, we only consider the proofs TT such that TT is obtained from the normal subproofs TT\ and TT2 by a cut with the cut-formula C. We divide the form of each subproof into five types: 1) the identity ax iom, 2) the substitution axiom, 3) ended with a rule of inference other than cut and C is its main formula, 4) ended with a rule of inference other than cut but C is not its main formula, and 5) ended with a cut. Accordingly, we need to consider 25 possibilities in total. We will cut down the number of those possibilities, step by step. S t e p 1 If one of the subproof is the identity axiom (type 1), we can use the axiom reduction to decrease the size of a proof and apply the inductive hypothesis. This cuts down the number to 16. If one of the subproof ends with a rule of inference but C is not its main formula (type 4), we can use the permutative reduction and apply the inductive hypothesis. Now, we have only 9 possibilities.
234
M
SHIRAHATA
S t e p 2 If both of the subproofs are substitution axioms (type 2/type2), then 7r itself is already a chain. If both of the subproofs end with rules of inference with C as their main formulas (type 3/type 3), then we can use the symmetric reduction to decrease the size and apply the inductive hypothesis. The number is now 7. S t e p 3 Let's consider the cases type 2/type 3 and type 3/type 2. For the former, the last inference rule of 7r2 must be the left comprehension. If 7Ti is critical, then n already forms a cluster. If 7Ti is not critical, then we can use the non-critical axiom reduction and use the inductive hypothesis to the immediate subproof of the result. For the latter, the last inference rule of 7Ti must be the right comprehension or the extensionality. If it is the comprehension, then we can use the same argument as we have just seen for type 2/type 3. Hence suppose that the rule is the extensionality. If 7T2 is critical, then we can use the critical axiom reduction to decrease the size and apply the inductive hypothesis. If -K^ is non-critical, then 7r already forms a cluster. The number is then 5. For the rest of 5 possibilities, the subproof -K\ or TT2 ended with a cut (type 5) is involved. We only argue for the case when -K\ is of type 5 since the other case is entirely similar. Since TT\ is normal, the cut must be part of a cluster. Then, the cut-formula C of the cut with TTI and 7r2 comes from the chain for the cluster or from one of the subproofs r of 7Ti which is used in one of the consecutive applications of cut with the chain. Suppose that C comes from r . Since r is part of the cluster, r is obtained from its immediate subproof T' by the comprehension or the extensionality. We then use the permutative reductions for cut to move the cut with 7T2 right after r ' , followed by the comprehension or the extensionalty. Since cr(r') + CT(7T2) < cr(iri) + a(iT2) = cr(7r), we can apply the inductive hypothesis to the proof obtained from r' and 7r2 by a cut. Note that it is still followed by the comprehension or the extensionality after the reduction. Hence, the result forms a cluster. Therefore, we can assume that C comes from the chain. S t e p 4 We now consider the cases (type 5/type 5), (type 2/type 5) and (type 5/type 2). For the first case, we can assume that C comes from the chains of TTI and 7r2. Then, we combine the two chains by the permutative reductions to create a new chain. For the second and the third cases, we similarly use the permutative reductions to combine the substitution axiom and the chain in the other subproof to create a new chain. S t e p 5 The last two cases are (type 3/type 5) and (type 5/type 3). For the former, the last inference rule of 7Ti is either the comprehension or the extensionality. Suppose that it is the extensionality. Then C is one of
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235
the gates in the chain of 7r2. If the gate is critical, then we can use the critical reduction to decrease the size and apply the inductive hypothesis. If the gate is non-critical, TT already forms a cluster. Hence suppose that the last inference rule is the comprehension. Then C is one of the endformulas of the chain of 7r2. If the end-formula is critical, then TT already forms a cluster. If the end-formula is non-critical, it is part of a non-critical segment of the chain. We take the maximal such segment in size. We then use the permutative reductions and Lemma 5 to move the non-critical segment to the inside of 7Ti. The last inference rule of this new subproof, call it Ti, is the comprehension according to Lemma 5. Note that there may be subproofs of 7r2 which are used in cuts with the gates of the non-critical chain. By the permuative reductions for cut, we move such subproofs above the last rule of inference of T\ , resulting in a subproof r 2 . Since all those transformations do not increase the size, we have a(r 2 ) <
• *i
e\-s = t,E
Thpp G go, A
[M #o,*,s = t,Po eqo\-pn+i egn+i
e,fro,#,p 0 ego h pn+1 G gn+i,S r , 0 , ^ o , ^ l - p n + i G g n +i,S,A
reduces to : T3 r , e , ^ o hp G g,s,A ^,pGgl-p n +i e gn+i r , 0 , ^ o , ^ t - p n + i Gg n +i,S,A where (*) is the maximal non-critical segment with s = t as one of its gates. For the case (type 5/type 3), the last inference rule of 7r2 must be the comprehension. We can then use the same argument as we have just seen for (type 3/type 5). I We introduce the ordering among clusters in a given proof TT. Suppose that the closed cluster a is formed around the chain r . By the definition of a cluster, none of the critical gates and non-critical end-formulas in r are used as cut-formulas in IT. However, they may appear as subformulas of
236
M
SHIRAHATA
formulas s 6 t or s = t which are used in cuts within other clusters. If one of the critical gates or non-critical end-formulas of the cluster a appears as a subformula of one of the cut-formulas in the cluster /?, we say that (3 is lower than a. In order to construct a formula which contains a gate or an end-formula A a s a subformula, some inference rules other than cut need to be applied to A. Since a cluster is a consecutive application of cuts, such a formula is formed outside and below the cluster itself. Therefore, it is not possible that a gate or an end-formula of the lower cluster appears as a subformula of a cut-formula in the upper cluster. Hence, the ordering thus defined among clusters does not contain a loop. Then, there are clusters which are minima under this ordering. Corollary 8 There is no proof of the empty sequent. Proof By Theorem 7, it suffices to consider normal proofs. Let TT be a normal proof. If 7r does not contain a cut, then the end sequent of 7r is not empty. Otherwise, choose one of the clusters, a, which is a minimum under the ordering. If the cluster is not closed at both ends, then one of the endformulas remains as a subformula in the end sequent. If the cluster is closed at both ends, then the chain contains at least one critical gate, which remains as a subformula in the end sequent. I
5
Arithmetic in STCOM
We now sketch the development of arithmetic in STCOM. Since the formal proofs in STCOM usually become very large, we only give informal proofs. Exponentials First of all, we may bootstrap STCOM by adding the ex ponential operators. For this, we extend the definition of formulas to include ! A and ? A but we do not allow comprehension over the formulas contain ing the exponentials. Hence, the terms of this extended system are exactly the same as before. The use of extensionality is restricted accordingly, and we have the standard inference rules for exponentials: A,T\- A !A,ThA
ThA,A rhA,?A ! T h ?A,A !T,h ?A, \A
!i,!i,rhA !A,ThA
r h A, ?A, 1A r h A , ?A
A, \T\- ?A ?A, ! T h ?A
where ! V and ? V are the multisets of formulas of the form ! A and ? A, respectively.
LINEAR S E T THEORY
237
The consistency of the extended system can be shown by the cut elim ination argument. In this case, we first remove all the cuts with the cutformulas ! A or ? A, and then we can proceed as before [9]. Note that if we allow the exponentials inside the set terms, then the Russell's paradox will be reproduced with the term {x : 3y[\ (x — y) <S> ? (y ^ y)]}. The constants 1 and _L can be defined as s = s and s / s, respectively. We write ! n A for the formula A
1. \s = t, \A[t/x]\-
\A[s/x],
2. \s = t, \A[s/x]Y-
\A[t/x],
and similarly for ? A[t/x] and 1 A[s/x}. P a i r s The unordered pairs {5, i) are defined as {x : x = s 0 x = t}. The singleton {5} is {s.,5}, which is equivalent to {x : x = s} by the extensionality. Then, the ordered pair (s,t) can be defined in the standard way as {{s},{s,£}}. Under this definition, we can show the standard property of the ordered pairs. Fact 10 The following sequents are provable in STCOM: 1. s = s' ®t = tl \- (s,t) =
(s',t'),
2. (s,t) = (s',t') ^ s = s', = (s1,?) \-t = t',
3. s = s' ® \2(s,t) I
\2(s,t)
= (s,t')\-t
= t'.
N u m e r a l s a n d N a t u r a l N u m b e r s The numerals n for the natural num bers n are defined in the standard way. We write Us for {x : 3y(x 6 y
Vy[! (6 e y&Nz{! z e y -o sc{z) G y)) - o x e y].
238
M
SHIRAHATA
Under this definition, we have the induction principle limited for the terms, and we can show that all the numerals satisfy N(x). P r o p o s i t i o n 11 The following are provable: 1. ! (6 G s&\/y( \y e s-o sc(y) G 5)) h \/x(N(x)
- o x G s),
2. \- N(h), for any natural number n. Proof 1. This is immediate from the definition. 2. We use the (external) induction on n. For n = 0, assume ! (6 € y8Nz( \z€y-o
sc(z) G y)).
Then, 0 G y. Hence, N(0). For n = m + l, assume ! (6 G y&Nz{ \z ey-o
sc(z) G y)).
Then, Vx(N(x) -ox G y). By the inductive hypothesis, N(m). Hence, m G y. So, ! m G y. By the assumption again, Vz(! z G y —o s(z) G y). Hence, m 4- 1 G y so that 7V(ra + 1). 1 Fact 12 The following are provable for any natural numbers n:
1. h n T l ± 6, jg. h 0 = n © 3x(iV(a;) 0 n = s(x)) We write 0 for the empty multiset. This is useful for the proof by contra diction, and we can delete the contradictory cases as follows. A1 1-0 Ai h B A2\~B Al®A2\-B where we use weakening with the formula B. P r o p o s i t i o n 13 The following are provable for any natural numbers n and m: 1. n G ra h n C rh, 2. n G m 0 7 7 i G p l - n G p ; 3. n G n h 0, ^. n = 77i h 0, /or n ^ m.
239
LINEAR S E T THEORY
Proof 1. We use the induction on ra. When m = 0, this holds vacuously. Suppose n G p + 1. Then, n G p o r (in the sense of 0 ) n = p. In the former case, nCpCp+lby the inductive hypothesis. In the latter case, n = p C p + 1. 2. Suppose h G ra
hep.
3. We^use the^induction on_n. When n = 0, this is obvious. Suppose p+1 G p + 1. Then, p+1 e p or p+1 = p. In the former case, p G p + 1 C p. Hence, pep. By the inductive hypothesis, we have then 0. In the latter case, p G p + 1 = p. Hence p G p s o that we have again 0. 4. Assume h = m. There are two cases. If n < ra, then h n G ra so that n en. Hence, we obtain 0. Similarly for the case ra < n. I For the proof of the next proposition, we recall that the multiplicative conjunction ® distributes over additive disjunction ©: A,C\-A®C B,C\-B®C A,Ch(A<8>C)e(B(8>C) B,C\-(A®C)®(B®C) (A 0 B) ® C h (A ® C) 0 (B ® C) A,Ch (A0B)(8>C B,Ch (AeB)(8>C A(g)Ch(AeB)<8)C B<8)Ch(A©B)®C (A ® C) 0 (B ® C) h (A 0 B) ® C This allows to combine cases by the multiplicative conjunction and makes the following inference admissible:
r h Ai e . . . e Am A h Bi e . . . e Bn r, A h (Ax ® Bi) 0 ... 0 (Am ® Bn) where the number of cases on the right is ra by n. P r o p o s i t i o n 14 The following sequents are provable for any m and n: 1. ].2 sc(x) = sc(m) h x = ra, 2. \nx e ra h \nx — 0 0 . . . 0 \nx = ra - 1. Proof 1. Since \- x e sc(x), we can derive sc(x) = sc(m) \- x G 5c(ra). Further more, x G 5c(ra) implies x = O 0 . . . 0 x = ra. Then, we have !2 5c(x) = sc(ra)
h
(a; = 0 (8) sc(x) = 5c(ra)) 0 . . . 0 ( x = ra — 1 (8) 5c(x) = sc(m)) 0 (a; = ra)
240
M
SHIRAHATA
and !2 sc(x) = sc(m) h 1 = sc(rh) © . . . (rh = sc(ra)) © (a; = rh). For n < ra, we can derive n = sc(ra) h 0 since n ^ ra + 1. Hence, !2 sc(aj) = sc(ra) h x = m 2. The proof is by induction on n. When n = 0, this holds obviously since x e rh implies x = 0 © . . . © : r = ra — 1. For n = p + 1, let a(f) be x = f. Then, \px em
(\pa(6)
© . . . ! p a ( r a ^ l ) ) 0 (a(6) © a ( r a ^ ~ l ) )
We distribute
■ Functions Since our terms are very much restricted, we should not ex pect to have terms represent numerical functions. Instead, we consider the numeralwise representation of n-ary functions / by n + 1-ary predicates P(x,y) Definition 15 The predicate P(x,y) numeralwise represents the function f : to —► UJ if for any n-tuple of natutal numbers rh and natural number p, we have f(m)=p
=>
\-P(rh,p)
h Vxy [ (P(rh, x)
241
LINEAR SET THEORY
Assume F(fh,y)
-
g(x)
=
/i(n,x,/(n,x)).
To represent such a function, we construct the three predicates Init(x,y,w,G) and Rec(x,y,w,H) as Rec(x,y,w,H)
<3> 3z [ \Q X = sc(z)
Init(x, y,w,G)
<=>
Rec(x,y,w,H)
<$
Fun(x,w),
<8>3u3v( !2 (z, u) G w <8> i7(z, y, it, v) (8) (x, v) G u>) ]
3Z[\QX
x = 6
= sc(z)
Then, let the predicate F(x,y,z)
be as follows:
3w [! \fu ( !n it G 5c(x) —o [(/m^(w, y, w, G) © Rec(u, y, w, H))
= p, £/ten h
F(m,n,p).
Proof It suffices to find a witness term s such that 1. h ( m , / ( m , n ) > G 5, 2. !7 it G sc{m) h Init(u, n, 5, G) © Rec(u, n, 5, iJ), 3. ?4 ^ G sc(ra) h Fun(u,
s).
Let 5 = d e / {x : x = (6, / ( 0 , n ) ) © . . . ® a ; = ( m ) / ( m , n))}. 1. Since h ( m , / ( m , n ) ) = (ra,/(ra,ri)), we have h (m, f(m,n)) immediately.
G 5
2. First, we have ?7 u G sc(ra) h ! 7 u = 6 © . . . \j u = m. Hence it suffices to show for each r < m, \7u = r h Init(u,h,s,G)
©
Rec(u,n,s,H).
242
M
SHIRAHATA
We consider two cases. Suppose r = 0. Let p = / ( 0 , ft) = g(n). Then, h (0,p) G 5 and h G(n,p). Hence, h 3^({0,z) G s
h
H(q,n,p,p')
and h (r,p') G 5. Hence, we have ! 7 w = f l - \ju — sc(q)<8>3v3w( !2 (g,^) G 5(8>i?(^,fi,^,ti;)(g)(f,?/;) G s). ^From this follows '.7 ii = f h Rec(u,h, s, if). 3. It suffices to show for each r < ra, •2 (F,V) G 5 h v =
f(r,n).
If g / r, then (f,v) = ( g , / ( g , n ) ) h 0. Hence, we have (f,i>) G s h (f,i>) = ( r , / ( r , n)). /,From this follows !2 (r,u) G s N = f(r,n).
I
L e m m a 18 For any natural numbers m and ft, F(m, n, x) ® -F(m, n,y) \- x = y. Proof The proof is by induction on m. Let p = f(m,n) formula '.7 it G sc(f) —o ((Init(u,
and D(u,r,w)
n, iu, G) © Rec(u, n, iu, if)) ® Fun(u,
be the w)).
Then, F ( m , n, x) is 3w [ ! VuD(u, ra, if) (8) !2 (ra, x) G « ) ] , and it suffices to show ! ViuD(u, ra, w)
Since h 0 G sc(0), we have
\/uD(u,6,w)
\- (Init(Q,n,w,G)
0 itec(Q,n,w,.H"))
Note that 0 = sc(z) implies z G 6, and then 0. Hence, Rec(0,n,w,H) So, VuD(u,0,w) h Init(0,n,w,G)
h 0.
Since g(n) = / ( 0 , n) = p, we have G(n, z)\- z — p and (0, z) G w <8>(2(ri, 2) h (6,p) G w. Hence, Init(0,n,w,G) h (6,p) G W. So, ! 2 VuZ}(u,6,u;)r- ! 2 (0,p) G w ® ! 2 F u n ( 6 , w ) . However, !2 (0,p) G w ® !2 (0, x) G u> <S> Fun(0, w) \- x = p. Hence, ! \/uD(u, 6, w)
243
LINEAR S E T THEORY
m = q + 1: First note that !n u G sc(q) implies !n u G 5c(m) since h sc(
h
\\/uD(u,q,w).
Since h m G 5c(7fi), we have VuD{u, 6, u>) h (Init(m,
n, W, G) © Rec(m, n, iu, if)) (8) Fun(m,
However, m = 0 implies q G 0, and then 0. So, Init(m,n,w,G) VuD(u, 0, iy) h Rec(m, n, iu, 77) (8) Fun(m, Since m = 5c(2;) is identical to sc(q) = sc(z), ! 3 £ = g. So,
w).
\- 0. Hence,
w).
we have \Qth = sc(z)
h
\Q rh = sc(z), !2 (z, v) e w <8> H(z, n, v, v') ® (m, u') 6 w I" !2 (q, v) €w <8> H(q, n, v, v')
So, we have \\/uD(u,m,w):Rec(m,n,w,H)
h (rh,p) G w. Hence,
WuD(u,m,w),
h !2 (m,p) G w.
!2 Rec(rh,n,w,H)
Then, \\/uD(u,rh,w),
\2VuD(u,m,w)
h !2 (TO,J3) GID(8I!2
However, !2 (rh,p) G u> (8> !2 (TO,X) G w
Fun(m,w).
\- x = p. Hence,
! "iuD(u, TO, it;) (8> !2 (TO, x) G w h a; = p.
■ Therefore, we can now conclude this brief development of arithmetic with the following theorem. T h e o r e m 19 The numeralwise primitive recursion.
representable functions
are closed under
244
6
M
SHIRAHATA
Concluding remarks
In set theory based on contraction-free logics, there seem to be two inter esting issues for further study. The first issue is the precise determination of the expressive power of such systems. Recently, Girard formulated a weak system of linear logic [3], light linear logic, in which the computation (normalization) through CurryHoward style encoding halts in the polynominal time of the size of input numerals (specific form of proofs). He noted that naive set theory can be carried out in this new system for the same reason as ours (normalization by induction on the size of proofs is possible). However, it does not seem to be clear how much extra expressive power is added by the set abstraction mechanism. On the other hand, we showed that one can explicitly construct a fixpoint in a linear (affine) set theory with the non-strict comprehension and the appropriate paring, and all total recursive functions are numeralwise representable in such a system [11]. We can expect the further progress of research in this direction. The second issue is the semantics of such systems. The standard ap proach to the semantics of set theoy based on a non-standard logic is to modify the Boolean-valued models with the appropriate algebra for the logic. The idea has been carried out in [9], but this construction does not yield the models for the unrestricted comprehension. The models for linear set theory with the unrestricted comprehension in terms of Kripke seman tics [7] and phase semantics have been studied. However, they are not easy to construct except as the term models. It is only very recent that we find a construction of a rather satisfactory model, combining the Scott-style in verse limit construction and coherence space semantics [12]. This result seems to be a good starting point for the further investigation of semantics.
References [1] F.B. Fitch. "A system of formal logic without an analogue of the Curry W-operator." Journal of Symbolic Logic, 1936. [2] J.Y. Girard. "Linear logic." Theoretical Computer Science, 50, 1987, 1-102. [3] J.Y. Girard. "Light linear logic." (manuscript). [4] V.N. Grishin. "A nonstandard logic and its application to set theory," (Russian). Studies in Formalized Languages and Nonclassical Logics (Russian), Izdat, "Nauka," Moskow. 1974, 135-171. [5] V.N. Grishin. "Predicate and set theoretic calculi based on logic without contraction rules," (Russian). Izvestiya Akademii Nauk SSSR
LINEAR S E T THEORY
245
Seriya Matematicheskaya, 45, no.l, 1981, 47-68. 239. Math. USSR Izv., 18, no.l, 1982, 41-59 (English translation). [6] L. Hallnas. On Normalization of Proofs in Set Theory. Dissertiones Mathematicae 261. Polska Akademia Nauk, Instytut Matematyczny, Warszawa, 1988. [7] Y. Komori. "Illative combinatory logic based on BCK-logic." Math. Japonica, 34, No. 4, 1989, 585-596. [8] H. Ono and Y. Komori. "Logics without the contraction rule." Journal of Symbolic Logic, 50, 1985, 169-201. [9] M. Shirahata. Linear Set Theory. Dissertation, Department of Philos ophy, Stanford University, 1994. [10] M. Shirahata. "A linear conservative extension of Zermelo-Fraenkel set theory." Studio, Logica, 56, 1996, 361-392. [11] M. Shirahata. "Fixpoint theorem in linear set theory." (in preparation). [12] M. Shirahata. "A coherence space semantics for linear set theory." (in preparation). [13] R.B. White. "A demonstrably consistent type-free extension." Mathematica Japonica, 32, 1987, 149-169. [14] R.B. White. "A consistent theory of attributes in a logic without con traction." Studia Logica, 52, 1993, 113-142. D e p a r t m e n t of M a t h e m a t i c s Keio University, Hiyoshi C a m p u s [email protected]
247
A Solution to a Problem of Marek and Truszcynski Su Kaile Inst. of Computer Science, Shantou Univ., Shantou 51603, PRC Chen Huowang Dept. of Computer Science, Changsha Inst.of Tech. Changsha, 410073, PRC Abstract We first show that every normal default theory is representable in the class of free normal default theories. This gives a new feature of extensions of normal default theories. By the above result an example of default theory is obtained which is not reprsentable in the class of normal default theories but has exactly infinitely countably many extensions and those extensions are pairwisely inconsistent. It gives a solution to a problem of W. Marek and M. Truszcynski and refutes a related assertion presented by them.
1
Introduction
Reiter's default logic is one of the most prominent formalization of nonmontonic reasoning. Recently the theory of representability for default logic was introduced by W. Marek and M. Truszcynski [2]. They introduced the notions of representability and equivalence for default logic. Two default theories (Di, W\) and (D2, W2) are said to be equivalent to each other, de noted by (DUW{) « (D2,W2), if ( £ > i , ^ i ) and (D2,W2) have the same extensions. A defaut theory (D, W) is said to be representable in a class of default theories if (D,W) is equivalent to a default theory in the class. We prove that every normal default theory is representable in the class of prerequisite-free normal default theories, and give a new essential fea ture distinguishing normal default theories from unrestricted ones. Using this feature we construct an example of default theory which has exactly infinitely countably many pairwisely inconsistent extensions, but is not rep resentable in the class of normal default theories. This gives a negative an-
248
K.
Su AND H.
CHEN
swer to the question of W. Marek and M. Truszcyriski ([2,p. 130]) whether the following theorem holds without the assumption that the extensions are finitely generated. T h e o r e m A [2, Corollary 5.10] If a defalt theory (D,W) has at least one extension and all the extensions for (D,W) are finitely generated, then the default theory is representable in M (the class of normal default theories). Moreover, the example mentioned above refutes an assertion of W. Marek and M. Truszcyriski([2,p.l29]) that if a default theory (D,W) has at least one and at most countably many pairwisely inconsistent extensions then the conclusion of the above theorem holds even without the assumption that extensions of (D,W) are finitely generated. For convenience, we introduce some definitions and notations of default logic. A detailed development can be found in [3,4]. A (closed) default theory (D,W) consists of a set W of first-order formulae, and a set D of defaults. A default is an expression of the ioimA:Bl£'Bk where A, C andf?;s are called the prerequiste, consequence and justifications of the de fault respectively. For a default S = A:B^-Bk, we set A = PRE(S) ,C = CONS(8). For a set D of defaults, we set CONS{D) = {CONS{6)\6 e D}. For an arbitrary default theory £ = (D, W), a set E of sentences is said to be an extesion of £ if E is a fixed point of the operator A^ which is defined as follows. For an arbitrary set S of sentences, A^(5) is the least set S' of formulae that satisfies:
(1)W c w {2)Th(S) = S (3) For every 6 = CeS1.
2
A:B
^-^
<E £), if A G S' and - £ l 5 . . . , -.£* g 5 , then
Normal default theories are representable in the class of free normal default theories
In this section we shall give the representablity result of normal default theories. In order to prove the result we first give the following definition and lemmas. Definition 2.1 For a default theory £ ;= (D, W) and a set S of sentences, a sequence {6i}(i < n) of defaults in D is said to be a 5-default proof in £, if 1. All the negations of justifications of 6{S are not in S.
A S O L U T I O N T O A PROBLEM O F M A R E K AND TRUSZCYNSKI
2. W U CONS({6j\j
< i}) \- PRE(Si)
249
for all i < n.
L e m m a 2.1 Let £ = (D,W) be a default theory, S a set of set of sentences. If {Si}i
2Th(THz(S))
cr^(S),
3 for every default then C G THz(S). 1. W C THz(S)
=
A:B
^-^
e D if A e TH^(S)
and - i # i , . . . , -.£* 0 5,
is trivially true.
2. Let y G THz(S). There exist Yi,..., yfc G TH^(S) {^i}t
and S-default proofs
yiA...Anhy, W u CCWS({^|z < m}) h yi,..., T^UCOiVS({^|z
250
K. Su AND H. C H E N
Since W U CONS({Si\i
< n) U {«'}) h C, we have C € T t f ^ S ) .
Our main result in this section is: T h e o r e m 2.1 Every normal default theory is representable in the class of free normal default theories (a default theory is called free normal if its all defaults are of the form ^ ). P r o o f Let (D,W) be an arbitrary normal default theory f. We shall prove that £ is equivalent to the free normal default theory £' = (D',W), where W =W, and D' is
{ : R B ° A A "; A R B ( n '" I 3({} - d. p. {Si}i
where "{}-d. p. {<$;} i< n " means "{}-default proof {^i}i
= Th(WU
{X\%
G D' and -.X' £ E}), it suffices to prove
that for all f G D ' if - I ^ £ , then X G A^{E). Suppose ''B^'.'AB^II and {8i}i
A S O L U T I O N T O A PROBLEM O F M A R E K AND TRUSZCYNSKI
251
A^(E) C Af(E). This gives that A^(E) = A^>(E) under the condition that A(.(E) = E or AC(E) =E.
