Lecture Notes in Computer Science Edited by G. Goos and J. Hartmanis
22 [[
Formal Aspects of Cognitive Processes Interdisciplinary Conference Ann Arbor, March 1972
Edited by Thomas Storer and David Winter
Springer-Verlag Berlin-Heidelberg • New York 1975
Editorial Board: P. Brinch Hansen. D. Gries C. Meter • G. Seegm011er • N. Wirth Dr. Thomas Storer Dr. David Winter The University of Michigan Department of Mathematics Ann Arbor, MI 48104/USA
Library of Congress Cataloging in Publication Data
Interdisciplinary Conference in the Formal Aspects of Cognitive Processes~ Ann Arbor, Mich., 1972. Formal aspects of cognitive processes. (Lecture notes in cc~iputer science ; 22) Includes index. i. Cognition--Congresses. I. Storer~ Thomas~ ed. ii. Wiinter~ David J., ed. !II. Title. IV. Series. BF311. !56 1973 001.53 74-32111 ISBN 0-387-07016-8
AMS Subject Classifications (1970): 68-02, 68A30, 6 8 A 3 5 , 68A45, 68A50, 68A55 CR Subject Classifications (1974): 3.6, 3.7 ISBN 3-540-07016-8 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-07016-8 Springer-Verlag New Y o r k . Heidelberg. Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1975. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PRE FACE The academic disciplines which have produced models of cognitive processes vary from psychology to sociology, linguistics, psyeholinguistics, mathematical biology, neurophysiology, communication sciences, mathematics and logic, each of which may embody s e v e r a l distinct and diverse subdiseiplines.
Further, the types of models involved range
from heuristic to empirical, pragmatic, programmatic, and (varying degrees of) formalistic with every combination of these also being represented. The "Interdisciplinary Conference in the F o r m a l Aspects of Cognitive P r o c e s s e s " , which was held in Ann Arbor, Michigan in March 1972, was primarily devoted to formal models of cognitive processes with particular emphasis on the explication of the assumptions and hypotheses underlying new and existing models and theories in the above areas.
As was
anticipated, the atmosphere and the talks presented at the conference stimulated a free flow of ideas among the participants which, it is hoped, has opened continuing channels of communication among investigators from the disciplines represented, thereby contributing to continuing studies in the a r e a of theoretical models of cognitive processes.
We a r e publish-
ing here a representative selection of the papers presented at the conference in further suppert of these studies. We take this opportunity to warmly thank those who helped us with the conference and these conference proceedings.
The conference was funded by the Institute of Science and
Technology at The University of Michigan. Zukowski.
Most of the organization work was done by Lee
The typing of the proceedings was done by Catherine Rader.
thank Springer Verlag for agreeing to publish the proceedings.
Thomas Storer, Buffalo and Ann Arbor David Winter, Ann Arbor November 1973
Finally, we wish to
TABLE OF CONTENTS
~LCO~
ABOARD
Thomas S t o r e r
................................................................
1
THEORIES OF THE BRAIN; 5F/~AVIOR, THE M]ZID, ROBOTS AND COGNITIVE PROCESSES, RESPECTIVELY John T. Lamendella ...........................................................
4
TWO CLASSES OF HOLOGRAPHIC PROCESSES EEALIZABIZ IN THE N E U I ~ REAI~ J: Patrick Cavanagh ..........................................................
14
SEMANTIC MEMORY RETFJEVAL: SOME DATA AND A MODEL Elizabeth F. Loftus ..........................................................
41
IMPLICATION AS AN A L T ~ T I V E TO SET-II~gLUSION AS THE SEMANTIC PP_IMiTIVE Arnold Lewis Glass ...........................................................
55
STRUCTUEED-S~ORAGE AFA (Abstract)Armen Gabrielian, Seymour Ginsburg ...........................................
70
PREDICATE C A L C ~ U S F E A T L ~ G F _ ~ T I O N David Rothenberg .............................................................. 72 A MATHEMATICAL MODEL FOR PERCEPTION APPLIED TO THE PERCEPTION OF PITCH David Rothenberg .............................................................. 126 MODELS OF SPEECH PRODUCTION Chin-W. Kim ................................................................... 142 TOWARDS A THEORY OF LINGUISTIC MEMORY Terrence J. Keeney ............................................................
159
THE G R ~ OF RELATIVE ADJECTIVES AND COMPARISON Renate Bartsch, ~ e o Verm~nann ................................................ 1 6 8 A SIMPLE HIERARCHICAL MODEL OF NATURAL SEIJZCTION David J. Winter ............................................................... 186 ON THE NOTION OF A RULE Thorns Olshewsky .............................................................. 1 8 9 EMPIRICAL RESTRICTIONS ON THE POWER OF TRANSFORMATIONAL GRAMMARS Royal Skousen ................................................................. 2 0 4
WELCOME ABOARD Thomas Storer T h e U n i v e r s i t y of M i c h i g a n W e l c o m e to the " I n t e r d i s c i p l i n a r y C o n f e r e n c e in the F o r m a l A s p e c t s of Cognitive Processes".
The w o r d " i n t e r d i s c i p l i n a r y " was put in the announced title in the advance
f l y e r , by the way, a s a hopeful e x p r e s s i o n of m y own p r e d i s p o s i t i o n t o w a r d s the c o n f e r e n c e participants.
Things w o r k e d out p r e t t y well in that r e g a r d , I think; the d i s c i p l i n e s r e p r e -
s e n t e d by o u r s p e a k e r s include c o m p u t e r and c o m m u n i c a t i o n s c i e n c e s , l i n g u i s t i c s , m a t h e m a t i c s , n e u r o p h y s i o l o g y , philosophy, psychology, s o c i a l r e s e a r c h , and s y s t e m s and i n d u s t r i a l e n g i n e e r i n g - - a n d the s u b d i s c i p l i n e s of e a c h of t h e s e h e r e r e p r e s e n t e d is equally v a r i e d . L e t m e s a y now t h a t the r e s p o n s i b i l i t y for the choice of the p a r t i c u l a r a r e a r e p r e s e n t a t i v e s for t h i s c o n f e r e n c e is e n t i r e l y m i n e - - t h e c o m m o n t h r e a d , o r " b o n d " , being only (as s u g gested) a n i n t e r e s t in cognitive p r o c e s s e s and a f o r m a l i s t i c a p p r o a c h . in t h a t r e g a r d a r e w a r r a n t e d a t t h i s t i m e .
P e r h a p s a few w o r d s
S p e a k e r s w e r e no__t.tc h o s e n on the b a s i s of r e p u -
tation o r a r e a but, r a t h e r , on the b a s i s of a d i s c i p l i n e d a p p r o a c h a s s e t f o r t h in t h e i r p r e liminary abstracts.
T h i s i s not to s a y that I n e c e s s a r i l y a g r e e with e i t h e r the p a r t i c u l a r
a p p r o a c h o r the c o n c l u s i o n s of any given s p e a k e r ; s i m p l y that e a c h h a s s o m e t h i n g i m p o r t a n t to be said within a f r a m e w o r k which allows for s o m e s o r t of f o r m a l s t a t e m e n t a n d / o r analysis.
And t h i s , I feel, could be p r o f i t a b l e for the whole c o m m u n i t y .
T h i s p a r t i c u l a r c o n f e r e n c e c a m e about in the following way: in the p a s t 2 - 1 / 2 y e a r s I have t r a v e l e d to s e v e r a l c o n f e r e n c e s in one o r m o r e of the a r e a s h e r e r e p r e s e n t e d - - s o m e t i m e s speaking, s o m e t i m e s only a t t e n d i n g - - a n d I found that s e v e r a l i n t e r e s t i n g o b s e r v a t i o n s w e r e s e e m i n g l y to be m a d e .
Among these are numbered:
1) T h e s e c o n f e r e n c e s w e r e s t a u n c h l y a n t i - f o r m a l i s t t e .
Most s p e a k e r s , g r a n t e d ,
b o r r o w e d f r e e l y f r o m the m o r e f o r m a l d i s c i p l i n e s , but often a b u s e d the language b o r r o w e d f r o m the p a r e n t d i s c i p l i n e with r e s p e c t to that d i s c i p l i n e .
Personal interchanges at these
m e e t i n g s left m e s o m e w h a t s k e p t i c a l a s to w h e t h e r o r not p r e c i s e t e r m s , u s e d i m p r e c i s e l y , w e r e being s u b s t i t u t e d f o r i m p r e c i s e t e r m s . 2) Unlike the m a t h e m a t i c s and logic c o n f e r e n c e s (the only o n e s of m y p r e v i o u s a c t i v e p a r t i c i p a t i o n ) the t a l k s w e r e often e x e r c i s e s in d e f e n s i v e n e s s ( p a r t i c u l a r l y of reputation) and developed into e m o t i o n a l i n v o l v e m e n t , s o m e t i m e s to the exclusion of p r o p o s e d content. 3) Often, at the v e r y u n i v e r s i t y which housed the c o n f e r e n c e , t h e r e w e r e people (who, it a p p e a r e d to m e , w e r e v e r y good, a n d not to be d i s m i s s e d lightly) who " n e v e r went to s u c h things" even though t h e r e w e r e in the a p p a r e n t m a i n s t r e a m of the d i s c i p l i n e in question, had s o m e t h i n g i m p o r t a n t and r e l e v a n t to say, and w e r e , quite obviously, v e r y i n t e r e s t e d in the
topics u n d e r d i s c u s s i o n .
T h o s e to whom I a m r e f e r r i n g felt t h a t the v e r y lack of a c o m m o n
f o r m a l g r o u n d w o r k w a s a b a r r i e r to p r o d u c t i v e d i s c u s s i o n s a n d r e s u l t s . 4) A d i s p r o p o r t i o n a t e l y s m a l l s e g m e n t of the r e s p e c t i v e c o m m u n i t i e s i n v o l v e d s e e m e d to m o n o p o l i z e the a d d r e s s e d . In view of t h e s e ( a d m i t t e d l y subjective) o b s e r v a t i o n s (on an a d m i t t e d l y s m a l l s a m p l e space), I r e s o l v e d to s o m e d a y t e s t t h e s e o b s e r v a t i o n s by o r g a n i z i n g a c o n f e r e n c e w h i c h 1) was of a f o r m a l i s t i c bent, 2) i n v i t e d s p e a k e r s without r e g a r d to r e p u t a t i o n , 3) a g g r e s s i v e l y a d v e r t i s e d to " t h o s e usually excluded f r o m such p r o c e e d i n g s " , and 4) excluded m o s t of the usual r u n of s p e a k e r s via 1), 2) and 3) above, and the i n j u n c tion t h a t the only thing a s p e a k e r m a y l e g i t i m a t e l y a s s u m e f r o m h i s a u d i e n c e is s o m e d e g r e e of f o r m a l s o p h i s t i c a t i o n . The U n i v e r s i t y of Michigan, u n d e r the a u s p i c e s of the D e p a r t m e n t of M a t h e m a t i c s , with the s p o n s o r s h i p of the I n s t i t u t e of Science and Technology has m a d e m y r e s o l u t i o n a reality at this time.
And t h i s c o n f e r e n c e was viewed as an e x p e r i m e n t - - a " t r i a l r u n " , a s i t
w e r e - - a n d the r e s p o n s e so f a r h a s b e e n e x t r e m e l y positive, a l m o s t o v e r w h e l m i n g .
I
(immodestly) a s c r i b e t h i s r e s p o n s e to the c o r r e c t n e s s of (at l e a s t s o m e of) m y o b s e r v a t i o n s c o n c e r n i n g the " u s u a l " c o n f e r e n c e of t h i s type, and t h a t t h e s e r e a c t i o n s a r e s h a r e d by s o m e s u b g r o u p of t h e b r o a d c o m m u n i t y . Now I, too (in conjunction with J o h n L a m e n d e l l a ) , h a v e a f o r m a l a p p r o a c h to cognitive p r o c e s s e s - - i e . , a m o d e l (which it is not m y p u r p o s e to d i s c u s s h e r e ) - - a n d have spoken about it s e v e r a l t i m e s to v a r i o u s c o n f e r e n c e s and " i n t e r e s t e d " g r o u p s .
U n f o r t u n a t e l y , it
s u f f e r s f r o m the defect of not p r o p e r l y belonging to any d i s c i p l i n e , p r e s u p p o s i n g s o m e n e u r o physiology, a little s e t t h e o r y , logic, g r a p h t h e o r y (damn little), and one t r i v i a l topological r e s u l t (which can be o m i t t e d if y o u ' l l take m y word that v a r i o u s s t r u c t u r e s w h i c h a r e i n t r o duced a r e well--defined)--all in all (with the e~ception of the n e u r o p h y s i o l o g y and the one "lifting t h e o r e m " f r o m topology) m a t e r i a l c o v e r e d in the f i r s t four weeks of our F r e s h m a n Honors Calculus c o u r s e .
The model a r o s e in conjunction with a n a t t e m p t to f o r m a l l y c h a r -
a c t e r i z e c e r t a i n l i n g u i s t i c p r o c e s s e s , but, r e c o g n i z i n g the g e n e r a l i t y of the t h e o r y , we couched it in the language of " g e n e r a l i z e d " cognitive p r o c e s s e s - - a s p e c i a l c a s e of which is the o r i g i n a l c h a r a c t e r i z a t i o n p r o c e s s u n d e r d i s c u s s i o n
L i n g u i s t i c j o u r n a l s say, "Too
f o r m a l ; b e s i d e s t h i s stuff is psychology. " Psychology j o u r n a l s devoted to cognitive p r o c e s s e s say, "Too f o r m a l ; b e s i d e s , t h i s stuff is m a t h e m a t i c s . " M a t h e m a t i c s j o u r n a l s say, "Too i n t e r d i s c i p l i n a r y ; and b e s i d e s , the m a t h e m a t i c s involved is t r i v i a l .
Write a book."
Some
of the f r u s t r a t i o n g e n e r a t e d by the above also s e r v e s a s p a r t i a l m o t i v e for t h i s c o n f e r e n c e .
T h i s c o n f e r e n c e i s , finally, a n a t t e m p t to obviate the above and s i m i l a r f r u s t r a t i o n s (which I know f o r a fact a r e not e x p e r i e n c e s unique to J o h n L a m e n d e l l a a n d myself) t h r o u g h the p r e s e n t a t i o n of a n i n t e r d i s c i p l i n a r y c o l l e c t i o n of p a p e r s devoted to the f o r m a l a s p e c t s of cognitive p r o c e s s e s a s a viable, valuable, p r o d u c t i v e d i s c i p l i n e in its own r i g h t .
The
~peakers h e r e a s s e m b l e d r e p r e s e n t m y choice for the s p o k e s m e n in s u p p o r t of t h i s viewpoint, and they will b e g i n p r e s e n t i n g the h a r d e v i d e n c e a t 1 P . M . t h i s a f t e r n o o n .
THEORIES OF THE BRAIN; BEHAVIOR, THE MIND, ROBOTS AND COGNITIVE PROCESSES, R E S P E C T I V E L Y J o h n T. L a m e n d e l l a San J o s e State College Many t r a n s f o r m a t i o n a l g r a m m a r i a n s have b e c o m e c o n v i n c e d t h a t the field of l i n g u i s t i c s i s , in r e a l i t y , a b r a n c h of cognitive psychology and t h a t a t r a n s f o r m a t i o n a l g r a m m a r should be viewed a s a cognitive t h e o r y of language.
T h i s position h a s led at l e a s t s o m e
l i n g u i s t s to w o n d e r how one m i g h t v a l i d a t e c l a i m s of cognitive r e l e v a n c e ; how one could c h o o s e between t h e o r i e s which equally well a c c o u n t e d for the e m p e r i c a l data but with d i f f e r e n t psychological i m p l i c a t i o n s .
Having a s k e d t h e s e q u e s t i o n s , l i n g u i s t s join the r a n k s of
p s y c h o l o g i s t s , p h y s i o l o g i s t s , p s y c h o - p h y s i o l o g i s t s , n e u r o - p s y c h o l o g i s t s , p h i l o s o p h e r s , and o t h e r s who a l s o do not know what they m e a n when they talk about cognitive p r o c e s s e s . In this p a p e r I will aLtempt to outline a f r a m e w o r k for d i s c u s s i n g q u e s t i o n s like " W h a t is a cognitive p r o c e s s ?" " W h a t would a f o r m a l t h e o r y of cognitive p r o c e s s e s be like ? " "What is the r e l a t i o n s h i p b e t w e e a t h e o r i e s of the b r a i n , t h e o r i e s of b e h a v i o r , t h e o r i e s of m e n t a l p r o c e s s e s , t h e o r i e s of r o b o t s , and t h e o r i e s of cognitive p r o c e s s e s ? "
The d i s t i n c t i o n s I m a k e
apply only to a n i d e a l i z e d w o r l d of s c i e n t i f i c i n q u i r y i n h a b i t e d by s t r a w m e n who c o n s t r u c t t h e o r i e s p u r e l y of one type r a t h e r than the m e s s y types of t h e o r i e s w h i c h have a way to t u r n ing up in the r e a l world.
W h a t I give h e r e should be t a k e n as a plea for the e s t a b l i s h m e n t of
a conceptual f r a m e w o r k in t e r m s of which we c a n u n d e r s t a n d what a g i v e n r e a l w o r l d t h e o r y is a t h e o r y of. The f i r s t c l a s s of t h e o r i e s I would like to d i s t i n g u i s h i n v o l v e s an a n a t o m i c a l and physiological d e s c r i p t i o n of the s t r u c t u r e s a n d p r o c e s s e s of the n e r v o u s s y s t e m .
The t h e o -
r e t i c a l c o m p o n e n t of such d e s c r i p t i o n s would for the m o s t p a r t be h y p o t h e s e s filling in the gaps of e m p i r i c a l o b s e r v a t i o n .
P h y s i c a l d e s c r i p t i o n t h e o r i e s would include d e f i n i t i o n s of
m o r p h o l o g i c a l u n i t s such a s the n e u r o n , s p i n a l c o r d , and the ~q c r a n i a l n e r v e .
In addition,
they would d e s c r i b e functional u n i t s such a s the a u d i t o r y pathways, the e x t r a - p y r a m i d a l m o t o r s y s t e m , and the t i m b i c s y s t e m .
The t e r m " f u n c t i o n a l " a s u s e d in t h i s c o n t e x t r e f l e c t s
a c o n c e r n with the s p a t i o - t e m p o r a l o r g a n i z a t i o n of a c t i v i t y in the a n a t o m i c a l u n i t s defined. The a u d i t o r y pathways a r e defined a s a " f u n c t i o n a l " e n t i t y on the b a s i s of s e q u e n c e d p a t t e r n s of n e u r a l a c t i v i t y t r a v e r s i n g s p e c i f i e d s t r u c t u r e s in the n e r v o u s s y s t e m o v e r t i m e .
F o r the
n e u r o p h y s i o l o g i s t , s t r u c t u r e and function a r e i n s e p a r a b l e a n d it would m a k e no s e n s e for the p h y s i o l o g i s t to talk of a u d i t o r y pathways a p a r t f r o m such s t r u c t u r e s a s the c o c h l e a r nuclei, m e d i a l geniculate body, etco
Hypothetical functional e n t i t i e s s u c h a s the s o d i u m pump i n -
volved in the conduction of the n e r v e i m p u l s e a r e p r o p e r l y p o s i t e d by the p h y s i o l o g i s t only
when they a r e c o n s t r u e d a s h a v i n g some r e a s o n a b l y d i r e c t p h y s i c a l r e a l i z a t i o n r a t h e r than m e r e l y being a way of looking at what h a p p e n s . The notion of functional
component
cognitive p r o c e s s e s i s quite d i f f e r e n t .
found in m a n y i n f o r m a t i o n p r o c e s s i n g m o d e l s of
T h u s , we m i g h t d i s c u s s a functional e n t i t y m o u s e t r a p
a p a r t f r o m any s p a t i o - t e m p o r a l p h y s i c a l m a n i f e s t a t i o n . it c a t c h e s m i c e .
Something i s a m o u s e t r a p p r o v i d e d
What p h y s i c a l f o r m it h a s is i r r e l e v a n t .
N e v e r t h e l e s s , it should be pointed
out t h a t while a p h y s i o l o g i s t ' s definition of function i n v o l v e s a p h y s i c a l m a n i f e s t a t i o n , p h y s i ological d e s c r i p t i o n s d o n ' t r e d u c e to a n a t o m i c a l s t r u c t u r e s in any s i m p l e m a n n e r .
T h u s , for
e x a m p l e , the a u d i t o r y pathways have no d i r e c t p h y s i c a l e x i s t e n c e a s a d i s t i n c t entity. The q u e s t i o n i s , could a physiological d e s c r i p t i o n of n e u r a l p r o c e s s e s count a s an explanation of cognitive p r o c e s s e s .
C e r t a i n l y some s c h o l a r s b e l i e v e it is only b e c a u s e we
d o n ' t know enough a b o u t the b r a i n t h a t we c a n n o t give adequate p h y s i o l o g i c a l a c c o u n t s of c o g nitive processes.
H o w e v e r , j u s t a s a p h y s i o l o g i c a l d e s c r i p t i o n r e d u c e s to an a n a t o m i c a l
d e s c r i p t i o n only in an oblique, a s yet f o r m a l l y undefined way, so a d e s c r i p t i o n of cognitive p r o c e s s e s r e d u c e s to physiological d e s c r i p t i o n in s o m e even m o r e oblique, even m o r e u n d e fined way. The c o n c e r n s of the i d e a l i z e d p s y c h o l o g i s t a r e quite d i f f e r e n t f r o m t h o s e of the i d e a l i z e d physiologist.
F o r e x a m p l e , while a physiological t h e o r y would be c o n c e r n e d with
the b i o c h e m i c a l a n d / o r s t r u c t u r a l m o d i f i c a t i o n s which u n d e r l i e m e m o r y s t o r a g e in the b r a i n , i t would not be c o n c e r n e d with the fact t h a t people s t o r e i n f o r m a t i o n about the w o r l d and u s e t h i s i n f o r m a t i o n in p a r t i c u l a r ways for p a r t i c u l a r r e a s o n s .
If r e v e r b e r a t i n g c i r c u i t s , f a c i l i -
t a t e d synaptie t r a n s m i s s i o n , n e t w o r k s t r u c t u r e s , p r o t e i n s y n t h e s i s , and glial p r o c e s s e s w e r e all involved in the s t o r a g e of i n f o r m a t i o n about the w o r l d , the p h y s i o l o g i s t , a s p h y s i o l o g i s t , would not notice.
T h e r e is no physiological b a s i s f o r identifying t h e s e d i v e r s e p h y s i c a l
s t r u c t u r e s a s functionally e q u i v a l e n t in the s e n s e in w h i c h they all function to s t o r e i n f o r m a tion.
T h i s i s the p~ychologist~s notion of function.
Even if a totally adequate d e s c r i p t i o n of
p h y s i o l o g i c a l p r o c e s s e s , e x i s t e d , a d e s c r i p t i o n of cognitive p r o c e s s e s would s t i l l have to be given by the field of psychology. A v e r y popular a p p r o a c h to the study of cognitive p r o c e s s e s within the r e a l w o r l d field of psychology h a s been to i g n o r e t h e m e n t i r e l y and d e s c r i b e i n s t e a d the e x t e r n a l b e h a v i o r w h i c h i s the r e s u l t of cognitive p r o c e s s e s . forms.
T h e o r i e s within t h i s a p p r o a c h can take s e v e r a l
E a r l y b e h a v i o r i s m and m u c h s o - c a l l e d n e o - b e h a v i o r i s m e x e m p l i f y a n a p p r o a c h I
will call b e h a v i o r i a l taxonomy.
Within this f r a m e w o r k , a d e s c r i p t i o n of a s p e c i f i e d domain
of h u m a n b e h a v i o r i n v o l v e s a s y s t e m of p r e d i c t i o n s of the a s s o c i a t i o n s a m o n g e x t e r n a l e v e n t s and s t a t e s ; i . e . ,
an i n v e n t o r y of s t i m u l u s - r e s p o n s e p a i r s .
The two m a j o r s o u r c e s of e v i -
dence for the c o n s t r u c t i o n of such t a x o n o m i e s a r e o b s e r v a t i o n of o v e r t b e h a v i o r in r e l a t i o n
to e x t e r n a l e v e n t s and e x p e r i m e n t a l l y d e r i v e d c o n c l u s i o n s about the c o r r e l a t i o n s between observed events. T h e s e t h e o r i e s a c c o u n t f o r i n t e r n a l p r o c e s s e s only in the s e n s e that the a s s o c i a t i o n between input s t i m u l i and output r e s p o n s e s of an o r g a n i s m a r e t a k e n to u l t i m a t e l y involve p r o c e s s e s in the n e r v o u s s y s t e m .
The t e r m cognitive p r o c e s s , often u s e d to show m e r e l y
that one i s not a n a i v e S-R b e h a v i o r i s t , h a s c o m e to be applied by naive S-R b e h a v i o r i s t s to things s u c h as c r e a t i v i t y , p e r s o n a l i t y , e m o t i o n , m o t i v a t i o n , and o t h e r p r e v i o u s l y u n m e n tionable topics.
The a t t e m p t h a s b e e n m a d e to " t h e o r e t i c a l l y " a c c o u n t for t h e s e p h e n o m e n a
solely in t e r m s of o v e r t b e h a v i o r without involving i n t e r n a l p r o c e s s e s and s t r u c t u r e s in any t h e o r e t i c a l l y r e l e v a n t way.
M o s t p s y c h o l o g i s t s have r e c o g n i z e d the f a i l u r e of t h e s e a t t e m p t s
and m a n y n e o - b e h a v i o r i s t s have s h i f t e d t h e i r t h e o r e t i c a l i n t e r e s t s beyond b e h a v i o r a l t a x o n o my.