3
A new feature of extensions of normal de fault theories and an example of default theories
It is well-known that, for every normal default theory, (1) there exists an extension and (2) two different extensions are inconsistent. W. Marek and M. Truszcynski showed that the two properties mentioned above, in several cases, are essentially the only two features distinguishing normal default theories from unrestricted ones. They gave the definition of finitely gener ated theory, and obtained some results such as Theorem A mentioned in the Introduction. But the question raised by W. Marek and M. Truszcynski, introduced at the beginning of this paper, remains open. This section we give a new essential feature of normal default theories. By this feature we will construct an example of default theory which has exactly infinitely count ably many pairwisely inconsistent extensions, but is not representable in the class of normal default theories. It gives a negative answer to the problem of W. Marek and M. Truzcynski and refutes the assertion of them in [2, p. 129] mentioned at the beginning of this paper. T h e o r e m 3.1 Let (D,W) be a free normal default theory, then for all extensions JE?I, E2 of (D, W) such that E\ / E2 there is a sentence a such that a € Ei and E2 U {a} is inconsistent, and for every extension E* of (D, W), either a G E* or E* U {a} is inconsistent. P r o o f Before we prove this theorem we first introduce a theorem, which can be found in [3]. T h e o r e m ( [ 3 , p.47]). For extensions E i , E2 there exist Di, D2 C D such that E{ = Th(W U CONS{Di)) ( i = 1,2). Suppose that E± ^ E2. Then E\ % E2, and, by the Theorem above, there is a default 6 = "jj- G D\ such that B 0 E2. Hence ->B G E2, otherwise B G A^(E2) = E2. So E2 U {a} is inconsistent. Moreover B G E\{ by ^ G D i ) , and for every extension E* of (D,W), B G E* if - . £ 0 E*, i.e. we have that B G E*, or -.B g E*. This completes the proof of the last claim. C o r o l l a r y 3.1 Let (D,W)
be a normal default theory, then for all
252
K. Su AND H. C H E N
extensions E\, E2 of (D,W) such that E± ^ E2 there is a sentence a such that a G E\ and E2 U {a} is inconsistent, and for every extension E* of (D, W), either a e E* 01 E* U {a} is inconsistent. P r o o f By the Theorem 2.1, every normal default theory is representable in the class of free normal default theories. E x a m p l e 3.1 The default theory (Do, Wo) is defined as follows: Wo = {qi -> Pi
^Pi\i^j}, O'ij
where #;S and pjS (i < j) and all a^s (i < j) are different atoms , and dji = -idij for i < j .
T h e o r e m 3.2 The default theory (Do, Wo) has exactly K0 (infinitely countably) many extensions and these extensions are pairwisely inconsis tent, however, is not representable in the class of normal default theories N. The above theorem can be proved directly by the two following lemmas and Corollary 3.1 . L e m m a 3.1 Let Ej = Th(W0 U {pj} U {a^i G u}) for all j G u. Then EjS are exactly all extensions of (D0,Wo) and E{ U Ej is not consistent if
^ZiP r o o f It is easy to prove that EjS are extensions of (Do, Wo) and EiUEj is not consistent if i ^ j . We only show that for an arbitrary extension E of (D0, Wo) there is a j such that E = Ej. First, we have ( by [3, Proposition 6.2.9]) E = Th(W0UCONS(GD(E))), where GD(E) is the set of all generating defaults for E, i.e. GD{E) = { 1 % G w, -np,- I E) U {?i-^\Pj € E, ^ a y i E,i,j € w}. Second, it is easy to see that there exists a j such that pj G E, otherwise -^f- (£ GD(E) for all i,j, and CONS('ff) $ Th(W0 U CONS(GD(E))), hence ^ 0 G D ( E ) for all j . Thus G£>(£) = {}, we have E = Th{W0), this contradicts that E is an extension. Finally, suppose pj0 G E, we shall show that E = £Jj0 by proving the following facts: p
Fact 1: 'fr £ GD(E) for j ± j 0 .
A S O L U T I O N T O A PROBLEM O F M A R E K AND TRUSZCYNSKI
If j ^ j 0 then qj —> ->pj0 G WQ C E. -i^- G E. Thus ^ £ GDE(E) is proved. F a c t 2: p , 0 £7 and hence ^ ^
Since Th(E)
253
— E, we have
g G£>(F) for all z ^ j 0 -
By Fact 1 E = Th(W0 U {p i o } U F ) where F C {a^|z, j G CJ}. If ^ G E for some j 7^ j 0 , then Wo U {pj 0 } U F \- pj. If we replace pj by 1_ (false), then W<5 U {Qi - - ± I* ^ j } U foj U F h l where WQ CWQ. Since gi —> -> _L is valid, we have WQ U {p JO } U F h l , this contradicts that E is consistent (since WQ is). Fact 3 : 'ff- G G£>(£), ^
^
G GL>(£) for all L
By Fact 1 and Fact 2, F C r/i(W 0 U te0} U { a ^ J i G UJ}). Since ^ 0 U {p j o } U {aijo\i eu}\/
^qjo,
and W0U{pjo}U{aijo\ieu}\/
-iCLij0 for all z,
we have and dij0 $ E for all i. Thus '-^
G GD{E) and
Pjo
' aijo
G GD(E) for all t,
^From these facts, it is easy to see that GD(E) = { ^ }
U {Pj0
Pjo
: aijo a
\i G a;}.
iJo
Hence F = Th(W0 U { f t o } U {aij0\i G w}) = F j o . L e m m a 3.2 Let EjS be as in Lemma 3.1, C an arbitrary sentence such that C G EQ and ->C G 2?i. Then there is an Ei such that C & Ei and
P r o o f Suppose that C be a sentence such that C £ EQ and ->C G E\. There exists a number k such that C contains no atoms in {Pj\k< j}U{dij\i
<j,k<j}
254
K.
Su AND H.
CHEN
Let j be a number greater than k. We shall prove WoU{pj}U{dij \i £ CJ} 1/ C. Otherwise, assume that W0U{Pj}U{ai:J\i<Ecu}\-C, i.e. Wo U {pj} U {a{j\i < j} U {^aj{\j
< i} h C.
If we substitute T for a^(z < j), and by _L for aji (j < i), we have Wo U {pj} h C. If substitute T for p j , we obtain that Wo U {% -> T|z # j } U { T } h C where ^ C W 0 . Then W0 H C. This contradicts that ->C 6 i?i. Thus we have that C g Ej. By the same reason, -iC £ JE?J- for all j >
k.
Our results show that in some cases existence of extensions and incon sistency of two different extensions are not essentially only features distin guishing normal default theories from unrestricted ones. References [1] R. Reiter, A logic for default reasoning, Artif. Intell. 12, 81-132, 1980. [2] W. Marek and M. Truszcynski, Nonmonotonic Logic: Context- De pendent Reasoning, Spring-Verlag, Heiderberg, 1993. [3] P. Besnard, An introduction to default logic, Springer-Verlag, Berlin, 1989. [4] D. W. Etherington, Reasoning with Incomplete Information, Pitman, London, 1988.
255
Credulous Reasoning About Defaults* Yao-Hua Tan*
Leendert W.N. van der Torre*
Abstract In this paper we propose a preference-based conditional logic for cred ulous reasoning about defaults. A conditional default "if f3 then by default a" is either formalized by the strong preference "(3 A -*a is not preferred to or equivalent to /3 A a", or by the weak preference "the preferred (3 is an a". We show t h a t these two expressions, instances of what we call the ordering and minimizing usages of preference orderings, can be considered as duals of each other. Moreover, we give a formalization of ordering and minimizing in Boutilier's modal logic CT40 and we show how to combine them in a two-phase default logic.
1
Introduction
C o n d i t i o n a l logic is a p o p u l a r framework t o formalize defeasible reason ing [Del88, A M 9 1 , Alc94, B o u 9 4 a , Mor95]. T h e c o n d i t i o n a l s e n t e n c e "if /3 ( t h e a n t e c e d e n t or condition) t h e n by default a ( t h e consequent or conclu sion)" is r e p r e s e n t e d in t h i s a p p r o a c h by t h e formula j3 > a , w h e r e ' > ' is t h e i m p l i c a t i o n of c o n d i t i o n a l logic. T h e p o p u l a r i t y of this framework is b a s e d on t h e lack of s t r e n g t h e n i n g of t h e a n t e c e d e n t of t h e c o n d i t i o n a l implica t i o n , which is used t o formalize t h e specificity principle. For e x a m p l e , from t h e ' b i r d s fly' default b > f t h e ' p e n g u i n s fly' default (p A b) > f c a n n o t b e derived. However, t h e lack of s t r e n g t h e n i n g of t h e a n t e c e d e n t also h a s d r a w b a c k s , in p a r t i c u l a r t h e so-called irrelevance p r o b l e m . For e x a m p l e , from t h e ' b i r d s fly' default b > f t h e 'red birds fly' default (r A b) > f c a n n o t b e derived. In t h e framework, t h e set of defaults derivable in t h e c o n d i t i o n a l logic is called t h e 'conservative core' a n d conditional formulas a r e a d d e d t o t h i s core by D e l g r a n d e ' s irrelevance principle [Del88] or m e c h a n i s m s equiv alent t o s y s t e m Z [Pea90] like r a t i o n a l closure [LM92], t h e m i n i m u m speci ficity principle [BDP92] or B o u t i l i e r ' s 'only k n o w i n g ' c o n s t r u c t i o n [Bou92]. *This research was partially supported by the ESPRIT III Basic Research Project No.6156 DRUMS II and the ESPRIT III Basic Research Working Group No.8319 MODELAGE. tErasmus University Research Institute for Decision and Information Systems (EuRIDIS). Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Nether lands. E-Mail: {ytan,ltorre}@@euridis.fbk.eur.nl. Tel: (+31)10-4082601. Fax: (+31)104526134. Http://www.euridis.fbk.eur.nl/Euridis/welcome.html.
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T A N AND VAN DER T O R R E
However, these solutions suffer from a specific instance of the irrelevance problem called the inheritance problem. For example, given that penguins are exceptional birds b > ->p, from the 'birds have wings' default b > w the 'penguins have wings' default (b Ap) > w cannot be derived by system Z. In this paper, we propose a conditional logic for credulous reasoning about defaults, i.e. a logic in which the two defaults T > p and T > ->p can consistently exist together, where T stands for any tautology. Because it is a credulous logic, we can accept strengthening of the antecedent to solve the irrelevance and inheritance problems. Obviously, we loose the possibility to formalize specificity by accepting unrestricted strengthening of the antecedent. However, the following example shows another problem with unrestricted strengthening of the antecedent. E x a m p l e 1 (apples and pears problem) Consider the defaults that (1) you normally buy apples or pears, and (2) you normally do not buy apples. It is counterintuitive to derive that if you buy apples, then you normally buy pears. Assume a conditional default logic that validates at least substitu tion of logical equivalents and the following Gentzen-style inference patterns Strengthening of the Antecedent (SA), Weakening of the Consequent (wc) and Conjunction (AND). P l > a
SA •
' w c
:
( A A A ) > Q 77-^7
x
P > {ax V a2) (3 > au(3 > a2
AND : ^
T-^-
p > ( a i A a2)
Furthermore, assume as premises the defaults T > ( a V p ) and T > ->a, where a can be read as "buying apples" and p as "buying pears". The intu itive default T > (->a Ap) can be derived by AND. $From the latter default, the default a > (-»a A p) can be derived by SA. Unfortunately, from this default, the counterintuitive default a > p can be derived by WC. This de fault is considered to be counterintuitive, because it is not grounded in the premises. If a is true, then the first premise is fulfilled and the second one is violated. This inference can be blocked by replacing unrestricted strength ening of the antecedent by the following version of restricted strengthening of the antecedent,1 in which O is a modal operator and 0 is true for all consistent propositional formulas (j).
ft >a,P(ftAftAa) ^ _ 1
(A A ft) > a
T h i s restriction in strengthening of t h e antecedent is uncontroversial, because t h e expression p > q with p A q inconsistent is counterintuitive for logics of defeasible rea soning. However, in some other usages of conditionals and preference logics like deontic logic, such expressions do have an intuitive (and i m p o r t a n t ) reading, see e.g. [TvdT96].
257
CREDULOUS REASONING
The default a > (->a A p) cannot be derived from the default T > (-ia A p) by RSA. Unfortunately, the counterintuitive a > p can still be derived in another way. $From the intuitive default T > (->a A p) the intuitive T > p can be derived by w c . $From this latter obligation, the counterintuitive a > p can be derived by RSA. Both derivations are represented in Figure 1.
T > (a V p) T > -na AND/
T > (a V p) \
T > -.a
AND,
T > (-ia Ap)
T > (naAp)
I SA
I WC
a > (->a Ap) I WC a >p
T > p I SA / RSA a> p
Figure 1: apples and pears problem The apples and pears problem can be solved by a technique, which might look odd at first sight, but which turns out to work well, namely to forbid application of RSA after WC has been applied. This means that in deriva tions first RSA has to be applied, and only afterwards WC may be applied. We call this the two-phase approach in default logic. Such a sequencing in derivations is rather unnatural and cumbersome from a proof-theoretic point of view. Surprisingly, the two-phase approach can be obtained very intuitively from a semantic point of view. In this paper we show that the two-phase approach can be obtained by combining two usages of a preference ordering in a preference-based seman tics of a default logic. For the two usages we define two different types of default conditionals, which we call type-1 and type-2 defaults. The two types of defaults correspond to two different ways to evaluate formulas in a preference ordering. Type-1 defaults are formalized by strong preferences and evaluated by what we call Ordering, a process in which the whole order ing is used to evaluate a formula. Type-2 defaults are formalized by weak preferences and evaluated by what we call Minimizing, in which the ordering is used to select the minimal elements that satisfy a formula. The minimiz ing approach is commonly taken in preferential semantics for non-monotonic logics, see for example [Sho88, KLM90, Mak93, Bou94a]. In this paper, we
258
T A N AND VAN DER T O R R E
consider credulous reasoning about defaults. In such reasoning, there can be several equivalence classes of preferred models. Hence, the definition of a type-2 default only considers truth in an equivalence class of preferred models instead of truth in all preferred models. In the two-phase approach, the first phase corresponds to ordering, and the second phase corresponds to minimizing. In semantic terms the two-phase approach simply means that first a preference ordering has to be constructed by ordering worlds, and subsequently the constructed ordering can be used for minimization. In this paper we formalize type-1 and type-2 defaults in Boutilier's modal preference logic CT40 [Bou92]. The logic CT40 and the minimizing ap proach are well-known, but the logic of ordering and the two-phase approach to defeasible reasoning introduced in this paper are new. This paper is organized as follows. In Section 2 we give the preference logic in which we formalize type-1 and type-2 defaults as strong and weak preferences. We show that these preferences are instances of the order ing and minimizing usages of preference orderings. In Section 3 we show how ordering and minimizing can be combined in a two-phase default logic and how this solves the problems of Example 1. Finally, in Section 4 we briefly consider the sceptical case and in Section 5 we mention some related research.
2
A logic for reasoning about defaults
Preference-based default logics are default logics of which the semantics contains a preference ordering (usually on worlds of a Kripke style possible world model). This preference ordering reflects different degrees of 'nor mality': a world is preferred to another world if it is, in some sense, more normal than the other world. For example, a value can be associated with each world; in such cases, the ordering is connected (for all W\ and w2 we have wi < w2 or w2 < w\). However, in general the preference ordering can be any partial pre-ordering. Hence, only reflexivity and transitivity are assumed. An expression "by default p" is expressed by a preference for p, which may mean that 1. "p is preferred to ->p regardless of other things", or that 2. "p is preferred to ->p other things being equal", or 3. some intermediate reading. Many authors (for example [TP94, Bou94a]) take the second (ceteris paribus) reading, because the first reading does not allow for two or more uncondi tional preference statements to exist consistently together, as observed by
CREDULOUS REASONING
259
von Wright in [vW63]. For example, the preferences for p and q will quickly run into conflict when considering the worlds p A ->q and ->p A q.
2.1
Ordering
In this paper, a strong preference for p means that "->p is not preferred to or equivalent to p, regardless of other things". Obviously, in a connected partial pre-ordering, the expression "-■/? is not preferred to or equivalent to p" is equivalent to "p is preferred to -ip". Hence, for connected orderings, which are quite popular in preferential semantics, our reading has the problem described by von Wright. However, we do not restrict ourselves to connected orderings, but we allow any partial pre-ordering. With such orderings, the preferences for p and q will not run into conflict when considering the worlds pA^q and -^pAq: these worlds are only incomparable. Notice that the whole ordering is taken into account when a default is evaluated. That is why we call it the ordering approach to default logic. Similarly, a conditional default "if q then by default p" is represented by "no -tp A q is preferred to or equivalent to some p A q, regardless of other things". The preferences are formalized in Boutilier's logic CT40, for the details and completeness results of this logic see [Bou94b]. CT40 is a bimodal propositional logic of inaccessible worlds. Boutilier notes that many appli cations of preference logics (like the one we describe below) do not need the complexity of inaccessible worlds, but it makes the definitions easier and the semantics clearer. Definition 1 (Syntax of C T 4 0 ) The logic CT40 is a bimodal system with the two normal modal connectives □ and □. The dual 'possibility' connectives are defined as usual: O a =def ~ , a - , a and O a =def ~" n "■a. Moreover, the two following modal connectives are defined: □ a =def HCYA □ a and O a =def OaV O a. CT40 is axiomatized by the following set of axioms and inference rules. K
D(a - > / ? ) - * (Da -► □/?)
K'
□ (a -> p) -> ( □ a - > □ p)
T 4 H
Da-^a Da^ DDa 5 (DaA B 0) -+U (a V p)
Nee MP
0From a infer □ a £ From a —> (3 and a infer f3
(Semantics of C T 4 0 ) Kripke models M = {W, <, V) for CT40 consist of
260
T A N AND VAN DER T O R R E
W, a set of worlds, <, a binary transitive and reflexive accessibility relation, and V, a valuation of the propositions in the worlds. The modal operator □ refers to accessible worlds and the modal operator □ to inaccessible worlds. M, w \= Da iff W M,w | = 6 a iff W
eWifw'<w, eWifw'^w,
then M,w' \= a then M,w' \= a
Given this modal preference logic, we define type-1 defaults as strong preferences. Definition 2 Type-1 defaults "if (3 then by default a", written as (5 > a, are defined as follows. P>a=defU((0Aa)^O(p^a)) Intuitively, a default q > p expresses a strict preference of all p A q over -ip A q. The following proposition shows that this preference is represented by a negative condition: no ->p A q is preferred to a p A g , P r o p o s i t i o n 1 LetM = (W,<,V) be a CT40 model. M,w\= p> a iff for all w\,W2 G W such that M,W\ f= p A ->a and M,W2 \= P A a, it is true that W\ -£ W2 ■ P r o o f => Assume a model M = (W,<,V) with two worlds wi,W2 G W such that M,wi (= P A -*a, M,w2 \= P A a and wi <w2. We have M, w2 \fc (P A a) —► D(/3 —> a)). Hence, M,w ^ p > a for some world w G W, because from the semantic definitions follows immediately that M, w )=□ cj> iff for all worlds w' £ W, M,w' |= (/>. <= Assume M,w ^ p > a for some world w. Hence, there is a world w2 eW such that M,w2 ^ (P A a) —► □(/? -> a)). It follows that M,w2 \= P f\a and there is a world w\ G W such that M, W\ |= P A -«a and w\ < w2 • Notice that the normality ordering is global (in the sense that the nor mality ordering is not relative to a world) and nested operators therefore do not have an intuitive reading, although they have a formal meaning in CT40. The following example illustrates the definition of type-1 defaults as strong preferences. E x a m p l e 2 Let \a\ denote a world that satisfies a. Assume models that consist of four worlds \p/\q\, \~>pAq\, \pA-*q\ and \^p/\->q\. A model satisfies the default T > p when neither \-*pAq\ nor |—ipA —>g| is preferred to \pAq\ or \pA~*q\. For example, the Kripke model \p A q\<\p A-*q\<\-*p A-*q\<\-*p A q\ in Figure 2 is a model for T > p but not for T > q. Note that T > q is not true, because |-rp A ^q\<\^p A q\ and \p A-*q\<\-*p A q\. This shows how in the ordering approach the whole ordering is taken into account in the evaluation of a formula, and not just the most preferred \p A q\ worlds.
CREDULOUS REASONING
261
Figure 2: Preference relation with four worlds In the beginning of this section, we mentioned von Wright's problem that absolute preferences for p and q are not mutually consistent, and the solution in our logic. This solution is illustrated in the following example. E x a m p l e 3 Let M be a model of CT40 that satisfies the two defaults T > Vi and T >p2 and contains \Pl A ^p2\ and \-*Pl Ap2\ worlds. The \Pl A ~-p22 and | -npx A p2 \ worlds of M are incomparable, because the first is more normal than the latter with respect to default T > Pl, but less normal with respect to default T > p2. An example of such a model M is represented in Figure 3.
Figure 3: Preference relation with von Wright's problem The following proposition gives several properties of the type-1 defaults. P r o p o s i t i o n 2 The logic CT40 validates the following theorems o/Strengthening of the Antecedent (SA), Conjunction (And) and Disjunction ( O r ) , and a version of Transitivity (Trans'). SA And Or Trans'
A (p (0 (7
> > > >
a-+(0! A fa) >a ax A 0 > a2) -+ 0 > (<*i A a2) ai A 0 > a2) -+ 0 > (a x V a2) 0 A 0 > a) -► 7 > (a A 0)
262
T A N AND VAN DER T O R R E
The logic CT40 does not validate the following theorems of Weakening of the Consequent ( W C ) , Transitivity (Trans) and the sceptical axiom (D). WC Trans D
/3>c*i - > / 3 > (c*i V a 2 ) (i>(3A/3>a)-+'y>a -.(/? > a A fi > -.a)
P r o o f T/ie (non)theorems
can easily he verified by proving
(un)derivability
in CT40. The validity of strengthening of the antecedent follows from the fact that a strong preference of p over ->p implies a preference of p A q over -ip A #. The most remarkable property of the logic is the invalidity of weakening of the consequent. Intuitively, the lack of weakening of T > p to T > (pV q) is the consequence of the fact that T > p expresses a preference of all p over -*p, because from such a preference does not follow that p V q is always preferred to ->p A -iq. This is illustrated by the following example. E x a m p l e 4 Reconsider the Kripke model in Figure 2. The model satisfies T > p but not T > (p\l q). This illustrates that the type-1 defaults do not have strengthening of the antecedent. Transitivity, expressed by the theorem T r a n s ' , is an unusual and re markable property for default logics. The following example illustrates this property. E x a m p l e 5 Let S he the set of conditional defaults {T > a , a > i), where a can be read as ca certain man going to the assistance of his neighbors' and t as 'telling the neighbors that he will come \ Hence, the two defaults can he read as 'a certain man normally goes to the assistance of his neighbors \ and l ifhe goes, then he normally tells them he is coming'. The default T > (aAt) can be derived from S with Trans', which expresses that normally, the man goes to the assistance of his neighbors and he tells them he is coming.
2.2
Minimizing
Type-2 defaults are defined as weak preferences in the modal preference logic. In this paper, we consider credulous reasoning about defaults. In credulous reasoning, there can be several 'extensions' or equivalence classes of preferred models. Hence, the definition of a type-2 default only considers truth in an equivalence class of preferred models instead of truth in all preferred models. For the details of this definition, see [Bou94b]. 2 2 Boutilier makes a conditional true if the antecedent is false, i.e. he defines (3 >a a =def □ -i/3V O (f3 A □(/? —> a)). Moreover, the definition can deal with infinite chains.
263
CREDULOUS REASONING
Definition 3 Type-2 defaults "if 0 then by default a", written as 0 > 3 a, are defined as follows. 0>3a=def$(0An(0->a)) The default q > 3 p is true in a model if p is true in an equivalence class of most preferred \q\ worlds of the model. Hence, the default q >3 p refers to the preferred worlds where q is true, and T > 3 p refers to the most preferred worlds. The following example illustrates the definition of type-2 defaults and compares it with type-1 defaults. E x a m p l e 6 Reconsider the Kripke model in Figure 2. The model satisfies T >3 p and T >3 q, whereas T > q is not true in this model. Since T >3 q is equivalent to O Uq it is clear that q has to be true in some most preferred | T | worlds, and also that less preferred | T | worlds do not effect the truth of O Uq. Hence, in the evaluation of T >3 q only preferred elements are taken into account and not the whole ordering. The main properties of the logic are given by the following proposition. It illustrates that ordering and minimizing are duals, as far as we consider the properties strengthening of the antecedent and weakening of the consequent. P r o p o s i t i o n 3 The logic CT40 validates the following WC3
0 >s CLI -> P >3 (ai V a2)
The logic CT40 does not validate the following SA3 AND3 DD3 D3
theorem.
theorems.
0X >3 a-+{0! A fo) >3 a 0 >3 ax A 0 >3 a2 -» 0 >3 («i A a2) 7 >3 0 A 0 >3 a -> 7 > 3 OL ^(0>3aA0>3^a)
P r o o f The (non)theorems in CT40.
can easily be verified by proving
(un)derivability
The logic validates weakening of the consequent, because the most pre ferred world that satisfies pi also satisfies pi V p2- However, the logic does not have strengthening of the antecedent of T >3 p to q >3 p, because the preferred | T | worlds may be different from the preferred \q\ worlds. This property is illustrated by the following example. E x a m p l e 7 Reconsider the Kripke model of Figure 2. The model satisfies T >3 q but not -ip >3 q. For example, ~>p >3 q is false because the preferred \->p\ worlds are the \->pA-^q\ worlds. Hence, >3 does not have strengthening of the antecedent.
264
3
T A N AND VAN DER T O R R E
Two-phase approach to default logic
3.1
Combining ordering and minimizing
The idea of combining ordering and minimizing is to combine formulas with > and >3 operators, where ordering should be strictly stronger than min imizing. However, this combination is not satisfactory in the logic CT40, because we cannot derive (3 >s a from (3 > a. The following proposition gives the relation between the two operators. P r o p o s i t i o n 4 The logic CT40 validates the following
theorem.
(3 > aA $ (0 A a) -> (3 >3 a P r o o f The theorem can easily be verified by proving derivability in CT4O. It is equivalent to the following formula: (B ((/? A a) -► U(p -+ a ) ) A O ((3 A a)) - + 0 (f3 A D(/3 -> a)) Hence, when O (aA/3) is false, then ordering is not stronger than minimizing. For example, it can easily be verified that the logic CT40 validates the theorem a > _L but it does not validate a >s _L. In the following definition, (3 >c a has an additional condition which works like a 'consistency check' to test whether a A (3 is possible. Definition 4 Consistent type-1 defaults "if (3 then by default a", as f3 >c a, are defined as follows.
written
(3>c a =def (3 > aA O (/3 A a) This new type of ordering is strictly stronger than minimizing, as is shown in the following proposition. P r o p o s i t i o n 5 The logic CT40 validates the following
theorem.
(3 >c a -► /3 >3 a P r o o f Follows directly from Definition 4 and Proposition 4The type-1 defaults (3 >c a validate weaker versions of the theorems of Proposition 2, like for example the following Restricted Strengthening of the Antecedent ( R S A ) and Restricted Conjunction (RAnd). RSA RAnd
ft
> c aA O (f31 A f t A a ) - ^ (ft A ft) > c a P>c ^Ap>c
a 2 A 5 (/? A ax A a2) -> P>c (^ Aa2)
CREDULOUS REASONING
265
We already saw restricted strengthening of the antecedent in Exam ple 1 in the introduction. We elaborate on Example 1 in Section 3.2. To strengthen the theorems above, we consider only models in which all propositionally satisfiable formulas (/) are true in some world. This can be 'axiomatized' with Boutilier's axiom scheme L P , see [Lev90, Bou94b] for a discussion. The axiom scheme L P states that every formula (j) without any occurrences of modal operators, which is propositionally satisfiable, is true in some world. Definition 5 The logic CT40* is CT40 extended with the following scheme: L P : 0 (j> for all satisfiable propositional
axiom
We write |= for logical entailment in CT4O*. The following example illustrates the logic CT40* and the idea of com bining ordering and minimizing. E x a m p l e 8 Let S be the set of defaults {T > c (->r A ~^g),g >c r,r >c g}. The intended model is given in Figure 4- We have S ^ T > c -»r, S ^ T > c -\g, S |= T >3 -ir and S (= T >3 ->g.
ordered less preferred situations
Figure 4: Preference relation
3.2
The two phases in a default logic
The two phases in a default logic correspond to the two different kinds of defaults > c and >g. Semantically, the first phase corresponds to ordering (> c ) and the second phase to minimizing (>g). /.From a proof theoretic point of view, the first phase corresponds to applying valid inferences of > c like RSA, RAND etc, and the second phase corresponds to applying valid inferences of >a like w c . The basic technique of default logic as a two-phase logic is that a conclusion of the form (3 >3 a can be derived either with or without P >c a. In the first case 0 > 3 a can be derived via 0 >c a with
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Proposition 5, which says that the latter formula implies the first one. If so, we say that (3 >3 a is derived in the first phase. In the second case we say that (3 >s a. is second phase derived. The important difference is that in the first phase we can apply RSA to (3 >3 a, because of the simultaneous occurrence of /3 >c a. We apply RSA to (3 >c a to obtain, for example, ((3 A 7) > c a, and then due to Proposition 5 we also obtain ((3 A 7) >s a. If (3 >c a does not occur simultaneously with (3 >3 a, then there is no way we can apply RSA to this formula. Being a minimizing formula it lacks RSA. Hence, once (3 >3 a has been derived in the second phase, we loose RSA permanently for subsequent derivations of this formula. Analogously, we can say that (3 >s a is first phase or second phase entailed by a set of premises, depending on whether S does or does not entail 0 >c a. The following example shows that the two-phase approach solves the apples and pears problem of Example 1 in the introduction. E x a m p l e 9 (apples and pears problem, continued) Let S = {T > c ( o V p ) , T >c -ia}, where ->a does not entail the negation of p. We have S | = 0 (-.a A p), S \= T > c (-.a A p) and S \= T > 3 (10 Ap), S fcT >c p and S \= T >3 p. The crucial observation is that a >3 p is not entailed by S. First of all, a > 3 p is not first phase entailed by S via T >g p, because T >3 p is not first phase entailed by S. Secondly, a >3 p is not second phase entailed by S via T >3 p either, because in second phase entailment >3 does not have strengthening of the antecedent at all. Thirdly, it is not second phase entailed by S via a first phase derivation of a >3 (~c (-iaAp) is not entailed by T > c (-iaAp) due to the restriction in RSA.
4
Further research
Preference semantics are used in several different areas. In fact, the notions 'ordering' and 'minimizing' were introduced in a deontic logic based on pref erences [TvdT96]. Although there are various subtle differences between these areas, see [Mak93] for a survey, we think that (at least) the following two discussions within deontic logic are relevant for default logic. First of all, the discussion about the validity of the deontic D axiom (see e.g. [TvdT96]) is relevant for sceptical reasoning about defaults. Secondly, defeasible de ontic logic [vdTT95] and its multi-preference semantics [TvdT95] can be relevant to formalize specificity. We can consider the two-phase approach with sceptical reasoning about defaults. The obvious candidate for the sceptical type-2 default is "a condi tional a is true in all most preferred \(3\ worlds", which we write as (3 >v OLThe following definition of this default in CT40* is from [Bou94b]. Definition 6 Type-2 sceptical defaults "if (3 then by default a ", written as
CREDULOUS
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REASONING
(3 >v ex, are defined as follows. P>W
=defQ (0 -> O(0 A D(/3 -> a)))
However, to combine > v with > c , the latter has to be strictly stronger than the other. The following example illustrates that this type-2 default cannot satisfactorily be combined with the type-1 default 0 >c a, because the ordering of worlds can be too weak. E x a m p l e 10 Consider the default T >c p. All models that satisfy \p\£\-*p\ are models of T > c p. Hence, \ p \ worlds and \ ->p \ worlds are either incomparable, or \p\ worlds are strictly preferred to | -ip| worlds. Let M be a model in which all \p\ and \->p\ worlds are incomparable. M satisfies T > c p, but it does not satisfy T > v p. Hence, T > v p is not entailed by T >c p. For minimization, we only want the models in which \p\ worlds are strictly preferred to \~>p\. A solution of the previous problem is to define a preference ordering on models, which prefers models which are maximally connected with respect to the partial pre-ordering < , i.e. with the most binary relations of <. The preferred models of this ordering are the only models which are used for minimization. 3 Definition 7 Let Ml = {Wu Ru < l 3 Vi) and M2 = {W2,R2,<2,V2) be two CT40* models. M\ is preferred to M2 for mapping r, written as M\ C r M2,
iff: 1. r is a one-to-one mapping of the worlds of W2 to the worlds of W\ such that the worlds satisfy the same propositions, and 2. If iv 1 <2 W2 for w\,w2
G W2 then T(WI) < I
T(W2).
We write Mi \ZT M2 iff Mi C T M2 and M2 g r - i M\. The ordering on models (C) should not be confused with the ordering on worlds ( < ) . The ordering on models is a technical trick to ensure that the worlds within a model are maximally connected, whereas the ordering on worlds expresses the normality ordering. Given the preference ordering on models, we can define a notion of preferential entailment, see [Sho88, KLM90]. 3 I n [Bou92], maximally connected models related t o system Z are models in which only t h e premises are known, a n d are formalized with Levesque's 'only knowing' (alias 'all-Iknow') o p e r a t o r [Lev90]. However, system Z defines a unique preferred model, whereas in our case t h e r e are m a n y distinct preferred models. Hence, we cannot simply copy this 'only knowing' concept.
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Definition 8 Let M = (W, R,<,V) be a model and S be a set of sentences. A world w e W of M preferentially satisfies S, written as M,w \=n S iff M,w \= S and there is not a model M' and a mapping r such that M',T(W) \= S and M' CT M (M is a preferred model of S). S preferentially entails (j), written as S \=^ $> iff for &M M and w, if M,w \=^ S then M,w \= (j). The following example illustrates the notion of preferential entailment. E x a m p l e 11 Reconsider the set of defaults S — {T > c (-ir A -*g),g >c r,r >c g} of Example 8. Without the preference ordering on models, the |-ir A -i01 and \r A g\ worlds could be incomparable. Such a model M would still satisfy M \= T >3 ->r but it would not satisfy M \= T >v - r . Preferential entailment is a typical mechanism from non-monotonic rea soning. The combination of ordering and minimizing is non-monotonic, as the following example illustrates. E x a m p l e 12 Let S = {T > c p) and S' = {T > c p, T > c -xp). We have S \=\2 T >v p and y ^ c T > v p . Hence, by addition of a formula we loose conclusions. Unfortunately, the previous example also shows that ordering is not stronger than minimizing, a condition for the two-phase approach. For this reason, we define a new type-1 default >r> for the sceptical case, which trivially satisfies a counterpart of Proposition 5. Definition 9 Type-1 sceptical defaults "if (3 then by default a", written as /3 >D a, are defined as follows.
/3 >D a =def /3 > a A ft > v & The two-phase approach with >D and >v works exactly like the twophase approach with > c and > 3 . The following example illustrates several properties of this new type-1 default. E x a m p l e 13 The set of defaults {T >D p, T >£> q) is consistent, but the set of defaults {T >£) p, T >D ^p} is inconsistent. Moreover, the set of defaults {T >£> p, q >D ~ip} is inconsistent too. The latter example shows that the specificity principle is not formalized in the proposed two-phase defeault logic. An interesting approach to this problem is the multi preference framework proposed in [TvdT95].
CREDULOUS REASONING
5
269
Related research
An example of a default logic with ordering and minimizing is the dynamic interpretation of defaults in Veltman's preference-based default logic [Vel91]. For example, in his logic the formula T [normally p] \= presumably (p V q) means that after ordering all worlds by preferring p to -ip, p V q is true in the preferred worlds. A distinction between the logic proposed in this paper and Veltman's logic is that in the latter, the order of the defaults influences the set of derivable formulas. Moreover, in Veltman's logic defaults are not conditionals and ordering and minimizing are not modeled in the same object language. Two phases can be traced in several popular non-monotonic logics like Reiter's default logic, autoepistemic logic and circumscription, which we illustrate by Reiter's default logic [Rei80]. Consider default theories con sisting of a set of normal default rules — , which express that a is part of an extension (a deductively closed set of formulas) if (3 is part of the exten sion and -ia is not, and a factual sentence (for simplicity we assume that the facts can be represented by a single formula). For example, the 'birds fly' default rule ^j- expresses that / is part of an extension if b is part of the extension and ->/ is not; hence, birds are assumed to fly unless there is knowledge of the contrary. Type-1 and type-2 defaults can be identified in Reiter's default logic as follows. 1. For type-1 defaults, assume a fixed factual sentence. The 'birds fly' default ^ is stronger than the 'red birds fly' default 6AT:^ in the sense that if a default theory contains the first default, then the second one can be added to the default theory without changing its set of extensions. 2. For type-2 defaults, assume a fixed set of normal Reiter defaults. If the factual sentence of a default theory is b and one of the extensions contains / , then this does not imply that the default theory with facts b A p has an extension that also contains / (e.g., with default rules ^ £ ->p
and
MZEZ). f '
In this paper we proposed a logic for credulous reasoning about both types of defaults. In the vocabulary of a logic for reasoning about type-1 defaults, the 'red birds fly' default is derivable from the 'birds fly' default. The example illustrates that ^ A J : a can be derived from ^ , which means that normal Reiter defaults validate strengthening of the antecedent. Simi larly, it can be shown that ^ " V 7 cannot be derived from ^ p , which means that normal Reiter defaults do not validate weakening of the consequent. In contrast, type-2 defaults do not validate strengthening of the antecedent (of b to b A p in the example above), but they do validate weakening of the consequent (because extensions are deductively closed). In this paper,
TAN AND VAN DER TORRE
270
we combined type-1 and type-2 defaults by the additional condition that type-1 defaults are strictly stronger than type-2 defaults. For Reiter's de fault logic, this additional condition means (in the credulous case) that for every default theory with default rule ^ and factual sentence f3, there is an extension that contains a. Hence, the reasonable additional condition means that for every default rule ^ , /3 A a is consistent.
Acknowledgement Thanks to Patrick van der Laag for several discussions on the issues raised in this paper.
References [Alc94]
C. E. Alchourron. Philosophical foundations of deontic logic and the logic of defeasible conditionals. In Deontic Logic in Computer Science: Normative System Specification, pages 43-84. John Wi ley & Sons, 1994.
[AM91]
N. Asher and M. Morreau. Common sense entailment: a modal theory of nonmonotonic reasoning. In Proceedings of the Twelfth International Joint Conference on Artificial Intelligence (IJCAV91), Palo Alto, 1991. Morgan Kaufman.
[BDP92]
S. Benferhat, D. Dubois, and H. Prade. Representing default rules in possibilistic logic. In Proceedings of the Second Inter national Conference on Principles of Knowledge Representation and Reasoning (KR'92), pages 673-684, Cambridge, MA, 1992.
[Bou92]
C. Boutilier. Conditional logics for defeault reasoning and be lief revision (phd thesis). Technical Report 92-1, Department of Computer Science, University of British Colombia, 1992.
[Bou94a] C. Boutilier. Conditional logics of normality: a modal approach. Artificial Intelligence, 68:87-154, 1994. [Bou94b] C. Boutilier. Unifying default reasoning and belief revision in a modal framework. Artificial Intelligence, 68, 1994. [Del88]
J.P. Delgrande. An approach to default reasoning based on a firstorder conditional logic: revised report. Artificial Intelligence, 36, 1988.
[KLM90] S. Kraus, D. Lehmann, and M. Magidor. Nonmonotonic reason ing, preferential models and cumulative logics. Artificial Intelli gence, 44:167-207, 1990.
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[Lev90]
H. Levesque. All I know: a study in autoepistemic logic. Artificial Intelligence, 42:263-309, 1990.
[LM92]
D. Lehmann and M. Magidor. What does a conditional knowledge base entail? Artificial Intelligence, 55:1-60, 1992.
[Mak93]
D. Makinson. Five faces of minimality. Studia Logica, 52:339-379, 1993.
[Mor95]
M. Morreau. Allowed consequence. In Proceedings of the Four teenth International Joint Conference on Artificial Intelligence (IJCAF95). Morgan Kaufman, 1995.
[Pea90]
J. Pearl. System Z: a natural ordering of defaults with tractable applications to default reasoning. In Proceedings of Theoreti cal Aspects of Reasoning about Knowledge (TARK), San Mateo, 1990. Morgan Kaufmann.
[Rei80]
R. Reiter. A logic for default reasoning. Artificial 13:81-132, 1980.
[Sho88]
Y. Shoham. Reasoning About Change. MIT Press, 1988.
[TP94]
S.-W. Tan and J. Pearl. Specification and evaluation of pref erences under uncertainty. In Proceedings of the Fourth Inter national Conference on Principles of Knowledge Representation and Reasoning (KR'94), pages 530-539, 1994.
Intelligence,
[TvdT95] Y.-H. Tan and L.W.N. van der Torre. Why defeasible deontic logic needs a multi preference semantics. In Proceedings of the ECSQARU'95. Lecture Notes in Artificial Intelligence 946. Springer Verlag, 1995. [TvdT96] Y.-H. Tan and L.W.N. van der Torre. How to combine ordering and minimizing in a deontic logic based on preferences. In Pro ceedings of the Third Workshop on Deontic Logic in Computer Science (keon'96). Springer Verlag, 1996. [vdTT95] L.W.N. van der Torre and Y.-H. Tan. Cancelling and overshad owing: two types of defeasibility in defeasible deontic logic. In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAF95). Morgan Kaufman, 1995. [Vel91]
F. Veltman. Defaults in update semantics. Technical Report ITLI Prepublication Series LP-91-02, Department of Philosophy, University of Amsterdam, 1991. To appear in the Journal of Philosophical Logic.
[vW63]
G.H. von Wright. The logic of preference. Edinburgh, 1963.
273
Computational complexity of infinite-valued Lukasiewicz propositional logic H. Wagner Department of Computer Science Fernuniversitat, D-58084 Hagen Abstract It is shown for infinite-valued Lukasiewicz propositional logic that the consequence relation is complete for coNP and that the conse quence problem is n 2 -complete. Using the NP-completeness of the existential theory of real addition a simplified proof for the NPcompleteness of the satisfiability problem is given.
1
Introduction
Many-valued, especially infinite-valued Lukasiewicz logic never belonged to the mainstream of logical research. Yet a lot of investigations concerning the logical foundations of infinite-valued Lukasiewicz logic has been done in the years from about 1920 to about 1965, culminating in Scarpellinis ([12]) proof of incompleteness of first order infinite-valued Lukasiewicz logic. It seems that Scarpellinis negative result breaked down the interest in these logics. However mainly caused by applications in computer science the importance of many-valued logics has grown during the last decade. So it seems natural to investigate the computational complexity of these logics. While it is quite obvious that most of the results of classical propositional logic carry over to finite-valued logics (for instance NP-completeness of the satisfiability problem, existence of a tableau calculus etc.) the situation for infinite-valued logics is much more intricated. A starting point of investigation was Ragaz dissertation ([11]). Consider ing the arithmetical complexity of infinite-valued logics he especially showed the decidability of infinite-valued Lukasiewicz propositional logic. Mundici ([10]) sharpened this result by showing the NP-completeness of the satisfia bility problem. In fact he only proved this result for the weak satisfiability notion, but his proof is easily adapted to the case of the strong satisfiability notion. In this paper we will show that the computational complexity of the satisfiability problem and of the consequence relation can be obtained
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H WAGNER
as a trivial consequence of the NP-completeness of the set of Ei-formulas in the theory of real addition. Furthermore we will consider the more general consequence problem: given a recursive enumerable set M of formulas and a formula A, decide wether A semantically follows from M. This problem is interesting, because the semantical consequence relation is lacking the finiteness property. We show that this decision problem is I n c o m p l e t e .
2
Preliminaries
We are defining the language IPL of infinite-valued Lukasiewicz propositional logic.
2.1
Syntax
We fix a finite alphabet E = {x, |, ->, A, V, 0 , (,)}. The set of variables and the set of formulas are defined inductively. Definition 1 (of the set V of propositional -xeV -ifyeV then y\ e V. We also write xm for x\...
variables)
| (m strokes).
D e f i n i t i o n 2 (of the set of formulas) - Every propositional variable in V is a formula. - If A and B are formulas then so are ->A, ( i A B ) , ( A V B) and (A 0 B). We will use A, B, C, A0, B0, C o , A i , . . . as syntactic variables for formulas. For a formula A we will also write A (xi,...,xn) if the only variables in A. The following logical connectives for bold conjunction, implication and for equivalence are introduced as Abbreviations:
(A ®B) := -.(-.Ae-iB) {A^B) :=(-iA0B) {A <-> B) := ((A ^B)A(B-+
A))
1 : = ( - 1 X 0 a;)
0:=-.l For technical reasons we use V and A as basic logical operators. Of course (A A B) and (A V B) could have been defined by {Ay B) : = ( n ( - n i 0 5 ) ® 5 ) .
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We adopt the usual conventions for omitting parentheses. So for in stance we also write A © B instead (A © B) and Ai A . . . A An instead (... ((Ai A A 2 ) A As) A . . . A An). For every formula A in IPL the length || A || of A is defined as the length of A as a string over the alphabet E. If each occurence of a variable p in a formula A is substituted by a formula B we denote the resulting formula as A [p/B].
2.2
Semantics
For defining the semantics in the infinite-valued case we use the interval [0,1] (this is the set of all reals r with 0 < r < 1) as the set of logical values. The semantics of IPL is based on the concept of interpretations. An interpretation 7 is a mapping 7 : V —> [0,1]. Let X denote the set of all interpretations. Definition 3 The value val (A, I) of a formula A with respect to the inter pretation I G X is defined inductively: val(y,I) := I(y) foryeV val (-.A,/) : = l - v a l ( A , 7 ) val (A A B, I) := min (val (A, 7 ) , val (B, I)) val (A V B, I) : max (val (A, I), val ( £ , 7)) val (A © 5 , 7 ) := min (1, val (A, I) + val ( 5 , 7)) A is a tautology (denoted by |= AJ ijff val (A, 7) = 1 for all I G X. If there is I G X such that val (A, 7) = 1 we say that A is (strong) satisfiable and that I satisfies A. A is weak satisfiable iff for some I G X val (A, 7) > 0. Let M be a set of formulas. An interpretation I G X satisfies M iff for all A G M I satisfies A. M is satisfiable iff there is some I € X that satisfies M. M \= A iff for all I G X : if I satisfies M then I satisfies A. In case M = {B} we also write B \= A. Some consequences: - val (A -+B,I) = min (1,1 - val (A, 7) + val (B, I)) - val (A 0 B, I) = max (0, val (A, 7) + val ( £ , 7) - 1) - val (A~BJ) = 1- |val (A, I) - val ( £ , 7)| For later we will introduce some further abbreviations: Definition 4 For all natural numbers n and formulas A G IPL: 0
J2A-.= O
n+l
A
n
, Y, --= ®Y,A n+l
0(n,A,a;):=^AA
A
/ n
\
[^2A^X)
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H WAGNER
We state without proof: L e m m a 1 For all natural numbers n, all formulas A G IPL pretations I:
and all inter
1. val [ Y^ A, I ] = min (1, n • val (A, I))
val (0 (n, A,x),I)
=
l=>I(x)>
71 + 1
Definition 5 To each formula A of IPL we define a formula A* and a formula FA by: (y)* := cy *-+ y for each propositional variable y (-.A)* := (c^A
:= {CAAB ^ c
A
A
cB)
A A* A 5 *
{A V B)* := (cAwB ++cAV cB) A A* A B* (A 0 B)" := ( c A e s ~ (
ofV.
Obviously the length of FA is linear in the length of A. L e m m a 2 Let A be an arbitrary formula of IPL. Let further I be an interpretation with I (eg) = val (B, I) for all subformula B of A. Then val (A, I) = val (FA,I). Proof: We prove by induction on the definition of A, that val (A*,/) = 1. Our claim is trivial in case A = y. Case A = ->£: By induction hypothesis we have val(jB*,I) = 1. By our assumption on I we further have val (c-,jg, I) = val (-»£, I) = 1 - val (B, I) = 1 — val (cB, I) = val(-iC5,7), so val(c-,B <-► ->CB,I) — 1, therefore val (A*, 7) = 1. Case A = B ®D: By induction hypothesis we have v a l ( # * , / ) = val(£>*,/) = 1. Further we have val (CB®DJ)
=
val (B @DJ)
=
min (1, val (cB,I)
Again we get val (A*, I) = 1.
= min (1, val (B, I) + val (D,
+ val(c£>,/)) = val (cB
I))
®cD,I).
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The remaining cases are proved in a similar way. So our induction is finished. But now we get val {FAJ)
= val (cA AA\I)=
val {cA,I)
= val (A, I ) .
□ Corollary 1 A formula A of I PL is (strong resp. weak) satisfiable if and only if its translation FA is (strong resp. weak) satisfiable.
3
Complexity of the satisfiability problem and the consequence relation
While it is well known by Cook's theorem ([2]) that the satisfiability prob lem for classical 2-valued propositional logic is NP-complete the complexity of the corresponding problem for the infinite-valued propositional logic of Lukasiewicz wasn't determined for a long time. In 1987 Mundici ([10]) proved the NP-completeness for this problem too, using the weak satisfia bility notion (but his proof can easily be adapted for the case of the ordinary satisfiability notion). The difficult part of his proof was showing contain ment in NP. Hahnle ([5]) and independently the author ([13]) then gave simplified proofs for the NP-completeness of the satisfiability problem by reduction to some linear programming problems. Developing a tableau cal culus for many-valued logics Hahnle was led to a polynomial time reduction of the (strong) satisfiability problem to a mixed integer programming prob lem, while in [13] a nondeterministic Turing reduction to (a special case of) linear programming was obtained. As mixed integer programming is NP-complete and linear programming has been proven to be P-complete by Khachian ([7]) and Kamarkar ([6]), both methods proved containment in NP. However, as we will see now, the classical proof of decidability of IPL, which belonged to the folklore of the logical community and which e.g. is given in [11] and in [4], contains nearly all necessary details for an even more trivial proof of NP-completeness. Furthermore this approach has the advantage that for many decision problems of IPL the computational complexity can be determined in an easy way. Let SATJPL denote the set of satisfiable formulas of IPL, SATIPL
:= {A e IPL \ 31 e l.val (A, I) = 1}
and TAUTITL
:= {A G IPL \
\= A}
the set of all valid formulas. Satisfiability problem and tautology problem mean the corresponding decision problems for these sets.
278
T h e o r e m 1 SATIPL
H WAGNER
is
NP-complete.
Proof: U SATIPL is NP-hard:" Let SAT denote the satisfiability problem in the classical two-valued case. We give a polynomial time many-one reduction of SAT to SATIPLTo any instance A (x\,..., xn) to SAT we assign F (A) := A (x\,..., xn) A (xi V -1X1) A . . . A(xn V ->xn) as an instance to SATIPLIt is easy to see that A(xu...,xn) e SAT iff F(A) e SATIPL and that F(A) is computable from A(xi,... , x n ) in polynomial time. So it follows that SATIPL is NPhard. U SATIPL is in NP:" We define a translation, which assigns to each formula A of I PL a E x formula T (A) in the theory TH(7£, + , < , 0 , 1 ) , the theory of real addition, in such a way, that this translation is computable by a deterministic Turing machine in polynomial time. For the language of TH(7£, + , < , 0 , 1 ) we are using the same symbols + , < , 0 and 1, for the individual variables we are using the same symbols as for the propositional variables in IPL. Further we introduce the following formulas as abbreviations in the theory of real addition: x = max (y, z) :<$ (x = y A z < y) V (x = z A y < z) x = min (y, z) :<$ (x = y A y < z) V (x = z A z < y) x = y :<£> x < y A y < x x < y :<$ -i (y < x) x = 1 — y :<& x + y = 1. Here the introduced max and min function symbols have the usual mean ings of the maximum and minimum functions. By Lemma 2 it suffices to define such a translation for the formulas FA ( = CA A A*), where A* is a conjunction A* = A i < i < n ^ °^ f ° r m u ^ s A{ of the form (c y <-► y), (c-lB <-► -.c#), {cBAD ^ cB A cDj, (cByD ^ cB V cD), and (cB<£L> <-► (CB © CD))- We use here = as symbol for the equality in the metalanguage of the theory of real addition. Let V {cy «-> y) := cy = y A 0 < y A y < 1 V [c^B <-► ^cB) := c^B = l-cBA0
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3x(cyi = 1 A 0 < Cyi A c^ < 1 A T' (^4*))- Obviously this translation is poly nomial time computable, and FA is satisfiable (in IPL) iff T (FA) is valid in the theory of real addition.However, by a well known result of von zur Gat hen and Sieveking ([3]), the set of Si-formulas in the theory of real addition is NP-complete. We therefore get SATIPL e NP.
□ Corollary 2 2. TAUTIPL
1. The weak satisfiablity problem is
NP-complete.
is complete for coNP.