Many n e o - b e h a v i o r a l a p p r o a c h e s now a t t e m p t to i n c o r p o r a t e c o v e r t i n t e r v e n i n g v a r i a b l e s
w h i c h m e d i a t e between e x t e r n a l s t i m u l i and r e s p o n s e s . B e f o r e d i s c u s s i n g the s t a t u s of t h e s e f o r m u l a t i o n s a s d e s c r i p t i o n s of cognitive p r o c e s s e s , it would p e r h a p s be helpful to c o n s i d e r a h i s t o r i c a l d e v e l o p m e n t in the field of l i n g u i s t i c s w h i c h b e g a n with a n o t h e r b r a n d of b e h a v i o r a l t a x o n o m y and a l s o ended with a c o n c e r n for d e s c r i b i n g c o v e r t s t r u c t u r e s , Although it bad no a s p i r a t i o n s of p s y c h o l o g i c a l r e l e v a n c e , s t r u c t u r a l i s m within the field of l i n g u i s t i c s adopted the m e t h o d o l o g i c a l s t r i c t u r e s of b e h a v i o r i s m and shunned a l l " m e n t a l i s m " in d e s c r i b i n g language b e h a v i o r .
The s t r u c t u r a l l i n g u i s t ' s m a i n c o n c e r n was
the d e v e l o p m e n t of a n a l y t i c a l t e c h n i q u e s w h i c h could be u s e d as d i s c o v e r y p r o c e d u r e s for o b j e c t i v e l y getting a t the s t r u c t u r e of language data. A p h r a s e s t r u c t u r e g r a m m a r d e s c r i b e s a language by a s s i g n i n g to e a c h s e n t e n c e in t h a t language a l a b e l e d b r a c k e t i n g w h i c h gives its s y n t a c t i c s t r u c t u r e . to be a s e t of s e n t e n c e s . pus of data.
The language is defined
F o r the s t r u c t u r a t i s t s , t h i s m e a n t the s e n t e n c e s found in t h e i r c o r -
During the 1950's in the field of l i n g u i s t i c s , t h e r e developed the r e a l i z a t i o n t h a t
j u d g m e n t s of native s p e a k e r s about the g r a m m a t i c a l i t y of s e n t e n c e s could p r o v i d e e v i d e n c e which was j u s t a s " e m p i r i c a l " and " s c i e n t i f i c " a s t h a t obtained by copying down w h a t e v e r native s p e a k e r s h a p p e n e d to say, taking t h e i r u t t e r a n c e s to be ipso facto g r a m m a t i c a l .
Per-
h a p s C h o m s k y ' s m o s t s i g n i f i c a n t c o n t r i b u t i o n to l i n g u i s t i c s w a s the r e i n t r o d u c t i o n of the notion t h a t n a t i v e s p e a k e r s m a k e m i s t a k e s and u t t e r u n g r a m m a t i c a l s e n t e n c e s . sigui£ieant in a t l e a s t two r e s p e c t s .
T h i s r e v e l a t i o n was
F i r s t , it led to a r e d e f i n i t i o n of what w a s a c c e p t a b l e data
for the l i n g u i s t m d a c o n c e r n f o r language s t r u c t u r e s which a r e n e v e r m a n i f e s t e d in o v e r t s p e e c h production.
Once the l i n g u i s t c o u l d n ' t r e l y on the n a t i v e s p e a k e r ' s s e n t e n c e production,
but i n s t e a d on what the n a t i v e s p e a k e r i n t u i t e d to be g r a m m a t i c a l , it was only a s h o r t step to a c o n c e r n f o r the n a t i v e s p e a k e r ' s i n t u i t i o n s about which s e n t e n c e s s e e m e d r e l a t e d and w h i c h
didn't.
F o r s e n t e n c e s such a s open the door, e m p i r i c a l e v i d e n c e could now be found for the
u n d e r s t o o d you w h i c h y o u r f o u r t h g r a d e E n g l i s h t e a c h e r told you was t h e r e .
This eventually
led to the d i s t i n c t i o n between s y n t a c t i c s u r f a c e s t r u c t u r e s a n d deep s t r u c t u r e s .
Secondly,
the e l i m i n a t i o n of m i s t a k e s f r o m the data and the addition of s e n t e n c e s w h i c h had n e v e r been u t t e r e d but whose g r a m m a t i c a l i t y the n a t i v e s p e a k e r would a f f i r m , led e v e n t u a l l y to the competence/performance distinction.
In i t s c u r r e n t f o r m the d i s t i n c t i o n s t a t e s that a t r a n s -
f o r m a t i o n a l g r a m m a r c h a r a c t e r i z e s the knowledge n a t i v e s p e a k e r s have of t h e i r language ( t h a t i s , l i n g u i s t i c c o m p e t e n c e ) r a t h e r t h a n the u s e to which t h i s knowledge i s put ( t h a t i s , linguistic p e r f o r m a n c e ) .
T h i s i s the p r i n c i p a l b a s i s for C h o m s k y ' s c l a i m t h a t l i n g u i s t i c s is
a b r a n c y of cognitive psychology. Should a t r a n s f o r m a t i o n a l g r a m m a r be c o n s t r u e d a s a t h e o r y of cognitive p r o c e s s e s b e c a u s e it p o s i t s a b s t r a c t s t r u c t u r e s which a r e n e v e r m a n i f e s t e d in o v e r t language b e h a v i o r ? Should n e o - b e h a v i o r i s t f o r m u l a t i o n s be c o n s i d e r e d to d e s c r i b e i n t e r n a l p r o c e s s e s b e c a u s e they p o s i t c o v e r t m e d i a t i n g r e s p o n s e s ? It is not by p u r p o s e h e r e to a r g u e q u e s t i o n s about p a r t i c u l a r t h e o r i e s , although m y b i a s e s should be c l e a r ,
W h a t I would like to do is c l a i m t h a t it i s f r u i t l e s s to d i s c u s s s u c h
q u e s t i o n s without s o m e g e n e r a l c o n c e p t u a l f r a m e w o r k w h i c h t e l l s one what kind of thing a cognitive p r o c e s s i s and, in a g e n e r a l fashion, what would c o n s t i t u t e a t h e o r y of such a thing. C o n s i d e r a type of theory, call it an a b s t r a c t c h a r a c t e r i z a t i o n t h e o r y , w h i c h i s still within the r e a l m of e m p i r i c a l d e s c r i p t i o n but goes beyond the b e h a v i o r a l t a x o n o m y a p p r o a c h . Such a theory, in addition to accounting for the s t r u c t u r e of o b s e r v e d b e h a v i o r , defines the i m p l i c i t r e l a t i o n s h i p s in the s t r u c t u r e of the data.
It c o n t a i n s a n e m p e r i c a l taxonomy t h e o r y
a s a c o m p o n e n t but u t i l i z e s o t h e r types of e v i d e n c e a s well.
In p a r t i c u l a r , it e m p l o y s e x p e r -
i m e n t a l e v i d e n c e b a s e d on intuitions and i n t r o s p e c t i o n s about the r e l a t i o n s h i p s holding b e tween the o v e r t s t r u c t u r e s .
By d e s c r i b i n g t h e s e data, i t m a y be said t h a t t h i s type of t h e o r y
c h a r a c t e r i z e s the f a c t s which a c c o u n t f o r the o v e r t f o r m of the data.
An a b s t r a c t s y n t a c t i c
s t r u c t u r e r u l e such a s " E v e r y s e n t e n c e c o n s i s t s of a noun p h r a s e followed by a v e r b p h r a s e " m a y c o r r e c t l y a s s e r t t h a t people p o s s e s s this bit of i n f o r m a t i o n a s p a r t ot t h e i r linguistic knowledge but in no way could this s t a t e m e n t be t a k e n to c o n s t i t u t e a d e s c r i p t i o n of t h i s k n o w ledge n o r of the way it i s i n t e g r a t e d into m e m o r y s t r u c t u r e s in e i t h e r a p h y s i o l o g i c a l o r psychological sense. We m i g h t c o m p a r e the types of t h e o r i e s d i s t i n g u i s h e d s o f a r in t e r m s of t h e i r t r e a t m e n t of the cognitive p r o c e s s e s u n d e r l y i n g h u m a n a r i t h m e t i c b e h a v i o r . F i r s t , the p h y s i o l o g i s t would have nothing s p e c i a l to s a y a b o u t the cognitive p r o c e s s e s involved when people add a n d s u b t r a c t u n l e s s , f o r e x a m p l e , t h e r e w e r e s p e c i a l i z e d c o r t i c a l n e u r o n s i n v o l v e d in the s t o r a g e and u t i l i z a t i o n of a r i t h m e t i c i n f o r m a t i o n .
While we m i g h t a g r e e that a b e h a v i o r a l taxonomy of a r i t h m e t i c b e h a v i o r c a t a l o g s i n t e r e s t i n g data, we would not be s a t i s f i e d that t h e s e data help explain the u n d e r l y i n g c o g n i tive p r o c e s s e s which w e r e r e s p o n s i b l e f o r the b e h a v i o r in question.
An a b s t r a c t c h a r a c t e r -
ization t h e o r y of a r i t h m e t i c knowledge a l r e a d y e x i s t s as a b r a n e y of n u m b e r t h e o r y and p o s i t s r u l e s such a s 1 + 1 = 2.
This t h e o r y not only d e s c r i b e s o v e r t human a r i t h m e t i c b e h a v i o r
( a f t e r e l i m i n a t i n g i r r e l e v a n t p e r f o r m a n c e v a r i a b l e s such as l a p s e s of attention, stupidity, etc. ) but c h a r a c t e r i z e s p e o p l e s ' knowledge of a r i t h m e t i c r e l a t i o n s h i p s . 10 + 2 = 7 + 5.
F o r e x a m p l e , that
It g e n e r a t e s an infinite s e t of a r i t h m e t i c e x p r e s s i o n s including novel e x p r e s -
sions which have n e v e r been produced.
While it is t r u e , in a s e n s e , that n u m b e r t h e o r y
a c c o u n t s for what people know about a r i t h m e t i c , no one i s t e m p t e d to say that it d e s c r i b e s a r i t h m e t i c knowledge in the mind and i s t h e r e f o r e a b r a n c h of cognitive psychology. An a b s t r a c t c h a r a c t e r i z a t i o n theory p o s i t s m a p s which r e l a t e f o r m a l o b j e c t s , but t h e r e a r e no p r o c e s s e s defined by the t h e o r y .
A transformational grammar describes a sen-
tence in p a r t by a s s i g n i n g it a set of syntactic p h r a s e m a r k e r s , r e l a t e d by a s e t of m a p s called t r a n s f o r m a t i o n s .
T h e r e i s no logical o r c h r o n o l o g i c a l o r d e r i n g between deep s t r u c -
ture t r e e s and s u r f a c e s t r u c t u r e t r e e s in a t r a n s f o r m a t i o n a l g r a m m a r ; t h e s e a r e s i m p l y two f o r m a l o b j e c t s which a r e p a r t of the s t r u c t u r a l d e s c r i p t i o n of the s e n t e n c e . c h a r a c t e r i z a t i o n t h e o r y also does not p o s i t any functional e n t i t i e s .
An a b s t r a c t
It i s s i m p l y a s y s t e m for
cataloging e m p i r i c a l data of both the o v e r t behavior type and the a b s t r a c t r e l a t i o n s h i p type. No p s y c h o l o g i c a l c l a i m s a r e made by such t h e o r i e s . The next c l a s s of t h e o r i e s to be d i s t i n g u i s h e d I wilt call functional d e s c r i p t i o n theories, t h e r e being at l e a s t t h r e e d i f f e r e n t l e v e l s of such t h e o r i e s which could be c o n s t r u e d as " m o d e l i n g " s o m e a s p e c t of the o r g a n i s m .
E a c h of t h e s e t h r e e subtypes of t h e o r y have d i f -
f e r e n t goals, a r e c o n s t r u c t e d on the b a s i s of d i f f e r e n t evidence~ and model the o r g a n i s m in a different sense.
The f i r s t c l a s s of t h e o r i e s , call it a b e h a v i o r a l i s o m o r p h i s m t h e o r y , c o n -
tains the o t h e r two; the second, call it a p.sychological i s o m o r p h i s m the or~, c o n t a i n s the third, what I will call a p s y c h o - p h y s i c a l i s o m o r p h i s m t h e o r y .
F i g u r e 1 s u m m a r i z e s the
t y p e s of t h e o r i e s p o s i t e d and t h e i r i n t e r r e l a t i o n s h i p s . ANATOMICAL DESCRIPTION
BEHAVIORAL TAXONOMY
PHYSIOLOGICAL DESCRIPTION
ABSTRACT CHARACTERIZATION BEHAVIORAL ISOMORPHISM PSYCHOLOGICAL ISOMORPHISM "~
>PSYCHOPHYSICAL ISOMORPHISM
Figure 1
It has been s u g g e s t e d that if one wanted to u n d e r s t a n d how a b i r d f l i e s , a good way to a p p r o a c h finding out would be to build one. a s a model of a bird.
However, not e v e r y t h i n g which can fly qualifies
At a m i n i m u m , the model would have to fly like a b i r d r a t h e r than a
jet plane or a h e l i c o p t e r .
In o r d e r to qualify a s a model of s o m e o r g a n i s m , the m i n i m a l c o n -
s t r a i n t on a t h e o r y m u s t be behavioral i s o m o r p h i s m within a s p e c i f i e d domain.
A model
w h o s e motive p o w e r was a r u b b e r band could qualify a s a b e h a v i o r i s o m o r p h i s m t h e o r y of a bird, fox" the domain "flying b e h a v i o r " , if the m o d e l flew the way a bird d o e s .
In o r d e r for
a c h e s s playing p r o g r a m to qualify a s a behavioral model of human c h e s s playing, it would have to show the s a m e p a t t e r n of developing skills, make an o c c a s i o n a l blunder and once in a while allow an opponent to r e t r a c t r e a l l y dumb m o v e s .
A b e h a v i o r a l i s o m o r p h i s m t h e o r y of
a b i r d would not have to have f e a t h e r s and be a biological o r g a n i s m , but h o w e v e r the b e h a v i o r was g e n e r a t e d , it would have to be i s o m o r p h i s with what a b i r d d o e s .
Notice, of c o u r s e , that
it is the g e n e r a t i o n or definition of the b e h a v i o r which i s at i s s u e r a t h e r than the production of the behavior.
A running c o m p u t e r p r o g r a m which p r o d u c e s b e h a v i o r may be f l a s h i e r than
the s a m e p r o g r a m w r i t t e n on a piece of p a p e r , but it i s the s a m e t h e o r y in e i t h e r c a s e . A b e h a v i o r a l i s o m o r p h i s m t h e o r y b e a r s a s p e c i a l r e l a t i o n s h i p to a b e h a v i o r a l taxonomy t h e o r y for the domain of definition in that the f o r m e r would i n c o r p o r a t e the l a t t e r by including in its r e p e t o i r e of b e h a v i o r all of the e m p i r i c a l data about the b e h a v i o r of the o r g a n i s m and i t s r e s p o n s e s to s t i m u l i . isomorphism theory.
This is r e a l l y the only c o n s t r a i n t on a b e h a v i o r a l
The i n t e r n a l p r o c e s s e s of the model could be d e v e l o p e d by p r a g m a t i c
t r i a l and e r r o r and it w o u l d n ' t m a t t e r what f o r m they took.
It m a k e s no s e n s e to a s k what
kind of evidence would v e r i f y the a p t n e s s of its i n t e r n a l s t r u c t u r e .
The only r e l e v a n t q u e s -
tion to be a s k e d is: Does it define the behavior of the o r g a n i s m in q u e s t i o n ? A b e h a v i o r a l taxonomy of a r i t h m e t i c b e h a v i o r would s i m p l y c o n s i s t of a l i s t of a s s o c i a t i o n s Of the s o r t : When Ss a r e p r e s e n t e d with the s t i m u l u s "How m u c h i s one and one?", the r e s p o n s e " t w o " i s e l i c i t e d with a . 83 probability.
Given this u n d e r s t a n d i n g of the
o v e r t behavior, the behavioral i s o m o r p h i s m t h e o r i s t would have the task of c o n s t r u c t i n g an automaton which would " b e h a v e " in e x a c t l y this way.
Such m o d e l s a l r e a d y e x i s t .
An adding
m a c h i n e i s a f a i r l y good b e h a v i o r a l i s o m o r p h i s m t h e o r y of human a r i t h m e t i c b e h a v i o r , e x cept i n s o f a r a s it f a i l s to make the r i g h t s o r t s of m i s t a k e s . What d i s t i n g u i s h e s a behavioral i s o m o r p h i s m t h e o r y f r o m a b e h a v i o r a l taxonomy is that the f o r m e r is an automaton containing a s e t of d e v i c e s e a c h of which e m b o d i e s one o r m o r e of the functions which account f o r e i t h e r the g e n e r a t i o n o r production of the d e s i r e d behavior.
Its functional e n t i t i e s a r e things like input d e v i c e , output device, m a t c h i n g p r o c -
e s s , m e m o r y s t o r a g e d e v i c e , etc.
In a b e h a v i o r a l i s o m o r p h i s m t h e o r y t h e r e a r e p r o c e s s e s
defined which a r e c h r o n o l o g i c a l l y o r d e r e d .
No one could build a r o b o t using only the i n f o r -
10 marion in a 0ehavioral taxonomy, has mathematical
but a behavioral isomorphism
or mechanical
theory is a robot whether it
form.
I doubt that any one would want to claim that the structures and processes behavioral isomorphism
theory of man
would constitute a theory of human
of a
cognitive processes,
although there are clearly people who would be content with this type of theory. Turing test, apart from missing the point of the controversy thesizes just such a behavioral isomorphism A concern for such models
The famous
about thinking, implicitly hypo-
model of man.
is a valid pursuit in its own right, but the cognitive
psychologist is interested only in a subset of the behavioral isomorphism
theories.
This
subset is defined by several further constraints which establish the class of psychological isomorphism
theories.
The functional components
which will generate the desired behavior, ally involved when people perform
processes
but are equivalent to the functions which are actu-
the behavior.
the model are interrelated in the same
of this type of theory are not just any set
Additionally,
the functional components
way the functional components
of people's cognitive
are actually related.
These theories incorporated
the insights and data from an abstract characterization
theory as well as a behavioral isomorphism people's knowledge description,
theory.
If, for example,
it were true that
of the syntactic structure of sentences involved two levels of grammatical
a psychological
isomorphism
theory of language might contain a functor which
changed a deep syntactic structure into a surface syntactic structure. that two robots, one constructed as only behaviorally isomorphic psychologically examine
isomorphic
to man
Notice, of course, and the other as
would be indistinguishable to an observer who was not free to
their internals workings. Many
of the psychological
have been computer
isomorphism
simulation models.
models which have been attempted thus far
For example,
the General Problem
Shaw and Simon attempts to simulate aspects of certain types of human also exemplifies the use of an additional type of evidence characteristic morphism
models.
Namely,
subject protocols of the thinking-out-loud
data which has only recently (again) become
respectable in psychology.
this type of data is information about the conscious mental experiences in some
of
cognitive process.
awareness
would,
Aspects of problem
of course,
always be taken at face value. report that he solves complex
Solver of Newell,
problem
of psychological
GPS iso-
variety, a type of What is gained from of someone
engaged
solving which are not available to conscious
not be derivable from protocol data and this sort of data cannot For example, problems
someone
with trick mathematical
by visualizing the problem
and then letting a piece of chalk write down the answer.
We
abilities might
on a mental blackboard
would consider the matter care-
fully before we build a blackboard and an animate piece of chalk into our model
problem solving.
solving.
of human
11 P s y c h o l o g i c a l i s o m o r p h i s m t h e o r i e s also c o n s t r u c t functional i n t e r p r e t a t i o n s of e x p e r i m e n t a l l y d e r i v e d c o n c l u s i o n s about c o v e r t s t a t e s and e v e n t s c a u s a l l y a f f e c t i n g b e h a v i o r in o r d e r to c o n s t r u c t i t s a c c o u n t of cognitive p r o c e s s e s .
In c o n s i d e r i n g v i s u a l i n f o r m a t i o n
p r o c e s s i n g , it s e e m s r e a s o n a b l e to conclude t h a t in o r d e r to get b i n o c u l a r d e p t h c u e s , the input f r o m one v i s u a l c h a n n e l m u s t be c o m p a r e d with the input f r o m the o t h e r channel.
Thus,
the psychological i s o m o r p h i s m t h e o r i s t m i g h t p o s i t the e x i s t e n c e of a single B i n o c u l a r C o m p a r i s o n C o m p o n e n t which o p e r a t e s to e x t r a c t t h e s e b i n o c u l a r c u e s on the b a s i s of input f r o m the left v i s u a l field a n d the r i g h t v i s u a l field.
Of c o u r s e , the p u r e p s y c h o l o g i c a l i s o m o r p h i s m
t h e o r i s t ig~aores physiological data so he would not be a w a r e o r c a r e how c o m p l e x the a c t u a l situation is; n o r would he find out t h a t his model i s not functionally e q u i v a l e n t to h u m a n v i s ual i n f o r m a t i o n p r o c e s s i n g beyond the b e h a v i o r a l i s o m o r p h i s m type of functional e q u i v a l e n c e . A r e a l l y p u r e psychological i s o m o r p h i s m t h e o r i s t m i g h t not even have noticed that m a n h a s two e y e s .
Of c o u r s e , it would be r e d i c u l o u s to c o n s i d e r v i s u a l p e r c e p t i o n without taking into
a c c o u n t a l l of the a v a i l a b l e physiological data.
No one would do s u c h a thing and e x p e c t h i s
model to a c t u a l l y d e s c r i b e h u m a n v i s u a l i n f o r m a t i o n p r o c e s s i n g .
One w o n d e r s , h o w e v e r ,
what t h e r e is about such t h i n g s a s p r o b l e m solving and language w h i c h allows " m a t e r i a l i s t " s c h o l a r s to feel c o m f o r t a b l e with m o d e l s of t h e s e p r o c e s s e s h a v i n g no b a s i s in p h y s i o l o g i c a l fact. Some people m i g h t b e t e m p t e d to define cognitive psychology such t h a t low l e v e l r e c e p t o r functions a r e viewed a s b e i n g m e r e l y physiological w i t h no r e l e v a n c e to the p s y c h o logist.
However, a t what point in the m e r e l y physiological v i s u a l pathways do cognitive p r o c -
e s s e s kick i n ?
Does v i s u a l i n f o r m a t i o n p r o c e s s i n g b e c o m e cognitive only a f t e r it h a s l e f t t h e
v i s u a l a r e a s of the c o r t e x and gone off into the m y s t e r i o u s " a s s o c i a t i o n a r e a s " w h e r e m e n t a l p r o c e s s e s a r e r e p u t e d to l u r k ?
U n l e s s we want to u s e the t e r m cognitive p r o c e s s a s a s y n -
onym for " c o n s c i o u s m e n t a l e x p e r i e n c e " , t h e r e i s no notion of " h i g h e r m e n t a l p r o c e s s " such t h a t we c a n functionally m o d e l v i s u a l cognition while i g n o r i n g the n e u r o p h y s i o l o g y of vision. P s y c h o l o g i c a l i s o m o r p h i s m t h e o r i e s a r e the f i r s t type of t h e o r y we have c o n s i d e r e d which in any s e n s e d e s e r v e s the l a b e l p s y c h o l o g i c a l and one w h i c h m i g h t r e a s o n a b l y be viewed a s telling us s o m e t h i n g about cognitive p r o c e s s e s .
Still, not e v e r y o n e would be s a t i s f i e d that
the i n t e r n a l p r o c e s s e s u n d e r y l i n g m a n ' s b e h a v i o r had been fully explained.
One p r o b l e m is
t h a t the i n t e r n a l s t r u c t u r e of the functional c o m p o n e n t s of a p s y c h o l o g i c a l i s o m o r p h i s m t h e o r y have no c o n s t r a i n t s o t h e r than t h e i r c a p a c i t y for p e r f o r m i n g the function.
Even if the
t h e o r y h a s one c o m p o n e n t c o r r e s p o n d i n g to e v e r y one p s y c h o l o g i c a l c o m p o n e n t ( a s s u m i n g a o n e - t o - o n e i s o m o r p h i s m w e r e d e s i r e d ) , the i n t e r n a l o p e r a t i o n s of the c o m p o n e n t s could n o t hope to r e f l e c t the way people p r o c e s s i n f o r m a t i o n while i g n o r i n g the physiological f a c t s entirely.
12 A m a j o r r e a s o n people a r e willing to a c c e p t m o d e l s of such things a s p r o b l e m solying w h i c h i g n o r e the b r a i n h a s to do with a r e a l w o r l d p r o b l e m for r e a l w o r l d i n v e s t i g a tors that doesn't trouble idealized, imaginary scholars.
It i s c l e a r that a cognitive p r o c e s s
s u c h a s p r o b l e m solving is i n t i m a t e l y tied up to v e r y m a n y o t h e r cognitive p r o c e s s e s .
An
explanation of the s t r u c t u r e of i n f o r m a t i o n in m e m o r y , the f o r m of i n c o m i n g p e r c e p t u a l data the l i m i t a t i o n s of s h o r t - t e r m m e m o r y , l e a r n i n g , m o t i v a t i o n , language, and a l m o s t e v e r y thing e l s e would be r e q u i r e d b e f o r e anyone could m a k e a n y r e a s o n a b l e c l a i m s about how h u m a n b e i n g s a c t u a l l y solve p r o b l e m s .
All of t h e s e p r o c e s s e s , in t h e i r t u r n , a r e i n t e r -
twined with p r o b l e m solving and the r e s t . The m o s t c o m m o n solution to t h i s p r o b l e m h a s been to a t t a c k only a s m a l l p a r t of one of the cognitive p r o c e s s e s and beg t h e r e s t .
T h i s i s c a l l e d m a k i n g " s i m p l i f y i n g assumptions'~
U s u a l l y the s m a l l p a r t which i s c h o s e n to be e x p l a i n e d i s the data f r o m a p a r t i c u l a r e x p e r i ment.
One finds an a r t i c l e e n t i t l e d s o m e t h i n g like, "A f o r m a l model of the s t r u c t u r e of
linguistic c o n c e p t s " w h i c h t u r n s out to be a c o m p u t e r s i m u l a t i o n t h e o r y w h i c h a c c o u n t s for the l e a r n i n g of six p a i r s of t h r e e - l e t t e r n o n s e n s e s y l l a b l e s u n d e r s p e c i f i e d c o n d i t i o n s of p r e s e n t a t i o n and r e h e a r s a l by five m a l e E n g l i s h - s p e a k i n g college s o p h o m o r e s f r o m M i d d t e b u r y , Connecticut.