3. The consequence relation of IPL,
i.e. the set
{{A, B) | A, B e IPL
and
A\=B}
is complete for coNP. Proof: In each case the proof of hardness for the corresponding com plexity class is trivial, so we prove only containment. 1. Just replace in the definition of T (FA) the equality CA = 1 by CA > 0. We get: FA is weak satisfiable (in IPL) iff T (FA) is valid in the theory of real addition. 2. Immediate consequence of (1). 3. Let T (A, B) := 3x(cA = 1 A 0 < cA A cA < 1 A V (A*) A c 5 < l A 0 < c B A CB < 1 A T' (#*)), where x is the sequence of individual variables occuring in V (A*) or in T (B*). (Observe, CA and c# are contained in x). We get A )f=B iff T (A, B) is valid in the theory of real addition. So containment in coNP follows for the consequence relation.
□ 4
Complexity of the consequence problem
In classical two-valued propositional logic we have the finiteness theorem: if a formula A semantically follows from an infinite set M of formulas, there is already a finite subset M' of M such that A already follows from M'. This property fails for the infinite-valued Lukasiewicz propositional logic. So it seems interesting to consider the following consequence problem: Given a recursive (enumerable) set M of formulas of IPL and a formula A of IPL, decide wether M \= A. The corresponding problem for classical twovalued propositional logic is Ei-complete (we assume here some knowledge of Kleene's arithmetical hierarchy).
280
H WAGNER
T h e o r e m 2 The consequence problem for infinite-valued Lukasiewicz propositional logic is complete for II2 (with respect to many-one reduction). Proof: "The consequence problem is in n 2 : " This follows already from the corresponding result of Ragaz ([11]) for inifinite-valued first order logic. "The consequence problem is hard for II2:" We will show this by reducing a n 2 -complete problem to the consequence problem. We will use special two-counter machines introduced by ([8], p.61). Such a special two-counter machine is an automaton with finite-state control and exactly two counters, which are memory units capable of storing any natural numbers. The finite control may modify its counters as it changes state. Formally we present special two-counter machines as a sequence ( P i , . . . ,P/t) of instructions P;, 1 < i < k, where each instruction P{ is one of the following three kinds: H ("halt") add(j) ("add 1 to the number in the second counter and go to instruction j " ) sub{j, I) ("if the number in the first counter is not zero, then subtract 1 from it and go to instruction j] otherwise interchange the counters and go to instruction /") We stipulate that P&, and only P&, is H, and that for each P;, 1 < i < k, if P{ is add(j) or sub(jj) then 1 < j,l
GFi}
is n 2 -complete. We reduce HQQ to the consequence problem. Given a code m of a special two-counter machine we assign to this ma chine m a pair ( r m , Bm) where Tm is an infinite recursive set of formulas of IPL and Bm £ I PL in such a way that machine m halts for infinitely many inputs n iff Tm (= Bm.
COMPUTATIONAL C O M P L E X I T Y
281
Assume machine m codes a sequence (Pi,...,Pk) of A: instructions. We introduce an abbreviation for some of the variables. Let () be a recursive bijective mapping from A/"4 in J\f . We write (a, b, c, d) instead () (a, b, c, d). Define for all natural numbers e, r, n Qr '• = x(r,0,0,l) C r,e "— x(r,n,e,2) ^r,e '— x (r,n,e,3) Pl,r '— x (r,n,0,4)
Pfc,r :— x(r,n,0,k + 3) •
We use the variable c™e to express that machine m after doing n steps for input r has stored the number e in its first counter: Similarly d™ e ex presses this property for the second counter. The variables p " r , 1 < i < k, determine which instruction is next to be executed when machine m for input r has done n steps. Further we define 6r := 6 (r,qr,x) for all r £ Af. T m contains for all natural numbers r, n, e, / the following formulas:
c°
for all i and j with 1 < i < k and 1 < j < k, such that P{ is add (j)
P"rA
-
p ^ A c ^ ' A ^
1
for all i, j and / with 1 < i < k and 1 < j , I < k, such that P* is sub (j, I) Plr - Or If m isn't the code of a machine let Tm := 0. jE?m is just the formula x. We show: m G i?oo iff F m |= Bm. "only if": By induction on n it is easily proved that for all natural numbers n and r: if machine m for input r after doing n steps without stopping reaches the configuration (i,u,v) then Tm \= pfr A c™u A d" v .
282
H
WAGNER
Now assume m G #00 • Then there exist infinitely many r with (ra,r) G Hi. So for every interpretation / , if / satisfies Tm the there exist infinitely many r, such that for some n I satisfies p%r. Therefore for infinitely many r I satisfies 6r. We get by by lemma 1, that i" satisfies x and therefore J- 771 f—1 ^ -
"if": Assume m £ HOQ and m is the code of a special two-counter machine. So there exists r' G Af such that for all r > r' (m,r) £ Hi. Therefore for all inputs r > r' machine m runs through an infinite sequence («0,r, Uotr, V0,r)
i
n
, ( Z i , r , U i > r , Vi jT .) , . . .
of configurations with (io,r,^o,r,^o,r) = (I,7*,7*)r ^ k for all n € Af. For all r G A/" we define
Especially we have
Z?r,o:={rf,r,
({P£1.<£/1.
if
A-,„ = {p*P, <£,,«£«} and
Pi = 1
1
^M^ .^ }
add(j)
if
{p]^\Cf\d^}
A-.n = { ^ , ^ , 0 , ^ } ^ d Pt = Sub(jJ) if £ r , n = { p ? r > c £ / + 1 , d ? , e } and
0
Pi = sub (j, I) if Dr,n = 0 or pJJ G £>r,n
A-.n+l := ^
Dr := ( J #r,n
r6Af
An interpretation J is defined by 1 1 r+l
i f p G £> if p = qr and 3n G A/". Pfc,r G A-
I(p):={ sup{4r|r6-A/',3n.p)fjr6^r} 0
ifp = x else
It is easy to prove that this interpretation I satisfies Tm and that I satisfies pfr A c™u A dj?v iff machine m for input r is in configuration (i,u,v) after executing n steps.
283
COMPUTATIONAL C O M P L E X I T Y
^From the definition of Dr we get that for all r > r' there is no n such that p £ r G Dr. Further
I(x) = sup j - ^ - | r e N,3n.plr
G Dr\ < - ^
< 1.
As / satisfies Tm it follows that Tm )f=x, i.e. Tm %=Bm. It is not difficult to prove that Tm is a recursive set and that (an index of) T m can be recursively computed from m. (We won't go into details here as we have omitted coding details.) So we get a recursive many-one reduc tion of HQQ to the consequence problem of IPL. Therefore the consequence problem is n 2 -hard.
□ References [1] I.M. Bomze, Optimierung - Theorie und Algorithmen, BI Wissenschaftsverlag, 1993 [2] S.A. Cook, The complexity of theorem-proving procedures, in Proc. 3rd Ann. ACM Symp. on Theory of Computing (1971) 151-158 [3] J. von zur Gathen, M. Sieveking, Weitere zum Erfiillungsproblem polynomial aquivalente kombinatorische Aufgaben, in: E. Specker, V. Strassen, Komplexitat von Entscheidungsproblemen, LNCS 43, Springer Verlag 1976, 49-71 [4] S. Gottwald, Mehrwertige Logik Eine Einfuhrung in Theorie und Anwendungen, Akademie Verlag 1989 [5] R. Hahnle, Automated deduction in multiple-valued logics, Oxford Uni versity Press 1993 [6] N. Karmarkar, A new polynomial-time algorithm for linear program ming, Combin. 4 (1984) 373-395 [7] G. Khachian, A polynomial algorithm in linear programming, Soviet Mathematics Doklady 20 (1979) 191-194 [8] H.R. Lewis, Unsolvable cases of quantificational formulas, AddisonWesley Publ. Comp., Inc., 1979 [9] R. McNaughton, A theorem about infinite-valued sentential logic, The Journal of Symbolic Logic 16 (1951) 1-13 [10] D. Mundici, Satisfiability in many-valued sentential logic is NPcomplete, Theoret. Comp. Sci. 52 (1987) 145-153
284
H WAGNER
[11] M.E. Ragaz, Arithmetische Klassifikation von Formelmengen der un endlichwertigen Logik, Dissertation ETH Zurich 1981 [12] B. Scarpellini, Die Nichtaxiomatisierbarkeit des unendlichwertigen Pradikatenkalkiils von Lukasiewicz, The Journal of Symbolic Logic 27 (1962) 159-170 [13] H. Wagner, Computational complexity of infinite-valued Lukasiewicz propositional logic, Technical Report 184 - 5/1995, Fernuniversitat Hagen
285
NDK and Natural Reasoning Mariko Yasugi *
Masahiro Nakata
Abstract A revised version of NDK, a system of first order predicate calcu lus formulated in the natural deductions where the rule of inference can be applied to any disjunctive component. NDK thus does not have axioms. iVDK-deductions are simpler than NK-ones in many cases. Various features of classical proofs are investigated by tak ing the advantage of the formulation of NDK. The major result is a conversion algorithm of TV JD If-deductions to iVif-type-deductions, which are regarded as more natural, preserving the logical structures. Introduction In [7], we proposed a new formal system, NDK, of first order classical predicate calculus. It is an extension of NJ, natural deduction system of constructive logic, where the rule of inference can be applied to a disjunctive component of a formula. Since an axiom of the excluded middle is a theorem of NDK, NDK does not need any axioms. The system NDK in this article is a revised version of the system given in [7]. For this reason as well for the reader's convenience, we will present definitions and statements in some detail. The system NDK is reviewed first (Section 1). Equivalence between NDK and NK is established in Section 2, giving explicit translations of deductions. Proofs (deductions) are compared for LK,NK and NDK, es pecially by proving Peirce axiom (Section 3). For many cases, it is easier proving in NDK than in NK. (See [2], for example, for the systems NJ, NK and LK.) In Section 4, Wang's algorithm for the propositional part of NDK was dealt with in 4 of [7]. A generalized version of it is stated here. For the reader's convenience, we prove it in detail. Let us break the introduction temporarily and explain the reasons why we came up with the system NDK. They are listed below. *This work has been supported in part by Grant-in-Aid for Scientific Researches (No. 07864012) and by Program of Okawa Foundations (No. 95-12).
286
YASUGI AND NAKATA
(1) The first author has worked on the termination problems of reduc tions in sequent-type calculi, by way of ordinal notations. The accessibility of ordinal notations would guarantee the termination of reductions of proofs. The accessibility proofs of ordinal notations were then carried out in sys tems of constructive arithmetic, which were in turn mapped to certain term systems. The last ones would act as tools of termination mechanism of the reductions of proofs. (The main references are [3], [4], [5], [6].) This is too an indirect method. We expect that, similarly to the case of NJ-type systems, there be term systems inherent in NDK-type systems. This is yet a theme to be worked out. (The normalization of NDK is defined in 3 of [7].) (2) It is well known that there is a neat corelation between L J and N J, but the mutual relations between LK and NK do not yield fruitful results. On the other hand, there is a beautiful relation between LK and NDK (2
of [7]). (3) Interesting properties which hold for sequent calculi such as the midsequent property and Wang's algorithm also hold for NDK (See Theorem 11 and 4 in [7] as well as Section 4 of this article). (4) In many cases, 7VZ) If-deductions are simpler than, for example, NK -deductions (Section 3), and hence more convenient in automatically constructing deductions. We plan to elaborate on this theme later on. Our major concern in this article starts at Section 5, where various notions concerning 7VZ} if-deductions are given. Among them are critical inferences and negative dual pairs. These definitions prelude subsequent three sections. In Section 6, NDK-inferences are classified into natural ones and artificial ones, and then the Oshiba's project is explained. Oshiba in [1] proposed a conversion algorithm to transform Lif-proofs to LJ-type proofs allowing the axioms of the excluded middle. We will carry out a similar task concerning iVZ} if-deductions We regard classical D I, ->I and VJ as artificial, and hence try to remove them, by adding the excluded middle type axioms, preserving thereby the logical structures of original deductions. The transformation algorithm is described in Sections 7 and 8, and some typical examples are given in Section 9. At the end, in Section 10, we remark on the characteristics of the critical D I as an annihilator. The authors would like to express their gratitude to T. Oshiba and N. Saneto for their interest in our work.
NDK
1
AND N A T U R A L R E A S O N I N G
287
System NDK
We introduce the system NDK, which is an extension of constructive logic NJ to classical logic. NDK has no axiom. Inference rules of NDK can apply to an arbitrary component of the disjunctive form of the premise. Here we give some notions and notations as well as the inference rules of NDK. R e m a r k 1 The system NDK was first introduced in 1 of [7]. We are presenting a revised version of NDK here. In particular, the symbol _L designating falsefood is not adopted. Instead, the empty conclusion is ad mitted. This streamlines the description of inference rules as well as of natural conversions between NDK and LK. Formulas of NDK are those of the first order predicate calculus, and hence no details will be given. Formulas are denoted by letters A, B, ■ ■ -, and we use the letters T, A, • • • to denote a formula or the emptiness, r v i will stand for honest r V A if r is a formula; it will denote A if T is empty. Definition 1.1 The notion of disjunctive components as follows. (1) If A is atomic, then the disjunctive component (2) If A is of the form B A C,B D C,->B,VxB disjunctive component of A is A itself. (3) If A is of the form B V C, then disjunctive disjunctive components of B, disjunctive components
is defined recursively of A is A itself. and 3xB, then the components of A are of C and A itself.
For example, disjunctive components of X V (-«X V Y) are X V (->X V Y),X, ->X V Y, -iX and Y. (X V ->X is not a disjunctive component.) R e m a r k 2 We can paraphrase this definition as follows. An occurrence of an expression in a formula A is said to be surface if it is not in the scope of any connective other than V within A. A subformula of A, say B, is called a disjunctive component of A if it is surface in A. (Notice that A is itself a disjunctive component.) Definition 1.2 An TVJDK-deduction is defined in the same manner as that of NJ, with generalized rules of inference. The rules of inference are listed below. They are Introduction and Elim ination (abbreviated to / and E respectively) of logical connectives except for the rule of contradiction, abbreviated to Con. The explicitly written V 's are surface, so that the auxiliary formulas and the principal formulas of inferences are disjunctive components. An assumption with label a will be denoted by [A : a], though the label will be omitted most of the time. (An assumption is never empty. That is, [ : n] is not admitted.) The labels of discharged assumptions are indicated at the inferences in parentheses.
288
YASUGI AND NAKATA
r x v A v r 2 rxvi?vr 2 i\ V (A A B) v r 2
ri v Aj v r 2 Ti V(Ai V A 2 ) V T 2
M
V7
i\\/(A 1 A A 2 ) v r 2 rx v Ai v r 2
A£
(z = 1,2)
(z = 1,2)
[A a] [B : b] i\ v (A v s) v r 2 D i\ v £> v r 2
D
(a,6)-VE
[A: a] Ti v # v r 2 ri v (A D 5) v r 2
(a)- D /
Ai V A V A 2
Ti V (A D B) V T 2
Ai v ri v B v A 2 v r 2
I)£
[A: a] rivr2 Ti v - A v r 2
(a) - - /
i\ v F v r 2 Ti V VxF v r 2 (eigenvariable
Ai V A V A 2 Ti V -nA V T 2 -.£ Ai V Ti V A 2 V r 2
rx v \/xF v r 2 \/E Tx V F[t/x] v r 2
v/ condition)
[F:a] Tx V F[t] v r 2 3/ Tx V 3xF V T 2
Tx V 3xF v r 2 Tx V D V r 2 (eigenvariable
-j— Con
b
(a) - 3 £
condition)
NDK
AND N A T U R A L R E A S O N I N G
289
The inferences VI and 3E satisfy the eigenvariable condition, that is, V7 has no free occurence of x in the assumptions, Ti and T 2 , and 3E has no free occurence of x in D and in the assumptions excepting [F : a]. An inference is called classical when one of the F i , r 2 , Ai and A 2 as above is not empty.
R e m a r k 3 (1) The empty premise in Con is introduced by the application of -*E that is not classical. (2) In D E, - J and -*E, the associations of components in the conclusion are arbitrary. For example, in D E, there are several possibilities. ((Ai V r i ) V J B ) V ( A 2 V r 2 ) a n d ( A i V r 1 ) V ( J B V ( A 2 V r 2 ) ) are some of the examples. This much of freedom makes the deduction forming easier. According to Proposition 2 of [7], we can claim some elementary prop erties with regards to NDK. For example, A V ~>A is a theorem of NDK] NDK is equivalent with NK; from r x V (D V D) V T 2 , can one derive Ti V D v r 2 , where the only elimination used will be the V-elimination applied t o D V D . The last fact suggests that we can introduce the following as a derived rule (named Contraction). Y1 V (D V D) v r 2 ^ . . ^ . . -p
Contraction
1 1VU V12
In 2 of [7], natural mappings between TV .D if-deductions and LK -proofs are defined. We can easily modify them to the present version of NDK. Reductions and normal forms of NDK -deductions are defined in 3 of [7] similarly to those of NJ. Proposition 9 there claims that a cut-free proof of LK is transformed to a normal deduction of NDK, and hence it holds that every 7V.DK-theorem has a normal NDK-piooL
2
Equivalence between NDK and NK
In this section, we give a revised proof of the fact that NDK is equivalent to NK. For this purpose, we show the following lemma as the first step. L e m m a 1 Let X V B and ->X be deducible from respectively V and A in NK. Then B is deducible from T U A in NK without adding the axiom of the excluded middle. (Here T and A denote finite sets of assumptions)
290
YASUGI AND NAKATA
Proof. A
XV B
B
[B:2]„
Remark 4 The thorem below has been shown in [7]. The proof that NDK is a subsystem of NK given here is an improved one, so that it renders a clear insight of where the axiom of the excluded middle is necessary. T h e o r e m 1 NDK is equivalent to NK. Proof. Inference of NK are those of constructive NDK the axiom of the excluded middle is deducible in NDK;
inferences and
A V -nA Therefore we obtain that NK is a subsystem of NDK. Next, in order to establish that NDK is a subsystem of NK, we show that any TVDif-deduction II can be translated into an TVif-deduction, by induction on the construction of II. Let I be the last inference in II. If I is not classical, then we can immediately apply the induction hy potheses. Let us thus assume that I is classical. If I is A/, then II is of the form
r 1 v i v r 2 ri v B V r 2 rx v (A A B) v r 2 Let IT be the iVif-deduction !
i\ v B vr 2
[A = 3]
[r: vr 2 :2]
[B:4]
At)B
r x v r 2 v B Vi v (A A B) V r 2 i\ v (A A 5) v r 2 , (2,4) ri v (A A B) v r 2
x
NDK
AND N A T U R A L R E A S O N I N G
II can be transformed into the following deduction in
rx v A v r2
NK.
[Ti v r 2 : i]
rivr2vi
^ v ( A A B ) V r 2 IT
r i V ( i A 5 ) v r2
(1.3)
If I is AE, then n is of the form
ri v (Ay A A2) v r2 II can be transformed into the following deduction in \
NK.
[A, A A 2 : 2 ]
r\ v (Ai A A2) v r2
fTivr 2 :i]
Ai
ri v r2 v (^x A A2) rt v Aj v r2 ri v A{ ri v A, v r2
V r2
(
If I is V.E, then II is of the form [A:l]
[B:2]
riv(iv5)vr2 b r1yD\zr2
b ^' 2 ^
Let IF be the deduction [A:l]
[B:2]
b [A V B : 4]
b
Tj V Z) V T 2
I \ V £> V T 2
i\ v L> v r2 II can be transformed into the following deduction in
Ti V (A V B) V T 2
(
' '
NK.
[r x V T 2 : 3]
r i v r 2 v ( A v 5 ) r i v i ) v r 2 IT ri v £> v r 2 ^'
j
292
Other cases except for D 7, ->/ and V7 can be dealt with similarly. If I is -
[A
rivr2 i?! v -.A v r 2
(i)
II can be translated into the following deduction in NK with the axiom of the excluded middle.
[A:l] ri v r 2 A v -.A
[-A : 2]
Ti v -A. v r 2 rx v -.'A v r 2 Tj v -.A v r 2
(1,2)
If I is D 7, then II is of the form \A Ti V B v r 2
ri v (A D B) v r 2
(i)
II can be translated into the following deduction in NK with the axiom of the excluded middle. Let II' be the deduction below.
[A: I] Tx V B V T 2
[-Ti : 5]
BVT2
[-r 2 : 3] B AD B (1)
ri v (A D B) v r 2
NDK
AND N A T U R A L R E A S O N I N G
293
By virtue of Lemma 1, this is an TVK-deduction. Using II', we obtain the following.
Fi : 4] :
[r2 : 2]
: ri v -.iTi rx v (A p B) v r2 IT(4,5) r 2 v^r 2 riV(Ap5)vr 2 TX\/(AZ>B)Vr2 (2 3) r!v(iDB)vr 2 ' Notice that the newly added axioms of the excluded middle are I \ V - T i and T 2 V - T 2 . If I is VJ, then II is of the form
ri v F v r2 ri v VxF v r2 Let IT be the deduction below.
I \ V F V T2
[-.ri : 4]
F v r2
[-ir 2 : 2]
ri v VxF v r2 By virtue of Lemma 1, this is an TVif-deduction. II can then be transformed as follows
r2v-r2
[r 2 : 1]
Fi : 3] !
! riWxFvr 2
ri v -.ri ri v V^F V r2 w(3,4) riWxFvr 2 (1 2)
Ti V VxF v r 2
This completes the proof of Theorem 1. R e m a r k 5 In the proof above, the use of the axioms of the excluded middle is essential in cases of classical D / , - J and VI, because they deduce the axiom of the excluded middle (or propositions which are equivalent to it). In the case of ->/, we have
294
YASUGI AND NAKATA
In the case of D 7, we have [A:l] AV(XA^X)
iV(iDlAnl)
l }
In the case of VI (assuming that A has no free occurrence of x), we have [Vx(A V F(x))] A V F(x) A V VxF(x) All other inferences can be transformed into iVJ-inferences.
3
Comparison of proofs in LK, NK and NDK
In many cases, iVDJf-deductions are simpler than JVK-deductions and LK- proofs. We will see this through proofs of Peirce Axiom in respective systems. Peirce Axiom is of the form ((A D B) D A) D A, and the system obtained from NJ by adding it is the classical logic. NDK: [A: I] AVB
(1)
A V (A D B)
w
[(ADB)DA:2]
^ A ((A NK:
Contraction
(2)
DB)DA)Z)A
Let II be the following deduction. [A: 4] [-.A: 5] — ~ [A:l] AVB
B
J 5 s AV^A
( 4 )
[B:S\ ADB Av(ApB)
[-4:2] AV(ADB)
AV(ADB)
A\/(ADB) (5,1)
AV(ADB) Then the deduction as required is the following. [ADB:6]
n
[A : 71 ■
((A
"
[(ADB)DA:S]
A , -A
DB)DA)DA) (8)
N
(7,6)
NDK
AND N A T U R A L R E A S O N I N G
295
LK: A A,B
:
=> A, AD B (ADB)DA,ADB^ (A: D B)D A^ A, A {A D B)D A=> A ((A
A
cut(A D B)
DB)DA)DA cut-free A => A
A^ A,B =* A, AD B (ADB)DA--
(ADB)DA
A^A => A, A =>A
{{ADB)DA)DA
4
Wang's algorithm
We consider Wang's algorithm, which is an algorithm deciding the valid ity of a given formula in the propositional part of NDK. The result in 4 of [7] regarding Wang's algorithm can be generalized to the question whether a formula is valid relative to given formulas, with a minor modification. Definition 4.1 Let E be a finite sequence of formulas Bi, • • •, Bn(il > 0), and let F be any formula. Then the figure S [Bi]"-[Bn]
is called a sequent (of F from E). [#i], • • •, [Bn] are said to be the as sumptions and F is said to be the conclusion of S. If F is true whenever Bi, • • • , B n are true, then the sequent S is said to be valid. One can also say that F is valid relative to E. We assume that assumptions are labelled. We can define the resolution rules, which generate new sequents from given ones. =>(* a ) (=>(* c )) means the resolution of an assumption (the conclusion) whose outermost logical connective is *. ■■■{AAB}---
■■■{A}[B}---
296
YASUGI AND NAKATA
TI v (.4 A B) v r 2 ==>(AC) i\ v A v r 2 ; ■■•[AVB]-..
--.[A].--
r
=>(vo)
---[Apg]---
r
r
;
r
r
... ;
rvi
r 2 =>(Dc) ri v B
V r2
-•■h4---
r
■■■[£]■■■
•••[£]•••
=> (Da)
T1\/(ADB)V
ri v B V r 2
■■■
=^MrvA
■ ■■
- 1 4 -
^ v-.A v r 2 =^(-,0) rxvr 2 Starting with a sequent S, the resolution process should continue until all assumptions become atomic and the minimal components of the conclusion become atomic. Such a sequent will be called terminal. The collection of sequents thus resolved, a tree begining with a given sequent 5, is called a resolution tree of S and is denoted by Ts or T^FT h e o r e m 2 (1) All resolution trees are finite. (2) A sequent S is valid if and only if, for each terminal sequent S' of Ts, there is a same atomic formula in an assumption and in the conclusion oiS'. (3) If a sequent of F from E is valid, then T^F induces an NDKdeduction of F from £ without live assumptions except E. Proof. (1) and (2) are easily examined. (3) When the condition holds, we can construct an iVDK-deduction of F from E from the resolution tree of S. It is defined as follows by induction on the construction of the tree Ts. A terminal sequent is of the form
"[*]"■
i\ v x v r 2 where X is atomic. The conclusion can be obtained from [X] by some V/'s. Let (*a) be the first resolution rule applied to S. (See Definition 4.1.) Let us
NDK
297
AND N A T U R A L R E A S O N I N G
assume the induction hypotheses, and construct the last step of a deduction. (Aa)
(Ac)
[A AB]
[A A B] AE B
AE
Ti v A v r2
ri v B v r2
I \ V (A A B) v r 2
A7
(Vo)
[A : a] [B : b]
[AW B]
r
r
Oc)
O*) [AD
(a,b)-VE
B]
r v i
[A: a]
D £
ryi?
rx v .B v r2 rx v (A D B) v r2 ( « ) - 3 /
r v r Contraction
(-*)
(^)
[,4:a] rvi
[-.A] r
r!vr2 (a) - - 7 rx v -.A v r2
"
This completes the proof of Theorem 2. Example. Let Ts be the following. [n(nAAnB):l] 4VB
.
a) *(-«)
(4VB)V(niAnB)
298
YASUGI AND NAKATA
=>(Ac) ( i V 5 ) V n i ; ( i v 5 ) V
[A: 2] AyB The coresponding NDK-deduction
[B:3] AMB is constructed below.
[A: 2} [B:3] AVB AVB ( 3 ) (2) (iV^)Vni w (AV5)Vi5 (iV^)V(niA^) ~
5
-£
[-.(-.A A-ȣ) : 1]
Notions on deductions
We will consider an NDK-deduction II, and define some concepts re garding formulas and inferences occurring in II. When we speak of a formula A, we thus mean an occurrence of A in II. Definition 5.1 1) Consider two disjunctive components in II, say A and B. A will be called a predecessor of B (in II) with respect to inference 7, if one of the following holds. In the figures below, C(X) will denote a disjunctive component containing X as a disjunctive compoment. (1) / is either an introduction or an elimination with one premise:
rvc(i)v A r V C(Y) V A
J
Y is obtained from X by an application of / . A is C(X) and B is C(Y), or A is a component of T (respectively A) in the premise and B is the corresponding component in the conclusion (with several exceptions). Ex ceptional cases are the following. (1.1) J is VI which derives XVZ from X , hence Y is of the form ( I V Z ) , and B is the Z in Y. In this case, B has no predecessor. (1.2) I is VJ which derives X V Z from X, A is X and B is X. In this case, A is the predecessor of B. (1.3) I is -ii", and B isY. In this case, X is empty, F is of the form ->Z and B has no predecessor. (2) / is A/:'
rvcppvA rvc(y)vA rvC(lA7)VA
7
NDK
AND N A T U R A L R E A S O N I N G
299
A is C(X) or C(Y) and B is C(X A F ) , o r A and B are the corresponding components in T or A. (3) IisD E: ]?! V X V Ax
r 2 V X D Y V A2 7
Ti v r 2 v y v Ai v A 2 A is X D Y and or A's. (Precise is Ti V T 2 , then conclusion, then (4) / is 3E:
5 is F , or A and 5 are the corresponding formulas in T's definition would be more complicated. For example, if B A can be either I \ or T 2 . If B is the entire formula of the A can be either of the premises.)
r V 3xF V A
b
rvDvA
1
There are three cases. (4.1) A and B are the corresponding components in D, T or A. (4.2) A is the 3xF in the major premise and B is the assumption F (4.3) Situations as were noted in (3). For example, if B is V V D, then A is either r or D in the premises. (5) / is WE:
m
PI
rv(ivr)vA b rvDvA
b
J
There are three cases. (5.1) A and B are the corresponding components in D, T or A. (5.2) B is the X in the assumption and A is the X in the major premise. Similarly with Y. (5.3) Similarly to (4.3). (6) I is - . £ :
rivivAi r 2 v-iivA 2 Ti v r 2 V Ai V A 2 A and B are the corresponding components in T's or A's, or the cases similar to (4.3) apply. 2) A disjunctive component A is said to be initial if A has no predecessor. This happens in one of the following cases. (2.1) A is an assumption which is not discharged by 3E or WE. (2.2) A is introduced by Con. (2.3) A is the principal formula of a -•/.
300
YASUGI AND NAKATA
(2.4) A is the newly introduced component of a V/. 3) A string of (occurrences of) formulas in P, say C\, C2,... , C n , is a sequence of formulas in P with the property that, for each i, 1 < i < n, C; is a disjunctive component in P and Ci+i is a predecessor of C{. 4) A string as above is called a path of Ci if Cn is initial. 5) Let C and D be occurrences of disjunctive components in II. D is said to be an ancestor of C (C is a descendant of D) if there is a string of C in II containing D. 6) A disjunctive component C is said to be irrelevant if there is a path of C whose initial formula is the newly introduced component of a Ml. 7) A path of a deduction II is a path of its conclusion. Definition 5.2 A classical D / or VJ is said to be critical if it is applied to an irrelevant (occurrence of a) formula. A classical - 1 / is regarded as critical. Definition 5.3 Excluded middle type axiom (cited from (6) of 3 in [1]) A pair of formulas E and F is called a negatively dual pair if it satisfies one of the following. (In this case F is called a negative dual of E, and vice versa.) (1) E = A and F =-*A for any formula A. (2) E = -iA and F = A for any formula A. (3) Suppose (Ei,Fi) is a negatively dual pair, i = 1,2. (3.1) (E,F) = (E1AE2,F1WF2) (3.2) (E,F) = (E1yE2,F1AF2) (4) Supposse (E',F') is a negatively dual pair. (4.1)
(£,F)EE(VX£',3XF')
(4.2) (E,F) = (3xE',VxF') When (E, F) is a negatively dual pair, E\JF is called an excluded middle type axiom. It is obvious that an excluded middle type axiom is a theorem oiNK. Definition 5.4 According to 3 of [1], we deal with the excluded middle type axiom instead of the pure excluded middle. (See Definition 5.3.) (1) The system obtained from NDK with excluded middle type axioms added will be called NDKE. (2) The system obtained from NK with excluded middle type axioms added will be called NKE. (3) The notions defined in Definitions 5.1 and 5.2 can be extended to NDKE. In particular, a disjunctive component of an axiom does not have any predecessor, and hence it is initial. The notion of a critical inference will remain invariant under addition of axioms. It is obvious that NK, NKE, NDK and NDKE are all equivalent.
NDK
AND N A T U R A L
REASONING
Definition 5.5 Let II be an NDK E-deduction. will be defined as follows.
The rank of II,
301
rank(U),
Let (3 be a path of II. rp is the number of critical inferences along (3. (See 7) of Definition 5.1 and Definition 5.2.) Then
rank(R) := Y_] rp (3-.path
Definition 5.6 A disjunctive component in a deduction is said to be essen tial if it is not irrelevant.
6
Natural reasoning
According to the comment made at the end of Section 1, it holds that, given a set of assumptions T and a conclusion A, if A be iVDK-deducible from T, then one can construct a normal iVDif-deduction from T to A. This fact could be made use of for automated theorem proving. Our present interest lies, however, elsewhere. In 5 of [7], we briefly explained Oshiba's project in [1], and outlined how to execute his idea in NDK. The purpose of the remaining sections is to give an accurate treatment to it. Oshiba speculates that automated theorem proving (of first order clas sical predicate calculus) in terms of cut-free proofs is, though efficient, far from natural human thinking. According to him, employment of excluded middle type axioms appeal to human thinking better. He first proves a sharpened version of the interpolation theorem for LK, and then applies it to automatical transformation of an LK-proof into an LJ-proof with excluded middle type axioms added. Dealing with the same subject in NDK reveals to us some features of classical deductions. We first speculate on the natural and artificial reasonings with regards to the classical inferences of NDK. Consider, as an instance, A/:
rvivA
rvffvA
TV(iAB)VA One can be easily convinced that this is a valid reasoning. That is, if T or A is true, then so is the conclusion. If A and B are true, then (regardless of the truth values of T and A) so is A A B, and hence so is the conclusion. On the other hand, D / of a certain kind is hard to be intuitively com-
302
YASUGI AND NAKATA
prehended. Consider the following. [A:l] 3xA [D] 3xA V C 3xA V D 3xA V (C A D) (l)-DJ 3xA V (A D C A D) The assumption [A : 1] is discharged to imply C AD, leaving 3xA alone. In order to justify this inference, one has to reason as follows. Assume A. If A is true, then so is 3xA, and hence the conclusion is true. Otherwise, A has to be false. In this case, A D C A D is true, regardless of C and D. This flow of reasoning can be formalized as follows.
[A::3] [^A : 4]
C [A: 2} 3xA AV-^A
3XAV(ADC
[D] CAD AD
AD)
,„,
CADW
3xA V (A D C A D)
3xAV(ADCA D)
(2
'4)
The assumptions [A : 2] and [A : 3] correspond to the assumption [A : 1] above. Various experiments in thoughts have thus convinced us that, all the inference rules of NDK except classical D I, -ii" and VI can be accepted as natural, while the latters are somewhat artificial. As was explained in the Remark 5 in Section 2, these two groups of inference rules are distinguished so that the natural ones can be converted to NJ-deductions (without the axiom of the excluded middle), while the artificial ones essentially need the axiom of the excluded middle in such a conversion. We are thus led to the Definition 6.1 Let NDKE* be the system NDKE classical inferences of D / , ->/ and VI admitted.
in which there are no
Our aim is then to define a conversion algorithm of iVDlf-deductions into NDKE*-deductions, preserving the structure of the original deduction as much as possible. What this last condition means will become clear later. N o t e A uniform transformation procedure of classical iV.Dif-inferences has been given in Section 2. Here we aim at a more delicate analysis of deduc tions.
NDK
AND N A T U R A L R E A S O N I N G
303
The transformation algorithms will be successively given in the subse quent sections. Let us state our objective of the next two sections explicitly. O b j e c t i v e Define a conversion algorithm of any TVDiif-deduction to an NDKE*-deduction, preserving the original logical structure.
7
Preparatory transformations
We first classify the artificial inferences, that is, the classical D / , - J and VJ, into four types. Let-the indicated inference in each figure below be called J . J is assumed to satisfy each of the conditions (1) through (4) respectively. Type I:
Type I :
[A:n]
rvcvA rv(iDC)vA
(n)-Dl
(1) C is essential. Type IH: D 7 where (3) C is irrelevant.
rvfvA r V VxF V A
VJ
(2) F is essential. Type IV: V7 where (4) F is irrelevant.
We first eliminate Type I and Type II by the transformations defined in Section 2. P r e p a r a t o r y t r a n s f o r m a t i o n s . ^,From any TVDif i^-deduction, elimi nate Type I inferences and Type H inferences with the transformation pro cedures in the proof of Theorem 1 in Section 2. We thus have only to eliminate inferences of Type IH and Type IV. We will henceforth deal with an NDKE-deduction II which has no inference of Type I or Type II
8
Conversion algorithm
What is left for us to do is to define an algorithm to eliminate critical inferences, that is, inferences of Type III and Type IV. For this purpose, we need some lemmas. L e m m a 2 Let II be an TVDif ^-deduction with the conclusion r V G V A. Suppose G is irrelevant. Then one can trnsform II to obtain a deduction II'
YASUGI AND NAKATA
304
of r V A satisfying the following conditions. (i) The assumptions and the axioms of II7 are those of II as well as a new assumption E, which is characterized as follows. E is of the form Vx m • • • \fx\E', where E' is the conjunction of the assumptions which are discharged by D / or ->I applied to the ancestors of G. (Such inferences D I and ->/ are critical.) xi, • • • ,xm are the eigenvariables in II occurring free in E'. (ii) II' is obtained from II by eliminating the inferences which are applied to the ancestors of G and by adding E as well as some VE's. In case a critical inference indeed applies to an ancestor of G in II, rank(U')
< rank(TV)
IT can be constructed by induction on the number of the inferences in II below the initial components of paths of G. Since the construction is a routine work, we will explain it with an ex ample so that the reader can get an insight. II: [A(x) : 1]
[B{y) : 2]
3xA{x)
3xA(x)
D B(y)
W) B W V C
(2)7, B(y) V (B(y) p C) B(y) V 3y(B(y) D C) VyB(y) V 3y(B(y) p C) VyBjy) V (A(x) P 3y(B(y) p Q) l \fyB(y) V \/x(A(x) D 3y(B(y) D C))
[Vy(A(x)AB(y)):3] A{x) A B(y) A(x) 3xA(x)
j
2 3
[My{A{x) A B(y)) : 3] A{x) A B{y) B(y) 3xA(x) D B(y)
W) VyB(y) G is Wx(A(x) D 3y(B(y) D C)). The critical inferences Iu I2 and I3 have been eliminated and the conjunction of A(x) and B(y), each discharged in
NDK
305
AND N A T U R A L R E A S O N I N G
It, is formed. Since y becomes an eigenvariable, \/y is prefixed to We thus obtain Vy(A(x) A B(y)) as E.
A(x)AB(y).
L e m m a 3 Under the same situation as in Lemma 2, we can construct an NDKE-deduction of G, say IT'', satisfying the conditions below. (i) The assumption of II" is the negative dual F of the E obtained in Lemma 2. (ii) II" is obtained from II by copying the inferences which derive G. In II", G is not irrelevant and there are no critical inferences. Let formula P be -^A(x) V ^B(y)
and A be the deduction below. [B(y):6]
[A(x):4]
hA(x):5]
hB(y):7] G
(fi)
B{y) D G 3y(B(y) [P : 8]
D G)
A(x) p 3y(B(y)
p G)
v
'
A(s) D 3y(B(y) D G)
n / ; for the II as above looks like this: [3y(-*A(x)V^B(y)):9] A (8) A(x) D 3y(B(y) D C) Vx(A(x) D 3y(£(y) 3 G))
Let us now consider a deduction II with conclusion H. Let T V G V A denote a formula in II, where G is irrelevant. Let \£ be the subdeduction of II whose conclusion is T V G V A. Among those, take a lowermost r v* G V A such that rank(^) > 0. By virtue of Lemma 2, we have a deduction
[£]
rv
A
satisfying the conditions (i) and (ii). Introducing G by VI or Gon, and copying II, we obtain the deduction Hi: [E]
rv A rvGv A H
306
YASUGI AND NAKATA
Notice that, since D / does not apply to a descendant of G below r v G v A in II, there is no critical inference there. So rankijli) < rank(TV), and hence III can be transformed into an NDKE*-deduction E^. Next, Lemma 3 yields a figure
[F] G In II, G is eliminated immediately, due to the condition on r V G V A. Simulate II below G, and introduce some components of H at the end if necessary. One then obtains a figure II2: [F\ G H rank(TL2) < rank(IV), and hence II2 can be transformed into an deduction IT*;. Finally, we obtain II*:
m EVF
': n* H H
NDKE*-
\F] ! 115 H
One can see that II* preserves the structure of II. We can thus conclude that II* is the desired deduction. It especially holds that no new Type I or Type H inference sneaks in. R e m a r k 6 The conversion algorithm described above for Type HI and Type IV inferences splits the original deduction II into two parts: the essensial part (IIi) a n d the irrelevant part (II*,). Each part simulates II and, at the end, merges with the other in terms of VE with an excluded middle type axiom as the major premise. We have thus attained our objective.
9
Examples of conversion
(1) Exampe of [1]
NDK
307
AND NATURAL REASONING
II: P(b)\fP(c) W P(b) V (P(6) D P(c)) (P(a) D P(b)) V (P(b) D P(c)) (P(a)DP(b))VVy(P(b)DP(y)) (P(a) D P(b)) V 3xVy(P{x) D P(y)) \fy(P(a) D P(y)) V 3xV;/(P(:r) D P(y)) 3xVy(P(x) D P(y)) V 3xVy(P(x) D P(y)) -, w , D , . D , .. Contraction 3xVy(P{x) D P(y))
nx = nj:
n 2 = n^:
[V^(^) = 1] P(b) P(a) D P(b) 3x\/y(P(x) D P(y))
hP(6):2]
[P(6):3]
P(c) P(b) D P(c)
{
[3z^P(z) : 4] 3xVy(P(x) D P(y)) (2) 3xiy{P{x) D P(y))
IP: [\/zP(z) : 1]
: nj VzP(z) V 3z-*P(z)
>
[3ziP(z) : 4]
in;
3x\/y(P(x) D P(y)) 3xVy(P(x) D P(y)) 3xVy(P(x) D P(y))
(2)H: [\/x(F(x) V C1)] P(x) V C WxF(x) V C
308
YASUGI AND NAKATA
II*: Wx(F{x) V C)] F(x) V C
CVnC
[C : 2] VxF(x) V C
VxF(x) VxF(x) V C (2,1)
VxF(x) V C
10
[-.C : 1]
Critical inference as an annihilator
At the end, we speculate on the algorithmic characterization of the crit ical D I. Consider [A:l]
B
Bye
(1)
_D/
By {A DC) This can be explained as follows. "If A is true, then so is B: otherwise (A is false and hence) A D C is true regardless of C." A D C thus has an active role only when A is not true, but then the validity of A D C is independent of C. We can thus take X A ->X, a contradiction, as C. This means that a critical D I annihilates the content of C. It will then be acceptable to call such an inference an annihilator.
References [1] T. Oshiba, An Approach to Human Reasoning in Automatic Theorem Proving, Jouhou Shod Gakkaishi(in Japanese), vol.35(1994), 222-231. [2] M. E. Szabo, The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam-London, 1969. [3] M. Yasugi, Construction principle and transfinite induction up to So, J. Austral. Math. S o c , vol.31(1982), 24-47. [4] M. Yasugi, Hyper-principle and the functional structure of ordinal di agrams, Comment. Math. Univ. St. Pauli, vol.34, no.2(1985), 227-263, the opening part; vol.35, no.1(1986), 1-38, the concluding part. [5] M. Yasugi, The machinery of consistency proofs, Annals of Pure and Applied Logic, vol.44(1989), 139-152.
NDK
AND N A T U R A L R E A S O N I N G
309
[6] M. Yasugi and S. Hayashi, A functional system with transfninitely de fined types, Logic, Languages and Computation, Lecture Notes in Com puter Science 792(1994), 31-60 (edited by N. Jones, M. Sato and M. Hagiya), Springer-Verlag. [7] M. Yasugi and K. Ryu, NDK, A New Classical System, The Bull, of the Inst. of Computer Science of Kyoto Sangyo Univ., vol. 11(1994), 1-25.
Faculty of Science, Kyoto Sangyo University Kita-ku, Kyoto 603, Japan e-mail: [email protected]. Graduate School of Science, Kyoto Sangyo University Kita-ku, Kyoto 603, Japan e-mail: [email protected].
311
D E F A U L T LOGIC A N D IT'S VARIANTS: A SEMANTICAL VIEW *
ZHANG M I N G Y I
Applied Mathematics Institute, Guizhou Academy of Sciences Guiyang, Guizhou, 550001, P.R.China Department of Computer Science, Guizhou University Guiyang, Guizhou, 550025, P.R.China Abstract In view of importance of well-understood semantics for knowledge repre sentation systems, various semantical views for default logic and it's variants have been presented. But they are different each other in form. This paper provides a semantical framework, which supports a uniform model theoretic semantics for Reiter's default logic, Lukaszewicz and Brewka's variants of default logic. 1. Introduction Reiter's default logic (DL)[1980]M is currently one of the most popular and widely used formalization of default reasoning, and has attracted atten tion as a formal system for nonmonotonic reasoning. People have tended to be satisfied with intuitive characterization of the extensions of default logic, together with the Tarskian semantics of individual extension. Even though default logic has intuitively been well understood it took several years until a model theoretic semantic was given. In order to get a semantic character ization of default as a whole, Etherington [1986,1987] ^'^ viewed defaults as extending the first order knowledge about incompletely specified world and selected restricted subsets of the models of the underlying first order theory. Lukaszewicz [1985] ^ formalized this idea for a restricted class of de fault theories. This work was generalized to cover the entire class of default * Supported by National Advanced Research and Development Project of China. Key Words: Semantical Framework, Compatible Frame, Partial Order > # , First Order Interpretation, Model.
312
ZHANG M I N G Y I
theories by Etherington [1986, 1987] [ 3 ^ 4 l Lukaszewicz [1988]t7], Guerreira and Casanova [1990] ^ presented alternative semantics respectively. In order to overcome drawbacks of Reiter's logic ( they are mainly lack ing cumulativity and joint consistency with the generated extension, as has been discussed in various papers, e.g. in [9,10,11]), Brewka [1991] M , intro duced a slight variant of default logic, called cumulative default logic (CDL). Since those semantics proposed for DL [Etherington 1986,1987; Guerreiro and Casanova, 1990 etc.] are not applicable any more, Schaub [1991]t12^ provided a semantic characterization of CDL. Considering how to solve the floating conclusion problem, Brewka t13^ further extended the idea underly ing CDL and presented a framework which is based on a fundamental shift of the role of defaults within the formalism, i.e. defaults are not considered as inference rules but elements of the language. This framework lacks a general model theoretic semantic. Zhang Mingyi [1992,1993]t 14 ^ 15 ^ 16 ] gave characterizations of the extensions for Reiter's DL and Brewka's CDL re spectively, which are based on a set of facts and a set of defaults. This motivates us to make the research reported in this paper. In what follows, we present a semantic framework to cover Reiter's DL, Lukaszewicz and Brewka's variants. We begin by providing a brief introduc tion to DL and it's variants. Afterwards a uniform semantical framework and notions of compatibility are introduced. Then we show how Brewka's LDL and Li extensions are characterized by the presented semantical frame work. Finally we argue that our framework supports the design of a model theoretic semantics of variants default logics by cope with Reiter's DL and Brewka's CDL and Lukaszewicz' modified DL. 2. Primary Let L be a first order language. A default is any expression of the form a : fa,..., Pn/lf, where a, ft(l < i < n) and 7 are closed first order formulas. A default theory, DT, is an order pair (D, W), where D is a set of defaults and W a set of first order formulas. Definition 2.1 (DL extension)^ Let (D,W) be a DT, S a set of formulas. Let T(S) be the smallest set satisfying the following conditions: l.WC T(S) 2. ThL(T(S))=T{S) 3. if a : ft, ...,/3„/7 eD,a£ T{S), -.ft g 5(1 < i < n), then 7 <E T{S). A set E of closed wffs is an extension of (D,W) iff E = T(E), i.e. E is a fixed point of the operator V. Definition 2.2 ^ Let D be a set of defaults, we define PRE(D)
=
{a\(a:p1,...,0n/1)eD}
DEFAULT L O G I C AND I T ' S VARIANTS
CON(D) = CCS(D)
{1\(a:(3u...,PJ1)eD}
= {ft|(a : Pu~Ml)
KERN(D) ASS(D)
313
= {de D\d = true :
eD,l
= {de D\d = true : ft, ...,/3 n /7}
The logic CDL is based on a shift from simple proposition to more com plicated structure called assertions. An assertion is a pair (p, X) consisting of a first order formula p and a (finite) set of formulas X, the support of p. Let K be an arbitrary set of assertions, we use the following notations: FORM(K), the set of formulas of K, is the set {p\(p, {ri, ...,r n }) G K) SUPP(K), the support of K, is the set { ^ ( p , {ru ...,r n }) G K, 1 < i < n} Given an assertional default theory, ADT, (D,W), where D is a set of defaults and W a set of assertions, a CDL extension is defined using a Reiter-like fixed point construction as follows: Definition 2.3 (CDL e x t e n s i o n ) ^ An extension of an AST {D,W) is a fixed point of the operator T which, given a set of assertions S, produces the smallest set of assertions S' satisfying the following conditions: l.WCS' 2. Ths(S') = S' 3. if a : 3/j G £>, (a, { n , ...,r n }) G S' and {/3,7} U FORM(S) U SUPP(S) is consistent then (7, { n , ...,r n ,/3,7}) G 5". Here Ths(A) is the smallest set such that A C Ths(A) and if
(^^...^X^GTM^) and a i , ...,0^ h a: (h stands for classical provability) then (a,XiU...UXk) G Ths(A). As in the framework for CDL gave by Brewka t13l, the language . D ^ F is the set of all defaults. A default theory, DTB, T is a set of defaults, i.e. T C DEF. There is no distinct treatment of facts and defaults as in DL: facts can be represented as a special type of defaults of the form true : true/a. Defaults of the form true : fa, ...,pn/j are called assumable. We will often use a as shorthand for the facts true : true/a as well as : /?i, ...,/3 n /7 f ° r the assumable true : /3i,..., Pn/j- Extensions will be generated in a stepwise manner. Let R be a set of monotonic inference rules and T a set of defaults. The closure of T under R is denoted CR(T) and called potential belief set. Definition 2.4^13^ (default logic) A default logic is a pair (R, Ext), where l.R is a set of (monotonic) inference rules on defaults.
314
ZHANG M I N G Y I
2. Ext: P(DEF) »-> P(P(DEF)) is a function mapping a set of defaults D to a set of subsets of D, the extensions of D. We usually are only interested in the value of Ext for sets of defaults which are closed under R. Extensions of the closure CR(T) will also simply be called extensions of T. It is worth to point out that Brewka defined skeptical provability. This makes trivial failure of cumulativity avoided in the limiting case, where no extension exit. That is, every instance of Brewka's framework satisfies cumulativity. To reconstruct Reiter's DL in this framework Brewka gave a logic LDL — {RDL,ExtDL,) in the following way. Definition 2.5t13^ RDL is the following set of rules: DL0 => true :
true/true
DL1 true : ft, . . . , f t / 7 , 7 h a => true :
ft,...,ft/o-
DL2
true : ft, . . . , f t / 7 , irue : ft+i,..., {3m/a => true : ft, ...,/3m/j
true : ft, ...,/? n /7,7 : Pn+i,-,Pm/o' 13
=> true :
Aa
ft,...,^m/tr
Definition 2.6t l Let T be a set of defaults, C the closure of T under RDL, E is an LDL extensions of T, E G ExtDL{C) iff 1. £ C ASS(C) 2. for each assumable d = true : ft, ...,/3 n /7 € A5 , 5(C): d € E iff {ft} U CON(E) is consistent for all i (1 < i < n). Brewka also gave an exact correspondence between DL extensions and consequents of LDL extensions t 13 l. T h e o r e m 2.1 ^ Let (D,W) be a Reiter DT, and T = D U {true : true/p\p e W}. The function CON is a bijective mapping from the set of LDL extensions of T to the set of DL extensions of (D,W). To avoid some of the drawbacks of DL he introduced the following non monotonic logic, which is quite well-behaved for semi-normal default theo ries. Definition 2.7^131 RLl consists of the following rules: i^o tautologies: =$■ true : true/true
D E F A U L T L O G I C AND I T ' S VARIANTS
315
R\ weakening: true : /fl/7,7 ^ & => ^ ^ e : /3/<7 i?2 combination: true : 0i/ji,
true : #2/72 => frue : ft A #2/71 A 72
Rs chaining: true : /3/7,7 : r/8 => true : ft A r / £ R4 equivalence: a : ^ / 7 , a = rj,/3 = T ^ rj : r / 7 i?5 disjoining: £r?ze : /3/7, fr-ue : r / 7 ==> true : /? V r / 7 #6 cases: a : (3/j, 7] : r/<5 => a V 77 : /? A r / 7 V (5 Definition 2 . 8 ^ Let T be a DTB, C it's closure under # L l . E is an Li extension of T ,i.e. E e ExtLl(C) iff E is a maximal subset of C such that 1. E C ASS(C), that is E consists of assumable in C. 2. KERN(C) C E 3. CCS(E) U CON(E) is consistent. Remark. If E = 0 is an LDL extension of a default theory ,then there is some f3g £ CC£({<5}) s u c n that { A } is inconsistent for all 6 G ASS(C). Without lose of generality we always suppose that an LQL extension E of T is not empty in this paper. T h e o r e m 2.2 f13^ Let T be a semi-normal DT. T has an extension iff CON(KERN(T)) is consistent. T h e o r e m 2.3 ^ (semi-monotony) Let T be a semi-normal DT, d = a : (3/y a default such that (3 h 7 and (3 is not eguivalent to true. For every extension E of T there is an extension E' of V = TU {d} such that E C E'. 3. T h e Semantics Framework In this section a uniform semantics framework and notions of compati bility of a frame are introduced, and properties of frames are studied. Let $ be a class of first order interpretations for the first order language L. Like those approaches introduced by Etherington and Schaub, we consider a pair of the form (\P, if) ,where ^ is a set of models and # a class of sets of models. For any set F of closed wffs we denote the set of all mod els of F by MOD(F), i.e. MOD(F) = {^ e $\il> |= F}. If F = 0 we define MOD($) = $ . Here 0 stands for empty set. In particular, MOD({false}) = 0, MOD({true}) = $.