If a n o t h e r i n v e s t i g a t o r c o m e s along and v a r i e s one of the p a r a m e t e r s ,
one
thing leads to a n o t h e r and a new ~ b f i e l d of psychology is b o r n . An additional r e a s o n for t h i s o v e r - s p e c i a l i z e d t r e a t m e n t i s the d e s i r e to avoid c l a i m s which do not a r i s e d i r e c t l y f r o m e x p e r i m e n t a l r e s u l t s in the s t a n d a r d f o r m a t .
Since the
m o d e l s developed a m o u n t to a s u m m a r y of p a r t i c u l a r data, t h e i r a p p l i c a b i l i t y always depends on the p a r t i c u l a r way the p r o b l e m was p r e s e n t e d to the s u b j e c t s , the a m o u n t and type of r e h e a r s a l , b a c k g r o u n d s of the s u b j e c t s , and so on.
T h e r e i s an e x c e l l e n t c h a n c e t h a t cognitive
psychology, a s b e h a v i o r a l psychology b e f o r e it, wilt end up c o n s i s t i n g of an u n s t r u c t u r e d heap of s u p e r f i c i a l , u n r e l a t a b l e e x p e r i m e n t a l data.
It does not s e e m r e a s o n a b l e to allow t h i s
type of s c i e n t i f i c i n v e s t i g a t i o n to continue to be the d o m i n a n t a c t i v i t y in the field of psychology. If psychology is e v e r going to get off the ground, s o m e way m u s t be found to i n c o r p o r a t e data into a g e n e r a l f r a m e w o r k w h e r e it c a n i n t e r a c t with a t t the o t h e r data being g a t h e r e d .
One
way to begin to tie t o g e t h e r the data which h a s been g a t h e r e d is to b r o a d e n the scope of the m a i n s t r e a m of psychology to include physiological facts. One m a y d e c i d e a s a m a t t e r of p r a c t i c a l fact t h a t it is u n r e a s o n a b l e to go beyond the p s y c h o l o g i c a l i s o m o r p h i s m l e v e l in d e s c r i b i n g cognitive p r o c e s s e s o r that one d o e s n ' t c a r e to t r y .
N e v e r t h e l e s s , it is not u n r e a s o n a b l e for s o m e o n e to w i s h for an e x p l a n a t i o n of c o g n i -
tive p r o c e s s e s which did explain exactly how h u m a n b e i n g s p e r f o r m t h e s e functional o p e r a tions.
A type of t h e o r y which would fully explain h u m a n cognitive p r o c e s s e s is the t h e o r y I
have called psycho-physical
isomorphism.
This theory contains the insights of physical
~3 description theories and psychological isomorphism theories. It presents an explicit answer to the question of the relationship between the mind and the brain. It explains how neurophysiological processes in human beings embody psychological functions. The development of such a theory is, in my opinion, the prime goal of cognitive psychology. Some psychologists might be tempted to avoid a psycho-physical orientation on the grounds that neurophysiology has not progressed far enough to say anything of significance about cog,aitive processes. I seriously doubt that this is true. There is an enormous amount of relevant information about the brain and its functioning. Even if one constructed a psychophysical theory on the basis of physiological data which turned out to be wrong, the theory would be e loser to an adequate theory than one which ignores the physiological data entirely. Neural net theories can be constructed at a level of generality which leaves them almost c e r tain to be correct models of information processing in the brain.
Further specificity could
a w a i t f u r t h e r data. Some p h y s i o l o g i s t s m i g h t be t e m p t e d to avoid a p s y c h o - p h y s i c a l o r i e n t a t i o n on the grounds that psychology h a s not p r o g r e s s e d f a r enough to s a y a n y t h i n g of s i g n i f i c a n c e about cognitive p r o c e s s e s .
However, the b e h a v i o r i s t s have a m a s s e d a prodigious q u a n t i t y of data
and o v e r the p a s t few y e a r s i n f o r m a t i o n p r o c e s s i n g m o d e l s have begun to develop v e r y i m p r e s s i v e c o m p u t e r s i m u l a t i o n s of a s p e c t s of cognitive p r o c e s s e s . The b e s t thing w h i c h could happen to both fields would be the d e v e l o p m e n t of a s t r o n g e r i n t e r e s t in p s y c h o - p h y s i c a l i s o m o r p h i s m t h e o r i e s .
The b e s t thing w h i c h could happen to a n
individual i n v e s t i g a t o r would be to have a c l e a r idea of what he is r e a l l y i n t e r e s t e d in e x p l a i n ing: the b r a i n , b e h a v i o r , the mind, r o b o t s , o r c o g n i t i v e p r o c e s s e s .
He should know w h e t h e r
the data he h a s u s e d is the kind of e v i d e n c e w h i c h l e a d s to m o d e l s of the s o r t he i s i n t e r e s t e d in.
And finally, he should c a r e f u l l y e x a m i n e his c o m p l e t e model to s e e what he h a s modeled.
TWO CLASSES OF HOLOGRAPHIC PROCESSES R E A L I Z A B L E IN THE NEURAL R E A L M J. P a t r i c k C a v a n a g h Carnegie-Mellon University Holography r e f e r s to a b r o a d c l a s s of s t o r a g e and r e t r e i v a l p r o c e s s e s b a s e d on the r e c o r d i n g of i n t e r f e r e n c e p a t t e r n s .
As a m o d e l for a n e u r a l m e m o r y s y s t e m , holography
p r o v i d e s both a s s o c i a t i v e and r e d u n d a n t s t o r a g e along with a m a t h e m a t i c a l l y e x p l i c i t d e s c r i p t i o n of the s y s t e m ' s o p e r a t i o n .
L a s h l e y (1929) w a s a m o n g the f i r s t to p r o p o s e t h a t
m e m o r y was coded in t e r m s of n e u r a l i n t e r f e r e n c e p a t t e r n s and was let to this c o n c l u s i o n by his d i s c o v e r y of r e d u n d a n t s t o r a g e in the b r a i n .
T w e n t y - s e v e n y e a r s l a t e r , B e u r l e (1956)
showed how s u c h i n t e r f e r e n c e p a t t e r n s m i g h t be s t o r e d so that the o r i g i n a l i n f o r m a t i o n could be r e t r i e v e d without d i s t o r t i o n .
In 1963, van H e e r d e n noted the s i m i l a r i t y between B c u r l e ' s
h y p o t h e s i s and the m o r e c o n c i s e r e p r e s e n t a t i o n of the i n t e r f e r e n c e p r o c e s s o f f e r e d by h o l o graphic theory.
Subsequently, a n u m b e r of a u t h o r s have outlined, to v a r y i n g d e g r e e s , the
analogy between h o l o g r a p h i c and n e u r a l p r o c e s s e s ( J u l e s z and P e n n i n g t o n , 1965; P r i b r a m , 1966, 1969, 1971; W e s t l a k e , 1967, 196o, 1970).
Of t h e s e , only W e s t l a k e (196~, 1 9 7 0 ) h a s
given a d e t a i l e d a n a l y s i s of the n e u r a l m e c h a n i s m s involved.
His h y p o t h e s i s , h o w e v e r , like
B e u r l e ' s , p l a c e s s e v e r e r e s t r i c t i o n s on the f i r i n g p a t t e r n s of n e u r o n s that a r e at odds with the known p r o p e r t i e s of n e u r a l codes. T h i s p a p e r will d e m o n s t r a t e t h a t h o l o g r a p h i c s t o r a g e c a n be o b t a i n e d when only the most general neural codes are assumed.
A recognition system employing transmission
holography is p r o p o s e d and the l i m i t e d c a p a c i t y of this type of s t o r a g e for multiple r e c o r d l a g s leads n a t u r a l l y to a m o d e l l i n g of s h o r t t e r m m e m o r y .
The s y s t e m p r e d i c t s a wide
r a n g e of b e h a v i o r a l data in r e c o g n i t i o n t a s k s b a s e d on a single a s s u m p t i o n c o n c e r n i n g the p r o p e r t i e s of individual n e u r o n s . B e f o r e d e t a i l i n g the n e u r a l h o l o g r a p h i c p r o c e s s e s , a b r i e f d e s c r i p t i o n of optical holography will d e m o n s t r a t e s o m e of the p r o p e r t i e s of this m e t h o d of s t o r a g e . Optical H o l o g r a p h y Optical holography, developed by G a b o r (1945), a l l o w s the r e c o r d i n g of both the p h a s e and a m p l i t u d e of a w a v e f r o n t of light.
Since p h o t o g r a p h i c p l a t e s a r e s e n s i t i v e to i n -
t e n s i t y but not to phase, the plate is exposed to two w a v e f r o n t s s i m u l t a n e o u s l y and the r e s u l t i n g i n t e r f e r e n c e p a t t e r n is r e c o r d e d .
If both w a v e f r o n t s a r e m o n o c h r o m a t i c and c o -
h e r e n t in phase, a s t a b l e i n t e r a c t i o n o c c u r s so t h a t the i n t e r f e r e n c e p a t t e r n codes both a m p l i t u d e and phase in t e r m s of i n t e n s i t y v a r i a t i o n s .
The r e c o r d e d i n t e r f e r e n c e p a t t e r n i s
the h o l o g r a m ; i l l u m i n a t i n g the h o l o g r a m with one of the o r i g i n a l w a v e f r o a t s c a u s e s the
15
r e c o n s t r u c t i o n of the o t h e r .
The h o l o g r a m is thus an a s s o c i a t i v e m e m o r y of the two w a v e -
fronts. If one of the original w a v e f r o n t s w a s the light r e f l e c t e d f r o m a t h r e e - d i m e n s i o n a l object, for e x a m p l e , the r e c o n s t r u c t i o n will be an e x a c t r e p l i c a of the o b j e c t such that no visual t e s t can d i f f e r e n t i a t e the two.
In o t h e r w o r d s , all of the phase and amplitude i n f o r -
mation has been r e c o v e r e d . F i g u r e i i l l u s t r a t e s the two s t e p s in the h o l o g r a p h i c p r o c e s s ,
a and b a r e w a v e -
f r o n t s that v a r y spatially in amplitude and phase, and A and. D a r e t h e i r t r a n s f o r m s at the photographic plate.
The photographic plate is s e n s i t i v e to r e c o r d i n g e n e r g y , i . e . the produci
RECORDING
b(x,y)
- b(x,y) e
i@(x,y)
A(u,v) • B(u,v)
photographic plate
RECONSTRUCTION __.b(X,y)
)gram
~(x,y) FIGURE i: OPTICAL HOLOGRAPHY
16
of the r e c o r d i n g i n t e n s i t y and the e x p o s u r e d u r a t i o n . E = t(A + B)(A + B)
(i)
A_, B__ - wavefront field vectors E - exposure energy t - exposure duration * - denotes complex conjugate After developing, energy;
the transmission
coefficient of the plate is a function of the exposure
if the function is linear, the resulting expression
is
T = BE = ~t(i A I2 + IB__[2 + A B * + A * + B )
(2) T - transmission coefficient ]3 - slope of transmission vs. exposure
relation If one of the original wavefronts
_~ or more
A__, is incident on the hologram,
the transmitted
A. T :
its transform
components
in the hologram
plane,
are
Ajt(IAI2+IBt2)+ tAAJ'% IA_[2B_
The first term describes the second represents plete reconstruction
precisely,
the transmission
a complex
of the a_ wavefront
wavefront
of the b wavefront
dependent
at an attenuated amplitude;
on both a and b; the third is a com-
attenuated by a real-valued
variable that can be con-
sidered constant if a and b are independent.
Under appropriate different directions
recording
geometries,
so that the reconstruction
the transmitted
components
propagate
of b is available uncontaminated
in
by the other
wavefrontso The hologram
is an associative
memory
in that storage of a and b allows recon-
struction of b_ upon reference
by a and vice versa.
(under certain transformation
between
part of the hologram
can reconstruct
The essential requirement are monochromatic
in the neural realm, interaction.
These
the first requirement
that code object information
properties
allow the wavefronts
is therefore to determine (1965) have assumed,
not only places unattainably
is that they to interfere
the possibility of holographic
of neural spike trains are all of a single frequency
This assumption
field) in that any
wavefront.
In demonstrating
Both Beurle (1956) and Wesflake
that wavefronts phase.
the complete
of the wavefronts
pattern.
or distributed memory
the object field and the recording
and phase coherent.
in a stable spatio-temporal
It is a redundant
the locus of the stable as in the optical case, and fixed in their relative
stringent requirements
of neural firing, but also is invalidated by the wide use of frequency stimulus attributes (e. g,, intensity, el. Perkel and Bullock,
1966).
storage
on the stability
in the brain for coding of
17 M o r e o v e r , such an a s s u m p t i o n is u n n e c e s s a r y in the d i s c r e t e n e u r a l s y s t e m a s opposed to the continuous optical c a s e ; the s t r u c t u r a l c o h e r e n c e i n h e r e n t in the fixed i n t e r n e u r a l c o n n e c t i o n s of the b r a i n a l r e a d y p e r m i t a s t a b l e s p a t i a l i n t e r a c t i o n of w a v e f r o n t s r e g a r d l e s s of the f r e q u e n c y a n d p h a s e of individual spike t r a i n s .
Since the i n t e r a c t i o n will
however, v e r y r a n d o m l y in t i m e ( a s s u m i n g r a n d o m v a r i a t i o n in i n t e r s p i k e i n t e r v a l s ) , it m u s t be shown that the m e a n spike r a t e i s the d o m i n a n t p a r a m e t e r in coding and s t o r a g e .
If this
is the c a s e , s p a t i o - t e m p o r a l s t a b i l i t y i s a c h i e v e d and the m a i n t e n a n c e of p r e c i s e spike a r r i v a l t i m e s , such as p r o v i d e d by f r e q u e n c y and p h a s e locked spike t r a i n s , is not r e q u i r e d . At p r e s e n t , t h e r e is a good deal of e v i d e n c e s u p p o r t i n g m e a n r a t e codes in v a r i o u s n e u r a l s t r u c t u r e s ( P e r k e l and Bullock, 196~).
The next s e c t i o n d e a l s with the n e u r a l h o l o g r a p h i c
p r o c e s s e s that a r e p o s s i b l e u s i n g such c o d e s . N e u r a l Holographic P r o c e s s e s L i n e a r Codes Since i n c o m i n g spike t r a i n s a r e m o s t likely of v a r y i n g f r e q u e n c i e s with s o m e c o m p o n e n t of r a n d o m fluctuation in each, it i s i m p r o b a b l e that a s t a b l e t e m p o r a l i n t e r a c t i o n can be a c h i e v e d at the level of individual spike a r r i v a l s .
The i n t e r f e r e n c e p a t t e r n m u s t
t h e r e f o r e be defined in t e r m s of s o m e t i m e - - a v e r a g e d p a r a m e t e r .
Mean r a t e , the a v e r a g e
spike a r r i v a l r a t e within s o m e t i m e i n t e r v a l , a p p e a r s to s e r v e the r e q u i r e d p u r p o s e .
The
f i r s t i m p o r t a n t a t t r i b u t e of the m e a n r a t e d e s c r i p t i o n of n e u r a l a c t i v i t y is t h a t a n e u r o n is a r e l a t i v e l y l i n e a r t r a n s d u c e r of t h i s p a r a m e t e r for i r r e g u l a r (Poisson) e x c i t a t o r y or i n h i b i t o r y input ( E n g e r , e t a l . ,
1969; P e r k e l , e t a l . , 1964).
The m e a n r a t e output of a n e u r o n thus
r e s p o n d s l i n e a r l y to the a l g e b r a i c s u m of the m e a n r a t e s of multiple inputs ( a s s u m i n g the inputs a r e tadependent--Segundo, et a l . , 196o) o v e r the r a n g e d e l i m i t e d by the s u p p r e s s i o n of the n e u r o n ' s output a n d i t s m a x i m u m f i r i n g r a t e .
If a d j a c e n t c e l l s a r e i n t e r r e l a t e d in s o m e
m a n n e r , the p r o c e s s i n g e l e m e n t s of the n e u r a l s y s t e m a r e m o r e a c c u r a t e l y the e n s e m b l e s of i n t e r d e p e n d e n t cells; the l i n e a r o p e r a t i n g r e g i o n s of such e n s e m b l e s m a y be c o n s i d e r a b l y g r e a t e r than t h a t of the c o n s t i t u e n t n e u r o n s , depending on the d i s t r i b u t i o n of the individual ranges.
( W h e t h e r the p r o c e s s i n g e l e m e n t s a r e individual c e l l s o r g r o u p s of i n t e r r e l a t e d
c e l l s , they can be r e p r e s e n t e d c o n c e p t u a l l y a s single n e u r o n s with no loss in g e n e r a l i t y , ) The p r o p e r t y of [ i n e a r i t y leads to g r e a t s i m p l i f i c a t i o n in the m a t h e m a t i c a l d e s c r i p t i o n of the s y s t e m - a s i m p l i f i c a t i o n t h a t i s a l s o enjoyed in optical holography by v i r t u e of the l i n e a r i t y of the wave equations d e s c r i b i n g the p r o p a g a t i o n of light t h r o u g h m o s t m e d i a . The second and m o s t i m p o r t a n t a t t r i b u t e of the m e a n r a t e p a r a m e t e r is that its use in v a r i o u s f o r m s a s a code for s t i m u l u s p r o p e r t i e s h a s b e e n d e m o n s t r a t e d in m a n y n e u r a l s t r u c t u r e s ( P e r k e l and Bullock, 1968).
The two f o r m s of m e a n r a t e coding m o s t p r e v a l e n t in the
b r a i n a r e f r e q u e n c y modulation o r m e a n r a t e modulation (MRM) and d i r e c t i o n a l r a t e change
18
DRC, most frequently called simply mean rate coding, but it will be labelled differently here to emphasize that frequency modulation is also a mean rate code . Although other codes have been identifiedI, the analysis of neural holographic processes will center on these two. Both involve a base rate of random firing; MRM implies a periodic oscillation of the mean rate about the base level and is a candidate code wherever periodicity is found-e . g . , the theta rhythm (4-7 Hz) in the hippocampus, the alpha (8-13 Hz) in the thalamus and cortex, and beta (20-30 Hz) in the cortex; DRC is the classical code for stimulus intensity and involves a continuing increment of decrement in the mean firing rate about the base level as shown by De Valois, et al. (1962) in the lateral geniculate. The MRM code can be represented by a cosine function:
(4)
mean rate = r+ M cos(2~rft+e) r f t 0 -
M
For the MRM generator Thus,
code to be meaningful
of the carrier frequency
f will be the same
in a neural holographic
(a pacemaker)
process,
that modulates
there must be a single
all the spike trains involved,
for all spike trains and 0, the phase of the modulation
train, will be with respect to a common The DRC
the base rate amplitude of deviation from base rate frequency of modulation time phase of the particular spike train relative to some reference
code can be expressed
senting the direction--increment
for each
reference. in terms
or decrement--of
of a binary function, the change in mean
denoted
evn,
repre-
rate.
eva0 = (_±)0
0 - the direction index, taking integer values
mean rate = r+ Mevn0
r, M - as before
(5)
Thus the mean rate increases by M if the direction index, 0, is even and decreases by" M if it is odd. The direction index is in some respects analogous to phase; unlike phase, howe v e r , it i s a d i s c o n t i n u o u s d i m e n s i o n .
i
Mean rate codes are, in general, the most suitable for holographic purposes. If more than one type of code supports storage and reconstruction, it might be possible to superimpose the various codes (assuming they are orthogonal) and simultaneously store and retrieve independent information wavefronts in the same storage area. Distribution coding, although it has been frequently observed (Perkel and Bullock, 1968), is an improbably candidate for significant holographic processing. First of all, with many weak, ~ndependent syaaptie inputs (which must be assumed for redundant storage to occur), the distribution of output firing bears no relation to input distributions (Segundo, et al., 1968). Second, even under the condition of a limited number of relatively strong inputs, storage and retrieval of distribution parameters other than the mean is possible only in a number of special cases.
19 The n a t u r a l o c c u r r e n c e of t h e s e two m e a n r a t e c o d e s m a k e s the r e q u i r e m e n t s for h o l o g r a p h i c s t o r a g e f a r l e s s r e s t r i c t i v e in the n e u r a l c a s e than in the optical.
The s t r u c -
t u r a l c o h e r e n c e of the fixed i n t e r n e u r a l c o n n e c t i o n s and the t i m e - a v e r a g e d p r o p e r t y of the c o d e s for w h i c h n e u r o n s a r e l i n e a r t r a n s d u c e r s p e r m i t a s t a b l e s p a t i o - t e m p o r a l i n t e r a c t i o n of w a v e f r o n t s t h a t have no fixed f r e q u e n c y o r p h a s e r e l a t i o n s .
It r e m a i n s to be d e m o n s t r a t e d ,
h o w e v e r , that the n e u r a l i n t e r f e r e n c e p a t t e r n s can be s t o r e d i s o m o r p h i c a l l y by s o m e p r o c e s s of n e u r o n a l change.
B e f o r e c o n f r o n t i n g t h i s t a s k , the n e u r a l s t r u c t u r e r e q u i r e d for h o l o -
g r a p h i c p r o c e s s e s will be analyzed. N e u r a t R e p r e s e n t a t i o n of Symbolic I n f o r m a t i o n The two codes d e s c r i b e d above c h a r a c t e r i z e single spike t r a i n s .
A neural wavefront
i s s i m p l y a s p a t i a l a r r a y of spike t r a i n s a n d will be c o n s i d e r e d the mode of r e p r e s e n t a t i o n for s y m b o l i c i n f o r m a t i o n in the b r a i n ( s t i m u l u s p a t t e r n s , c o n c e p t s , m o t o r c o m m a n d s , e t c . ) . F i g u r e 2 d e p i c t s a p o s s i b l e s t r u c t u r e for a n e u r a l h o l o g r a p h i c s y s t e m ,
a and b a r e
input w a v e f r o n t s t h a t m a p t h r o u g h t r a n s f o r m s X a n d Y, r e s p e c t i v e l y , onto the s t o r a g e field
feedback j . m - - - - - - m
/ / I
OUTPUT
I-----t.
© ©
FIELDS
a m
© STORAGE
iNPUT
--O-U
FIELD
FIELDS
b
/
/
/
11 feedback
FIGURE 2: A p o s s i b l e s t r u c t u r e f o r a h o l o g r a p h i c n e u r a l s y s t e m .
¸
20
neurons.
7 i s the d i a g o n a l m a t r i x of t r a n s m i s s i o n c o e f f i c i e n t s e x p r e s s i n g the l i n e a r
r e l a t i o n b e t w e e n the a l g e b r a i c s u m of the inputs to e a c h n e u r o n and the m e a n output r a t e of each neuron--i, e.,
• r e p r e s e n t s the e f f i c i e n c y with w h i c h input r a t e v a r i a t i o n s a f f e c t o u t -
put r a t e v a r i a t i o n s .
W and Z m a p t h i s output e_ onto u and v.
The transforms X,Y,W,
and Z a r e s u c h t h a t with no i n f o r m a t i o n r e c o r d e d in the s t o r a g e a r e a ,
a a n d b a r e a b l e to
c r o s s t h r o u g h e a c h o t h e r and end up s e p a r a t e l y in u and v, r e s p e c t i v e l y , w o r d s , p r o d u c e s no output in v a n d b no output in u.
a__, in o t h e r
S i m u l t a n e o u s l y e x p o s i n g the s t o r a g e
field to a a n d b a l t e r s the t r a n s m i s s i o n c o e f f i c i e n t s of t h e s e n e u r o n s in p r o p o r t i o n to the m
i n t e r f e r e n c e p a t t e r n of a a n d b.
T h e s p a t i a l v a r i a t i o n s of T w h i c h code t h i s i n t e r f e r e n c e
p a t t e r n now allow w a v e f r o n t a to p r o d u c e output in v which, it will be shown, i s a r e c o n s t r u c t i o n of b ' s output to u.
T h u s , the s p a t i a l v a r i a t i o n s of 7 a r e the n e u r a l h o l o g r a m .
Expressing these relations mathematically: u = We
(6)
v =
(7)
zc
c = r +T(Xa+Yb) --
(8)
C
A = Xa
(9)
B = Yb
(i0) - denotes that the elements are coded spike trains a, b - input wavefronts (column crete pulse trains u, v - wavefronts arriving (column vectors)
of the variable
vectors)
of dis-
at the two output fields
c - w a v e f r o n t output f r o m the s t o r a g e field neurons W , X , Y, Z - t r a n s f o r m m a t r i c e s of s y n a p t i c c o u p l i n g coefficients A,B__- the input w a v e f r o n t s a t the s t o r a g e field ( c o l u m n vectors) r
e
- c o l u m n v e c t o r of the b a s e f i r i n g r a t e s of the s t o r a g e field n e u r o n s
7 - d i a g o n a l m a t r i x of t r a n s m i s s i o n c o e f f i c i e n t s of s t o r a g e field n e u r o n s
I n i t i a l l y 7 i s a s c a l a r m a t r i x ( a l t h o u g h t h i s is not a n e c e s s a r y a s s u m p t i o n , it s i m p l i f i e s analysis).
21
T = kI u = W r +W~(Xa+Yb) = W r + W k l X a + W k l Y b --
e
c
u = W r +X(WXa+WYb) --
e
-
--
--
(Ii)
;
-
similarly, v = Zr
+X(ZXa+
ZYb_)
(12)
C
F o r a to produce output only in u and b only in v, WY = 0 ,
zx = 0
(13)
A f t e r r e c o r d i n g , the t r a n s m i s s i o n c o e f f i c i e n t s have been a l t e r e d in s o m e m a n n e r . lkv's a r e v a r i a b l e a c r o s s the s t o r a g e field.