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Definition 3.1 Call a pair ( # , # ) a frame ,where # ^ 0, # G P ( $ ) , # G P ( P ( $ ) ) . A frame ( * , ^ ) is distributed compatible iff * D ft ^ 0 for all Q e ^ . A frame ( # , * ) is joint compatible iff * n fl{fi|fi 6 # } # 0. For any frames (tf, $ ) and ( 0 , A), (*, # ) = (fi, A) iff * = fi, $ = (l. Definition 3.2 Let 6 = a : /3i, ...,/3 n /7 be a default. We define a relation >§ on frames as follows: for any frames ( ^ 1 , ^ 1 ) and ^2,^2)5 ( * 2 , * 2 ) >s ( * i , * i ) holds iff 1. ^ 1 |= a, i.e. for any ip e ^I,I/J
\= a
2. # 2 = {V> G *ih/> 1=7}
3. *2 = * i U Ui< i <„{MO J D({A})} We can extend this relation to a sequence of defaults and a set of defaults respectively. Definition 3.3 Let (\£i, ^1) and (\£2) ^2) be any two frames and (Si) a a-type (a is an ordinal number) sequence of defaults. The induced relation >(Si) is defined as follows: (^2* ^2) >(«5i) ( ^ 1 , ^ 1 ) holds iff there is a sequence ((Q,i,Qi)) of frames such that l.(flo,fto) = ( * i , * i ) 2. for a successor ordinal i: (0,^,0,^ >6i_1 (fi;_i,fi;_i) 3. for a limit ordinal i: Q,i = f l j ^ f i ^ f i ; = Uj<;fij
4. (*2,*2) = (a,,n a ) For {}, the empty sequence of defaults, we define (^,\i/) >{} (\P, if) for any frame (\P, if). Definition 3.4 Let D be a set of defaults . The part order >D is defined as follows: (\&2, ^2) >D ( ^ 1 , * i ) holds iff there is a sequence (Si) of defaults in D ( possibly, the empty sequence of defaults) such that ( ^ 2 ^ 2 ) >(<5;) L e m m a 3.1 Let Du D2 be sets of defaults and D± C D 2 . If (^, if) >DX (fi,fi) then ( * , * ) >D2 (0,,£l). In particular ( * , $ ) >D ( * , ^ ) for any set of defaults D and any frame (\I>, if). L e m m a 3.2 Let S be a default and D a set of defaults. The relation >$ is anti-symmetric and transitive.( In particular, if ( ^ 1 , ^ 1 ) >s ( ^ 2 , ^ 2 ) ><5 ( * 3 , # 3 ) then ( ^ i , # i ) = ( # 2 , ^ 2 ) ) . The relation >D is reflexive, anti symmetric and transitive, i.e. >D is a partial order. Proof. Reflexivity and transitivity are obvious. For the anti-symmetry, it is clear that # 2 Q #1 and ^ 1 C if2 if ( ^ 2 , ^ 2 ) >D ( * i , * i ) . So, anti symmetry is true. R e m a r k It is worth noting that the definition of >D here is differed from that in [4] and [12]. As in [4], Ti >D T2 iff there is S G D such that Ti ><5 r 2 , where T is a set of models and D a set of defaults and F i , r 2 G P(T). We point out that the definition is not applied to the soundness and completeness results for a default theory. In fact, if D = (: A/B,B : C / F ) ,
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W = 0 then E = Th({B,F}) is the extension of (D,W). But there is no any subset D' of D such that V >D, MOD(W), since there is no any default S G D such that T >6 MOD(W), where r = { G $|<£ \= F}. Similarly the definition of >D given by Schaub t12^ is incorrect for an assertional theory. Definition 3.5 Let D be a set of defaults. A frame (*&,$) is >Dmaximal iff there is no any frame (ft, A) such that (Q,A) >D (^ r ,^ r ). (i.e. (Q,n) >D ( * , $ ) and (ft, ft) ± ( * , * ) ) . ( * , * ) is >p-maximally dis tributed compatible iff there is no any distributed compatible frame (DCF) (ft, ft) such that (ft, ft) >£> ( * , * ) . ( * , * ) is >r>-maximally joint com patible iff there is no any joint compatible frame (JCF) (£2,0) such that (ft,ft)>D (¥,*)• Definition 3.6 A set of defaults D is consistent ifiCON(D) is consistent. L e m m a 3.3 Let T be a DTB, C it's closure under RDL and T* = T f l C T* is consistent iff so is C. In particular, if T is consistent then so is C. Proof. "If Part". Trivial. "Only If P a r t " . Since T* is consistent there is i/; G \P such that ip \= CON(T*). We inductively show that * |= C07V({(5}) for any 6 G C. Base. Clear. Step. 1. If (5 = £r?ze : true/true then ^ |= £r^e. 2. If -0 |=
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(1) if 5i_i = true : Pi,...,0nh & T> t h e n (*i.*<) = ( W e * i - i | ^ N 7 } , * 4 - i U {M02?({&})|1 < i < n}) (2) if I. ^ _ i = true : true/true; or II. ^ _ i = £rue : ft,..,ft/a-, true : ft,.., ft/7 £ {^'U < (* ~ !)} a n d M O D ( { a } ) C M O D ( { 7 } ) ; or III. 6i-x = true : ft, . . , f t n / 7 A cr, £rue : ft,.., ft/7 G {^|j < (i - 1)}, and true : ft+i, ...,pm/a G {^|j < (i - 1)} then ( * i , * i ) = ( * i - i , * * - i ) (3) if 6i-i = true : ft, ...,0m/a, true : ft,..., ft/7 G {^|j < (z - 1)}, and 7 : pn+1,...,pm/a G C, then ( # ; , $,) = ( { ^ G ^ - i l ^ |=
0 M ) > c (*,{*}). Clearly, ( * , $ ) > c ( $ , { $ } ) • If there is (**,**) such that ( $ * , # * ) >s (\I>, ^ ) for some S = a : ft, . . . , f t / 7 G C then \I> |= a. So there are defaults d i , . . . , d m G ASS(C) such that COiV({di, ...,d m }) h a by compactness of first order logic. Repeatedly using rules DL2 and DL3 we have d = true : ft,..,ft/7 G A 5 S ( C ) , where { f t + 1 , . . , f t } = CCS({di, ...,d r o }). As above we know that each default in ASS(C) is in (Si), which implies that d is in (Si). Hence ^ \= 7 and {MO£>({ft})|l < i < n] C ^ . This shows that ( # * , * * ) = (\P,^), a contradiction. L e m m a 3.6 Suppose that the sequences ( ( ^ i , ^ ) ) and (Si) are defined as that in the proof of Lemma 3.5. Then the model (^I/,^) of ASS(C) is > T -maximal such that ( ^ , ^ ) > T ( $ , { $ } ) . Proof, we inductively show that for each j > 0: if S3, £ T then ( * , ¥ ) > ( f f r ) ( * • , * • ) > W ) ( * , { * } ) , where («?•) = (*) - {61,...6j} and (6tf=Tn{61,...,6j}. Base, j = 0, then So = true : true/true. Clearly, the induction conclu sion holds and (S*) = 0. Step. Suppose that the induction proposition holds for all k < j , i.e. if Sk # T, then ( * , * ) > ( , r ) (**,**) > W ) ( * , { * » > where (£**) = (£<) {<$A:|£ < j } , (S*) =Tf] {Sk\k < j}. Now assuming Sj $ T, it is easy to prove the induction proposition is true by induction on structure of Sj.
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Base. Trivial. Step. It is needed to consider the case where 6j = true : Pi,...,Pm/(T, true : ft,..,ft/7 G (<$*) and 7 : 0n+u...,0m/
(tf i + 1 ,tf i + 1 ) = (**,**). Therefore there is a subsequence (<$•) of (^) such that (<$•) C T and ( ^ 1 ^ ) >(Si) ( $ , { $ } ) • Since ( # , £ ) is >c-maximal then it is > T -maximal by Lemma 3.5. T h e o r e m 3.7 Let T be a DTB, C it's closure under RDL. ( # , # ) is a > T -maximal element such that ( # , $ ) > T ( $ , {$}) iff (*, ^ ) is > c -maximal one such that (*&,$) > c ( $ , { $ } ) . Proof. "Only If Part" We inductively prove that if ( $ * , # * ) >6 ( # , * ) then (^*,^*) = ( * , $ ) for any S 6 C. First, We note that for any 6 = a : /?i,...,/9 n /7 G C a n d ( # , v j > ) i f # |= a then {^ G ^ | ^ I" 7} # 0 by consistency of T and Lemma 3.3. So, ( { ^ G * | ^ |= 7 } , ^ U { M O D ( { f t } ) | l < i < n}) >6 Base. Clear. Step. 1. If £ = true : true/true then it is obvious. 2. If 5 = £r?xe : ft, ...,0n/j and d = £rwe : ft,...,ft/cr
G T,
({V € **|V N * } , **) >rf (**,**) >* (*, *)• So, (W G * |V N * } , **) ^ ( # , # ) . By the induction hypothesis we get ({ip G \P|^ |= a}, $*) = (*, * ) . But {^ G * | ^ |= a} = {ip G **\ip \= a } , so (**,**) = (tf, 4>). The other two cases are similar. "If Part". It is immediate by Lemmas 3.5 and 3.6. Corollary 3.8 Let T be a DTB, C it's closure under RDL. ( * , * ) is a >T-maximally (distributed or joint) compatible frame such that (\I>, ^ ) >T ( $ , { $ } ) iff it is a > c - maximally (distributed or joint) compatible frame such that ( # , if) >c ( $ , {$}). 4. Semantics for L ^ ^ and L\ extensions In this section we give semantical characterizations of Brewka's LDL and L\ extensions. Definition 4.1 Let T be a DTB, C it's closure under RDL and E it's LDL extension. Then TE is the generating set of E iff 1. TE C T 2. for each d G T:d G T# iff d G E or there are defaults true : ft,...,/3m/7, true : ft,...,ft/cr G £ such that d = a : /? n +i>—>/W7By the above definition we have
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Corollary 4.1 TE is consistent. L e m m a 4.2 Let E be an LDL extension of a DTB T, TE the generating set of E. If F is the closure of TE under RDL then F = ASS(F). P r o o f First we inductively show that d G E for all d G ASS(F). Base. If d G TE then rfe^. Step. 1. If d = true : true /true then d € E 2. If d = ^r^e : ft,...,ft/cr then there is d! = true : ft,..., ft/7 G F with 7 h cr. By the hypothesis we obtain d' G E. So d G F . 3. If d = true : ft, ...,/3 m /7 A cr then there are defaults di = trite : ft,...,ft,/7, d2 = tr^e : (3n+u ...,Pm/a G F . By the hypothesis dud2 G F . So d G F . 4. If d = true : ft, ...,ft n /cr then there are defaults di = true : ft,..., ft/7, ^2 = 7 : Pn+u-iPmlv G F . By the hypothesis d : G F . Since d2 G TE we have d £ E. Hence A S S ( F ) C F . Vice versa, we can prove F C ASS(F) similarly. Assume that d is any default in E. Base. Clear. Step. 1. d — true : true /true, it is obvious. 2. If d = true : ft, ...,ft/
ft,...,ft/7
GC
with 7 h cr. Since {ft} U CON(E) is consistent for all 1 < i < n then di G F . By the hypothesis dx G ASS^F). So d G A S ^ F ) . 3. If d = true : ft,...,/?m/7 A a then there are defaults d\ — true : ft,...,ft/7, d2 = true : ft+1, ...,/3 m /cr G C. Since {ft} U CON{E) is consistent for 1 < i < m then d\,d2 £ E. By the hypothesis d i , d 2 G ASS(F). So d G ASS(F). 4. If d = true : ft, ...,/^ m /cr then there are defaults di = true : ft,...,ft/7, d2 = 7 : ft+i,...,/?m/cr G C. Similarly, di G F and d2 G T. By the hypothesis dx G A S S ( F ) . So d G ASS(F). Definition 4.2 Let ( ^ , ^ ) be a distributed compatible frame and 8 = a : ft, . . . , f t / 7 a default. ( ^ , ^ ) is stable w.r.t. 8 iff if \I> \= a and there is i/ji G ^ such that i/>; |= 7, ft for each i : 1 < i < n then ({-0 G \P|^ |= 7 } , * U { M O F ( { f t } ) | l < i < n}) is distributed compatible. L e m m a 4.3 Let E be an LDL extension of a DTB T, T# the generating set of E. There is a >T-maximally distributed compatible frame (^, 4f) such that
1. oM)> T ($,W) 2. ( * , * ) is the model of E. 3. (*, # ) is stable w.r.t. 8 for all 8 £T. Proof. By Corollary 4.1 and Lemma 3.5 there is a >^-maximal frame ( # , # ) such that ( # , $ ) >TE ( $ , { $ } ) and that ( * , * ) is the model of
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ASS(F), where F is the closure of TE under RDL. So ($,\I>) is the model of E by Lemma 4.2. Clearly, ( $ , £ ) is distributed compatible. Now we show that ( * , * ) is stable w.r.t. 8 for all 8 e T. Assume that 8 = a : /?i,...,/?„/7 € T. If tf (= a, i.e. CON(E) f- a, then there are defaults ^1 J •••5 Om
G TE with CON({8u...,8m}) h a by compactness of first order logic. Repeatedly using rules DLi,DL2 and DL% we obtain true : 0n+1, ...,f3p/a G ASS(F) = E 8' = true : A , ...,/3 n ,/? n + 1 , . . . , ^ / 7 G ASSfC) where {/3n+1,...,/3p} = CCS({8U ...,6m} and C is the closure of T under RDL. Note that CON(E) U {A} is consistent for a l i i : n + 1 < z < p. If there is ^ G ^ such that ^ f= # , 7 then CON(E) U {A} is consistent for all z : 1 < z < p. So (5' G £ . Hence ( { ^ G * | ^ |= 7 } , ^ U {MOD({Pi})\l > i>n} = (\£,^), i.e. it is distributed compatible. From Lemma 4.3 it is easy to get the following proposition. T h e o r e m 4.4 (Correctness) If E is an LDL extension of a DTB T then the model (\I>, \£) of E is a >T-maximally distributed compatible frame such that
1. 0 M ) > T (*,{*}) 2. (*, $) is stable w.r.t. 6 for all 8 eT. T h e o r e m 4.5 (Completeness) Let T be a DTB. maximally distributed compatible frame such that
If ( # , # ) is a > T -
1. ( M ) > T ( M * » 2. ( # , $ ) is stable w.r.t. 8 for all 8 eT then there is an LDL extension E of T such that (\I/, ^ ) is the model of E. Proof. Let (Si) be a sequence of defaults in T with (*, ^ ) > (ff .) ( $ , {$}). Suppose F is the closure of (<$*) under RDL and £ = ASS(F). Clearly, (\I>, ^ ) is the model of E by Lemmas 3.5 and 3.7. Now we show that E is an LDL extension of T. Assume that 8 = true : /?i, ...,/3 n /7 G ASS(C), where C is the closure of T under RDL. If 8 G E then CON(E U { # } ) is consistent for all z : 1 < i < n. Vice versa, if CON(E) U {Pi} is consistent for all z : 1 < i < n then ({V> G * | ^ |= 7 } , * U {M0£>({A})|1 < z < n}) is distributed compatible by stability of (^^) w.r.t. 8. On the other hand,
({^ G #|V> N 7 } , * U {M0L>({A})|1 < % < n}) >6 (*,¥),
and
CM)
is >c-maximally distributed compatible by Corollary 3.8. So, \£ |= 7, {MOD({Pi})\l < i < n} C £ , which implies that £ G £ . In a similar method we can get a semantic characterization of L\ exten sions. Comparing the definition of L\ extensions with that of LDL and CDL extensions, we observe two basic differences: 1. The set of generating rules of closure increase two rules: disjoining and cases. 2. The joint consistent condition is applied to a set of assumable de-
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faults such that it is possible that applied default does not belong to the corresponding extension. In addition restricting ourselves to defaults with single justification makes the recorded justifications more complex. These are sketched by following examples: E x a m p l e 4.1 The default theory ({A : C/D, B : E/F}, {A V B}) (ADT ({A : C/D, B : E/F}, {(A V B, 0)})) has one DL extension ThL({A V B}) (one CDL extension Ths({(A V J3,0)}). The corresponding DTB has one Li extension which contains A V B and : C A D/D V F. Clearly, the applied defaults A : C/D and B : E/F are not contained in L\ extension. E x a m p l e 4.2 ({: A/C A D,: B/^(C A £>)}, 0) has no DL extension. As a ADT, it has two CDL extensions Ths({{C A D, {A})}) and Ths({(^(C A D), {B})}). The corresponding DTB has two extensions, one of which con tains :A/C, another contains :A/D. And neither Li extension contains the applied default : A/C A D by R3. E x a m p l e 4.3 ({: A/B,B : C / D } , 0 ) has no DL extension, but it has one CDL extension Ths({(B, {^4})}. The corresponding DTB has two ex tensions, which contain : A/B and : A A Cj-^B respectively. This added our difficulties. In a sense, we can only get a weaker semantics characterization of Iq extensions. But this just suggests useness of our semantics framework. As that we will see in section 5, the framework is also applied to CDL extensions. And it is not easy to reconstruct CDL extensions in Brewka's framework. T h e o r e m 4.6 (Correctness) Let T be a DTB, C its closure under RLXIf E is an L\ extension of T, then there is a >c-maximally joint compatible frame ( * , * ) such that
i. ( * , * ) > c ( M * » ; 2. tf = MOD(CON(E))3. MOD({/3}) G V for any 0 G CCS(E). P r o o f (outline) In a method similar to that in Lemma 3.5, we can define a frame ( # , * * ) such that $ = MOD(CON(E)),$* = {MOD({p}\/3 G CCS(E)} (using the well ordered set of E). Here we simply apply each de fault in order of E to constrict the sequence ((\I>;, \£;)). Since CON(E) U CCS(E) is consistent, it is easy to see that (^I/,^*) is a >^-maximally joint consistent frame such that ( * , ^ * ) >E ( $ , { $ } ) . Let # = ^ * U {MOD({(3})\a : 0/y G C - E,V (= <*,{/?, 7} U CON(E) U CCS(E) is consistent }. It is easy to show that (SI/,1^) is >c- maximally joint com patible such that ( ^ j ^ ) >c ( $ , { $ } ) . In fact, if there exists a default 6 = a : 0 / 7 G C - E such that ( { ^ G ^\I/J \= 7 } , * U {MOD({(3})}) >6 ( ^ 5 $ ) and that the left of the inequality is joint compatible, then ^ J= a, {£,7} U CON(E) U CCS(E) is consistent. Hence MOD({p}) G #. By compactness there are defaults in E : /^/7;(1 < i < n) such that 71 A ... A 7 n h o:. Therefore : fii A ... A (3n/a G i£. This implies that
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: ft A ... A /3n A Ph € E. So, {^ G *|V \= 7} = * T h e o r e m 4.7 (Completeness) Let T be a £>!#, C its closure under RLl and KERN(T) consistent. If ( * , * ) is a >c-maxima!ly joint compatible frame such that ( $ , $ ) >c ( $ , {$}), then there is an L x extension E such that $ = MOD(CON(E)) and that MO£>({/3}) G * for each 0 G CCS(E). P r o o f (outline) Let E = {: 0 / 7 G C | * |= j,MOD({(3}) G $ } . It is clear that E C ASS(C),KERN(C) C £ . We easy show that E is an L r extension of T by >c-maximally joint compatibility. 5. Application to DL and it's variants In this section we illustrate the expressiveness of our semantics frame by applying it to Reiter's DL, Lukaszewicz' modified extension and Brewka's CDL. So it seems interesting that this frame supports a model theoretic semantic for the design of a particular logic. A semantics of a Reiter DT, though it can be made similarly to Theorems 4.4 and 4.5 by the characterization of extensions for DL (Theorem 2.11 of [14]), is easy got by Theorem 2.1 in section 2. In fact it is sufficient to note that ( * , * ) >D ( $ , { $ } ) iff ( $ , * ) > T (MOD(W),{§}), where (D,W) is a DT and T = D U {true : true/p\p G W}. So we have T h e o r e m 5.1 (Correctness) If E is a DL extension of (D,W) then there is a >£>-maximally distributed compatible frame ( ^ j ^ ) satisfying the fol lowing conditions: 1. ( * , * ) >D (MOD{W),{$}) 2. (\P, * ) is stable w.r.t. 8 for all 8 G D 3. * = MOD(E) and * = {MOD({P})\(3 G CCS(GD(E))} U {$}, where GD(E) is the generating set of defaults for E. T h e o r e m 5.2 (Completeness) Let (D,W) be a default theory. If ( # , * ) is a frame satisfying the following conditions: 1. ( $ , # ) >D (MOD{W),{$}) 2. ( ^ j ^ ) is >£)-maximally distributed compatible and is stable w.r.t. 8 for allSeD then there is a DL extension E of (D,W) such that \P = MOD(E) and £ = {MOD({(3})\(3 G CCS(GD(E))}\J{$} where GD(E) is the generating set of defaults for E. In what follows, we give a semantics characterization of CDL extensions. To do this, the following notations and propositions in [16] are needed, and we always suppose that any ADT (D,W) is well-based, i.e. FORM(W) U SUPP(W) is consistent. Definition 5.lt 16 ] Let A = (£>, W) be an ADT. For any subset D' of D, A(D',A) = Ui>oD'i(A) where DJ(A) = {a : / ? / 7 G D'\FORM{W) h a} D'i+1(A) = {a : / 3 / 7 G D'\FORM{W) U CON&<{£)) h a} for i : i > 0
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Whenever A = (D,W) is clear from the context we may write A(D') instead of A(£>', A). Definition 5.2 Let E be a CDL extension of A = (D,W). Then the generating set of defaults of E, GD(E, A), is the set of all a : f3/j G D such that a G FORM(E), {£,7} U FORM(E) U SUPP(E) is consistent. Definition 5 . 3 ^ Let A = (D, W) be an ADT and D' a subset of D. We say that D' is A-robust (or just robust, if A is understood), if the following conditions are satisfied: 1. D' is joint compatible w.r.t. A 2. A(D') = D' 3. D' is a maximal subset of D w.r.t. the properties 1 and 2, i.e. there is no D" C D such that D' C D"', D' ^ D" and that D" is joint compatible and A(D")=D". T h e o r e m 5 . 3 ^ Let A = (D, W) be an ADT. If E is a CDL extension of A then GD(E,A) is robust. Vice versa, if D' C D is a robust set then there is a CDL extension E of (D,W) with GD(E, A) = D'. Suppose that A = (£>, W) be an ADT. Let T = D U {true : n , . . . , r n / p | ( p , { n , ...,r n }) G W) and T a DTjg. The following properties are easy obtained by Corollary 3.8, Definitions 5.2, 5.3 and Theorem 5.3. L e m m a 5.4 If D' is a robust set w.r.t.(D,W) then there is a > # maximally joint compatible frame (\P, \£) such that * = M O £ ( F O f l A f (WO U 4f = {MOD({q})\q
CON(D')),
G ST7PP(W0 U CCS(D')
U {te}}
and that ( * , * ) >D (MOD(FORM(W)),{MOD({q})\ q G S*7PP(W0} or q = trite). Proof, (outline) In a way similar to that in Lemma 3.5 we use the well ordered set D' = {di,d 2 , •••} and get sequences (Si) and ( ( ^ i , ^ i ) ) such that 1. (Si) consists of all defaults in D', 2- ( * , * ) > ( t f 0 ( M O 2 ? ( F O B M ( W ' ) ) , { M O D ( { 9 } ) | g € 5 l 7 P P ( W r ) or q = true}). 3. (\£,$) = (r\i>o^i,^Ji>o^fi) is >D-maximally joint compatible. L e m m a 5.5 If (\J>,\£) is a >£>-maximally joint compatible frame such that (tf, $ ) > # (MO£>(FOi*Af (WO), {MO£>({?})|? G S77PP(W0 or g = true}) then there is a robust set D' C D such that # = M O £ > ( F O # M ( W 0 u CON(D')), 4f = {MOL>({g})|g G S V P P ( W 0 u CCS(D') U {true}}.
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Proof, (outline) Let (di) is a sequence of defaults in D such that ( * , * ) >(*) (MOD(FORM(W)),{MOD({q})\q G SUPP{W) or q = true}). Let D' be a set of defaults which are in (di) . Then it is easy to show that D' is a desired set. T h e o r e m 5.6 Let A = (D,W) be a well-based ADT, . If E is a CDL extension of (D,W) then there is a >£>-maximally joint compatible frame ( # , * ) such that ( ¥ , $ ) >D (MOD(FORM(W)),{MOD({q})\q G SUPP(W) or q = true}) and that * = MOD(FORM(E)), * U {MOD({J})\J
e (CON(GD(E,
A))}
= {MOD({q})\q G SUPP(E) or q = true}. Vice versa, if ( * , $ ) is >D-maximally joint compatible such that ( # , # ) >D (MOD(FORM(W)), {MOD({q})\q G SUPP(W) or q = true}) then there is a CDL ex tension E of (D,W) such that MOD(FORM(E)) = # , {MO£>({g})|g G SUPP(E)} = ${J {MO£>({ 7 })|7 G CON(GD (E, A))}. Proof. It is obvious by Theorem 5.3 and Lemmas 5.4, 5.5. It is clear that Schaub's Correctness and Completeness Theorems (a se mantics characterization of Brewka's CDL) are an immediate conclusion of Theorem 5.6 here. Finally we can similarly get a semantics characterization of Lukaszewicz' modified extensions ^ in term of the semantics frame. For sake of simplicity we will only state the results since their proofs are easy. l e m m a 5.7 Let $ be any class of first order interpretations and D any set of defaults. A default 8 G D is $-applicable w.r.t. D iff the model (*, 4f) of {6} is distributed compatible and (*, £ ) >6 ( $ , {MOD({/3})\/3 G CCS(D)}). A subset D' of D is ^-applicable iff the model ( * , * ) of D' is distributed compatible and (vP,*) >D ( $ , { $ } ) . D' is maximally $applicable w.r.t. D iff the model (^J,^) of D' is maximally distributed compatible and (#,4>) >D ( $ , { $ } ) . T h e o r e m 5.8 Let (D,W) be a default theory, * the class of all models of W, i.e. $ = MOD(W). S is a modified extension of (D,W) iff there is a >£>-maximally distributed compatible frame (fi,Q) such that (Q, H) >D ( # , { $ } ) and il = MOD(S).
6. Conclusion In this paper a uniform semantical framework of various default logics by extending Etherington's semantics has been presented. We have shown
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ZHANG M I N G Y I
how our framework capture various (potential) application conditions of a default by a preference relation and notions of compatibility of a frame. That apart, we derive semantics characterizations of various default logics, which are corresponded to their syntactic characterizations. By the same method we can deal with Sandewall's accessibility of a BJ-pair t5l, Etherington's semantic ^ , as well as Giordano and Martelli's commitment to assumption default logic and quasi-default logic, etc. Compared to Besnard and Schaub's possible worlds semantics, we have not used a modal system to provide a semantical framework. This seems to be natural and simple. Furthermore, it appears to be interesting that the semantics presented in this paper can support the design of a semantical characterization of a particular nonmonotonic reasoning system , which we will discuss in another paper. References 1 Reiter R. A Logic for Default Reasoning, Artificial Intelligence, Vol.13, No.1-2, 1980. 2 Etherington D.W. Formalizing Non-monotonic Reasoning System, Ar tificial Intelligence, Vol.31, 1987 3 Etherington D.W. Reasoning from Incomplete Information, Pitman Research Notes in AI, Pitman Publishing limited, London, 1986 4 Etherington D.W. A Semantics for Default Logic, in Proc. of IJCAI-87, Milan, Italy, 1987 5 Sandewall E.J., A Functional Approach to Non-monotonic Logic, Proc. of IJCAI-85, Los Angels, 1985 6 Lukaszewicz W. Two Results on Default Logic, Proc. IJCAI-85,Los Angles, 1985. 7 Lukaszewicz W. Considerations on Default Logic-An Alternative Ap proach, in Proc. of Nonmonotonic Reasoning Workshop, New Paltz, N.Y, 1984 also Computational Intelligence, 4, 1988. 8 de Guerreiro R.A. and Casanova M.A. An Alternative Semantics for Default Logic, 3-rd International Workshop on Nonmonotonic Reasoning, South Lake Tahoe, CA, 1990 9 Pool D. What the Lottery Paradox Tell Us about Default Reasoning, in Proc. of 1-st ICPKRR, Toronto, 1989 10 Delgrande J.P. and Jackson W.K. Default Logic Revisited , Proc. of 2-nd International Conference on Principles of Knowledge Representation and Reasoning, April, 1991, by Morgan Kaufmann Publishers,Inc. 11 Brewka G., Cumulative Default Logic: in Defense of Nonmonotonic Inference Rules, Artif. Intell. 50, 1991. 12 Schaub T. Assertional Default Theories: A Semantical View, in Proc. of 2-nd International Conference on Principles of Knowledge Representation and Reasoning (KR-2), 1991
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13 Brewka G. A Framework for Cumulative Default Logic, TR-92-042, International Computer Science Institute, Berkeley, CA, 1992 14 Zhang Mingyi A Characterization of Extensions of General Default Theories, in Proc. of 9-th Canadian Society for Computational Studies of AI, Vancouver, BC(1992) 15 Zhang Mingyi, On the Existence of Extensions for General Default Theories, Science in China (English Edition), Series A, 10, 1993. 16 Gottlob G. and Zhang Mingyi cumulative Default Logic: Finite Char acterization, Algorithms and Complexity, Tech. Report, CP-TR- 93-54, Institute for Informationssystem, Tech. Univ. of Wien, Austria, 1993. also see Artificial Intelligence, vol.96, 1994, pp.829-845. 17 Giordano L. and Materlli A., On Cumulative Default Logic, Artif. Intell., 66, 1994. 18 Besnard P. and Schaub T,, Possible Worlds Semantics for Default Logics, Fundamenta Informaticae, 21,1994.
329
A D D I N G EVENTUALLY DIFFERENT REALS
Yi
ZHANG
Institute of Sorftware, Academca Sinica Beijing, China Department of Philosophy, Rutgers University New Brunswick, New Jersey 08903, USA cyzhang@math. rutgers. edu A b s t r a c t . In this paper, we will mainly study two cardinal invariants of continuum, a and a c , which we will define in the definitions 1.1 and 1.2. Among them, a is well-known, ac looks similar to a. Futhurmore, a and ae have some important "same-looking" features, for example, ac stays small in the Cohen forcing model as a does. In the second section, we will compare a, ac together with some other well-known continuum invariants as and b in a forcing model of ZFC. Our forcing terminologies are standard, people can find them in either [Kun] or [Jech]. 1. Definitions A n d S o m e Properties of Cardinals W e Study. Definition 1.1. If x,y C u, x and y are almost disjoint (a.d.) iff | xOy |< cu. An a.d. family is an A C p(ui) such that for any x G A, \x\ = UJ and any two distinct elements of A are a.d.. Let a be the least A such that there exists a maximal almost disjoint (m.a.d.) family T C p{u) of size A. Definition 1.2. Following A. Miller, we say that two functions f,g € U(J0 are eventually different (e.d.) iff \{neu>\f(n)=g(n)}\. Let ae be the least A such that there exists a maximal eventually different (m.e.d.) set of reals of cardinality A. The following ressults about a are well-known: (1) Any m.a.d. family F C p(u) is uncountable. The research is partially supported by a grant from "Chinese Climbing Project Foun dation for Natural Science" and CNSF-6957303. 1991 Mathematical Subject Classification 03E35, 20A15,20B07,20B35.
330
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(2) ZFC + MA implies that a = 2" (3) Let M f= (ZFC + -»CiJ), and let « be a cardinal in M such that CJI < K < 2U. Then M can be generically extended to an M[G] which preserves 2W, and there is, in M[G], a m.a.d. family A C p(u) of cardinality «. Hence Con(ZFC)
-> Can(ZFC
+ a < 2W).
(4) Let M |= (ZFC + OH"). There is a m.a.d. family JF C p(u) of size CJI in M such that for any Cohen generic G over M , T remains to be a m.a.d. family in M[G]. For the proof of (1), (2) and (4), see [Run]; for the proof of (3), see [Sh:P]. We can prove the corresponding results for ac by using the following c.c.c. p.o.set (A similar version of the forcing notion was studied by A. Miller in
M). Definition 1.3. For any A C ^CJ, define a partial order P ^ which consists of all conditions of the form (s,F), such that (1) 5 is a finite partial function from LJ to LJ, and (2) F is a finite subset of A, where ( s i , F i ) < (s 2 ,F 2 ) iff (5 2 C 5i) A (F2 C Fi) A V / G F2(f f] Sl C s 2 ).
T h e o r e m 1A(ZFC).
Any m.e.d. set i c ^ w is
uncountable.
T h e o r e m 1.5 (MA(K)). Let A C UUJ, where \ A \< K, and to < K, < 2U. Then there exists a g G "co such that
VfeA(\gnf\
oc = 2 W .
T h e o r e m 1.7. Let M \= (ZFC + -
V r / , c < ^ / C ^ | p r 7 n ^ c \).
A D D I N G EVENTUALLLY D I F F E R E N T R E A L S
331
Now we proceed with a system of iterated forcing of length ui with finite support as follows. At step a we assume that we have constructed a sequence (QTI • V < K + a) of pairwise e.d. functions in UUJ. We take A
* = {#77 | V < « + &}
and we use at this step the forcing notion IP^ a . Then we can prove the theorem by standard method.
□ Corollary 1.8. Con(ZFC)
-> Con(ZFC
W
+ ae < 2 ).
In the following we prove that ae has the similar character as a in Cohen forcing, i.e., we force with a c.c.c. partially ordered set Fn(I, 2) = {p | p is a finite function, dom(p) C 7, rang(p) C 2}. on the ground model M which satisfies ZFC -f GCH. We first state a well-known Lemma about Cohen forcing. L e m m a 1.9. Suppose 7, S G M. Let G be Fn(1,2)-generic over M, and let X C S with X e M[G\. Then X e M[G n Fn(I0,2)] for some 70 C 7 such that 70 <E M and (|7 0 | < | 5 | ) M . Proof. See [Kun].
□ By the above Lemma, it is sufficient to construct some m.e.d. family A C ^u in extensions via 70 = u which are countable in M. T h e o r e m 1.9. Let M \= (ZFC + CH). There is a m.e.d. function set A of size LJ\ in M such that for any Cohen generic G over M, A remains to be a m.e.d. function set in M[G]. Proof. Since Fn(co,2) has c.c.c. and M \= CH, there are at most UJU = 2" = u\ different antichains, there are at most (2^)^ = u)\ nice names for reals. In M , define a m.e.d. function set A of size u)\ as follows: Let (pa,Ta) for u < a < u\ enumerate all pairs (p,r) such that p € Fn(u, 2) and r is a nice name for a function of u. By recursion, pick function fa G "w as follows: Let {fn | n < u} be any e.d. function set in UCJ. If LJ < a < uj\ and we have functions fp for (5 < a, choose a function fa so that (1) V J 8 < a ( | { n | / a ( n ) = / / 9 ( n ) } | < ( j ) ,
332
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ZHANG
(2) if Pa ll~ ija G ^CJ), and V/3 < a(pa lh Ta and /^ are e.d.) then pa \\~ ra and / a are not e.d.. To see that fa maybe so chosen, let Aa = {fp | P < a}. Consider the c.c.c. p.o.set lPUa5 which we defined in Definition 1.3, in M. We consider the following dense sets:
Df = {(s,F)erAa En = {(s,F)€WAa
1/eF}, \n£
dom(s)},
Cntq = {(s, F) G I?Aa I 3x G co3m > n3r < q(s(x) = ra A r lh ra(x) = ra)}, where q < PaIt is easy to see that Df and En are dense in FA* • We prove that G n , g is dense in FA* Given (s,F) G F ^ a , n G w and 9 < pa. By assumption of (2), since | F | < UJ and q
<pa,
q lh ((3t G u)(\/z > t)(ra(z)
± f0(z),
for all 0 G F ) ) .
Then there exist q0 < q and t G w with n
+ f0(z),
for all /? e F ) ) .
Therefore, there exist r < qo and x > t and x 0dom(s) and m > n such that r lh r a ( x ) = ra. Also, this gives ra(x) ^ fp(x), for all (3 € F. Hence, if s' = 5 U {(x,ra)}, then ( s ' , F ) < (s,F) and ( s ' , F ) G Cn%q. Thus C n , g is dense in P ^ . Let £> = {£>/ I / G A a } U {En | ra G u} U {G n , q | n G a; A g < pa}. Then I L> |< to. By M A ( C J ) , there is a filter G a C ff^ such that for any d G D,
Ga n d ^ 0 . Let / a = U { 5 I ( s , F ) G G a } . Then / a satisfy (1) and (2). Now, let A = {fa I a <
ujf}.
Let G be IP-generic over M. Suppose towards contradiction that A is not maximal in M[G]. Then there is a {pa,Ta) such that pa G G, Pa lh (r a is a function of a;) and pa lh V/ G A(\ ra H / |< u).
A D D I N G EVENTUALLLY D I F F E R E N T R E A L S
333
Thus the condition of (2) holds at a and pa Ih (| ra D fa |< w), this implies 3n G u(pa Ih (r a n fa) C (n x n)). But this contradicts that Pa II" (Ta and fa are not e.d.).
□ By results in [M], we can easily have the following: Corollary 1.10. It is consistent with ZFC that ac < Cover
(meager).
2. A d d i n g Eventually Different Reals. In this section we will prove that it is consistent with ZFC, that a < ae. We will force UJ2 times with finite support by using the following p.o.set on a c.t.m. M of ZFC + GCH. Definition 2.1. Let IP = {(s,F) (suF1)<(82,F2)iff
\ s G u
(s2 C si) and (F2 C Fi) and u
Vn G CJV/ G u)((n G d o m ( 5 l ) \ d o m ( 5 2 ) ) A ( / G F 2 ) -> ( 5 l ( n ) ^ / ( n ) ) ) . Obviously, L e m m a 2.2. P iias c.c.c. Note that IP had been well studied by A. Miller in [M]. Especially the Lemmas 2.3., 2.7.,2.8., 2.10., 2.11., and 2.12 had been stated and proved in [M]. The reason we state them here is just for clarity of our proof of main result in this section. L e m m a 2.3. Let G be IP-generic over M. Then in M[G], there is a g G "to such that
Vf
euvnM(\fng\
L e m m a 2.4. Let M (= (ZFC + GCH). Then M can be generically ex tended to an M[G*] such that M[G*] \= (a e = UJ2 = 2 W ). Proof. We proceed with a system of iterated forcing of length u>2 with finite support as follows: Let IP be the p.o. set in Definition 2.1. Define P a for a < co2 as follows: (1) Po = IP M ; (2) P a = U/?
334
Yi ZHANG
It is easy to see that M[GU2] 1 = 2 ^ = ^ 2 We know that ZFC h (there is no m.a.d. family in uu has cardinality u>). We claim that there is no m.a.d. A* C ucu in M[GU2] such that A* has cardinality UJI. Assume that there exists one m.a.d. A* C "u in M[GU2\ which has cardinality o^. Consider the nice name of g G A*. Since for each n G u, there is a maximal antichain # £ which decides g(n), where g is a name of y. PU;2 is c.c.c, then | £?£ | = UJ. Let
g£A* n6w
Obviously | JB* |< UJ\. For each p G 5 * , supt(p) is a finite subset of u^Hence there is an a < UJ2 such that VpG £*(supt(p) C a ) . If G a is the component of GU2 in the iterated forcing up to (not including) a, then we have A* G M[Ga]. But this implies that A* is not maximal by our construction of forcing. Therefore we proved that M[GU2] \= at = UJ2 = 2W. D The idea of proving the following Lemma 2.5. was given by A. Miller in [M], the proof we give here is just a clearer version of the proof in [M]. L e m m a 2.5. Suppose lhP r G co, s G u
I # G [LU]
("v)n.
Give CJ the cofinite topology and give "LU and (uuj)n the product topology. Then by Tychonoff's theorem, ("to)71 is compact. Now we will prove that GH is open. For any (g0, ...,gn-i) P= {t,F) < (5,{po,...,0n-i}) and p II- re H, then let UH = { ( / o , . - . , / n - i ) | Vm G a; (m G dom(t) \ dom(s) —>
VA: < n(t(m) + fk(m))}.
G GH, if
A D D I N G EVENTUALLLY D I F F E R E N T R E A L S
By Definition 2.1, we know that (g0, ...,p n _i) G UH. since for any ( / 0 , . . . , / „ _ i ) G E/JT,
335
Also, UH C G # ,
p' - ( t , F U {/o, ...,/„_i}> < (5,{po,.-,Pn-l}>, by 7? lh r G iJ, and p' < p, we know that p' lh r G # . And £/# is open in (uu)n. Hence GH is open. Since GH U G K C GHUK,
therefore, there exists # G M < u , 5 such that
We get the conclusion. D We do not actually need the following Lemma 2.6 to prove the main results in this section. For clarity, this is just a warm-up exercise for Lemma 2.9. L e m m a 2.6. Let M (= (ZFC + GCH). Let G be ^-generic over M. Then, in M, we can construct a m.a.d. family T C p(u>) of size u)\, and T remains to be maximal in M[G]. Proof. Within M , we shall define a m.a.d. family T of size CJI such that T remains to be maximal in M[G] when G is IP-generic over M. From now on, all forcing terminology refers to the p.o. set P. Within M , do the following. By CH, let E = {{pi,ri)
\U)
enumerate all pairs (p, r ) such that p G IP and r is a nice name for subset of to. By recursion, pick infinite F^ C LJ as follows. Let {Fn | n < to} be any almost disjoint sets in P(CJ). If u < £ < u)\, and we have Fv for rj < f, choose F^ so that (1) V r / < $ ( | F r ? n ^ | < ^ ) , and (2) if Pi
lh (|r € | = a;) and Vr? < £(p c lh (|r e n F„| < w)),
then Vn G c^Vg < p$3r < q3m > n((m G F^) A (r lh m G r^)).
336
Yi
ZHANG
To see that F^ maybe so chosen, assume that the condition of (2) holds, since if it fails then only (1) need to be considered, and we can simply apply the fact that there is no m.a.d. family of size u (see [Kun]). Let En, for n G a;, re-enumerate {Fv | rj < £}. We shall build an F^ such that Vn G cj(|F^OE n | < u ) and p^ lh ( f o n i ^ l = u>). We shall build F^ in stage as follows:
Ft = | J Kn, n
where Kn G p(w) and if n is finite. When we choose each Ki, for i G a;, we shall make sure that Vm > n(Kmr\En = 0), therefore for any n, \F^C\En\ < u. Also, we shall arrange that Vn G oSip < p^3q < p3m G F^(m > n and (/ lh m G T$), i.e.,p€H-(|Fenr^|=cj). We list all (sn,ln,mn) G oo
Fi=\J
Kn.
n
By construction, for any n < CJ, F^ n En C Ui
7$,
i.e., p lh Ff Pi r^ C m + 1. Let p = (sn,F), where \ F \< ln and ran = m. By Lemma 2.5, there exists some r < p such that there exists t G Kn, and £ > m n , r lh (£ G r^). We get contradiction. Now let T={Fi
U < ^ } .
We claim that J 7 is a m.a.d. family in M[G].
A D D I N G EVENTUALLLY D I F F E R E N T R E A L S
337
Let G be IP-generic over M. If T failed to be maximal in M[G], there would be a (p^r^) such that p^ G G, Pt Ih \T(:\ = LJ and
p^ Ih Vx G .F(|T£ C\x\<
LJ).
Thus, the condition of (2) holds at £, but also p^ lh (|r^ f)F^\ < LJ). SO there is a q < p£, and there is a m G CJ such that glh (T^ f l i ^ C ra + 1). This implies that ^ 3 r < qit > m(t G F^ -► r Ih t G T$). But this is a contradiction. Therefore .T7 is a m.a.d. family in M[G]. D L e m m a 2.7. Vp G Pu,23
injective 3na < LJ
lhPa (q(a) = (sa, G) for some G G ["LJ]71*). Call such q canonical, from now on, we assume that, for any p G P ^ , p is canonical, since the canonical ones are dense. L e m m a 2.8. Suppose lh r G CJ, and given F G [^2]
= naAsqa
= sa) -> 3p < q(p Ih r G # ) ) .
Using Lemmas 2.7. and 2.8., we can prove our crucial Lemma: L e m m a 2.9. Let M \= (ZFC + GCH). There is a m.a.d. family T in M such that Va < LJ\ if Ga is IP a -generic over M, T remains to be maximal in M[Ga]. Proof. Within M , we shall define a m.a.d. family. By CH, let E = {fa,Tt,t)
| u ; < f
enumerate all triples (p, r, a) such that p G fUJl and r is a IP^ -nice name for subset of LJ. Without lost of generality, we may assume that in our list, for each £ < CJI, sup(supt(p^) < £) and T£ is a P^-name. By recursion, pick infinite F^ C LJ as follows: Let {Fn | n < LJ} be any almost disjoint set in P(LJ). If LJ < £ < LJI, and we have Fv for rj < £, choose F$ so that (1) \/rj
|
338
Yi
ZHANG
(2) if Vi lhp€ (I n |= LJ) and Vr? < (fa
lh P J r^ n F„ |< LJ),
then for any n G w and for any q < p^ with g G IPf there exists some r < q with r G IPf and there exists some m >n such that (m G F{) A (r thp€ ra G r^). To see that F^ maybe so chosen, assume that the condition of (2) holds, since if it fails then only (1) need to be considered, and we can simply apply the fact that there is no m.a.d. family of size LJ. Let En,
for n < LJ, re-enumerate {Fv | r\ < £}.
We shall build an F^ such that Vn G LJ(\F^ C\ En\ {\nnF(\=Lj). We shall build F^ in stage as follows:
< LJ) and p^ Ihp^
F( = U K" n
where Kn G p(u>) and Kn is finite. When we choose each K{, for i G LJ, we shall make sure that, for any p G IPf with p < p^, there exists a g G P^ with q < p, such that 3ra G F^(m > 77, and q lhpe m G r^), i.e.,p e lhPe (| F e O r ^
\=u).
We may assume s u p t ( ^ ) = {a^ G CJI | 1 < i < m^, for some m^}. Let 5 = (sai,...,sak)
and Z~ = (lai, ...,lak)
with
{ a f , . . . , a 4 j C {ai,...,a*}, here all a; < £. We list all (an, sn, Z~n, m n ) G (
A D D I N G EVENTUALLLY D I F F E R E N T
Vn G uVa\ G s u p t ^ ) ^ ^ C # ,
REALS
339
if af = a j for some 1 < j < k).
Let r n = min{r^ \ m a x { r € n U i < n ^ i , T h n , U < < n ^ i } } . To set Kn, we con sider ( 5 ^ , . . . , 5 ^ , / ^ , . . . , / ^ , m n ) . P i c k g < p e withsupt(gr) = {au...,ak} such that Ih (q(ai) = ( s ^ , F ) for some F G [^CJ]'"' where 1 < i < k). Then by Lemma 2.8, there exists a Hn G [CJ]
3p
^ = U *»• n
By construction, for any n < u, F^ n F n C U;< n ^ ' i-e-> l-^i fl i?n| < CJ. We show that Vp < p^(p lhp£ F^DT^ is infinite). Suppose otherwise, there exist some p < p^ and some m < u such that there is no r < p such that for any t, if m < t G F^, then r lhpe t G r$, i.e., p Ihp^ ^ D r ^ C m + 1. We may assume that supt(p) = {cti | 1 < i < A;}, and If- ( p ( a i ) = (s%,F)
for some F G M ^ ) ,
and
m = mn. Then there exists some r < p such that there exists t > mn and t G K n and r lhpc (t G TJ). We get contradiction. Now let J ^ - {F^ U < ^ M } . We claim that .F is an m.a.d. family in M[Ga] for any a
sucn
tnat
P£ ll~pa |r^| = cj, and V?7<€(p e lhp a | r e n i g < C J ) . Thus the condition of (2) holds at £, but also Pi lhpa | r ^ n F e | < c j . This is a contradition. Therefore, F is a m.a.d. family in M[Ga]. D
340
Yi
ZHANG
L e m m a 2.10. Suppose M C TV are transitive models of ZFC Then if G is ¥N-generic over TV then G D P M is FM-generic over M. Proof. We claim that for any A C P M , if A is a maximal antichain in P M , then A is an maximal antichain in FN. This is because of that being maximal antichain of a c.c.c. p.o. set is a II} property, therefore, it is absolute. If G is P^-generic over TV, then for any maximal antichain A in P M in M , A is a maximal antichain in P ^ in TV, therefore G 0 A / 0 in TV, we have (G fl P M ) n A ^ 0 in M . Hence G n P M is P M -generic over M .
□ Call a p.o. P absolute just in case it is definable (possibly with parameters in M) and given any TV D M, a transitive model of ZFC if G is P^-generic over TV, then G fl P M is P M -generic over M . Suppose P a for a < p is a finite support iteration of sbsolute partial order over M , i.e. P a + i = P a * P a where P a is some name for an absolute partial order in M [ G a ] . Given X C 0(X e M) define the iteration P*(X) for a < /3 as follows: For a g X , let P * + 1 = P* * 1 (where 1 is the one element order). For a e X , let P * + 1 = ff£ * (W<*)MlH«] where F * is P^-generic over M . For G a P a -generic over M define G* = Ga n P* . L e m m a 2.11. For any o: < (3, if Ga is P a -generic over M, then Ga is P* -generic over M. L e m m a 2.12. If lh r C a;, then there exists a X C co2 countable, X G M and for any G fUJ2-generic over M, TQ 6 M[G*]. Thus we conclude that M[G W2 ] H p(w) = | J { M [ f r a ] n p(u) \a
M[GU2],
and Ha is P a -generic over M } . Therefore we proved that T h e o r e m 2.13. It is consistent with ZFC that ac = u2 = 2" and a = u\. Now we try to compare a and ac with some other wellknown continuum invariants in our forcing model. Definition 2.14. Let / , g be functions in wu;, / <* g iff f(n) < g(n) for all but finitely many n G u. A subset of ww is called unbounded iff it is unbounded in (wo;, <*). We define b to be the least A such that there exists an unbounded subset of UUJ which has size A. It is well-known that b < a. Therefore, in our forcing model b = UJ\. It is easy to prove the following:
A D D I N G EVENTUALLLY D I F F E R E N T R E A L S
Remark (ZFC).
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b < ae
Proof. Assume that ac < b. Then there exists a m.e.d. set A C UUJ such that | .4 |= K < b . Let . 4 = {fa e uu | a < K}. Then .4 is domited, i.e., there is a function g G "u, for any f £ A, f <* g. Therefore, V/ G A3Nf
G uVn > Nf(f(n)
£ g{n)).
Thus, A is not a m.e.d. set of functions. We get contradiction. D Definition 2.15. Let as be the least A such that there exists a maiximal family of almost disjoint subsets of UJ x UJ that are graphs of partial functions from UJ to UJ. It is well-known that a < a5. Shelah provided a model in which a < as (see [S:207]). We can easily prove that in our forcing model, a5 — UJ2. T h e o r e m 2.16. M[GU2] \= a5 = UJ2. Proof. Assume that in M[GU2], there exists a maximal almost disjoint fam ily As of partial functions from UJ to UJ, | As \= UJ\. For any fa G As, here a < CJI, we can extend fa to a function ga G WCJ, such that p a (n) = fa(n), if n G dom(/a); ga(n) = 0, otherwise. Then we get As = {ga e^ujla
< ui}.
Let p a G A in M[G W2 ]. We know that there exists an a < UJ2, such that A G M[Ga]. Then for any (5 such that a < (3 < UJ2, the P^-generic function gp is e.d. from any function in A. Hence, in M[GU2], we can find some fuction gp which is e.d. from every fuction in A. This is contradicting that As is maximal in M[GW2\. Therefore, we proved the theorem.
□ Thus we conclude that it is consistent with ZFC that a = b = UJ\ and a e = a s = UJ2 = 2".
3. Q u e s t i o n s . We define ap as the following:
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Definition 3.1. Two permutations / , g G Sym(uj) are a.d. iff | {neuj\f(n)=g(n)}
\
Let dp be the least A such that there exists a m.a.d. set of permutations of cardinality A. We can prove that MA implies that ap = 2^. It is also not hard to prove that Con(ZFC) -» Con(ZFC + ap < 2") and in Cohen foring a p stays small. Using a similar (but much harder, since the space Sym(cu) is not compact, the method of proving Lemma 2.5 does not work) argument, we can prove that T h e o r e m 3.2. It is consistent with ZFC that a < a p . Proof. Force with the following c.c.c. p.o.set in Definition 3.3. For a detailed proof, see [Z].