These
F o r r e c o n s t r u c t i o n , a is again input, but b i s
z e r o (ignoring b a s e r a t e s ) . v = ZTXa = Z(~I+AT)Xa = Z~Ia+ ZATXa = ~ZXa+ ZA~Xa - average A7-diagonal
(14)
value of 7 matrix of variations
of ~- around
ZX=0 (~5)
V = ZATXa
Any t e r m in 7, t h e r e f o r e , that i s c o n s t a n t o v e r all s t o r a g e n e u r o n s d o e s not g e n e r a t e any output in the r e c o n s t r u c t i o n field.
Thus, A~-Xa m u s t contain Yb (i. e . ,
A~A m u s t contain
B) if an a c c u r a t e r e c o n s t r u c t i o n i s to be produced. The r e l a t i o n s between W and Y and Z and X e x p r e s s e d in equation (13) a r e e s s e n t i a l to the p r o c e s s i n g of the n e u r a l w a v e f r o n t s and a r e analogous to the r e c o r d i n g g e o m e t r i e s in optical holography that allow s e p a r a t i o n of output i m a g e s .
It i s p o s s i b l e ,
b e c a u s e of t h e i r i m p o r t a n c e to p r o c e s s i n g , that t h e s e r e l a t i o n s have developed through e v o lution; c o n v e r s e l y , u and v m a y s i m p l y be the s u b s e t of output fields of the s t o r a g e n e u r o n s whose t r a n s f o r m s obey t h e s e r e l a t i o n s . The feedback loops shown in F i g u r e 2 i l l u s t r a t e t h a t this s t r u c t u r e could be two c i r c u l a t i n g n e u r a l n e t s that i n t e r s e c t at a s e t of modifiable n e u r o n s .
The a s s o c i a t i o n b e -
tween the v a r i o u s " c e l l a s s e m b l i e s " (Hebb, 1949) r e a l i z a b l e in the n e t s thus o c c u r s at t h e i r intersection. Finally, in analyzing the s t o r a g e and r e c o n s t r u c t i o n , only the w a v e f r o n t s at the s t o r a g e field, A and B, n e e d be c o n s i d e r e d a s t h e s e a r e the w a v e f r o n t s that d i r e c t l y p r o duce the i n t e r f e r e n c e p a t t e r n to be s t o r e d .
The e x p r e s s i o n s for A and B a r e , for MRM
coding, w h e r e the s u b s c r i p t i i n d i c a t e s input activity at the i th s t o r a g e n e u r o n :
22
A = Xa = ~rAi+Aicos(2~rft+0i)~
(16)
]3 = Yb = [rBi + Bieos(2~rft+~i)]
(17)
and for DRC coding,
A__= Xa_ = [rAi+AievnOi~
(18)
]3-- Yb_= [rBi+,ievn i]
(19)
Models of Neuronal Change At present, the basis of neuronal change is unknown, but there are a number of hypotheses of the factors that might lead to such change (Kupfermann and Pinsker, 1969) and these fall into two main classes. The first states that the degree of usage of the synaptic terminals or of the postsynaptic cell somehow affects their subsequent efficiency, The second proposes that the concurrence of action potentials at the pre- and postsynaptic membranes and the consequent permeability- of both, allows some form of molecular communication that effects a change in transmission efficiency; the probability of concurrence at each synapse will depend both on the output rate of the postsynaptic cell and on the input rate at the synapse. To allow holographic storage, the neuronal change must code the linear interaction of the two inputs A and ]3 (i.e., A+D.
This requirement is met by the postsynaptic use
and concurrent use hypotheses, both of which reflect the output rate (A+ B) of the postsynaptic ceil; the synaptic use hypothesis, however, reflects only the input to each particular synapse and so cannot support
holographic processes.
Furthermore, since the inhibitory
and excitatory components of an input cancel in their effect on the output of the postsynaptic cell, isomorphic coding requires that their effects on the transmission efficiency of a cell also cancel. This is satisfied by the postsynaptic use hypothesis (simple use) where neuronal change is a function only of the output rate of the postsynaptic cell. In the case of concurrent use, the agent of neuronal change is the substance passing between pre- and postsynaptic processes and this interchange is a function not only of postsynaptic firing but also of the input rates at each synapse.
For the molecular agents of inhibitory and excita-
tory inputs to counteract each other, they must be mutually antagonistic and must be able to diffuse and interact, even if only on a local basis (assuming random spatial distribution of inputs), after passing across the postsynaptic membrane.
If input to each cell is either all
inhibitory or all excitatory, this cancellation requirement is, of course, unnecessary and the transport can be in either direction across the synaptic cleft (this form of the concurrence hypothesis is typically called specific use).
23 F i n a l l y , it can be shown that when the function e x p r e s s i n g n e u r o n a l change (the t r a n s m i s s i o n function) c o n t a i n s l i n e a r t e r m s in A o r B_, s i g n i f i c a n t n o i s e r e s u l t s in the r e c o n s t r u c t i o n field.
Since MRM coding is p e r i o d i c , any l i n e a r ' t e r m s will a v e r a g e to z e r o
o v e r one period; DRC coding, h o w e v e r , i s not t i m e v a r y i n g and l i n e a r t e r m s c a n only be s u p p r e s s e d in the c o n c u r r e n c e h y p o t h e s e s with c e r t a i n r e s t r i c t i o n s and c a n n o t be s u p p r e s s e d a t all in the s i m p l e use c a s e ( r e q u i r i n g , t h e r e f o r e , the a b s e n c e of l i n e a r t e r m s , and, in fact, a l l o d d - p o w e r e d t e r m s , in the t r a n s m i s s i o n function). To s u m m a r i z e the p r o p e r t i e s of the v a r i o u s h y p o t h e s e s : h o l o g r a p h i c s t o r a g e of MRM coded input i s s u p p o r t e d by s i m p l e and c o n c u r r e n t u s e and by specific use when input to e a c h cell is e i t h e r all e x c i t a t o r y o r i n h i b i t o r y ; h o l o g r a p h i c s t o r a g e of DRC coded input is p o s s i ble for s p e c i a l c a s e s of s i m p l e and s p e c i f i c use; the c h a n g e in n e u r o n a l t r a n s m i s s i o n e f f i c i e n c y (7) i n v o l v e s the cell a s a unit in s i m p l e use, local a r e a s of the cell in c o n c u r r e n t use (in both c a s e s , the c h a n g e a f f e c t s a l l input to the c e l l equally), and individual s y n a p s e s in specific use. The n e u r a l h o l o g r a p h i c p r o c e s s will be a n a l y z e d for the c a s e of c o n c u r r e n t u s e w h e r e e a c h c o n c u r r e n c e c o n t r i b u t e s equally to the change in t r a n s m i s s i o n efficiency.
Although any
n u m b e r of o t h e r m o d e l s could have been c h o s e n for a n a l y s i s , this p a r t i c u l a r model h a s a n u m b e r of a d v a n t a g e s : it o f f e r s m o r e s t r u c t u r e a n d f e w e r r e s t r i c t i o n s than the s i m p l e use model; it a p p l i e s equally to p r o c e s s i n g units of single c e l l s o r e n s e m b l e s of c e l l s ; the e q u a l i t y of the e f f e c t of e a c h c o n c u r r e n c e i s i n t u i t i v e l y a t t r a c t i v e ; the r e s u l t s g e n e r a l i z e r e a d i l y to the specific use model; and b e h a v i o r a l data d i s c u s s e d in the l a t t e r half of t h i s p a p e r show t h a t the t r a n s m i s s i o n function m u s t be a n e x p o n e n t i a l (Cavanagh, 1972) and s u c h a function is a f u n d a m e n t a l c h a r a c t e r i s t i c of the model to be c o n s i d e r e d . The s e l e c t e d model p o s t u l a t e s that c o n c u r r e n t u s e l e a d s to change, but change in what d i r e c t i o n ? Most e x p e r i m e n t a l e v i d e n c e , e s p e c i a l l y in the p e r i p h e r a l n e r v o u s s y s t e m , s u g g e s t that use w e a k e n s , r a t h e r than s t r e n g t h e n s , the t r a n s m i s s i o n e f f i c i e n c y of a c e l l when input i s e x c i t a t o r y ( S h a r p l e s s , 1964).
Such a c h a n g e i s c o m p a t i b l e with the p h e n o m e n a of
habituation, extinction, and s p o n t a n e o u s r e c o v e r y , a m o n g o t h e r s .
G r i f f i t h s (1966) h a s
d e m o n s t r a t e d how this w e a k e n i n g a c t s a s n e g a t i v e f e e d b a c k to s t a b i l i z e n e u r o n output; if, c o n v e r s e l y , use led the s t r e n g t h e n i n g , the c o n c o m i t a n t p o s i t i v e f e e d b a c k would produce i n s t a b i l i t y o r r u n a w a y in the c e l l ' s output. No e x p e r i m e n t a l work has been done on the e f f e c t s of i n h i b i t o r y input.
However, to
obtain the s a m e s t a b i l i t y , c o n c u r r e n c e of the posts)~aaptic a c t i o n potential with the a r r i v a l of i n h i b i t o r y i m p u l s e s m u s t lead to s t r e n g t h e n i n g of the e f f e c t i v e n e s s of i n h i b i t o r y input. w e a k e n i n g (i. e . , r e l e a s e f r o m inhibition) w e r e to r e s u l t , the c e l l ' s output would a g a i n i n crease autonomously.
If
24 The c o n c u r r e n c e of action p o t e n t i a l s a t a s y n a p s e m u s t a f f e c t not only the s y n a p s e involved but a l s o n e i g h b o r i n g s y n a p s e s which may be of s i m i l a r o r d i f f e r e n t n a t u r e .
This can
be a c c o m p l i s h e d if the m o l e c u l a r agent r e l e a s e d by an inhibitory c o n c u r r e n c e s t r e n g t h e n s , and that r e l e a s e d by an e x c i t a t o r y c o n c u r r e n c e w e a k e n s , the r e c e p t o r s e n s i t i v i t y of e i t h e r type of s y n a p s e . quired.
T h e s e two actions a r e then also mutually a n t a g o n i s t i c , a s p r e v i o u s l y r e -
Since the model is a d d r e s s i n g s h o r t t e r m m e m o r y p r o c e s s e s with a time s c a l e in
the o r d e r of s e c o n d s , such c h a n g e s in r e c e p t o r s e n s i t i v i t y ( a s s u m i n g e x c i t a t o r y input) might be s e e n on a g r o s s level as fatigue or a c c o m m o d a t i o n , a c o n s i s t e n t l y o b s e r v a b l e c e l l u l a r phenomenon. V a r i o u s a s s u m p t i o n s have been made c o n c e r n i n g the m e c h a n i s m s of n e u r o n a l change and the a n a l y s i s of holographic s t o r a g e and r e t r i e v a l will now be undertaken b a s e d on t h e s e assumptions.
It m u s t be e m p h a s i z e d that the f e a s i b i l i t y of holographic p r o c e s s e s does not
depend on t h e s e a s s u m p t i o n s and that equivalent d e r i v a t i o n s r e s u l t f r o m a v a r i e t y of o t h e r neuronal m o d e l s . F i g u r e 3 shows a single neuron (chosen a r b i t r a r i l y so that no s u b s c r i p t s will be used) w h e r e A r e p r e s e n t s the a l g e b r a i c s u m of input f r o m the a w a v e f r o n t and B the s u m f r o m the b w a v e f r o n t .
A
B
m
m
"t
÷
FIGURE 3: A single neuron in the s t o r a g e field. A and B a r e the s u m s of all the input e f f e c t s f r o m a and b r e s p e c t i v e l y .
25
Considering the MRM code, when the total synaptic effect is positive, then the output of the cell is in phase with the input.
With multisynaptic input, the input and output p r o c e s -
ses can be c o n s i d e r e d independent t with l i n e a r l y r e l a t e d means.
The expected n u m b e r of
c o n c u r r e n t p r e - and postsynaptie action potentials (hits) per unit t i m e is then d i r e c t l y p r o portional to the product of the mean input and output r a t e s . E(hit) = E {(re + 7(A+ B)). (A+ B)} .
(20) E - expected value
Assuming that the duration of exposure, t, is a random v a r i a b l e unrelated to the period of the modulating wavefront, the expected number of hits can be e s t i m a t e d by a v e r a g i n g o v e r one period,
E(hit) =
• ( r A + r B + A c o s ( 2 ~ t + 0) + B c o s ( 2 ~ t + ~)) d(2~ft) E(hit) = T(rA+ rB)2+ r c ( r A + r B) + 7(A2+ B2+ 2AB cos ~)
(21)
w h e r e ~/= 0 - ~ . "~ - phase difference between the two inputs If each hit contributes an equal amount to the weakening of the t r a n s m i s s i o n efficiency, then the r a t e of change of 7 will be l i n e a r l y r e l a t e d to E(hit).
A s s u m i n g an e q u i l i b r i u m
p r o c e s s that maintains T at a steady level when there is no input other than the base r a t e s , the e x p r e s s i o n is d~-dt = - aE(hit) + a ( 7 ( r A + r B) 2 + r c ( r A + rB)~ = -~'r(A2+ B2+ 2AB cos ~
(22)
Integrating o v e r t, the total t i m e for which the storage neuron is exposed to A and B_
=
~te- at(A2 + B2 + 2AB cos
(23) a, ~t - constants.
1
While an a r r i v i n g spike may initiate an action potential (or suppress an i m m i n e n t one in the c a s e of inhibitory input) at or n e a r the r e c e i v i n g postsynaptic site, this s p a t i o - t e m p o r a l input-output dependence only o c c u r s with significant frequency for the few synapses n e a r the axon hillock. With the number of input p r o c e s s e s in the o r d e r of thousands (10,000 synapses is typical in the cortex), the m a j o r i t y of s p a t i o - t e m p o r a l l y simultaneous p r e - and p o s t s y n aptic action potential a r r i v a l s will o c c u r as the postsynaptic action potential sweeps through the dendritic t r e e . Thus, for determining the rate at which these even a r e achieved, the input and output p r o c e s s e s a r e effectively independent.
26 If the total synaptic effect is inhibitory, the output c will d e c r e a s e when input (A+B) i n creases.
Therefore E(hit) = E { ( r c - v(A+ B.))"(A+ B)}
(24)
and the average, as in equation (21), over one period is E(hit) = -7(rA+ rB)2 + r c ( r A + r B) - "r(A2 + B2+ 2AB cos 3')
(25)
However, each hit now contributes to the i n c r e a s e of 7. Again, with an e q u i l i b r i u m process, dtd7 _ + O~E(hit)+ ~(7(rA+ r B) - r c ( r A + rB)~ = -~T(A2+ B 2 + 2AB cos 3') ,"
again, 7 = ke - a i ( A 2 + B 2 + 2 A B e ° s 3 ` ) As expected, both excitatory and inhibitory input (and therefore any combination of the two) lead to the same storage c h a r a c t e r i s t i c .
~ r e p r e s e n t s the storage p a r a m e t e r , the
strength with which incoming information a l t e r s the cell. k is simply the base level of 0with no information stored and t r e p r e s e n t s the exposure time of the input. The e l e m e n t s of [7], the diagonal m a t r i x of t r a n s m i s s i o n coefficients that codes the n e u r a l hologram, are given by 2 2 -c~t(A. + B. + 2A.B.cos 3`.) I
I
I
I
1
ii
For the DRC code, there a r e limitations on the i n t e r n e u r a l connections and base firing rates that will allow useful storage.
The r e s t r i c t i o n s a r e that the total synaptic effect
must be inhibitory and that the base r a t e s a r e r e l a t e d as follows: r e = 2(rA+rB)
.
(26)
In e l e c t r i c a l engineering t e r m s , equation (26) places the quiescent point midway in the o p e r ating range of the neuron, tt is the optimum operating point in that it allows the m a x i m u m signal input without distortion; n e v e r t h e l e s s , it is a definite r e s t r i c t i o n on the generality of the model.
The expected rate of hits in this case is
E(hit) = -~-(rA + r B) 2+ r c ( r A + rB ) + ( r c _ 2~r(rA + rB)~ "(Aevn0 + Bevn~) - 7(Aevn0 + Berne) 2 Because of the r e s t r i c t i o n of equation (26), this reduces to E(hit) = -T( rA+ r B) 2 + rc (rA+ rB ) _ T(Aevn0 + Bevu~) 2
(27)
27 Thus the c o m b i n e d effect of the two r e s t r i c t i o n s is to e l i m i n a t e the f i r s t o r d e r t e r m s of evn0 and evn~.
It c a n be shown t h a t t h e s e t e r m s m a k e any useful r e c o n s t r u c t i o n of i n f o r m a -
tion i m p o s s i b l e . T h e r a t e of change of ~ i s now d--'TT ' dt = v~E(hit) + ~ (T(r A + r B) 2 _ r c ( r A + rB)~ = _ ~-r(Aevn0 + Bevn~) 2 ," since
(evnO)2,
(evn¢) 2 = 1
and even0 • evn¢ = e v n ( 0 - ~), d7 - - a'r(A2+ B2+ 2ABevn'?) dt
(23)
w h e r e "t' = 0 - ~ ; thus
~- = )te-~t(A2+ B2+ 2ABevnT) 2
(29)
2
-a't(Ai+ Bi + 2AiBievn~i) and
~..
=Xe
Ii
Both codes have led to the c o n c l u s i o n of an e x p o n e n t i a l s t o r a g e c h a r a c t e r i s t i c .
That
is, the t r a n s m i s s i o n c o e f f i c i e n t s of the n e u r o n s which encode the i n t e r f e r e n c e p a t t e r n of the two w a v e f r o n t s v a r y e x p o n e n t i a l l y w i t h the input s i g n a l s .
It h a s been shown o p t i c a l l y
( F r i e s e m and Zelenka, 1967) t h a t r e c o n s t r u c t i o n is p o s s i b l e with n o n l i n e a r r e c o r d i n g c h a r acteristics.
The m a i n effect of the n o n l i n e a r i t y is to produce the h i g h e r o r d e r i m a g e s a n a l -
ogous to the h i g h e r o r d e r d i f f r a c t i o n s f r o m an optical g r a t i n g .
T h e s e unwanted i m a g e s c a n
t y p i c a l l y be s e p a r a t e d f r o m the d e s i r e d r e c o n s t r u c t i o n . A t p r e s e n t , no a n a l y s i s of the p o s s i b i l i t y of d e c a y of s t o r e d i n f o r m a t i o n will be m a d e , a s it does not play an i m p o r t a n t r o l e in the b e h a v i o r a l t a s k s to which the m o d e l will be applied.
The m e c h a n i s m of s t o r a g e i m p l i e s that t h e r e m a y be a n e x p o n e n t i a l d e c a y although
this is not s t r i c t l y t r u e .
The i n f o r m a t i o n m i g h t r e m a i n u n c h a n g e d o v e r t i m e until s o m e e x -
t e r n a l c o n t r o l signal e f f e c t i v e l y c a u s e s it to be e r a s e d .
In a s i m i l a r m a n n e r , a c o n t r o l s i g -
nal m i g h t a l s o g o v e r n the r e c o r d i n g of i n f o r m a t i o n , i n c r e a s i n g the value of a (equations (23) a n d (29)) to s t o r e i n t e r f e r e n c e p a t t e r n s and d e c r e a s i n g o~ to i g n o r e t h e m . Reconstruction Having developed e x p r e s s i o n s f o r % the f o r m of the r e c o n s t r u c t e d w a v e f r o n t c a n now be d e t e r m i n e d .
Initially, the a m p l i t u d e moduli, A and B, will be a s s u m e d c o n s t a n t for
e a c h n e u r o n to s i m p l i f y the d e r i v e d e x p r e s s i o n s ; the effect of r e l a x i n g t h i s a s s u m p t i o n i s a n a l y z e d at the end of this s e c t i o n .
28 To obtain r e c o n s t r u c t i o n after the interference pattern of A and B has been stored, A alone or B alone is input to the storage field. M R M Code. Considering reconstruction with the input of the A wavefront, output at
an a r b i t r a r y neuron is 2 2 _e = re + h e - t~t(A + B + 2AB cos T) • (rA+ rB+ A cos(2~Trft+ 0)) .
(30)
As demonstrated by Tokarski (196d) in the optical c a s e , the t r a n s m i s s i o n coefficient can be expanded in a F o u r i e r cosine s e r i e s to determine the image producing t e r m ,
C : rc+(rA+rB+ACos(27rft+0)~.
~ . . ~ T cos(m 0 m /
Tm , m = 0,1,2,...
(31)
- F o u r i e r coefficients.
Only the t e r m for m = I will contain a r e p l i c a of the B wavefront, all t e r m s for m > 1 produce higher o r d e r images.
The m = 0 t e r m is a constant.
The f i r s t two F o u r i e r coef-
ficients a r e 27T
1 TO= ~
I
e
-~t(A2+ B2+ 2ABcos
dT = e-°~t(A2+ B2)
• 10(2~AB)
(32)
0
10 - zero order modified
Tt=
Bessel function,
1 1 27r e - err(A2+ B2+ 2AB cos T) 7r • cos TdT = -2e-e~t(A2+ B2). i i(2~tAB) 0
11 - first order mofified Bessel function; thus (3O
C_= r c + (rA+ rB) X ( ~ 0
TmCOS(mT)~ + A cos(27rft+ 0) he-a~:(A2+ B2) 10(2~AB )
- A cos(27rft+ 0) @ ~ e - ~ t ( A 2 B2). 1 l(2a~tAB)~ cos T+ higher o r d e r t e r m s
(34)
In equation {34) : rc will produce the base firing r a t e s at the output field; rA+ r B and cos mT a r e independent and so w{ll generate a randomly distributed noise field that is subsumed in the base r a t e s at v; the third t e r m is constant for all the neurons of the storage field so that there is no output in v as a r e s u l t of this t e r m (equations (14) and (15)); the fourth t e r m is the reconstruction term; the higher o r d e r t e r m s propagate to v, but, analogous to the optical case, can be s e p a r a t e d from the d e s i r e d reconstruction by the t r a n s f o r m Z. The r e c o n -
29 struction t e r m is then c = -Xae -at(A2+ B2) r
11(2 tAB) (2 cos(2 t+ 0)- cos
but "y= 0 - ~ c = -kAe-a~t(A2+ B2) t (2~tAB) (eos(27rft + ~) + cos(2~ft+ 20- ~)) . r 1
(35)
Since amplitude moduti were assumed constant, only the phase of the B wavefront need be recovered for reconstruction.
This is the case in equation (35), the f i r s t cosine t e r m being
the phase of B_. The leading t e r m s r e p r e s e n t the amplitude of the ]3 reconstruction; the second cosine term is the complement of B and is s i m i l a r to the conjugate image in optical holography. Thus a linear reconstruction of the B wavefront has been obtained (the linearity is partially a result of the assumption of constant amplitude moduli) allowing the conclusion that neural holography can be achieved using this very general mean rate modulation code. The complementary and higher o r d e r reconstructions are equivalent to the multiple images produced in optical holography. It is reasonable to assume that these can be separated from the desired reconstruction given the appropriate anatomy of the neural structure. DRC Code. The analysis of reconstruction with the DRC code leads to a rather significant result.
Considering again input of the A wavefront alone, equations (6) and (29)
give c = r c - ( rA+ r B+ Aeven0) k e - c~t(A2+ B 2+ 2ABevn~ Expanding the variable part of the exponential,
c_ = r e - (rA+ rB+ hevn0)
ke-~t(A2+ B 2)
( - l i - 2 c~tABevn~/+
(2o~tABevnT)2 2'
}
but (e~Y) 2 n = 1 and (evaT) 2n+l = evn~,
n - any integer
c_ = r c - (rA+ rB+ Aevn0) k e - at(A2+ B2) • %1 + (2~AB)22,-----~--+ (2aCAB)44------~-+ " "
-(2a'tAB)evn7- ( 2 ~ B ) 3 evn~- (2atAB)5 3~ 51 evnT- . . . t
c = r c - (rA+ rB+ Aevn0) ke" tit(A2+ B2). (cosh(2a,tA B) - sinh(2atAB) evnT)
(36)
30 2 2 _c = r c - ( r A + r B+ Aev n 0)~.e - c~t(A + B ) cosh( 2~t AB)
+ ( rA+ r B) he - v~t(A2+ B 2 ) sinh( 2atA B) e v n 7 2 2 +~Ae-at(A +B )sinh(2~AB)evn0evn7
(37)
r
i s a g a i n the b a s e r a t e ; the s e c o n d t e r m c o n t a i n s the d i r e c t t r a n s m i s s i o n c o e f f i c i e n t which c is c o n s t a n t for all n e u r o n s and t h e r e f o r e allows p r o p a g a t i o n to u but not to v_j in the t h i r d
t e r m rA+ r B a n d evn~/ a r e i n d e p e n d e n t v a r i a b l e s and thus produce r a n d o m l y d i s t r i b u t e d n o i s e in v ; the fourth t e r m i s the r e c o n s t r u c t i o n t e r m .
c
r
= ~Ae-
at. 2+ 2. (A B ) sinh(2a'tAB)evnOevn7 ;
but evn0 • evaT = ewa(0 - 7) and 7 = 0 -
c
r
=
kAe_at(A2 +
2 B )sinh(2a~tAB) . e v n ~
(38)
The d i r e c t i o n d i m e n s i o n of the B- w a v e f r o n t is r e c o v e r e d (eraS) and s i n c e a m p l i t u d e moduli w e r e a s s u m e d c o n s t a n t , B h a s been l i n e a r l y r e c o n s t r u c t e d .
The i m p o r t a n t r e s u l t i s t h a t
only one w a v e f r o n t c o m p o n e n t is g e n e r a t e d ; the c o m p l e m e n t of B is not p r o d u c e d n o r a r e h i g h e r o r d e r i m a g e s , even though the r e c o r d i n g is n o n l i n e a r ( t h i s r e s u l t will hold f o r any f o r m of n o n l i n e a r i t y ) .
T h i s s i m p l i c i t y ia the output space, which c a n n o t be o b t a i n e d in the
optical o r n e u r a l MRM c a s e s , p e r m i t s a g r e a t deal of g e n e r a l i t y in the n e u r a l a n a t o m i e s a p p r o p r i a t e for h o l o g r a p h i c p r o c e s s e s . T h e Effect of V a r i a b l e A m p l i t u d e Moduli, The p r e v i o u s s e c t i o n s have d e a l t with s t o r a g e and r e c o n s t r u c t i o n with the a s s u m p t i o n of a c o n s t a n t a m p l i t u d e of input to e a c h neuron.
The i n f o r m a t i o n s t o r e d and r e c o v e r e d was that of p h a s e o r d i r e c t i o n .
Allowing the
a m p l i t u d e moduli, A and B, to v a r y o v e r the s t o r a g e n e u r o n s i n c r e a s e s the a m o u n t of i n f o r m a t i o n t h a t can be r e p r e s e n t e d , while still p e r m i t t i n g the faithful r e c o v e r y of p h a s e o r direction.
It is not p o s s i b l e , h o w e v e r , to r e c o v e r the a m p l i t u d e t e r m s without a c e r t a i n
a m o u n t of n o i s e a n d d i s t o r t i o n .
Since the condition of s t r i c t l y c o n s t a n t moduti i s unlikely in
the b r a i n , it is i m p o r t a n t to d e t e r m i n e the f o r m of the r e s u l t i n g d i s t o r t i o n . E x a m i n a t i o n of e q u a t i o n s (34) and (37) shows t h a t t h e r e a r e t h r e e s o u r c e s of d i s tortion.
F i r s t , the t e r m s
[~te - a t ( A 2 + B 2 ) 10(2v~tAB) },J f r o m equation (34), and
~Xe - a t ( A 2 + B2)cosh(2o~tAB)}, f r o m equation (37), a r e the d i r e c t t r a n s m i s s i o n c o e f f i c i e n t s .
31 As the constant terms in 7, they allow propagation to u but not to v. With variable moduli, these expressions are no longer constants. Their spatial variation is independent of 0 and and so produces a uniform noise output to v that increases in proportion to the variance of A and B. The effect of the noise depends on the ratio of the "area" in v to which the B wavefront propagates, to the total "area" reached by the noise. This ratio is determined by the output transform Z which could theoretically shrink the reconstruction "area" to minimize the effect of the noise. The second source of distortion arises from the encoding of the amplitude in the product of an exponential and a Bessel (equation (34)) or a hyperbolic sine (equation (37)) function. To determine if the amplitude of B is recoverable, the products are expanded; from (34)
~tA •e
-atA2 F
f l -OttA2\3 F
+
(39)
and from (37)
The expansions do contain first order terms in B. The first nonlinear term is cubic; however, the coefficient of this term can be made to equal zero with appropriate values of ~, t and A. The third source of distortion can be seen in equations (39) and (40). The terms to the left of the braces should be constant for a faithful reconstruction; the value of A, however, is variable. The variance of A around its mean value thus superimposes a random field of noise on top of the reconstructed B__wavefront. Againthere is a tradeoff between the increased information capacity with variable moduli and the resulting increase in noise. These sources of noise and distortion are similar to those faced in optical holography where a careful choice of recording geometries and techniques usually suffices to minimize their effects. It is evident that if the brain were to use holographic storage, it too would have to make careful choices of neural transforms and signal codes; these choices could be made by natural selection in evolution. In addition, the brain might have another mechanism to aid in noise suppression. If the neural holographic structure were two intersection networks as suggested previously, the self-seeking reverberations of such networks could act as feedback to suppress nonlinearities and random variability.
32 Storage of m o r e than one i n t e r f e r e n c e p a t t e r n It is p o s s i b l e for the n e u r a l h o l o g r a m to r e c o r d m o r e than one i n t e r f e r e n c e p a t t e r n while still allowing s e p a r a t e r e t r i e v a l of w a v e f r o n t s f r o m individual p a t t e r n s ,
The e q u i -
v a l e n t optical c a s e ( m u l t i p l e s t o r a g e on t w o - d i m e n s i o n a l h o l o g r a m s ) h a s been a n a l y z e d by C o l l i e r and P e n n i n g t o n (i967); t h e i r study showed that r e c o n s t r u c t i o n with no c r o s s t a l k i s p o s s i b l e when the w a v e f r o n t s b e i n g s t o r e d a r e uniquely c o d e d - - i , e , , t h e i r c r o s s - c o r r e l a t i o n s with o t h e r w a v e f r o n t s to be s t o r e d a r e m i n i m a l o r z e r o .
T h i s r e s u l t g e n e r a l i z e s r e a d i l y to
the n e u r a l c a s e as the w a v e f r o n t s A and B v a r y uniquely o v e r the s t o r a g e field and c a n be a s s u m e d to h a v e little c o r r e l a t i o n to s u b s e q u e n t s e t s of w a v e f r o n t s b e i n g s t o r e d .
Collier
and P e n n i n g t o n r e p o r t e d that the c r o s s t a l k in this c a s e is t r a n s f o r m e d into u n i f o r m l y d i s t r i b uted n o i s e t h a t does not i m p a i r r e c o n s t r u c t i o n . Storage of m u l t i p l e p a t t e r n s in the n e u r a l c a s e i s a c h i e v e d by the s e q u e n t i a l e x p o s u r e of the s t o r a g e field to p a i r s of input w a v e f r o n t s .
F o r the MRM code then,
N 7=kexp
'
2
2
(-~_~t.(A.+B.+2A.B.cos~) \ j=l j J J JJ
, N - n u m b e r of p a t t e r n s s t o r e d .
A s s u m i n g t h a t the w a v e f r o n t s have the s a m e a v e r a g e a m p l i t u d e moduli and that e x p o s u r e times are equal,
N :~ = k exp
c~Nt(A2+ B 2) - 2~tAB
, cos j=l
Expanding a s the p r o d u c t of N F o u r i e r c o s i n e s e r i e s (equations (32) and (33)), N 7 = k e -~Nt(A2+ B 2 ) . - ~ {10(2~tAB) - 2 t l ( 2 C ~ A B ) c o s 7 j + . . .} j=l
7 = k e - ~'Nt(A 2+ B 2 ) (10 (2a~AB)) N N
B2). (10(2 AB))N-1.
-
COS ~/~ j=l
+ i n t e r m o d u l a t i o n and h i g h e r o r d e r t e r m s
(4~)
The f i r s t t e r m i s the d i r e c t t r a n s m i s s i o n coefficient; the s e c o n d is the r e c o n s t r u c t i o n t r a n s .th m i s s i o n c o e f f i c i e n t - - e a c h cos % c o n t a i n s the i n f o r m a t i o n for r e c o n s t r u c t i o n of the j waveJ front. I n t e r m o d u l a t i o n or c r o s s t a l k is produced, on input of the i th w a v e f r o n t , by the c o s i n e t e r m s for j ~ 1 and by the t e r m s c o n t a i n i n g p r o d u c t s of two o r m o r e c o s i n e s .
33 S i m i l a r l y for the DRC code, N v = Xexp
cgNt(A2+
-2atAB
, evn j=±
Expanding as a product of N T a y l o r s e r i e s (from equation (36)), T = ~. e-C~Nt(A2+ B2) •
T = k e -aNt(A2+ B2)
N ~-~ {cosh(2~AB) - sinh(2atAB)" e v n ~ } j=1
cosh(2~tAB)) N N j=l
+ intermodulation terms
(42)
The terms are equivalent to those for the MRM code. Notice in equations (41) and (42) that the reconstruction transmission coefficients have the same value for each of the N patterns. That is, the order of storage does not affect the amplitudes of the reconstructed wavefronts. This lack of a serial position effect would not have resulted if some form of storage decay had been assumed.
Noticealso that
the amplitude of reconstruction decreases exponentially with the number of patterns stored. This implies that the transmission type neural hologram is a limited capacity store. Phase holography (Upatnieksand Leonard, 1970) does not attenuate amplitude as increasing numbers of patterns arc stored and so might be a candidate model for a large capacity store such as long term memory. Conclusions A mathematical analysis has shown that holographic storage can be achieved in the neural domain for information coded in terms of mean rates; as this is a time-averaged parameter, it is possible for individual spike trains in a wavefront to vary randomly in instantaneous frequency and phase--a striking contrast to the strict frequency and phase requirements of optical holography. The properties of the neural holographic system were investigated for two classes of codes assuming, for convenience, a linear effect of concurrent pre-and postsynaptic impulses.
Meanrate modulation (MRM) appeared more reason-
able in t e r m s of operating r e q u i r e m e n t s , although the directional r a t e change code (DRC)
did p e r m i t m o r e flexibility in neural t r a n s f o r m s .
Finally, the neural h o l o g r a m was shown
to be capable of multiple storage of associations such that information f r o m each individual association can be r e t r i e v e d separately.
34 T h e r e could be m a n y s t r u c t u r e s in the b r a i n t h a t a r e able to s u p p o r t t h e s e h o l o g r a p h i c processes.
The c o n f i g u r a t i o n of input c o n v e r g i n g f r o m two a r e a s is, of c o u r s e , w i d e s p r e a d .
It is only r e q u i r e d t h a t the n e u r o n s a t the i n t e r s e c t i o n be fatiguable for s h o r t t e r m s t o r a g e to o c c u r .
In addition, the two c o d e s a n a l y z e d a r e found in m a n y a r e a s of the b r a i n .
On the
o t h e r hand, h o w e v e r , it is a v e r y difficult t a s k to d e t e r m i n e w h e t h e r the a p p r o p r i a t e t r a n s forms are available,
T h u s , while holograph}, i s a f e a s i b l e s t o r a g e p r o c e s s in the b r a i n , the
c r i t i c a l physiological t e s t i s beyond r e a c h a t p r e s e n t . On the b e h a v i o r a l level, h o w e v e r , a r a n g e of powerful t e s t s of the h o l o g r a p h i c h y p o t h e s i s is p o s s i b l e and t h e s e a r e d e s c r i b e d in d e t a i l in a r e c e n t p a p e r (Cavanagh, 1972). In b r i e f , the p a p e r p o s t u l a t e s a s h o r t t e r m m e m o r y s y s t e m that s t o r e s the i n t e r f e r e n c e p a t t e r n s between the w a v e f r o n t of s e n s o r y i n f o r m a t i o n r e c e i v e d f r o m a s t i m u l u s and the w a v e f r o n t that codes the i n t e r n a l i n f o r m a t i o n or " m e a n i n g " of the s a m e s t i m u l u s .
This arrange-
m e n t allows r a p i d a c c e s s r e c o g n i t i o n of s e n s o r y e v e n t s and i m a g i n g of i n t e r n a l s y m b o l s (see F i g u r e 4} t The p a p e r i n v e s t i g a t e s r e a c t i o n t i m e (RT) p r e d i c t i o n s of the s y s t e m for a r e c o g n i t i o n t a s k developed by S t e r n b e r g (1966) in which a l i s t of i t e m s i s m e m o r i z e d and the r e a c t i o n t i m e to c l a s s i f y a t e s t i t e m a s to l i s t m e m b e r s h i p i s m e a s u r e d .
Since a t e s t s t i m u l u s whose
a p p r o p r i a t e i n t e r f e r e n c e p a t t e r n h a s b e e n s t o r e d ( m e m o r i z e d ) g e n e r a t e s a r e c o n s t r u c t i o n of its i n t e r n a l r e p r e s e n t a t i o n while a t e s t s t i m u l u s with no s t o r e d p a t t e r n does not, the h o l o g r a p h i c s y s t e m can c l a s s i f y s t i m u l i as positive (stored) o r n e g a t i v e (not stored) by m o n i t o r lag output to the r e c o n s t r u c t i o n a r e a
1
( F i g u r e 5).
A, the s e n s o r y r e p r e s e n t a t i o n , is a s s u m e d to be the end r e s u l t of s e n s o r y p r o c e s s i n g , w h a t e v e r the f e a t u r e e x t r a c t i o n s o r t r a n s f o r m a t i o n t h a t that m a y include. F o r e x a m p l e , in the v i s u a l s y s t e m , A m i g h t be the output of a r e a 19 of the v i s u a l c o r t e x . The t r a n s f o r m s r e q u i r e d by the h o l o g r a p h i c s y s t e m a r e s i m p l y t h o s e t h a t allow s e p a r a t i o n of r e c o n s t r u c t e d i m a g e s ; no o t h e r p r o p e r t i e s a r e n e c e s s a r y to s u p p o r t s t o r a g e and r e g r i e v a l . Additional p r o p e r t i e s a r e r e q u i r e d , h o w e v e r , to a c h i e v e the p a t t e r n r e c o g n i t i o n ability d e m o n s t r a t e d by the b r a i n . K a b r i s k y , et al. (±971) and G i n s b e r g (197i) have a n a l y z e d the v a r i o u s p o s s i b l e n e u r a l t r a n s f o r m s that a r e in a c c o r d with h u m a n p a t t e r n r e c o g n i t i o n a n d p e r c e p t u a l i l l u s i o n s . I t i s not n e c e s s a r y t h a t t h e s e be the t r a n s f o r m s that m a p A onto the s t o r a g e field n e u r o n s , they could equally well p r e c e d e the production of the A w a v e f r o n t . F i n a l l y , the a n a l y s i s of the h o l o g r a p h i c r e c o g n i t i o n s y s t e m a s s u m e s that X, Y, W, and Z a r e l i n e a r t r a n s f o r m s with r e s p e c t to the a v e r a g e a m p l i t u d e of a w a v e f r o n t . T h i s i m p l i e s that the column s u m s of e a c h t r a n s f o r m m u s t be c o n s t a n t a n d allows the input ( a and b_) and output (u and v) w a v e f r o n t s to be r e p r e s e n t e d by the v a l u e s of A, B, and c a t a single a r b i t r a r y n e u r o n in the s t o r a g e fieid. T h i s a s s u m p t i o n s i m p l i f i e s a n a l y s i s but i s not e s s e n t i a l in any way for h o l o graphic storage.
35
STORAGE to External label•
v
a
processes
to
Internal label•
cognitive
LTM
b
Hologram
RECONSTRUCTION a
to
v
cognitive processes
u B
N O RECONSTRUCTION -- a NOT STORED a"
--
FIGI~RE 4:
LTM
An holographic short term memory system. External information is represented by wavefront propagation from input (a.) through the storage field (A_) to the output field (u_), internal information, by propagation from b through B to v.
't A log I/I o I
~
Receptor/sensory transform
LI
ts
%!
1
!
ICo
I
Detectors
;I do !
r
I tdr "td° [
~.ol+r ~i d 1 Storage field
IH tR
FIGURE 5: S c h e m a t i c of s y s t e m a c t i v i t y upon p r e s e n t a t i o n of a t e s t s t i m u l u s . The gate s i g n i f i e s that a r e c o n s t r u c t i o n will o c c u r only if the i n t e r f e r e n c e p a t t e r n a p p r o p r i a t e to the t e s t s t i m u l u s has been s t o r e d . The i n h i b i t o r y input to the d i r e c t w a v e f r o n t d e t e c t o r p e r m i t s a posiLive r e s p o n s e if the r e c o n s t r u c t e d w a v e f r o n t i s d e t e c t e d b e f o r e the d i r e c t w a v e f r o n t . A, c , and c_ a r e r w a v e f r o n t a m p l i t u d e s , I i s the s t i m u l u s i n t e n s i t y and I 0 is the a~solute threshold.
Stimulus
I _
i
Reaction time components
Negative response
Positive response
37
Sternberg's (1966) original results have been replicated many times, over a wide variety of conditions. Typical data for this item recognition task are shown in Figure 6.
NEGATIVE RESPONSES
500
ss~Ct SS
4OO
~
POSITIVE
S~SS~,~ ~'
I
RESPONSES
300
1
2
3
4
N FIGURE 6: Typical r e a c t i o n t i m e r e s u l t s for the i t e m r e c o g n i t i o n task. N is the n u m b e r of i t e m s m e m o r i z e d . The i m p o r t a n t f e a t u r e s of the data a r e the l i n e a r i n c r e a s e in r e a c t i o n t i m e with i n c r e a s i n g length of the m e m o r i z e d list, the equality of the s l o p e s (which a r e u s u a l l y on the o r d e r of 30 to 40 m s p e r item) for positive and n e g a t i v e r e s p o n s e s - - i , e . , r e s p o n s e additivity, and a h i g h e r i n t e r c e p t for n e g a t i v e i n s t a n c e s than positive.
In the h o l o g r a p h i c model of the t a s k ,
the locus of r e a c t i o n t i m e v a r i a t i o n is taken to be the d e p e n d e n c e of w a v e f r o n t d e t e c t i o n t i m e (tdo, tdr,
s e e F i g u r e 5) on w a v e f r o n t a m p l i t u d e .
The a m p l i t u d e of the s t o r a g e field w h i c h
was found f r o m n e u r o l o g i c a l d e r i v a t i o n s to be e x p o n e n t i a l l y r e l a t e d to N, the n u m b e r of i t e m s s t o r e d in m e m o r y .
A m a t h e m a t i c a l d e r i v a t i o n b a s e d on the e x p e r i m e n t a l l y o b s e r v e d
r e s p o n s e additivity a l s o s u p p o r t s the exponential r e l a t i o n , justifying the c o n c u r r e n c e model of n e u r a l change u s e d in t h i s p a p e r .
In addition, the additive effect of n o i s e on r e a c t i o n
38 t i m e ( S t e r n b e r g , 1967) l e a d s to the c o n c l u s i o n of a l o g a r i t h m i c I d e t e c t i o n function;
this
c o n c l u s i o n is c o n f i r m e d by the r e l a t i o n between s i m p l e R T and s t i m u l u s i n t e n s i t y ( 9 9 . 9 p e r cent of the v a r i a n c e of four s i m p l e RT s t u d i e s - - C a t t e l l , 1866; P i ~ r o n , 1920; B a r t l e t t and Macleod, 1954; Minueci and C o n n o r s , 1964--is a c c o u n t e d for by the l o g a r i t h m i c d e t e c t i o n function in c o m b i n a t i o n with a log e n e r g y t r a n s f o r m at the r e c e p t o r - - H a r t l i n e and G r a h a m , 1932).
The c o n c a t e n a t i o n of the l o g a r i t h m i c d e t e c t i o n t i m e vs. a m p l i t u d e r e l a t i o n and the
exponential a m p l i t u d e vs. N r e l a t i o n p r e d i c t s that r e a c t i o n t i m e i s a l i n e a r function of the n u m b e r of i t e m s m e m o r i z e d and t h a t the slope of the function is the s a m e for p o s i t i v e and n e g a t i v e r e s p o n s e s with p o s i t i v e s b e i n g u n i f o r m l y f a s t e r than n e g a t i v e s .
Furthermore,
the
a s s u m p t i o n t h a t the b r a i n o p t i m i z e s its p r o c e s s i n g c a p a c i t y p e r m i t s the slope of the R T function to be e x p r e s s e d in t e r m s of the m e m o r y span for the type of m a t e r i a l b e i n g t e s t e d . Data f r o m the m e m o r y span and m e m o r y s e a r c h l i t e r a t u r e s u p p o r t e d the p r e d i c t e d r e c i p r o cal r e l a t i o n between span a n d p r o c e s s i n g r a t e (Cavanagh, in p r e s s ) . T h u s , on the b e h a v i o r a l level, the h o l o g r a p h i c h y p o t h e s i s is s u p p o r t e d without exception on a v a r i e t y of m e a s u r e s .
I n v e s t i g a t i o n of the n e u r a l h o l o g r a p h i c p r o c e s s is u n -
d e r w a y at p r e s e n t on t h r e e additional l e v e l s : the p o s s i b i l i t y of long t e r m m e m o r y b a s e d on phase holography o r a t w o - s t a g e " b l e a c h e d " t r a n s m i s s i o n holography; single cell r e c o r d i n g s of the fatigue o r h a b i t u a t i o n of n e u r o n s to c o n f i r m the b e h a v i o r a l l y d e r i v e d exponential r e l a tion~ and, finally, the p r o p e r t i e s of c o m p l e x i n f o r m a t i o n p r o c e s s i n g s y s t e m s b a s e d on s t r u c t u r e s of i n t e r c o n n e c t e d h o l o g r a p h i c m e m o r i e s . Acknowledgments T h i s r e s e a r c h w a s s u p p o r t e d in p a r t by P u b l i c H e a l t h S e r v i c e R e s e a r c h G r a n t MH-07722 f r o m the National I n s t i t u t e of Mental Health. I a m v e r y t r a t e f u l to Dr. W i l l i a m Chase, J o h n P a r k m a n , Susan P e t e r s , and R i c h a r d Young for v a r i o u s c o m b i n a t i o n s of e m p e r i e a l , t h e o r e t i c a l , and s p i r i t u a l help and e s p e c i a l l y to Dr. T o m C a l v e r t for invaluable a s s i s t a n c e in developing the n e u r o p h y s i o l o g i c a l model reported here.
I The logarithmic detection is very significant from another viewpoint. The neural holographic system that has been described is linear in terms of mean rates; a fundamental characteristic of random (Poisson) spike trains is, however, that the variance of the spike rate is linearly related to the mean. Thus, while the amplitude description of a wavefront is invariant over different absolute levels of firing (e. g., at different base rates, which change as "r changes during storage, or different directions of change--increase or decrease), the variance in which the amplitude information is submerged is variable. The logarithmic function is the one function that transforms the dependence of the variance on the mean to an invariance. The detectability of wavefronts is thus independent of absolute firing levels only' in the case of logarithmic detection.
39 Reference s Baker, F . H . , E.R. Sanseverino, Y. Lamarre, and G. F. Poggio (1969), "Excitatory Responses of Geniculate Neurons of the Cat", J. Neurophysiology, 32_,_, 916-929. Bartlett, N.R. and S. Maeleod (1954), "Effect of Flash and Field Luminance Upon Human Reaction Time", J. Optical Soc. A m e r . , 4_~4, 306-311. Beurle, R.L. (1956), "Properties of a Mass of Cells Capable of Regenerating Pulses", Philos. Trans. Royal Soc. London, Series B, 24___00,55-94. Bracey, G.W. (1969), "Two Operations in Character Recognition: A Partial Replication", Perception and Psychophysics, 6, 357-360. Cattell, J. McK. (1886), "The Influence of the Intensity of the Stimulus on the Length of the Reaction Time", Brain, 9, 512-515. Cavanagh, J . P . (1972), "Holographic Processes Realizable in the Neural Realm: Prediction of Short Term Memory Performance", Unpublished doctoral dissertation, CarnegieMellon University. Cavanagh, J . P . (in press), "The Relation Between Memory Span and Memory Search Rate", Psychological Rev,, to appear. Collier, R.J. and K.S. Pennington (1967), "Multicolor Imaging from Holograms Formed on Two-dimensional Media", Appl. Optics, 6_., 1091-1095. Creutzfeldt, O.D. (1970), "Some Principles of Synaptic Organization in the Visual System", in Francis O. Schmitt (Ed.), The Neurosciences, Second Study Program (New York: Rockerfeller University Press). DeValois, R . L . , G.H. Jacobs, and A.E. Jones (t962), "Effects of Increments and Decrements of Light on Neural Discharge Rate", Science., i36___,986-98d. Enger, P . S . , J.K.S. Jansen, and L. Wall~e (1969), "A Biological Model of the Excitation of a Second Order Sensory Neurone", Kybernetik, 6, 141-145. F r i e s e m , A.A, and J.S. Zetenka (1967), "Effects of Film Nonlinearities in Holography", Appl. Optics, 6, 1755-1759. Gabor, D.A. (1945), "A New Microscopic Principle", Nature, 16__~.1,777. Ginsburg, A . P . (1970, "Psychological Correlates of a Model of the Human Visual S stem", Unpublished m a s t e r ' s thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio. Griffiths, J.S. (1966), "A Theory of the Nature of Memory", Nature, ~
1160-1±63.
Hartline, H.K. and C.H. Graham (1932), "Nerve Impulses from Single Receptors in the Eye", J. Cellular and Comparative Physiology, t_.., 277-295. Hebb, D.O. (1949), Organization of Behavior (New York: Wiley). Julesz, B. and K.S. Pennington (1965), "Equidistributed Information Mapping: An Analogy to Holograms and Memory ~', J. Optical See. A m e r . , 55__, 604. Kabrisky, M., C.F. Hall, L. Goble, R.A. Gill, and J.W. Carl (1970, "Realization of a Data Independent Pattern Recognition System", Annual Symposium Record, Systems, Man, and Cybernetics, I . E . E . E . , 233-240. Kupfermann, I. and H. Pinsker (1969), "Plasticity in Aptysia Neurons and Some Simple Neuronal Models of Learning", in Jack T. Tapp (Ed.), Reinforcement and Behavior (New York: Academic Press).
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Lashley, K.S. (1929), Brain Mechanisms and Intelligence (Chicago: University of Chicago Press). Latour, P.L. (1967), "Evidence of Internal Clocks in the Human 10glen , 2__77,341-348.
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Minueci, P.K. and M.M. Conners (1964), "Reaction Time Under Three Viewing Conditions: Binocular, Dominant Eye, and Nondominant Eye", J. Experimental psychology, 6_%7, 268-275. Perkel, D.H. and T.H. Bullock (1968), "Neural Coding", Neurosciences Bulletin, 6, 221-348.
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Perkel, D.H., J. Schulman, T.H. Bullock, G.P. Moore, and J.P. Segundo (±964), "Pacemaker Neurons: Effects of Regularly Spaced Synaptie Input", Science, !45, 61-63. .I
Proton, H. (1920), "Nouvelles r~cherches sur l'analyse due temps de faience sensorielle et sur la loi qui relie ce temps }t l'intensite de l'excitation", Annie Psyehologique , 2_22, 58-142. PSppel, E. (1970), "Excitability Cycles in Centra[ Intermittency",
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K.H.
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Segundo, J.P., D.H. Perkel, H. Wyman, H. Hegstad and G.P. Moore (1968), "Input-Output Relations in Computer-Simulated Nerve Ceils. Influence of the Statistical Properties, Strength, Number, and Interdependence of Excitatory Presynaptic Terminals", Kybernetik , _4, 157-171.