□ Definition 3.3. Let ¥ = { (s,F) | 5 G u
is injective and F G [Sym(u;)]
where ( s i , F i ) < (s 2 ,F 2 ) iff (s 2 Q. -5i) and (F2 C i<\) and Vn G wVf G Sym(w)((n G d o m ( 5 l ) \ d o m ( 5 2 ) ) A ( / G F2) -> (ai(n) ^ / ( n ) ) ) . Interesting questions would be what the relationships are among ae and Op and a5. It is also interesting to know that if any of ac and ap and as has an upper-bound, which is a cardinal invariants of continuum. Acknowledgments. I would like to thank: Professor Boban Velickovic, for initially suggesting me to investigate a c , also for his hospitality during the spring and summer, 1995 when I was visiting Paris University 7, Paris, France; Professor Renling Jin, for various discussions I had with him about this paper; my friends Luca Bonatti, Martine Gueguen, Aitze Peng and Ming Xu, for all kinds of helps they offered me; last but most, Professor Simon Thomas, for his consist ant and patient helps.
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REFERENCES [vD]
E. van Douwen, The integers and topology, in Handbook of Set Theoretic Topol ogy (ed. K.Kunen and J. Vaughan), 1984, pp. 111-167, North-Holland, Ams terdam. [Jech] T. Jech, Set Theory, Academic Press. [Kun] K. Kunen, Set Theory. An Introduction to Independence Proofs, North Holland, Amsterdam. [M] A. Miller, Some properties of measure and category, Transactions of the Amer ican Mathematical Society 266, Number, 1 (July 1981), 93-114. [Sh:P] S. Shelah, Proper Forcing, Lecture Notes in Mathematics 940, Springer Verlag. [Sh:207] S. Shelah, On cardinal invariants of the continuum., in Axiomatic Set Theory (ed. J. E. Baumgartner, D. A. Martin, and S. Shelah), Contemporary Mathe matics 31., 1984, pp. 183-207, American Mathematical Society, Providence. [V] J. E. Vaughan, Small uncountable cardinals and topology, Open Problems in Topology (ed. J. van Mill and G. M. Reed), 1990, pp. 197-218, North-Holland, Amsterdam. [Z] Y.Zhang, Cofinitary Groups and Almost Disjoint Families.
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Asian Logic Conference:ALC Mariko Yasugi * The Mathematical Society of Japan (MSJ) is playing a more prominent role in international activities in Asia. I thus feel that it is a good time to report on The Asian Logic Conference (ALC) , whose aims and objectives are along the same line as those of the MSJ. Chong (Singapore), Tamthai (Thailand), Yang (China), Motohashi (Tsukuba:Japan) have provided the author with valuable materials in preparing for this article. These will also be kept for future reference.
1
The Beginning
In the year 1980, Crossley (Australia), Chong, Midler (Germany) and Mo tohashi were participating in Logic Colloquium 80 in Greece. They were concerned with the progress of logic in Asia, and thought that an inter national conference held in that region would be useful. Later that year Chong was spending his sabbatical leave at MIT, and had discussions with Anil Nerode who was also visiting MIT then. Nerode suggested, on the occasion of a visit by Sacks to Singapore scheduled for the following year, that a workshop be organized in Singapore. Chong approached various organizations for possible support. Several responded, offering coopera tion and financial aid. The Department of Mathematics at the National University of Singapore also agreed to host the conference. The plan then developed into an international meeting. Less than one year had lapsed before the vision of the four logicians who met in Greece became a reality. The communication within Japan was handled by Motohashi. In the beginning, the name of the conference was a temporary one, and it was not certain if there would be a second meeting. In fact it was succeeded by a conference in Thailand three years later, and one in China three years after that. The ALC has since evolved into the major meeting ground for specialists in logic and the foundations of mathematics in Asia. This was not antici pated by those such as Chong and others who organized the first meeting in 1981. * Translated from the original Japanese article in "Sugaku," edited by the Mathemat ical Society of Japan, vol 46, No.1(1994), 57-62.
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The Outline of the Conference Series
ALC aims at promoting activities of mathematical logic in the Asia-Pacific, as mentioned above, so that logicians both from within Aisia and elsewhere would get togeother and exchange information and ideas. Its purpose is therefore to establish a system of mutual acquaintance and collaboration among Asian logicians. There is no formal superorganization; the detailed plans of activities are discussed each time during one of the meetings, or through e-mail. We are still groping. The tacit assumption is that ALC be a conference on mathematical logic 'by the Asians and for the Asians,' as Chong put it at the Beijing meeting. The host country would be in Asia, and the organizing commitee members would be mostly from the host coun try. Active logicians from the region are invited to give talks, and special encouragement, in the form of financial assistance if necessary, is given to young logicians in the region. These are unwritten agreements. As Crossley emphasized at the beginning of his lecture in Beijing (1987), it is no less important for the participants to be heard than to listen to lectures. The contributed papers are accepted without filtering for this reason. Papers are refereed only during the publication process of the Proceedings. ALC accepts papers from all areas of mathematical logic. While mathematical logic originally grew from the study of the foundations of mathematics, an essentially intellectual pursuit, it has, however, developed and expanded into other areas and now bears close relationship with computer science (through computability theory), and so there are many conferences on the border areas of the two subjects. ALC has that tendancy too, but there are different opinions on how far we should be involved in computer science. What follows is an up-to-date report on the ALC. I will include the following items. 1 Name 2 Time and Place 3 Organization 4 Cooperating bodies 5 Finantial Aid 6 Invited Speakers 7 Number of Contributed Papers 8 Number of Participants 9 Publication 10 Prominent features 11 Miscel laneous
3
Record of Conference: from The First through The Third
The First Meeting 1 The First Southeast Asian Conference in Mathemat ical Logic 2 November 9-13, 1981 National University of Singapore (NUS), Sin gapore 3 Organizing Committee C. T. Chong, H. H. Teh, M. J. Wicks (NUS), J. N. Crossley (Monash)
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4/5 Singapore Mathematical Society, National University of Singapore, Department of Mathematics (NUS), Association for Symbolic Logic (ASL), International Union of History and Philosophy of Science, Division for Logic, Methodology and Philosophy of Science (IUHPS) 6 C. Ash, J. N. Crossley (Monash), M. Lerman (U. Conn), A. Mclntyre (Yale), A. Nerode (Cornell), G. E. Sacks (Harvard-MIT), M. Tamthai (Chulalongkorn) 7 9 (30 minutes each) 8 35 (From Japan: Susumu Hayashi, Toru Kawai, Nobuyoshi Motohashi, Koji Nakatogawa, Kanji Namba, Makoto Takahashi) 9 Proceedings, Southeast Asian Conference on Logic, Studies in Logic and the Foundations of Mathematics, vol. I l l , editied by C. T. Chong and M. J. Wicks, 1983, North-Holland 10 The themes of the tutorial sessions were recursion theory and model theory. Many of the papers were in these subjects. 11 In 1981, formal letters from Chong were sent to Motohashi, Crossley, Miiller and B.F.Nebres (the Phillipines). The letter said: 'I consulted people at MIT and NUS acording to our conversation in Greece. I have got feeling that we had better start with a small scale meeting. I will set the the dates in accordance with the visit of G.Sacks.' This was the start of ALC. Chong later recalls: 'It grew bigger than I had thought.' Even then, I hear, it was a more relaxed gathering than recent conferences. One of the participants, Hayashi, who had just got his degree, tells me a story which proves it. One of the speakers switched from English to Chinese in the course of his talk, and Chong interpreted the lecture into English for him. At trie end there was a big applause. We would like to keep up such an atmosphere. The Second Meeting 1 The Second Southeast Asian Logic Conference 2 October 29 - November 2, 1984 Bangkok, Thailand 3 Chair of the Organizing Committee Mark Tamthai (Chulalongkorn) 4 ASL 5 IUHPS, IBM (Thailand), Southeast Asian Mathematical Society, Math ematical Association of Thailand, Chulalongkorn University 6 C. T. Chong, J. N. Crossley, J. L. Loyd, A. Nerode, R. Soare, G.Takeuti 7 13 9 Annals of Pure and Applied Logic (APAL): Special Issue, vol. 31, 1986, Guest Editor: M. Tamthai, North-Holland. The proceedings of the conference series has continued to be in this journal up till 1993. This is a journal of high quality in mathematical logic, and the standard of review is strict. Reverse mathematics, hierarchy of sets, recursion theory, logic programming, extensions of Peano arithmetic are covered. The authors are from Singapore, Japan, USA, Britain, Canada, Australia and New Zealand.
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10 The theme of the conference was logic as well as its applications to mathematics and computer science. The themes of the tutorial session consisted of extensions of ZFC set theory and the programming language of the Fifth Generation Computer. 11 According to the letter by Tamthai, until then the conference had been organized in Southeast Asia, and they were not sure if anybody in Asia outside this district would be interested in organizing such a meeting. So the conference was named after Southeast Asia. The Third Meeting 1 The Third Asian Conference in Mathematical Logic 2 October 26 - 30, 1987 Beijing, China 3 Sponsor Institute of Software, Academia Sinica (Director: K. Xu) Organizing Committee K.Xu, D. Yang (Secretary) (Beijing), C. T. Chong, J. N. Crossley 4 ASL 5 IUHPS, International Centre for Theoretical Physics, Nature Science Foundations Academia Sinica, Institute of Software Academia Sinica, Na tional Science Foundation of China 6 R. I. Soare, R. Solovay, A. Nerode, S. G. Simpson, M. B. Pour-El, (all USA), C. T. Chong, G. Miiller, M. Yasugi (Japan) 7 72 Due to a large number of participants, there were many con tributed papers and the meeting was a lively one. Although the majority of the participants was from the host country, there were a fair number of Japanese and Euro-Americans. 9 APAL, vol.44,1989. Guest Editor.D.Yang The contents cover most subjects of mathematical logic. I was a bit concerned that the authors were from outside Asia except by Chong and two from Japan. 10 Thanks to the large number of participants, talks in various areas were presented. 11 I attended the meeting for the first time. The conference was much larger than the preceding two. This fact made us realize the size of China. I heard that Yang had worked for preparation almost all by himself. During the conference, his students helped him a lot. The site of the conference as well as the lodging of the overseas participants was a hotel near Beijing University. It was clear that a lot of hard work had been put into organizing the conference.
4
Towards the Session in Japan
Just prior to the Beijing Conference, a new committee called Committee on Logic in Japan and East Asia was formed by ASL. ASL was one of the co-operating organizations of ALC. Tugue (Nagoya University) chaired the
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committee, and Yasugi and Chong served as committee members. I guess that the possibility of holding a Logic Conference in Japan after Beijing had been tapped through Tugue. Chong related that Tugue had raised the possibility of hosting the fourth ALC during a conversation they had in Kyoto Hotel in 1987, just before the conference on Mathematical Logic and Its Applications held at the Research Institute of Mathematical Sciences in Kyoto that year. Considering the economic power of Japan (to the overseas eyes), it was a natural course of events. Japan was prosperous, maybe, but the community of Japanese mathematical logicians was rather indigent. Some of us, therefore, expressed anxiety over this. In Beijing, Tugue invited several of us, Yang, Chong, Yasugi and others to a meeting. There we discussed the possibility of the next conference. It was then that Japan accepted the responsibility. All the details were decided to be left to the host country. Chong emphasized the need of financial support. I was silently worried about the whole thing. At the meeting table in Beijing, I expressed my concern on the absence of invited speakers from the host country this time. According to Chong, it had been the same with the preceding two conferences. "It's perhaps due to oriental modesty, but you can break that rule in Japan," he said. Having returned from Beijing, we had to discuss a concrete measure. I happened to be on a committee in ASL, and at the same time a council of the Mathematical Society of Japan (MSJ). For this reason I was placed to start the matter. In December of 1987, I called Tugue as well as Shirai (Shizuoka University), then another council of MSJ, and Motohashi, a council of MSJ of the next term, to get together. We thus held the first consultation. There was no prospect of financial support at that time. At any rate, it was decided to put the base on the Section of the Foundations of Mathematics in MSJ. We circulated the news to the members thereof, and held the meeting involving persons interested on the occasion of the 1988 Spring Meeting of the MSJ at Rikkyo University. There Simauti (Rikkyo University) was elected the chair of the organizing committee. The con ference site was chosen to be Tokyo or the surrounding area, in view of manpower, transportaion and visa application considerations, concerning which Tokyo seemed to have the greatest advantage.. In July 1988, the organizing committee was formed, chaired by Simauti. The committee members were selected according to the following princi ple. The major force be from Asia, especially from Japan with emphasis on Tokyo. There would also be one from Oceania and one from Euro-America. We included computer scientists in the committee. Hirose together with Kakehi and Kikyo, of Waseda University, volunteered to take the respon sibility for administration. The date of the conference was set to be the beginning of Septemer, taking into account the need to travel to Tokyo from the site of ICM'90 in Kyoto, which was to be held in late August. The restriction of "South East Asia" area had been lifted as of the
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preceeding conference, and the management of the meetings had more or less stabilized. Considering these, a permanent name for the conference was proposed: Asian Logic Conference, Ajia Ronri Kaigi. This name is convenient. It has a nice rythm in English, and can be shortened to ALC. The practice of holding the conference every three years became definite from then on. We did not set any central subject, while we took into consideration that the invited speakers come from a variety of regions. As for the invited speakers, we asked all the program committee members for suggestions, and then contacted the candidates. All in all, things went quite smoothly. Each speaker kindly came on his/her own grant except one. We had reached a general agreement to start a precedent that henceforth the organizers do not suffer from too much financial load. That is, the most critical issues be taken care of, and things not of direct bearing to the conference be left aside. This would help to reduce the total costs. In our case, therefore, the party was of buffet type, and there was no organized sightseeing. Nevertheless, it seemed, the participants toured Tokyo and the surrounding areas by themselves. The conference site, especially the accomodation, was a big problem. It has to be a place which minimizes the financial burden of the partic ipants, and yet has to have the capacity of accomodating approximately 150 people. Eventually we succeded in securing the use of the CSK Edu cational Center in Tama-Shi. We owe it to the effort and the kindness of Hirose. Throughout, we enjoyed tremendous service by Director Kazuo Hirono of CSK. During the period of the conference, he even supplied us with manpower service. Thanks to this, we were able to operate the conference efficiently without much expence. We received heartfelt admiration from the overseas participants with words like "It has been a wonderful confer ence." Such a service was due to the policy of the CSK management as well as to Hirono's principle: "We need not be remembered by the participants; we only hope that they feel the visit to Japan has been rewarding upon returning home." The registration fee was payable at the conference site in Japanese Yen in cash. Those who were affiliated with educational or research institutions would pay 15,000yen, while those who were affiliated with companies would pay 30,000yen. We set such a distinction not to discriminate against com panies, but to respond to the wishes of some companies that they would like to contribute in this way. The speakers, and only speakers, were ex empt of meal and hotel expence. This policy was intended to encourage the participants to read papers. Modest amount of honororium was offered to invited speakers. The first announcement was sent out in September 1988, the second in September 1989 and the last one in June 1990. Tugue and Simpson respectively asked ASL and AMS so that they would announce the event.
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It was also the time when e-mail usages had spread, and so the efficiency of communications was greatly upgraded. It was of help that we were able to manage the trouble of visa in real time thanks to e-mail. The committee members residing in Japan forwarded preparation,meeting regularly in Simauti's office at Rikkyo University. Simauti's health had been deteriorating, and, in December 1989, we experienced the misfortune of his sudden death. Inspite of that, we preserved the organization with Hirose as the acting chairman. We bore in mind to realize Simauti's long cherished wishes to 'esteem Asian mathematicians and observe proprieties.' We thus proceeded to the Fourth Asian Logic Conference.
5
Record of Conference:The Fourth
The Fourth 1 The Fourth Asian Logic Conference (ALC) 2 September 3 - 6, 1990 CSK Educational Center, Tama-Shi, Tokyo 3 General Chair Tugue, Tosiyuki (Aichi Inst. of Tech.) Organizing Committee Simauti, Takakazu (Rikkyo U.,Chair), Hirose, Ken (Waseda U.,Secretary, later Acting Chair), Namba, Kanji (U.of Tokyo, Program Committee Chair), Motohashi, Nobuyoshi (U. of Tsukuba,Financial Affairs), Yasugi, Mariko (Kyoto Sangyo U.,External Affairs), Hirono, Kazuo (CSK), Chong, Chi Tat (NUS), Crossley, John N. (Monash U.), Simp son, Stephen G. (Penn.State U.), Tamthai, Mark (Chulalongkorn), Xu, K. (Academia Sinica), Yang, Dongping (Academia Sinica) Program Committee All the members of the organizing committee (Chair Namba), Doi, Norihisa (Keio U.), Kakehi, Katsuhiko (Waseda U.) 4 The Mathematical Society of Japan, Japan Society for Software Sci ence and Technology, Japan Association for Philosophy of Science, ASL 5 The Commemorative Association for the Japan World Exposition, CSK Corporation, International Information Science Foundation, Kawai Institute for Culture and Education, Calbee Corporation 6 S. R. Buss (U. of California, USA), R. G. Downey (Victoria U. of Wellington, New Zealand), J-Y.Girard (CNRS, U. Paris VII, France), M. Gitik (Tel Aviv U., Israel), S. Koppelberg (Freie U.,Germany), S.G.Simpson, A. Tsuboi (U. of Tsukuba, Japan), D. Yang (absent) 7 69 8 140 (100 from Japan) Overseas participants are from (then)Soviet Union, Autsralia, France, New Zealand, Singapore, Israel, Germany, China, Yugoslavia, Taiwan, Holland, Canada, Thailand, United Kingdom. It was regrettable that the organizers of the previous conferences, Chong, Tamthai, and Yang were absent for various reasons. From within the coun try, researchers of wide range areas, including analysis and computer sci ence, made the conference a lively one. 9 APAL, vol. 59, 1993, Guest Editors: Ken Hirose and Kanji Namba
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10 Wide range area was covered. 11 It was our wish that all the participants, foreign or not, would feel it enjoyable. One of the functions of the ALC is that the participants make new acquaintances so that future communications on research would be easier. We paid attention so that there would be nobody who came alone left without meeting and interacting with others. Since we all stayed at the same place, we were able to enjoy Karaoke-party and to converse over beer. It was a friendly meeting. We decided to reduce the number of invited talk in order t a emphasize the importance of the contributed talks. For this reason we scheduled just one invited talk at each session. We sent invitation letters to our seniors. Unfortunately, each had al ready some engagement. Professor Iyanaga, however, kindly attended the conference party. His reunion there with an old acquaintance Kurepa was a moving scene. We had asked Maehara (University of the Air) to give a party speach, offering an interpreter. He gave a talk on a history-let of the Foundations of Mathematics in post-war Japan. He told us that most of the Japanese logicians were disciples or grand disciples of Iyanaga, who was present then. We had the honor to have lyanaga's speech as well. There were addresses of thanks by Girard and unscheduled persons. Otherwise, the party continued with pleasant chats. Maehara attended the conference everyday in good humor. In March 1992, we were all astounded by the news of his death. Even now, I feel as if I could see his smiling face. Hirose continued to work to form a new department. Just after his efforts had borne fruit, he suddenly passed away from our world in August this year, having faught against his illness only for two months. I recall the occasion last year when we discussed about the 5th ALC. We have lost three persons who contributed to ALC Tokyo.
6
Towards the Fifth Conference
Even during the preparation for the Tokyo Conference, we were anxious about the viability of holding the next one. We anticipated difficulties, judging from the statistics of participations from Asian countries. To host a conference, manpower, economic resources, policital stability etc. are necessary. During the Tokyo Conference, we held a preparatory meeting involving individuals closely linked to organizing the ALC series, as well as a business meeting addressed to all the participants. At the business meeting, the invitatin by Chong saying "Welcome to Singapore once again!" was communicated. His invitation was accepted unanimously. The organizing committee of the 5th ALC consisted of several members of NUS. They set a policy that one hour invited talks were all from outside Asia, and forty minutes invited talks were from Asian countries. As for the
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latter, we on the advisory committee were asked for recommendations. I recommended a few candidates, and our opinions were well taken care of. At first, I felt that the distinction as above was rather strange. As a matter of fact, this is a kind of oriental consideration. Being a person of too much rationality, I felt urged to reconsider. I was struck with admiration afresh for the human capacity of Chong. He is naturally capable of well balancing the traditional oriental spirit and maintaining of academic standard.
7
Record of Conference: The Fifth
1 The Fifth Asian Logic Conference 2 June 14th - 17th, 1993 National University of Singapore, Singapore 3 Organizing Commitee C. T. Chong (Chair), Q. Feng, C.H. L. Ong, Y. N. Sun, Y. C. Tay, M. J. Wicks (NUS) Advisory Board J.N.Crossley, N.Motohashi, M.Tamthai, T.Tugue, D.P.Yang, M.Yasugi, 4 National UniverSity of Singapore 5 Lee Kong Chian Centre for Mathematical Research, Lee Foundation, Singapore Mathematical Society, Professor Tosiyuki Tugue 6 One hour H.P.Barendregt (Nijmegen), L.Blum (ICSI), H.J.Keisler (Madison), T.A.Slaman (Chicago), W.J.Mitchell (Florida), van den Dries (Urbana), H.Woodin (Berkeley) Forty minutes D.Ding (Nanjing), S.Hayashi (Ohtsu), J.Shinoda (Nagoya), K.Tanaka (Tohoku), S.P.Tung (Taiwan), J.Wang (Beijing) 7 50 8 99 About half the members of Department of Mathematics of NUS attended. There were 52 overseas participants, among them 11 from Japan and USA each and 8 from China. Besides the countries listed in the previ ous conference, there were participants from Turkey, Kazakstan, the Philip pines, Vietnam, Switzerland, Italy, and Poland. 9 APAL Volume 84, No.l, 1997 10 Although the subjects covered a wide range, many of the method ologies were in modern styles. This made us feel the new era in logic had arrived. 11 We met in a new Conference Hall. Everything was administered with efficiency and in good order. The food supplied during the morning and afternoon breaks was fine. All in all, I felt the economic and academic power of this young nation. Some of the old acquaintances had returned from their studies in Europe and USA, and some had turned from being students to faculty members. I was pleased with such changes. It was regrettable that there were less participants from Japan compared with those in Beijing. It could not be helped, as June is a particularly busy season in Japanese universities. It is a big problem to secure 6 days off,
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including travelling time. Another problem is that we cannot force others to pay themselves the travel expence. In my case, I was able to get subsidized by our university. I wish we could use at least part of the "General C" category of the Science Grants of Japan for overseas travel expence. x
8
Towards the Sixth Conference
At a dinner hosted by Chong, we discussed the ALC of 3 years later. The members present at the dinner were young Chinese from various countries except Crossley, Tugue and myself. Even though those Chinese belonged to different countries now, they exhibited an atmosphere of a family gathering. They respect traditions, while going through modernization. A message from Yang saying "We would like to invite you to Beijing next time" was communicated. Chong whispered to me "How about Japan?" I replied that we had just hosted the last conference. I broke into cold sweat. In fact we had no perspective of hosting another ALC. It might be that in China they may get subsidized by the state through the Academia Sinica. Even in that case, I felt I had caught a glimse of the power of China. In our case, we had totally depended on private funds. I do not know which was better. At any rate, it seems to be mistaking the means for the end to sacrifice research for collecting funds. It was agreed that China would be hosting the Sixth ALC in 1996, with the venue to be either in Beijing or Nanjing, to be decided at a later date by the hosts. In 1994, we heard that it had been decided that the Institute of Soft ware, Academia Sinica, would again organize the Sixth ALC in Beijing. There was concern expressed on the issue of funding. Chong wrote to the organizers that he would contact various sources for some additional funding.
9
Record of the Sixth Conference
1 The Sixth Asian Logic Conference 2 May 20th-24th, 1996 Academia Sinica, Beijing, China 3 Organizing Committee Yang, Dongping (Academia Sinica, Chair), Ding Decheng (Nanjing),Gao Hengshan (Academia Sinica), Huang Qieyuan (Academia Sinica), Li Wei (Beijing University of Aeronautics and Astronautics), Li Xiang (Guizhou), Lu Yizhong (Nanjing Aeronautics and Astronautics University), Moh Shaokui (Nanjing), Shen Engshao (Shanghai Jiaotong ), Shen Fuxing (Beijing Nor mal), Wang Shiqiang (Beijing Normal) 1 Remark added at translation: The new system of the Science Grants is more flexible in its use.
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Advisory Committee C.T.Chong (NUS), J.N.Crossley (Monash), T.Tugue (Konan Women's Junior College), M.Yasugi(Kyoto Sangyo) 4 Institute of Software, Academia Sinica 5 Institute of Software, Lee Foundation of Singapore 6 K.Ambos-Spies(Heidelberg), S.Buss(UC San Diego), S.B.Cooper(Leeds), D.Ding(Nanjing), Q.Feng(NUS), K.Hauser(UC Berkeley), A.Kechris(Caltech), S.Lempp(Wisconsin-Madison), W.Li(Beijing Univ of Aeronautics and As tronautics), K.Ono(JAIST), G. Sacks (Harvard), M.Yasugi(Kyoto Sangyo) 7 73 8 150 There were many participants from China. Japan and Singapore were also represented. There was also one participant from Taiwan. 9 World Scientific Publishing Company 10 The most significant event was that this was the first time that a representative from Taiwan attended a logic meeting in Beiing. Towards the end of the conference, a meeting was held and it was agreed that Taiwan would host the 7th ALC in Taipei in 1999. We see this as an indication of hope that there will be close scientific cooperation between experts across the Straits of Taiwan, and a symbol that the next century will bring further progress to logic in Asia. A few years from now, the political situation may change, and trips from China to Taiwan may become common. More specialists may grow in other countries. There may be more candidates for the host country. There will occur change in generation. I think that we need various kinds of effort in order to make ALC a real common property of Asia. The most important is to nurture mathematical logicians in each country. It is also desirable if we can establish a cooperative system as a result of our conferences. It is unfortunate that there has been no participant from Korea. I sent the announcement of the Fourth to several institutions in Korea. In a later day, I was introduced to the President of Korean Mathematical So ciety by Professor Hong of Nihon University. He was visiting Japan then. He promised me to distribute the announcement of the Fifth ALC within Korea. In fact, there were several entries from Korea for the Fifth, but nobody showed up. According to Chong's guess, the subjects of ALC are too science-techno-oriented, which does not appeal to the Korean logicians, who are more oriented towards natural languages and philosopy. The con tent of a conference shoud be, however, made up by the participants. We have to promote a better understanding of that. Finally, I would like to express my hope that ALC will continue to be not too stiff, while keeping up quality, and that any interested individual can attend the conference at ease, feeling it worthwhile to have been there. (First drafted September 14, 1993; revised and updated by the editors
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YASUGI
August 15, 1997) (Yasugi,Mariko Kyoto Sangyo University, Faculty of Science)