Sharpless, S.K. (1964), "Reorganization of Function in the Nervous System--Use and Disuse", in Victor E. Hall (Ed.), Annual Review of Physiology, 26_,_2,357-388. Sternberg, S. (t966), "High-speed Scanning in Human Memory", Science, 153, 652-654. Sternberg, S. (1967), "Two Operations in Character Recognition: Some Evidence from Reaction-time Measurements", Perception and Psychophysics, 2_, 45-53. Tokarski, J . M . J . (1968), "The Effect on the Hologram Record of a Nonlinear Relationship Between Amplitude T r a n s m i s s and Exposure", Appt. Optics, 7, 989-990. Upataieks~ J. and C.D. Leonard (1970), " C h a r a c t e r i s t i c s of Dielectric Holograms", I . B . M . J. Research and Development, 1_.~4,527-532. vanHeerden, P . J . (1963), "Theory of Optical Information Storage in Solids", AppI. Optics, 2, 393-400. Westlake, P.R. (1967), "Towards a Theory of Brain Functioning: The Possibilities of Neural Holographic Processes", Conference Proceedings of the 20th Annual Conference on Engineering in Medicine and Biology , I, E. E. E. Westlake, P.R. (1968), "Towards a Theory of Brain Functioning: A Detailed Investigation of the Possibilities of Neural Holographic Processes", Doctoral dissertation, University of California, Los Angeles (Ann Arbor, Michigan: University Microfilms; No. 68-12477). Westlake, P.R. (1970),"The Possibility of Neural Holographic Kyberuetik, 7, 129-153.
Processes Within the Brain",
SEMANTIC MEMORY R E T R I E V A L : SOME DATA AND A MODEL E l i z a b e t h F. Loftus U n i v e r s i t y of W a s h i n g t o n Introduction If I w e r e to a s k the a v e r a g e E n g l i s h - s p e a k i n g adult to n a m e an a n i m a l , o r a yellow f r u i t , o r a flower beginning with the l e t t e r " D " , he would o r d i n a r i l y be able to give m e a c o r r e c t a n s w e r in l e s s t h a n a few s e c o n d s .
P r o d u c i n g a piece of i n f o r m a t i o n t h a t one h a s
l e a r n e d s o m e t i m e ago and knows v e r y well r e q u i r e s v e r y l i t t l e e f f o r t indeed. we do i t ?
But, how do
By what p r o c e s s do we r e a c h into the huge s t o r e of i n f o r m a t i o n in m e m o r y and
produce a r e s p o n s e t h a t i s a p p r o p r i a t e to a given q u e s t i o n .
T h e r e i s no doubt t h a t we a r e
v e r y good a t doing t h i s , but at the m o m e n t we know r e l a t i v e l y l i t t l e about how we do it. Although it is too e a r l y to p r o v i d e d e t a i l s of the r e t r i e v a l m e c h a n i s m , we c a n m a k e c e r t a i n g e n e r a l d i s t i n c t i o n s a m o n g p o s s i b l e types of p r o c e s s e s .
F o r e x a m p l e , we can c l a s -
sify p r o c e s s e s of m e m o r y - a c c e s s in t e r m s of the e x t e n t to w h i c h they involve s u c c e s s i v e a s opposed to s i m u l t a n e o u s c o n s u l t i n g of the m e m o r y s t o r e .
This basic distinction has many
i m p o r t a n t c o n s e q u e n c e s : one of t h e m c o n c e r n s the effect of s i z e of the a r r a y to be s e a r c h e d on s e a r c h i n g t i m e .
To the e x t e n t t h a t the r e t r i e v a l p r o c e s s i n v o l v e s s u c c e s s i v e c o n s u l t a t i o n
of the m e m o r y s t o r e , r e t r i e v a l t i m e should be a function of the m e m b e r of i t e m s in the a r r a y to be s e a r c h e d .
C o n v e r s e l y , to the e x t e n t t h a t s i m u l t a n e o u s c o n s u l t a t i o n ( o r " p a r a l l e l
p r o c e s s i n g " ) i s involved, r e t r i e v a l t i m e should be r e l a t i v e l y unaffected by the n u m b e r of i t e m s in the a r r a y . Previous Research In m o s t of the e a r l i e r r e s e a r c h dealing with the e f f e c t of a r r a y o r c a t e g o r y size on r e t r i e v a l t i m e , s u b j e c t s have been given a r e c o g n i t i o n o r i d e n t i f i c a t i o n t a s k .
Typically they
m u s t decide w h e t h e r a g i v e n i n s t a n c e i s a m e m b e r of a p a r t i c u l a r a r r a y , c l a s s o r c a t e g o r y . Most of t h i s w o r k h a s d e a l t with r e l a t i v e l y s m a l l , newly l e a r n e d c a t e g o r i e s t h a t a r e t y p i c a l l y defined s o l e l y by an e n u m e r a t i o n of i n s t a n c e s (e. g . , N e i s s e r , 1963; P o l l a c k , 1963; R a b b i t t , 1959; S t e r n b e r g , 1966).
F o r e x a m p l e , S t e r n b e r g ( 1 9 6 6 ) gave s u b j e c t s a g r o u p of d i g i t s (e. g . ,
4, 8, 3, 5) and a s k e d t h e m w h e t h e r a n o t h e r digit (e. g . , 3) was a m e m b e r of t h a t set.
This
r e s e a r c h on " s h o r t - t e r m m e m o r y " h a s g e n e r a l l y found r e s p o n s e t i m e for b o t h p o s i t i v e a n d n e g a t i v e i n s t a n c e s to be a d i r e c t function of the size of the c a t e g o r y .
This result has been
i n t e r p r e t e d a s e v i d e n c e for the e x i s t e n c e of a s u c c e s s i v e and p e r h a p s e v e n e x h a u s t i v e s c a n ning process for retrieval from short-term memory. W h a t happens when l a r g e r and b e t t e r - l e a r n e d c a t e g o r i e s a r e u s e d ?
For example,
suppose we a s k a s u b j e c t to decide w h e t h e r a "dog" i s a n " a n i m a l " ? L a n d a u e r and F r e e d m a n
42
(1963) found that s u b j e c t s took l o n g e r to decide that an i t e m was not a m e m b e r of a l a r g e c a t e g o r y than to decide that it was not a m e m b e r of a s m a l l c a t e g o r y .
F o r e x a m p l e , it took
l o n g e r to decide that a " d e s k " was not an " a n i m a l " than to decide that it was not a "dog". T h e r e was a slight, but n o n s i g n i f i c a n t , d i f f e r e n c e in d e c i s i o n t i m e s for the p o s i t i v e i n s t a n c e s . Collins and Quillian (1969) and M e y e r (1970), using s o m e w h a t d i f f e r e n t p r o c e d u r e s , a l s o r e p o r t that r e t r i e v a l t a k e s l o n g e r for l a r g e r c a t e g o r i e s . Although L a n d a u e r and F r e e d m a n , Collins and Quillian~ and M e y e r all find an e f f e c t of c a t e g o r y s i z e , the e f f e c t is v e r y s m a l l r e l a t i v e to the s i z e of the c a t e g o r i e s u s e d .
For
e x a m p l e , L a n d a u e r and F r e e d m a n r e p o r t that the c o r r e c t identification of n e g a t i v e i n s t a n c e s took 53 m s e c . l o n g e r for l a r g e than f o r s m a l l c a t e g o r i e s .
arrays to be searched,
in order for such small differences to be caused
time it takes to scan the larger categories, mously
Given the r e l a t i v e s i z e of the
rapid, on the order of 1,000 words
the successive
by the additional
scanning would have to be enor-
per sec., which seems
improbable.
The authors
accordingly conclude that the small differences should probably not be interpreted as evidence for the existence of a successive scanning process. some
Thus,
effect of category size on retrieval from semantic
been interpreted in terms of successive processing. small that they seem processing of some
to argue
for a mechanism
previous research
memory,
has sho~a
but it has generally not
On the contrary,
the differences are so
consisting largely if not entirely of parallel
sort.
The Original Experiment In 1969 I conducted an experiment Freedman
at Stanford University.
on this problem
in collaboration with Jonathan
Since subsequent experiments
have used a semilar meth-
odology, and have in general been designed to test a model of retrieval that we proposed the publication growing out of that research
(Freedman
to describe the original experiment
detail.
The experiment mechanism
was designed to provide more
with particular emphasis
sing is involved.
in some
to decide whether
memory
information about the retrieval
One major departure from previous work is that rather than study identifi-
ject could actually produce a word himself,
it is simple,
1971), it is appropriate
on the question of the extent to which successive proces-
cation time as had been done before, this experiment
duce a member
and Loftus,
in
it was a member
concerned
the speed with which a sub-
Instead of giving him a stimulus and asking him
of a category,
of it. The identification procedure
he was given a category and asked to prohas been used in the past largely because
convenient and yet obviously involves the retrieval of information from the
store.
The production procedure
is more
complicated
and less convenient because
|n a sense it involves a higher level of retrieval--the individual must not only reach into the store and find information relating to a stimulus,
he must actually produce an item from that
43 store.
D e s p i t e its d i f f i c u l t i e s , the use of t h i s p r o c e d u r e s e e m e d to be j u s t i f i e d b e c a u s e it is
an i m p o r t a n t f o r m of r e t r i e v a l a n d p r o v i d e s additional i n f o r m a t i o n about the r e t r i e v a l p r o c e s s . The b a s i c p r o c e d u r e was to p r e s e n t the s u b j e c t with a s t i m u l u s c o n s i s t i n g of a noun c a t e g o r y p a i r e d with e i t h e r a l e t t e r o r an a d j e c t i v e , and a s k h i m to p r o v i d e a w o r d t h a t b e longed in the o v e r l a p defined by the p a i r .
F o r e x a m p l e , s u b j e c t s who w e r e p r e s e n t e d with
the p a i r " f r u i t - P " m i g h t s a y " p e a c h " , " p e a r " , o r " p l u m " , a m o n g o t h e r p o s s i b i l i t i e s .
A
c o r r e c t r e s p o n s e would be any word b e g i n n i n g w i t h " P " t h a t n a m e s a kind of fruit. R e s p o n s e s s u c h a s " a p p l e " o r "pony" would be i n c o r r e c t . As d e s c r i b e d above, the m a i n focus of the study was on the e x t e n t to w h i c h the r e t r i e v a l m e c h a n i s m involved s u c c e s s i v e scanning.
The effect of size of the a r r a y to be
s e a r c h e d on r e t r i e v a l t i m e was an i n d i c a t i o n of the p r e s e n c e o r a b s e n c e of s u c c e s s i v e p r o c e s sing.
The e f f e c t of a r r a y size w a s i n v e s t i g a t e d by s e l e c t i n g m a n y noun c a t e g o r i e s t h a t r a n g e d
in s i z e f r o m v e r y s m a l l (e. g . , s e a s o n s , gems) to e x t r e m e l y l a r g e (words, f i r s t n a m e s ) . A n o t h e r c o n s e q u e n c e of s u c c e s s i v e p r o c e s s i n g c o n c e r n s the effect of the a r r a y defined by the o v e r l a p of the noun c a t e g o r y with the a d j e c t i v e or l e t t e r r e s t r i c t o r , o r e q u i v a l e n t l y , the n u m b e r of p o s s i b l e c o r r e c t r e s p o n s e s .
The s i z e of the o v e r l a p v a r i e d f r o m a m i n i m u m
of one ( " a n i m a l - Z " , with " z e b r a " b e i n g the only c o r r e c t r e s p o n s e in m o s t p e o p l e ' s m e m o r y s t o r e ) to v e r y l a r g e ( " f i r s t n a m e - J " , with dozens of n a m e s being p o s s i b l e r e s p o n s e s ) .
To
the e x t e n t t h a t s u c c e s s i v e s c a n n i n g plays a r o l e in r e t r i e v a l , it s e e m s l i k e l y t h a t the l a r g e r the n u m b e r of p o s s i b l e r e s p o n s e s , the s o o n e r a s u b j e c t will find and p r o d u c e one. The study thus p r o v i d e s two ways of a s s e s s i n g the e x t e n t to w h i c h s u c c e s s i v e s c a n n i n g o c c u r s in the r e t r i e v a l p r o c e s s : the effect of s i z e of the c a t e g o r y on r e a c t i o n t i m e a n d the e f f e c t of o v e r l a p of the c a t e g o r y a n d m o d i f i e r on r e a c t i o n t i m e . Method F o r t y s u b j e c t s w e r e individually told t h a t we w e r e conducting a study of how m e m o r y worked, t h a t they would s e e i t e m s c o n s i s t i n g of c a t e g o r i e s and e i t h e r a d j e c t i v e s o r l e t t e r s , and t h a t they should r e s p o n d with a w o r d that was an a p p r o p r i a t e m e m b e r of the c a t e g o r y . They w e r e given e x a m p l e s , and told to r e s p o n d a s quickly a s p o s s i b l e , but to avoid e r r o r s . D u r i n g the e x p e r i m e n t , the s u b j e c t s a t in f r o n t of a s c r e e n in w h i c h w a s a window c o v e r e d by h a l f - s i l v e r e d g l a s s .
An index c a r d c o n t a i n i n g the s t i m u l u s w a s p l a c e d in a d a r k
e n c l o s u r e behind the m i r r o r and was p r e s e n t e d by i l l u m i n a t i n g the e n c l o s u r e .
A microphone
was p l a c e d in f r o n t of the s u b j e c t a n d he r e s p o n d e d by s p e a k i n g into it. A t r i a l c o n s i s t e d of the following.
As a c a r d with t h e i t e m p r i n t e d in l a r g e type was
p l a c e d in the d a r k e n e d e n c l o s u r e behind t h e h a l f - s i l v e r e d m i r r o r , t h e e x p e r i m e n t e r s a i d " R e a d y " , a n d p r e s s e d a button t h a t i l l u m i n a t e d the f i r s t m e m b e r of the s t i m u l u s p a i r .
After
a n i n t e r v a l of a t l e a s t . 5 s e c . , the second m e m b e r of the s t i m u l u s p a i r w a s a u t o m a t i c a l l y
44 illuminated, and s i m u l t a n e o u s l y an e l e c t r i c t i m e r with a DC clutch was s t a r t e d .
The s u b -
j e c t ' s v e r b a l r e s p o n s e a c t i v a t e d a voice key that stopped the clock and t e r m i n a t e d the t r i a l . E a c h s u b j e c t r e c e i v e d a r a n d o m s e q u e n c e of 96 s t i m u l i with the c a t e g o r y p r e c e d i n g the r e s t r i c t e r on half the t r i a l s and following it on the o t h e r half. Results The two r e s u l t s that b e a r on the s u c c e s s i v e s c a n n i n g i s s u e a s a s follows. 1) T h e r e was a n o n s i g n i f i c a n t c o r r e l a t i o n of -. 22 between c a t e g o r y s i z e and r e a c t i o n t i m e . Thus, c o n t r a r y to e x p e c t a t i o n s f r o m a s u c c e s s i v e scanning model, the amount of t i m e to r e t r i e v e a w o r d f r o m a l a r g e c a t e g o r y i s not g r e a t e r than for a s m a l l c a t e g o r y - - r a t h e r t h e r e is little r e l a t i o n s h i p b e t w e e n s i z e and l a t e n c y and what t h e r e is i n d i c a t e s that l a r g e r c a t e g o r i e s take l e s s t i m e than s m a l l c a t e g o r i e s . 2) T h e r e was a n o n s i g n i f i c a n t c o r r e l a t i o n of . 02 b e t w e e n the n u m b e r of p o s s i b l e c o r r e c t r e s p o n s e s and r e a c t i o n time~ a l s o c o n t r a r y to e x p e c t a t i o n s f r o m a s u c c e s s i v e s c a n n i n g model. Although size of c a t e g o r y and o v e r l a p do not affect r e a c t i o n t i m e , o t h e r v a r i a b l e s do. 1) R e c a l l t h a t e a c h s t i m u l u s c o n s i s t e d of a noun and e i t h e r a l e t t e r o r an a d j e c t i v e ; and that the noun c a m e e i t h e r f i r s t o r s e c o n d . R e a c t i o n t i m e was significantly f a s t e r when the noun c a m e f i r s t r a t h e r than s e c o n d (1.87 vs. 2.12 s e c . ) .
R e a c t i o n t i m e w a s a l s o significantly f a s -
t e r when the s t i m u l u s included an adjective r a t h e r than a l e t t e r ( 1 . 8 4 v s . 2.15 s e c . ) . 2) A s e c o n d v a r i a b l e involves the f r e q u e n c y in E n g l i s h of the p o s s i b l e c o r r e c t r e s p o n s e s . Specifically the f r e q u e n c y in E n g l i s h of the one c o r r e c t r e s p o n s e that had the h i g h e s t such f r e quency w a s r e l a t e d to r e a c t i o n t i m e . have f a s t e r r e a c t i o n t i m e s .
Stimuli that have h i g h e r f r e q u e n c y r e s p o n s e s tend to
This finding i s a f u r t h e r d e m o n s t r a t i o n of the f r e q u e n c y - r e a c t i o n
t i m e r e l a t i o n s h i p found by M a r b e quite long ago (Thumb and M a r b e , 1901; c i t e d in Woodworth and S c h l o s b e r g , 1954). 3) A n o t h e r i m p o r t a n t v a r i a b l e involves the likelihood that a p a r t i c u l a r r e s p o n s e will be given when s u b j e c t s a r e a s k e d to n a m e w o r d s that fit a p a r t i c u l a r c a t e g o r y . n a n c e " of a r e s p o n s e within a c a t e g o r y .
We call this the " d o m i -
R a t h e r than the f r e q u e n c y in the E n g l i s h language in
g e n e r a l , " d o m i n a n c e " r e f e r s to the f r e q u e n c y with which a w o r d i s given a s an e x a m p l e of a category.
I n f o r m a t i o n on dominance was obtained f r o m Battig and Montague (1969).
c l e a r that high dominance p r o d u c e s f a s t e r r e a c t i o n t i m e s .
It is
Within the " f r u i t " category~ for
e x a m p l e , "apple" i s m o r e d o m i n a n t than " l e m o n " ; thus t h e r e i s a t e n d e n c y to r e s p o n d to " f r u i t - A " m o r e quickly than to f r u i t - L " . Discussion The m a j o r focus of this study was the question of w h e t h e r the p r o c e s s of r e t r i e v a l f r o m s e m a n t i c m e m o r y involved s u c c e s s i v e scanning to any a p p r e c i a b l e e x t e n t .
To the d e g r e e that
45 s u c c e s s i v e s c a n n i n g of i t e m s in the m e m o r y s t o r e o c c u r s , r e a c t i o n t i m e should be l o n g e r when m o r e i t e m s have to be s c a n n e d .
When a s u b j e c t is a s k e d to p r o d u c e an i t e m f r o m an
a r r a y o r c a t e g o r y , the l a r g e r the c a t e g o r y , the l o n g e r the r e a c t i o n t i m e should be.
T h u s , if
r e a c t i o n t i m e s a r e l o n g e r f o r l a r g e r c a t e g o r i e s , it s u g g e s t s t h a t s u c c e s s i v e s c a n n i n g i s o c c u r r i n g ; if r e a c t i o n t i m e s a r e not l o n g e r for l a r g e r c a t e g o r i e s , i t s u g g e s t s t h a t s u c c e s s i v e s c a n n i n g i s not o c c u r r i n g .
The p r e s e n t study did not find a p o s i t i v e r e l a t i o n s h i p between
c a t e g o r y s i z e and r e a c t i o n t i m e .
F u r t h e r m o r e , to the e x t e n t t h a t s u c c e s s i v e s c a n n i n g plays
a r o l e in r e t r i e v a l , r e a c t i o n t i m e should b e s h o r t e r when t h e r e a r e a l a r g e n u m b e r of p o s s i b l e c o r r e c t r e s p o n s e s (the m o r e c o r r e c t r e s p o n s e s , the e a s i e r it should be to find one). The p r e s e n t study did not find a n e g a t i v e r e l a t i o n s h i p b e t w e e n the n u m b e r of p o s s i b l e e o r r e e t r e s p o n s e s and r e a c t i o n t i m e .
T h u s , all of the e v i d e n c e i n d i c a t e s t h a t the p r o c e s s of r e t r i e v a l
i n v o l v e s l i t t l e o r no s u c c e s s i v e s c a n n i n g of the m e m o r y s t o r e . We m i g h t m e n t i o n a t t h i s point t h a t the i t e m s b e i n g r e t r i e v e d w e r e e x t r e m e l y w e l l learned.
It m a y well be t h a t with r e l a t i v e l y o b s c u r e i t e m s , o r with e a s y i t e m s t h a t for s o m e
reason are
not found quickly, a s u b j e c t e v e n t u a l l y r e s o r t s to s u c c e s s i v e s c a n n i n g .
However,
it a p p e a r s t h a t with t h e s e o v e r l y - l e a r n e d i t e m s , the n o r m a l , s u c c e s s f u l p r o c e s s of r e t r i e v a l does not involve s u e e e s s i v e s c a n n i n g of the m e m o r y s t o r e . A R e t r i e v a l Model If the r e t r i e v a l p r o c e s s d o e s not involve s u c c e s s i v e s c a n n i n g , of w h a t does it c o n s i s t ? One m o d e l t h a t a p p e a r s to fit m u c h of the data i n v o l v e s a h i e r a r c h i c a l o r g a n i z a t i o n , b y w h i c h we m e a n a s y s t e m t h a t is divided into a n u m b e r of i n t e r c o n n e c t e d s u b s y s t e m s , e a c h of the latter being hierarchical itself.
Specifically, o u r b a s i c conception i s t h a t of a m e m o r y o r g a n -
ized into a c o m p l e x h i e r a r c h y c o m p o s e d of c a t e g o r i e s (e. g . , a n i m a l s ) with s u b s e t s of e a c h (e. g . , b i r d s , dogs) and s u p e r s e t s (e. g . , l i v i n g things).
The s e a r e h p r o c e s s n e e d not b e g i n
a t the top and w o r k down until the a p p r o p r i a t e s u b s e t is found ( a s s u g g e s t e d , for e x a m p l e , by G r e e n , e t a l . , 1963 a n d L i n d s a y , 1963). can be e n t e r e d d i r e c t l y .
R a t h e r , e a c h c a t e g o r y h e a d s i t s own h i e r a r c h y t h a t
T h u s , to find a b i r d the c a t e g o r y " b i r d s " i s e n t e r e d ; to find a n a n i -
m a l , the c a t e g o r y " a n i m a l s " i s e n t e r e d ; and so on. In o r d e r to explain the l a c k of e f f e c t of c a t e g o r y s i z e , additional a s s u m p t i o n s a r e necessary.
Within e a c h c a t e g o r y it s e e m s l i k e l y t h a t a v a r i e t y of s u b s e t s e x i s t .
Some a r e
noun c a t e g o r i e s t h a t a r e s p e c i a l i z e d m e m b e r s of the l a r g e r s e t (e. g . , b i r d s and dogs a r e s u b s e t s of a n i m a l s ) . b e r of r e a s o n s .
Some a r e c l u s t e r s of w o r d s t h a t a r e highly a s s o c i a t e d for any of a n u m -
T h e y m a y have q u a l i t i e s in c o m m o n (e. g . , a l l begin with the l e t t e r " S " o r
all have long n a m e s o r all r h y m e ) .
They m a y be a s s o c i a t e d with e a c h o t h e r in the i n d i v i d u a l ' s
e x p e r i e n c e (e. g . , all be a n i m a l s in the P o o h books).
These clusters are probably more
i d i o s y n c r a t i c than the m a i n , noun c a t e g o r i e s , but a r e s o m e w h a t c o n s i s t e n t a c r o s s i n d i v i d u a l s .
46 And finally, u n d e r e a c h c a t e g o r y i s an u n d i f f e r e n t i a t e d e n u m e r a t i o n of e x e m p l a r s .
Within all
s u b s e t s , c l u s t e r s and the e n u m e r a t i o n , the i n s t a n c e s will be l i s t e d in a m o r e o r l e s s c o n s t a n t o r d e r , a c c o r d i n g to t h e i r f r e q u e n c y in the language o r d o m i n a n c e in the c a t e g o r y . U s i n g t h i s as a g e n e r a l m o d e l , we c a n begin to d e s c r i b e how a p e r s o n m a n a g e s the t a s k t h a t he h a s b e e n g i v e n i n t h e a b o v e e x p e r i m e n t . ginning with the l e t t e r " P " .
He m u s t , f o r e x a m p l e , find a f r u i t b e -
It s e e m s r e a s o n a b l e to a s s u m e t h a t the p r o c e s s of r e t r i e v a l h a s
a t l e a s t two m a j o r s t e p s : 1) e n t e r i n g an a p p r o p r i a t e c a t e g o r y - - " f r u i t s " ; and 2) finding an a p p r o p r i a t e m e m b e r of t h a t e a t e g o r y - - " p e a c h " (plus, of c o u r s e , the t i m e r e q u i r e d to p r o duce the r e s p o n s e v e r b a l l y ) .
T h e s e two s t e p s m a y e a c h be divided into s u b s t e p s , but for the
m o m e n t we have no i n f o r m a t i o n on t h a t . It c a n be s e e n t h a t nothing in the model would i m p l y t h a t r e t r i e v M would take l o n g e r for l a r g e r c a t e g o r i e s .
Step 1 i s a s s u m e d to be equally e a s y for l a r g e and s m a l l c a t e g o r i e s ,
s i n c e the p a r t i c u l a r c a t e g o r y is e n t e r e d d i r e c t l y . step 2 need not be affected by c a t e g o r y s i z e . the c a t e g o r y that fits the r e s t r i c t i o n i m p o s e d .
S i m i l a r l y , but s o m e w h a t l e s s obviously,
The c r u c i a l p r o b l e m is finding an i n s t a n c e of Once the c a t e g o r y is e n t e r e d , the next s t e p
( p r e s u m a b l y a substep) is to find an a p p r o p r i a t e c l u s t e r of i t e m s .
U n d e r f r u i t , a c l u s t e r of
" f r u i t s b e g i n n i n g with P " e x i s t s , is e n t e r e d , and the f i r s t e n t r y is r e a d out.
If i n s t e a d the
t a s k w e r e to find a g e m whose c o l o r was g r e e n , the s m a l l e r s e t " g e m s " would be e n t e r e d d i r e c t l y , the s u b s e t " g r e e n g e m s " found a n d the f i r s t e n t r y r e a d out, a huge s e t with a huge s u b s e t , the s a m e p r o c e s s i s r e p e a t e d .
And to find " N o u n - Y " ,
In all c a s e s , it c o n s i s t s of
finding a m a i n s e t and a s u b s e t , and does not involve s e a r c h i n g t h r o u g h au a r r a y of i n s t a n c e s . T h i s m o d e l of the r e t r i e v a l p r o c e s s a c c o u n t s f o r the l a c k of effect of c a t e g o r y s i z e , and, it s e e m s to us, is quite p l a u s i b l e . c l u s t e r s exist.
It s e e m s unlikely t h a t e v e r y c a t e g o r y c o n t a i n s s e p a r a t e c l u s t e r s for e a c h
l e t t e r and a d j e c t i v e . clusters.
The one i m p l a u s i b i l i t y i s the notion t h a t so m a n y
However, t h e r e a r e two a r g u m e n t s in f a v o r of the e x i s t e n c e of s u c h
F i r s t , when a s u b j e c t i s a s k e d to n a m e s e v e r a l f r u i t s b e g i n n i n g with a p a r t i c u l a r
l e t t e r , he h a s l i t t l e difficulty in p r o d u c i n g m a n y c o r r e c t r e s p o n s e s , v e r y r a p i d l y (if, in fact, m a n y c o m m o n r e s p o n s e s exist). p a u s e and then say " p l u m " .
T h a t is, he does not s a y " p e a c h " , p a u s e and then say " p e a r "
The typical p a t t e r n we have found in i n f o r m a l t e s t i n g i s a s p u r t
of s e v e r a l r e s p o n s e s , p e r h a p s followed by a p a u s e and then s e v e r a l m o r e r e s p o n s e s , etc. This s u g g e s t s t h a t t h e s e w o r d s a r e c l u s t e r e d in s o m e s o r t of functional r e l a t i o n s h i p .
The
s e c o n d a r g u m e n t in f a v o r of t h e e x i s t e n c e of c l u s t e r s is t h a t when m u l t i d i m e n s i o n a l s c a l i n g and c l u s t e r i n g p r o c e d u r e s a r e applied to c a t e g o r y r e t r i e v a l data, c e r t a i n c a t e g o r y m e m b e r s t h a t have q u a l i t i e s in c o m m o n tend to c l u s t e r t o g e t h e r (Shepard, 1972). In t h i s model, we have m a d e the a s s u m p t i o n t h a t the m e m o r y s t o r e is o r g a n i z e d p r i m a r i l y into noun c a t e g o r i e s .
It a l m o s t c e r t a i n l y i n c l u d e s g r o u p s of i t e m s t h a t have q u a l i -
47 t i e s in c o m m o n (e. g . , w a r m , r e d , e t e . ) and mac- e v e n have g r o u p s that have initial l e t t e r s in c o m m o n (e. g . , all w o r d s s t a r t i n g with P).
In fact, we can handle the fact that r e t r i e v a l
t i m e is f a s t e r when a noun is p a i r e d with an adjective r a t h e r than a l e t t e r by postulating that the main c l u s t e r s a r e i t e m s that have q u a l i t i e s in c o m m o n and within t h e s e c l u s t e r s a r e groupings of i t e m s that have initial l e t t e r s in c o m m o n .
The m a j o r , useful o r g a n i z a t i o n ,
h o w e v e r , is thought to be in t e r m s of noun g r o u p i n g s .
The significance of t h i s a s s u m p t i o n
i s that the f i r s t s t e p of the r e t r i e v a l p r o c e s s would c o n s i s t of locating and e n t e r i n g the a p p r o p r i a t e noun c a t e g o r y .
The p r e s e n t e x p e r i m e n t p r o v i d e s s o m e data that a r e c o n s i s t e n t with
this model and that provide an e s t i m a t e of the duration of the f i r s t s t e p in the p r o c e s s . E a c h s t i m u l u s p a i r was p r e s e n t e d with the noun e i t h e r f i r s t o r s e c o n d . When the noun c o m e s s e c o n d , the total r e t r i e v a l p r o c e s s begins only a f t e r its p r e s e n t a t i o n . When the noun c o m e s f i r s t , s t e p 1 can be begun b e f o r e the s e c o n d half of the p a i r , any d i f f e r e n c e due to o r d e r can be a s s u m e d to be c a u s e d by that p a r t of the p r o c e s s that i s c o m p l e t e d b e f o r e the s e c o n d half i s shown.
Thus, the d i f f e r e n c e b e t w e e n noun f i r s t and noun s e c o n d i s an i n d i c a -
tion of the e x i s t e n c e of s t e p 1 and of its duration. When nouns a r e p r e s e n t e d f i r s t , the m e a n r e a c t i o n t i m e is 1.87 s e c . , with noun second it i s 2.12 s e c . ditions.
T h i s d i f f e r e n c e of . 25 i s quite stable a c r o s s s o m e w h a t d i f f e r e n t c o n -
With n o u n - l e t t e r p a i r s , it i s . 27 s e c . ; with n o u n - a d j e c t i v e p a i r s it i s . 23 s e c .
T h e s e data s u p p o r t the idea that step 1 c o n s i s t s of e n t e r i n g the noun c a t e g o r y , and s u g g e s t that the duration of s t e p 1 is a p p r o x i m a t e l y . 25 s e c . f o r the p r o c e d u r e e m p l o y e d h e r e . To s u m m a r i z e , we have p r e s e n t e d a h i e r a r c h i c a l s t o r a g e m o d e l , and have p r o p o s e d that the r e t r i e v a l p r o c e s s f r o m s e m a n t i c m e m o r y c o n s i s t s of two d i s t i n c t s t e p s ( e n t e r i n g the a p p r o p r i a t e c a t e g o r y and then finding an a p p r o p r i a t e
m e m b e r of that c a t e g o r y ) .
This g e n e r -
al model is c o n s i s t e n t with the lack of r e l a t i o n s h i p b e t w e e n c a t e g o r y s i z e and r e t r i e v a l t i m e , and is given s o m e s u p p o r t by the d i f f e r e n c e in r e a c t i o n t i m e s when the noun is p r e s e n t e d f i r s t and second.
That d i f f e r e n c e o f . 25 s e c . a l s o s e r v e s a s an indication of the d u r a t i o n of s t e p
1 of the r e t r i e v a l p r o c e s s . E x p e r i m e n t a l Support for the Model Now let me p r e s e n t s o m e e v i d e n c e supporting t h r e e d i f f e r e n t a s p e c t s of the model. 1) The s e m a n t i c m e m o r y s t o r e i s o r g a n i z e d p r i m a r i l y into noun c a t e g o r i e s . The a s s u m p t i o n that the m a j o r o r g a n i z a t i o n i s in t e r m s of noun c a t e g o r i e s h a s s e v e r a l important implications.
One o b s e r v a b l e c o n s e q u e n c e c o n c e r n s the n u m b e r of w o r d s that a
s u b j e c t can produce in a given amount of t i m e u n d e r v a r i o u s c o n d i t i o n s . If we p r e s e n t a noun c a t e g o r y and give a s u b j e c t , say, one minute to w r i t e down any i t e m s that belong in the c a t e gory, he should be able to produce m o r e i t e m s than if we p r e s e n t him with an adjective and ask him to w r i t e any i t e m s w h i c h have the i n h e r e n t quality of that a d j e c t i v e .
A c c o r d i n g to the
48 model~ when a s u b j e c t i s p r e s e n t e d with a noun c a t e g o r y , he e n t e r s that c a t e g o r y in his s e m a n t i c m e m o r y and begins to n a m e m e m b e r s of the c a t e g o r y that a r e s t o r e d t h e r e .
When
he s e e s an a d j e c t i v e , h o w e v e r , he m u s t s t i l l e n t e r the m e m o r y s t o r e at s o m e noun c a t e g o r y , w h e r e u p o n he can n a m e m e m b e r s of that c a t e g o r y which have the i n h e r e n t quality of the p r e s e n t e d a d j e c t i v e . When he has n a m e s a s many m e m b e r s a s he can, he m u s t shift to a n o t h e r c a t e g o r y and begin to n a m e i t e m s f r o m it.
Thus, when p r e s e n t e d with an a d j e c t i v e f e w e r
i t e m s should be p r o d u c e d for two r e a s o n s : (a) The s u b j e c t m u s t decide which c a t e g o r y to e n t e r f i r s t , and this d e c i s i o n takes t i m e , and (b) he m u s t shift f r o m one c a t e g o r y to a n o t h e r w h e n e v e r he n e e d s to, and shifting takes t i m e An e x p e r i m e n t w a s d e s i g n e d to d e t e r m i n e w h e t h e r the n u m b e r of r e s p o n s e s given to a noun i s d i f f e r e n t f r o m the n u m b e r of r e s p c a s e s given to an a d j e c t i v e .
Two hundred s t u d e n t s
w e r e told that they would s e e s t i m u l i c o n s i s t i n g of e i t h e r a noun o r an a d j e c t i v e .
If the s t i m -
ulus w e r e a noun c a t e g o r y , they w e r e to w r i t e down a s many i t e m s a s they could that b e longed to that c a t e g o r y ,
F o r e x a m p l e , s u b j e c t s who w e r e p r e s e n t e d with the s t i m u l u s " s e a -
food" might say " o y s t e r ~', " c l a m " , and " s h r i m p " , a m o n g o t h e r p o s s i b i l i t i e s . If the s t i m u l u s w e r e an a d j e c t i v e , the s u b j e c t was to w r i t e down a s many i t e m s a s he could that had the i n h e r e n t quality of the a d j e c t i v e .
F o r e x a m p l e , if the s t i m u l u s w e r e " h a r d " , the s u b j e c t m i g h t
say " b r i c k " , ~'roek", " w o o d ' , e t c .
Details about the p r o c e d u r e can be found in Loftus (1972).
The m e a n n u m b e r of r e s p o n s e s given in one minute to noun s t i m u l i w a s 12.03 and to adjective s t i m u l i it w a s 9.15.
M o r e d e t a i l e d a n a l y s e s of the r e s p o n s e p r o t o c o l s indicated
that when the s t i m u l u s w a s a noun, a l m o s t all the r e s p o n s e s w e r e i n s t a n c e s of that noun c a t e gory.
However, when the s t i m u l u s was an a d j e c t i v e , s u b j e c t s tended to n a m e a n u m b e r of
i t e m s f r o m one c a t e g o r y , shift to a n o t h e r c a t e g o r y and n a m e a n u m b e r of i t e m s f r o m that c a t e g o r y , shift again, and sG on.
One s u b j e c t ' s r e s p o n s e p r o t o c o l to the s t i m u l u s " s m a l l "
i l l u s t r a t e s this tendency: " r a t , m o u s e , c a n a r y , p a r a k e e t , toe, f i n g e r , eye, e a r " . j e c t , i t can be s e e n , n a m e d two a n i m a l s , two b i r d s , and four p a r t s of the body.
This s u b -
These re-
stilts a r e c o n s i s t e n t with a m o d e l that d e s c r i b e s m e m o r y a s being o r g a n i z e d p r i m a r i l y into noun c a t e g o r i e s .
Only a f t e r a s u b j e c t has e n t e r e d a noun c a t e g o r y d o e s he find c l u s t e r s of
i t e m s that have qualities in c o m m o n .
In o t h e r w o r d s , a f t e r c o m p l e t i n g the f i r s t s t e p of the
r e t r i e v a l p r o c e s s {category e n t r y ) , the s u b j e c t may find a c l u s t e r of i n s t a n c e s to which a given adjective a p p l i e s . The second piece of evidence in s u p p o r t of the a s s u m p t i o n that the m e m o r y s t o r e is o r g a n i z e d p r i m a r i l y into noun c a t e g o r i e s g r e w out of a c o n v e r s a t i o n with Allan Collins at a 1971 m e e t i n g of the E a s t e r n V e r b a l I n v e s t i g a t o r ' s League.
We w e r e d i s c u s s i n g the finding
that r e a c t i o n t i m e is significantly f a s t e r when a noun c a t e g o r y c o m e s b e f o r e the a d j e c t i v e o r letter restrietor.
This finding is s o m e w h a t s u r p r i s i n g in view of the fact that E n g l i s h language
49 h a b i t s f a v o r an a d j e c t i v e - n o u n o r d e r .
We talk about a "yellow b i r d " but n e v e r a " b i r d yellow".
The q u e s t i o n a r i s e s a s to w h e t h e r t h e r e a r e s p e c i a l a d j e c t i v e s s u c h t h a t r e a c t i o n t i m e in a production t a s k would be f a s t e r if the a d j e c t i v e p r e c e d e d the noun.
P e r h a p s if we could find
a d j e c t i v e s t h a t w e r e m o r e c l o s e l y r e l a t e d to p a r t i c u l a r i n s t a n c e s than w e r e the s u p e r o r d i n a t e s of t h o s e i n s t a n c e s , a r e v e r s a l would o c c u r .
F o r e x a m p l e , if " s o u r " i s m o r e c l o s e l y
r e l a t e d to " l e m o n " than i s " f r u i t " , then p e r h a p s p r e s e n t i n g " s o u r " f i r s t and " f r u i t " s e c o n d would r e s u l t in a f a s t e r " l e m o n " . We u s e d the Kent and R o s a n o f f (1910) n o r m s a s one c o u r s e of a d j e c t i v e s . n o r m s a r e the a s s o c i a t i o n s of 1000 s u b j e c t s to 100 f a m i l i a r E n g l i s h w o r d s .
These
In the Kent and
R o s a n o f f n o r m s , the listing of r e s p o n s e s to " s o u r " i n d i c a t e s t h a t " l e m o n " o r " l e m o n s " i s the second most common response.
The only w o r d g i v e n m o r e often i s " s w e e t " .
In the B a t t i g
and Montague (1969) n o r m s , h o w e v e r , the r e s p o n s e s to " f r u i t " i n d i c a t e t h a t " l e m o n " is the ninth most common response. " l e m o n " than i s " f r u i t " .
T h u s , in s o m e s e n s e , " s o u r " i s m o r e c l o s e l y r e l a t e d to
Even in c a s e s s u c h a s t h e s e , h o w e v e r , r e a c t i o n t i m e is s h o r t e r
when the noun i s p r e s e n t e d f i r s t .
The r e t r i e v a l model p r o p o s e d a c c o u n t s for t h i s fact by
a s s u m i n g t h a t m e m o r y i s o r g a n i z e d p r i m a r i l y into a c o m p l e x h i e r a r c h y of noun c a t e g o r i e s and t h a t the f i r s t step of the r e t r i e v a l p r o c e s s is to e n t e r one of t h e s e noun c a t e g o r i e s . 2. E a c h c a t e g o r y in the h i e r a r c h i c a l o r g a n i z a t i o n c a n be e n t e r e d d i r e c t l y . A portion of a h y p o t h e t i c a l m e m o r y s t r u c t u r e m i g h t c o n s i s t s of " l i v i n g t h i n g " with " a n i m a l " and " v e g e t a b l e " a s s u b s e t s of it, the s u p e r s e t " b i r d " and " s n a k e " a s s u b s e t s of " a n i m a l " , a n d " c a n a r y " and " r o b i n " a s i n s t a n c e s of the s u b s e t " b i r d " .
Suppose f o r a m o m e n t
t h a t a c a t e g o r y n a m e c a n n o t be l o c a t e d d i r e c t l y , but m u s t be found by b e g i n n i n g a t the top of the s e m a n t i c h i e r a r c h y and s e a r c h i n g don w a r d t h r o u g h the h i e r a r c h y .
I t follows t h a t the
t i m e r e q u i r e d to r e t r i e v e an i n s t a n c e of any c a t e g o r y should r e f l e c t the n u m b e r of s u p e r s e t s t h r o u g h which the s u b j e c t m u s t move b e f o r e finding the a p p r o p r i a t e c a t e g o r y .
T h u s , we
would e x p e c t the t i m e t a k e n to r e t r i e v e an i n s t a n c e of the c a t e g o r y " b i r d " to be g r e a t e r t h a n the t i m e to r e t r i e v e an a n i m a l , b e c a u s e the s u b j e c t m u s t move t h r o u g h at l e a s t one e x t r a s u p e r s e t to l o c a t e the c a t e g o r y " b i r d " .
Note that we have o p e r a t i o n a l l y d e s c r i b e d the h i e r -
a r a c h y by a s s u m i n g that, f o r a s n p e r o r d i n a t e - s u b o r d i n a t e p a i r of c a t e g o r i e s s u c h a s a n i m a l b i r d , the s u p e r o r d i n a t e (which i n c l u d e s e v e r y t h i n g t h a t belongs in the s u b o r d i n a t e category) will b e h i g h e r on the h i e r a r c h y .
F o r any s u p e r o r d i n a t e - s u b o r d i n a t e p a i r , t h e n , r e t r i e v i n g
an i n s t a n c e of the s u p e r o r d i n a t e c a t e g o r y should take l e s s t i m e , a c c o r d i n g the view t h a t e a c h c a t e g o r y c a n n o t be locate d i r e c t l y . A l t e r n a t i v e l y , suppose that when a s e a r c h p r o c e s s b e g i n s with a p a r t i c u l a r c a t e g o r y n a m e , t h a t n a m e is d i r e c t l y a c c e s s i b l e by s o m e c e n t r a l p r o c e s s o r , i n s t e a d of b e i n g a c c e s s i b l e only via a s e a r c h along a h i e r a r c h i c a l path.
In t h i s c a s e , t h e r e i s no r e a s o n to e x p e c t
50 any d i f f e r e n c e in the t i m e t a k e n to r e t r i e v e a n i n s t a n c e of the c a t e g o r y " b i r d " and the t i m e t a k e n to r e t r i e v e an a n i m a l . We have d e s i g n e d au e x p e r i m e n t to d e t e r m i n e w h e t h e r the t i m e r e q u i r e d to n a m e an i n s t a n c e of a c a t e g o r y is d e p e n d e n t upon the position of the c a t e g o r y in a s e m a n t i c h i e r a r c h y . In d e s i g n i n g the e x p e r i m e n t , a c e n t r a l p r o b l e m was to e n s u r e that c a t e g o r i e s u s e d could be c l a s s i f i e d a c c u r a t e l y a c c o r d i n g to t h e i r r e l a t i v e position in the h i e r a r c h y .
This was accomp-
l i s h e d by u s i n g p a i r s of n e s t e d c a t e g o r i e s in w h i c h the s u p e r o r d i n a t e c a t e g o r y included by definition e v e r y t h i n g t h a t belonged in the s u b o r d i n a t e c a t e g o r y .
F o r e x a m p l e , the c a t e g o r y
" b e v e r a g e s " c o n t a i n s all a l c o h o l i c b e v e r a g e s , " m u s i c a l i n s t r u m e n t s " c o n t a i n s all s t r i n g e d i n s t r u m e n t s , and " a n i m a l s " c o n t a i n s a l l b i r d s .
In t h i s e x p e r i m e n t the s u b j e c t was p r e s e n t e d
w i t h both a s u p e r s e t n a m e (e. g . , b e v e r a g e ) , and r e q u i r e d to give a m e m b e r of t h a t c a t e g o r y , and a s u b s e t n a m e (e. g . , alcoholic b e v e r a g e ) and r e q u i r e d to give a m e m b e r of it.
This
allowed a d i r e c t c o m p a r i s o n of the t i m e t a k e n to produce m e m b e r s of s u p e r s e t s and s u b s e t s . F o r a d d i t i o n a l d e t a i l s , s e e Loftus, e t al. (1970). Subjects took a n a v e r a g e of 1.60 s e e . to p r o d u c e a m e m b e r of a s u p e r o r d i n a t e c a t e g o r y , and 1 . 4 9 s e c . to p r o d u c e a m e m b e r of a s u b o r d i n a t e c a t e g o r y .
This nonsignificant
d i f f e r e n c e is i n c o n s i s t e n t with the view t h a t a s e a r c h p r o c e s s b e g i n s at the top of the h i e r a r c h y and follows pathways downward t h r o u g h the n e t w o r k .
Such a view would p r e d i c t l o n g e r
t i m e s for a s u b s e t c a t e g o r y than for a s u p e r s e t c a t e g o r y s i n c e the s u b s e t is l o c a t e d c l o s e r to the b o t t o m of the h i e r a r c h y . opposite d i r e c t i o n .
The p r e s e n t r e s u l t a c t u a l l y i n d i c a t e s a s l i g h t d i f f e r e n c e in the
T h e s e data a r g u e in f a v o r of a m o d e l in which e a c h c a t e g o r y can be
located directly. 3,
The r e t r i e v a l p r o c e s s c o n s i s t s of two m a j o r s t e p s .
In t h i s s e c t i o n , we d i s c u s s two e x p e r i m e n t s t h a t w e r e o r i g i n a l l y d e s i g n e d to p r o v i d e m o r e i n f o r m a t i o n a s to why s o m e i t e m s a r e r e s p o n d e d to quickly while o t h e r s r e q u i r e a c o n siderably longer response time.
F o r e x a m p l e , s u b j e c t s take l e s s than 1.00 sec. to n a m e a
c o l o r , but they take n e a r l y twice a s long to n a m e a building (Loftus, et a l . , 1970).
If e a c h
c a t e g o r y i s e n t e r e d d i r e c t l y , the r e a s o n t h a t s o m e c a t e g o r i e s a r e l e s s a c c e s s i b l e than o t h e r s c a n n o t be due to the n e c e s s i t y of s e a r c h i n g t h r o u g h m o r e of the h i e r a r c h y to r e a c h t h o s e categories. One possible answer
to the question of why the speed of producing instances of some
categories is faster than others is suggested by the effect of word frequency on identification time.
It is well established that reaction times to identify high frequency words
than for low frequency words nition may
(Rubenstein,
et al., 1970).
are faster
This difference in ease of recog-
account for difference in the speed of producing instances of a category.
51
C l e a r l y , s t i m u l u s r e c o g n i t i o n i s a l o g i c a l l y n e c e s s a r y s t e p in p r o d u c t i o n of a r e s p o n s e from memory.
In o t h e r w o r d s , when a s u b j e c t i s a s k e d to n a m e a m e m b e r of a p a r t i c u l a r
c a t e g o r y , he m u s t " r e c o g n i z e " the s t i m u l u s w o r d (in t h i s c a s e , the c a t e g o r y n a m e ) b e f o r e h e c a n find and produce a m e m b e r of the c a t e g o r y .
T h u s , the p r o d u c t i o n of the r e s p o n s e " p e a c h "
to the s t i m u l u s " f r u i t " depends not only on the e x i s t e n c e in m e m o r y of a link b e t w e e n " p e a c h " and '*fruit", but a l s o upon the s e m a n t i c i d e n t i f i c a t i o n of " f r u i t " when it is p r e s e n t e d .
The
r e c o g n i t i o n o r s e m a n t i c i d e n t i f i c a t i o n of a c a t e g o r y n a m e m a y be e q u i v a l e n t o r a t l e a s t a p a r t of w h a t we have b e e n c a l l i n g " c a t e g o r y e n t r y " o r s t e p i of the r e t r i e v a l p r o c e s s . If the f i r s t s t e p in the r e t r i e v a l p r o c e s s is indeed r e c o g n i t i o n of the c a t e g o r y n a m e and if r e c o g n i t i o n s p e e d is a function of the f r e q u e n c y in the language of t h a t n a m e , then t h i s step of the p r o c e s s should c l e a r l y take l o n g e r for c a t e g o r i e s whose n a m e s a r e low f r e q u e n c y words.
O t h e r things b e i n g equal, then, the t o t a l t i m e to n a m e a m e m b e r of a c a t e g o r y with
a h i g h - f r e q u e n c y n a m e should be s h o r t e r than the t i m e to n a m e a m e m b e r of a l o w - f r e q u e n c y category. An e x p e r i m e n t was conducted in w h i c h s u b j e c t s w e r e p r e s e n t e d w i t h a s e r i e s of c a t e g o r y n a m e s , and had to r e s p o n d with the f i r s t word t h a t they could think of belonging to the c a t e g o r y .
We will r e f e r to t h i s as E x p e r i m e n t I; m o r e c o m p l e t e d e t a i l s c a n be found in
Loftus and F r e e d m a n (1972),
The m e a n t i m e to produce a m e m b e r of a c a t e g o r y w h o s e ~ a m e
is a high f r e q u e n c y word w a s i . 64 sec° and to p r o d u c e a m e m b e r of a c a t e g o r y w h o s e n a m e is a low f r e q u e n c y word was i . 89 s e c .
The d i f f e r e n c e was highly s i g n i f i c a n t , i n d i c a t i n g
c l e a r l y t h a t the f r e q u e n c y of the c a t e g o r y n a m e is a s s o c i a t e d w i t h s p e e d of p r o d u c i n g a n i n s t a n c e of t h a t c a t e g o r y .
We have s u g g e s t e d t h a t t h i s is due to the g r e a t e r t i m e r e q u i r e d
m e r e l y to identify the l o w e r f r e q u e n c y n a m e .
The p o s s i b i l i t y r e m a i n s , h o w e v e r , t h a t the
f r e q u e n c y of the n a m e a l s o a f f e c t s the s e c o n d p a r t of the r e t r i e v a l p r o c e s s - - t h a t p a r t w h i c h o c c u r s a f t e r the n a m e h a s b e e n identified.
In o t h e r w o r d s , once the s t i m u l u s h a s b e e n fully
" r e c o g n i z e d " , it m a y s t i l l take l o n g e r to n a m e a m e m b e r of a c a t e g o r y with a low f r e q u e n c y name.
A s e c o n d e x p e r i m e n t , w h i c h we r e f e r to a s E x p e r i m e n t H, w a s d e s i g n e d to t e s t t h i s
possibility. In the s e c o n d e x p e r i m e n t , the p r o b l e m w a s to d e s i g n a p r o c e d u r e t h a t would e l i m i n a t e d i f f e r e n c e s due to the f i r s t ( r e a d i n g and identification) p a r t of the r e t r i e v a l p r o c e s s but t h a t would e n a b l e us to m e a s u r e the s p e e d of the r e s t of the p r o c e s s (finding and producing the correct response).
The f i r s t r e q u i r e m e n t w a s m e t by giving the s u b j e c t the c a t e g o r y n a m e a
full 3 sec. b e f o r e he had to m a k e any r e s p o n s e .
Since the f i r s t step t a k e s c o n s i d e r a b l y l e s s
than 3 s e c . p r e s u m a b l y t h i s p r o v i d e d a m p l e t i m e for the s u b j e c t to r e a d and identify any c a t e g o r y n a m e , and thus e l i m i n a t e d any d i f f e r e n c e s in r e s p c u se t i m e due to d i f f e r e n c e in r e a d i n g speed.
It was e s s e n t i a l a t t h i s point to p r e v e n t the s u b j e c t f r o m i m m e d i a t e l y b e g i n -
52 ning the r e s t of the r e t r i e v a l p r o c e s s until we in e s s e n c e told him to s t a r t .
This w a s
a c c o m p l i s h e d by r e q u i r i n g a v a r i e t y of r e s p o n s e s to the c a t e g o r y n a m e so that the s u b j e c t did not know ahead of time which r e s p o n s e he would have to give and a c c o r d i n g l y would not know for which to begin s e a r c h i n g .
The idea was that u n d e r t h e s e c i r c u m s t a n c e s the s u b j e c t
would wait until he was a s k e d for a p a r t i c u l a r r e s p o n s e to begin the a p p r o p r i a t e r e t r i e v a l process.
The s u b j e c t was given a c a t e g o r y n a m e and w a s r e q u i r e d to a n s w e r s e v e r a l q u e s t i o n s about that c a t e g o r y .
The q u e s t i o n s ( i n s t r u c t i o n s ) w e r e : (a) " f i r s t l e t t e r " , in which the s u b -
j e c t r e s p o n d e d with the f i r s t l e t t e r of the c a t e g o r y n a m e ; (b) " l a s t l e t t e r " , in which c a s e the s u b j e c t r e s p o n d e d with the l a s t l e t t e r of the c a t e g o r y n a m e ; (c) " l e n g t h " , in w h i c h c a s e the s u b j e c t r e s p o n d e d with the n u m b e r of l e t t e r s in the c a t e g o r y ; and (d) " m e m b e r " , in w h i c h c a s e the s u b j e c t r e s p o n d e d with the f i r s t w o r d that he could think of that n a m e d an object that belonged in the c a t e g o r y .
We took p r e c a u t i o n s to e n s u r e that the s u b j e c t s would not always
e x p e c t to produce a m e m b e r of any given c a t e g o r y .
F o r d e t a i l s the r e a d e r is r e f e r r e d to
Loftus and F r e e d m a n (1972). The data of i n t e r e s t w e r e the r e s p o n s e t i m e s of c o r r e c t r e s p o n s e s to the " m e m b e r " question.
The m e a n t i m e to p r o d u c e a m e m b e r of a c a t e g o r y
w h o s e n a m e is high f r e q u e n c y
was 1.48 s e c . and to produce a m e m b e r of a c a t e g o r y w h o s e n a m e is low f r e q u e n c y was 1.67 s e c . , a highly s i g n i f i c a n t d i f f e r e n c e .
Thus e v e n a f t e r the s u b j e c t has had a m p l e t i m e to r e a d
and identify the c a t e g o r y than a high f r e q u e n c y c a t e g o r y . What do we make of t h e s e r e s u l t s ?
R e c a l l that in the o r i g i n a l e x p e r i m e n t , s u b j e c t s
w e r e p r e s e n t e d with a noun c a t e g o r y plus a r e s t r i c t i n g l e t t e r o r a d j e c t i v e , and p r o d u c e d an i n s t a n c e of the c a t e g o r y w h i c h s a t i s f i e d the r e s t r i c t i o n i m p o s e d . When the c a t e g o r y n a m e was p r e s e n t e d f i r s t , the m e a n r e a c t i o n t i m e was 1.87 s e c . ; with the c a t e g o r y n a m e p r e s e n t e d second, it was 2.12 s e c .
The d i f f e r e n c e o f . 25 s e c . was quite stable a c r o s s s o m e w h a t d i f -
f e r e n t conditions, and was taken as e v i d e n c e that the d u r a t i o n of c a t e g o r y e n t r y , or s t e p 1 of the r e t r i e v a l p r o c e s s , was a p p r o x i m a t e l y . 25 s e c . The two e x p e r i m e n t s just d i s c u s s e d a l s o provide an e s t i m a t e of the duration of s t e p 1. In E x p e r i m e n t 1 the m e a n r e a c t i o n t i m e was 1.89 for low f r e q u e n c y c a t e g o r i e s and t . 64 s e c . f o r high f r e q u e n c y c a t e g o r i e s . 1.48 s e c . , r e s p e c t i v e l y .
In E x p e r i m e n t II the c o r r e s p o n d i n g t i m e s w e r e 1.67 s e c . and
Since E x p e r i m e n t I included identification t i m e (i. e . , included c a t e -
g o r y entry') ~hile E x p e r i m e n t II did not, the d i f f e r e n c e s in r e a c t i o n t i m e between the two e x p e r i m e n t s provide s o m e indication of identification t i m e .
F o r low f r e q u e n c y c a t e g o r i e s the
d i f f e r e n c e is . 22 s e c . (1.89 - 1.67) and for high f r e q u e n c y c a t e g o r i e s it i s . t6 s e c . (1.64 1.48).
A s e x p e c t e d , identification o r e n t r y of a low f r e q u e n c y c a t e g o r y t a k e s l o n g e r than for
a high f r e q u e n c y c a t e g o r y .
The m e a n for all c a t e g o r i e s is . t9 s e c .
Thus, although the
53
p r e s e n t p r o c e d u r e s a r e quite d i f f e r e n t f r o m t h a t u s e d in the o r i g i n a l e x p e r i m e n t and involve r e l a t i v e l y s i m p l e ( u n r e s t r i c t e d ) r e s p o n s e s , the e s t i m a t e s of t h e d u r a t i o n of s t e p t o b t a i n e d i n the two p a p e r s is quite c o m p a r a b l e (. 25 sec. v s . .
19 s e e . ) .
Conclusion Our b a s i c conception of h u m a n m e m o r y i s t h a t of a s t o r e c o n s i s t i n g of a l a r g e n u m b e r of i n t e r c o n n e c t e d and c r o s s - r e f e r e n c e d a s s o c i a t i v e and c a t e g o r y n e t w o r k s .
According
to the model, m e m o r y is o r g a n i z e d into a complex h i e r a r c h y c o m p o s e d of c a t e g o r i e s (e. g . , a n i m a l s ) with s u b s e t s of e a c h (e. g . , b i r d s , dogs) and s u p e r s e t s (e. g . , living things).
Fur-
t h e r m o r e , when a s e a r c h b e g i n s with a p a r t i c u l a r c a t e g o r y n a m e , t h a t c a t e g o r y c a n be e n t e r e d d i r e c t l y , without f i r s t s e a r c h i n g t h r o u g h o t h e r c a t e g o r i e s .
T h u s , to find a b i r d , the
c a t e g o r y " b i r d s " is e n t e r e d ; to find an a n i m a l , the c a t e g o r y " a n i m a l s " i s e n t e r e d , and s o on. A g e n e r a l m o d e l of the r e t r i e v a l p r o c e s s a s s u m e s t h a t the e n t i r e p r o c e s s c o n s i s t s of at l e a s t two m a j o r s t e p s : (a) e n t e r i n g an a p p r o p r i a t e c a t e g o r y and (b) finding and p r o d u c i n g a n a p p r o p r i a t e m e m b e r of t h a t c a t e g o r y , We have p r e s e n t e d the r e s u l t s of s e v e r a l e x p e r i m e n t s to s u p p o r t v a r i o u s a s p e c t s of the model.
F o r e x a m p l e , s u p p o r t for the view t h a t the r e t r i e v a l p r o c e s s c o n s i s t s of at l e a s t
two m a j o r s t e p s c o m e s f r o m the r e d u c t i o n in t o t a l r e a c t i o n t i m e o b t a i n e d w h e n a s u b j e c t h a s a l r e a d y r e a d and " i d e n t i f i e d " the c a t e g o r y n a m e .
T h i s r e d u c t i o n r a n g e s f r o m . 19 to . 25,
depending on the t a s k , and it s e r v e s as an i n d i c a t i o n of the d u r a t i o n of the f i r s t s t e p o2 the process. F u r t h e r d e v e l o p m e n t and t e s t i n g of t h i s model is badly needed, model is m o r e like a l o o s e outline t h a n a s p e c i f i c t h e o r y ,
At t h i s point t h e
Much m o r e r e s e a r c h will be
n e e d e d b e f o r e we c a n a c t u a l l y t r a n s l a t e t h i s outline into a c o n c r e t e model.
Even more re-
s e a r c h will be n e c e s s a r y b e f o r e we c a n build a r e a l i s t i c m o d e l t h a t t r u l y d e s c r i b e s t h e c h a r a c t e r i s t i c s of h u m a n m e m o r y .
One fruitful d i r e c t i o n would involve a r e f i n e d a n a l y s i s of d a t a
f r o m individual s u b j e c t s r a t h e r t h a n a n a l y s i s of d a t a a v e r a g e d a c r o s s s u b j e c t s , b e c a u s e the h i e r a r c h i c a l o r g a n i z a t i o n s u r e l y v a r i e s a c c o r d i n g to individual u s a g e and e x p e r i e n c e f r o m s u b j e c t to s u b j e c t .
We plan to t u r n to s u c h individual a n a l y s e s in f u t u r e w o r k .
References Battig, W . F . and W . E . Montague (1969), " C a t e g o r y N o r m s for V e r b a l I t e m s in 56 C a t e g o r i e s : A R e p l i c a t i o n and E x t e n s i o n of t h e C o n n e c t i c u t C a t e g o r y N o r m s " , J o u r n a l of E x p e r i m e n t a l Psychology, 80__, 3. C o l l i n s , A . M . and M.R. Quillian (1969), '~Retrieval T i m e f r o m S e m a n t i c M e m o r y " , J o u r n a l of V e r b a l L e a r n i n g and V e r b a l B e h a v i o r , 8, 240-247. F r e e d m a n , J . L . and E. F, Loftus (1971), 'TRetrteval of W o r d s f r o m L o n g - T e r m M e m o r y " , J o u r n a l of V e r b a l L e a r n i n g and V e r b a l B e h a v i o r , 10, 107-115.
54
Green, B . R . , J r . , A.K. Wolf, C. Chomsky, and K. Laughery (1963), "Baseball: An Automarie Question Answerer", in E.A. Feigenbaum and J. Feldman (eds.), Computers and Thought, New York: McGraw-Hill Company. Kent, G.H. and A.J. Rosanoff (1910), "A Study of Association in Insanity", American Journal of Insanity, 67, 37-96. Landauer, T. K, and J.L. Freedman (1968), "Information Retrieval from Long-Term Memory: Category Size and Recognition Time", Journal of Verbal Learning and Verbal Behavior, 7, 291-295. Lindsay, R.K. (t963}, "Inferential Memory as the Basis of Machines which Understand Natural Language", in E.A. Feigenbaum and J. Feldman (eds.), Computers and Thought, New York: McGraw-Hill Company. Loftus, E . F . (1972), "Nouns, Adjectives and Semantic Memory", Journal of Experimental Psychology, 1972, 96, 213-215. Loftus, E . F . and J, L. Freedman (1972), "Effect of Category-Name Frequency on the Speed of Naming an Instance of the Category", Journal of Verbal Learning and Verbal Behavior, 11, 343-347. Loftus, E . F . , J.L. Freedman, and G.R. Loftus (1970), ' ~ e t r i e v a l of Words from Subordinate and Superordinate Categories in Semantic Hierarchies", Psyehonomie Science, 2__1, 235-236. Meyer, D.E. (1970), "On the Representation and Retrieval of Stored Semantic Information", Cognitive Psychology, 21, 242-300. Neisser, U. (t963), "Decision-Time Without Reaction Time", American Journal of Psychology, 7_6, 376-385. Pollack, I. (1963), "Speed of Classification of Words into Superordinate Categories", Journal of Verbal Learning and Verbal Behavior, 2, 159-165. Rabbitt, P. M.A. (1959), "Effects of Independent Variations in Stimulus and Response Probability", Nature, 183, 1212. Rubenstein, H., L. Garfield, and J.A. Millikan (1970), "Homogr phic Entries in the Internal Lexicon", Journal of Verbal Learning and Verbal Behavior, 9, 487-494. Shepard, R. (1972), "Some Illustrative Applications", Handout prepared for Multidimensional Scaling Workshop, University of Pennsylvania, June. Sternberg, S. (1966), "High-Speed Scanning in Human Memory;' Science, 153., 652-654. Woodworth, R.S. and N. Schlosberg (1954), Experimental Psychology, New York: Holt, Rinehart, and Winston. Acknowledgment The preparation of this paper was supported by the National Institutes of Health under Grant No. MH-20280.
IMPLICATION
AS AN ALTERNATIVE TO SET-INCLUSION AS THE SEMANTIC PRIMITIVE
Arnold Lewis Glass Stanford U n i v e r s i t y
The c o m p o n e n t i a l a n a l y s i s of language a s an a p p r o a c h to s e m a n t i c s h a s a long, r i c h h i s t o r y s p a n n i n g s e v e r a l d i s c i p l i n e s , including philosophy, l i n g u i s t i c s , and psychology. The b a s i c idea i s t h a t a w o r d c a n be b r o k e n down into a s e t of p r i m i t i v e p r o p e r t i e s ( o r i d e a s , f e a t u r e s , m a r k e r s , e t c . ) t h a t c a n then be said to c o n s t i t u t e its m e a n i n g .
For example, man
m a y be i n t u i t i v e l y a n a l y z e d a s the s e t of p r o p e r t i e s ( m a l e , human); and p e r h a p s the p r o p e r t y " h u m a n " m a y be f u r t h e r a n a l y z e d a s ( i n t e l l i g e n t , m a m m a l ) , etc.
T h e s e p r o p e r t i e s ( a s they
will h e n c e f o r t h be called) a r e u l t i m a t e l y r e p r e s e n t e d a s a t o m i c , u n i t a r y , and i n d e p e n d e n t particles.
T h e y m a y be thought of a s a c o l l e c t i o n of uniquely c o l o r e d m a r b l e s , w h e r e a n y
a s s o r t m e n t is p o s s i b l e . The a s s u m p t i o n of i n d e p e n d e n t p r o p e r t i e s i s c o m m o n to a wide v a r i e t y of s u b s t a n t i a l l y d i f f e r e n t t h e o r i e s w h i c h have a p p e a r e d in d i f f e r e n t d i s c i p l i n e s .
Katz and F o d o r (1963)
use m a r k e r s and d i s t i n g u i s h e r s to d i s a m b i g u a t e the m e a n i n g of a w o r d in the context of a particular sentence.
Schaffer and W a l l a c e (1970) and M e y e r (1970) have p r o p o s e d s e t -
i n c l u s i o n m o d e l s to explain the c o r r e l a t i o n b e t w e e n the t i m e i t t a k e s to v e r i f y w h e t h e r a w o r d is a m e m b e r of a c e r t a i n s e m a n t i c c a t e g o r y and the " s e m a n t i c d i s t a n c e " between t h a t w o r d and the c a t e g o r y .
F o r e x a m p l e , it t a k e s l o n g e r for a s u b j e c t to judge " t r u e " in r e -
s p o n s e to the s t a t e m e n t " a c a n a r y i s a n o r g a n i s m " than i t t a k e s to v e r i f y " a c a n a r y i s a bird".
The logical d e c i s i o n in t h e s e s e t - i n c l u s i o n m o d e l s r e s t s on the p r e s e n c e of c e r t a i n
p r o p e r t i e s in the i n t e r s e c t i o n of the s e t s of p r o p e r t i e s defining w o r d s . is s e a r c h e d for a p r o p e r t y in the i n t e r s e c t i o n . with the a m o u n t of o v e r l a p b e t w e e n s e t s .
The union of the s e t s
Hence, s p e e d of v e r i f i c a t i o n i s c o r r e l a t e d
As a final e x a m p l e of c o m p o n e n t i a l a p p r o a c h e s ,
p r o g r a m s t h a t a t t e m p t to s i m u l a t e h u m a n language c o m p r e h e n s i o n (e. g . , Shank, t972; W i n o g r a d , 1972) utilize a p r i o r i defined p r i m i t i v e s o r p r o p e r t i e s . The L i m i t a t i o n s of a C o m p o n e n t i a l A n a l y s i s While c o m p o n e n t i a l a n a l y s i s h a s b e e n widely applied, i t h a s l i m i t a t i o n s . q u e s t i o n is a s k e d , "Is a n e l m a t r e e ? " ,
If the
the a n s w e r is sought in a c o m p o n e n t i a l model by
s e a r c h i n g a s e t l a b e l l e d e l m for a p r o p e r t y l a b e l l e d " t r e e " , o r e q u i v a l e n t l y , following a n a r r o w f r o m a node l a b e l l e d e l m to a p r o p e r t y l a b e l l e d " t r e e " . sufficient.
T h i s is c e r t a i n l y l o g i c a l l y
But f r o m a d e v e l o p m e n t a l point of view, how did t h e s e n o d e s get f o r m e d and
t h e i r s e t - r e l a t i o n s get s p e c i f i e d in the f i r s t p l a c e ? this question,
T h e r e a r e s e v e r a l p o s s i b l e a n s w e r s to
56 One a n s w e r is that the c o m p o n e n t s stand for independent p e r c e p t u a l f e a t u r e s .
In
this v e r s i o n of the t h e o r y , a v e r y s m a l l s e t of p e r c e p t u a l f e a t u r e s c h a r a c t e r i z e the s e t defining the c o n c e p t " p l a n t " , a s u p e r s e t of it c h a r a c t e r i z e s " t r e e " , and a s u p e r s e t of " t r e e " c h a r a c t e r i z e s '~elm".
Since a s m a l l s e t of p e r c e p t u a l f e a t u r e s a r e contained in the s e t f o r
" p l a n t " , a l a r g e s e t of o b j e c t s wilt have all the p e r c e p t u a l f e a t u r e s of " p l a n t " .
Some of the
o b j e c t s having all of the p e r c e p t u a l f e a t u r e s of "plant" will have all of the p e r c e p t u a l f e a t u r e s of " t r e e " .
Hence, it can be deduced f r o m t h e s e definitions that the s e t of o b j e c t s c a l l e d
plants includes the s e t of o b j e c t s c a l l e d t r e e s ; o r , c o n v e r s e l y , all t r e e s a r e plants. logical i m p l i c a t i o n is defined in t e r m s of p e r c e p t u a l f e a t u r e s .
Thus,
However, while the p e r c e p -
tual r e l a t i o n s h i p that e x i s t s between t r e e s and plants is c e r t a i n l y an i n s t a n c e of the a p p l i c a tion of logical implication, t h e r e a r e p r o b l e m s with making it the definition of logical i m p l i cation, s i n c e t h e r e a r e o t h e r u s e s of logical i m p l i c a t i o n w h e r e p e r c e p t u a l s i m i l a r i t y does not a p p e a r r e l e v a n t .
It d o e s not a p p e a r that a whale i s c l a s s e d with a dog (as a mammaD
i n s t e a d of a s h a r k (as a fish) on the b a s i s of p e r c e p t u a l s i m i l a r i t y .
B e n c h e s a r e c l a s s e d with
c h a i r s on the b a s i s of t h e i r function a s s o m e t h i n g to sit on r a t h e r than with t a b l e s on the
basis of perceptual similarity. implies p" seem reason,
And the statements
to have no interpretation in terms
defining components A second answer
solely in terms
"All men
of perceptual similarity at all.
for such primitives have run from "object" and "property"
what the theory can't explain.
which express logical
at a person's disposal due to innate mechanisms.
Postulating a primitive,
of course,
to "agent",
Asserting the existence of a primitive is only an empirical
something which underlies
is to make
the sets of properties more
abstract so that they can
represent any relationship which can be stated in natural language. be stated which generate this representation This is the approach
No concepts like the ones mentioned
above will be assumed
sentation on which logical decisions can be made how this representation
in the narrowest
that will be taken in this paper. to be innate.
A formal repre-
will be defined; and it will briefly be shown
can be generated by experience
one point of view, it may
In this case a system of
and relate it back to sensory data
which reference the external world.
models
is well-
thought.
A third answer
From
and
does not explain it, but only describes
defined and it is seriously argued that the primitive represents
rules must
Candidates
"causality",
claim to the extent which the relation of the primitive to the rest of the system
all human
For this
of perceptual features appears inadequate.
is to define a set of "primitive" components
relationships and are somehow
"implication".
are mortal" and "p and q
and related to natural language.
be considered that this model is a refutation of componential
sense of the term,
tation go beyond set-theoretic axioms.
since the axioms
However,
the model
used to generate the represenmay
also be seen as an extension
57 of t r a d i t i o n a l c o m p o n e n t i a l work, taking up a q u e s t i o n w h i c h p a s t t h e o r i s t s w e r e n ' t a l w a y s c o n c e r n e d with: how the r e p r e s e n t a t i o n i s g e n e r a t e d . Logical I m p l i c a t i o n B e t w e e n P r o p e r t i e s . A l m o s t any c o n c e p t can be f u r t h e r d i s c r i m i n a t e d into an a r b i t r a r y n u m b e r of s u b concepts.
" C o l o r e d " can be d i s c r i m i n a t e d into " r e d " and " g r e e n " , " g r e e n " in t u r n m a y be
d i s c r i m i n a t e d into " k e l l y " and " f o r e s t " . " T e e t h " m a y be d i s c r i m i n a t e d into " m e c h a n i c a l t e e t h " a n d " a n i m a l teeth", " a n i m a l t e e t h " m a y be d i s c r i m i n a t e d into " s h a r k t e e t h " and " h u m a n t e e t h " , and " h u m a n t e e t h " m a y be d i s c r i m i n a t e d into " m o l a r s " and " i n c i s o r s " .
It
does not s e e m likely t h a t when a w o r d i s used, i m p l i c i t in the u n d e r l y i n g c o n c e p t i s all the s u b c o n e e p t s which c a n l a t e r be thought up. biguous, but it is vague. tion.
F r o m t h i s point of view, a concept i s not a m -
E a c h s u b c o n c e p t is r e l a t e d to i t s s u p e r o r d i n a t e by logical i m p l i c a -
Thus, " m o l a r " i m p l i e s " h u m a n t e e t h " , " h u m a n t e e t h " i m p l i e s " a n i m a l t e e t h " , and
"animal teeth" implies "teeth".
A c o n c e p t t r e e i s shown in F i g u r e 1.
An e n t i r e t r e e will be
r e f e r r e d to a s a c o n c e p t while e a c h node on the t r e e will be r e f e r r e d to a s a p r o p e r t y .
t
e
e
t
animal teeth
teeth
teeth
/\
/\ molar
~
meci~anical teeth
teeth
incisor
For
inner teeth
FIG. 1.
outer teeth
teeth
teeth
teeth
teeth
teeth
The R e p r e s e n t a t i o n of the Concept T r e e for " T e e t h " .
e x a m p l e , in F i g u r e 1, the concept " t e e t h " i n c l u d e s the e n t i r e t r e e .
The c o n c e p t " a n i m a l
t e e t h " i n c l u d e s the p r o p e r t i e s " a n i m a l t e e t h " , " s h a r k t e e t h " , " h u m a n t e e t h " , " i n n e r t e e t h " , " o u t e r t e e t h " , " m o l a r s " , and " i n c i s o r s " .
The c o n c e p t " h u m a n t e e t h " i n c l u d e s the p r o p e r -
t i e s " h u m a n t e e t h " , " m o l a r s " , and " i n c i s o r s " .
The u s e of the t e r m p r o p e r t y h e r e for a node
on a t r e e i s e n t i r e l y c o n s i s t e n t with the e a r l i e r u s e a s the c o m p o n e n t of s e m a n t i c a n a l y s i s , s i n c e they will e v e n t u a l l y be shown to be the s a m e thing. Two c o n c e p t t r e e s c a n i n t e r s e c t and have a p r o p e r t y in c o m m o n .
The c o n c e p t t r e e
for " m a l e " and the c o n c e p t t r e e for " h u m a n " i n t e r s e c t at the p r o p e r t y " m a n " .
Then, s i n c e
5B
both trees include the property, property
"animal
grammed
"man )' implies both )'male ') and )'human".
teeth" implies both ~'animal" and "teeth".
These
Similarly,
examples
the
are dia-
in Figure 2.
male
anima~
teeth
man FIG. 2.
The R e p r e s e n t a t i o n of Intersection Concept T r e e s .
A single p r o p e r t y may have m o r e than one concept t r e e a s s o c i a t e d with it. example,
the properties
"animal",
"bird",
and "fish" all imply the property "animal".
and "fish" are part of one concept tree while "animal"
of another concept tree. ordinates within the same fish.
"pet", "bird",
This can be seen from
For But
and "pet" are part
the fact that two properties which are sub-
concept tree are mutually exclusive,
But a bird can be a pet, thus their underlying
concepts
i.e., a bird can not be a must
be distinct.
Logical Decision Making in the F o r m a l Model. It witl be apparent that many of the definitions of r e l a t i o n s a r e r e a l l y t h e o r e m s which can be proved f r o m a few simple axioms which define the model.
But it is not the
purpose of this paper to p r e s e n t mathematical proofs, so all r e l a t i o n s will simply be defined without attempting to distinguish which ones can be derived. Axioms. 1.
T h e r e exists a set P of p r o p e r t i e s , such that P = (Pl .
. . . .
pn ), where Pi is a
property. 2.
A concept_ t r e e Tx is a lattice like the one shown in F i g u r e 1, w h e r e if pj is
below Pi (Pj c Tx(Pi)) , then Pi~ implies Pi" F o r example, in F i g u r e 1, since "human teeth" is below " a n i m a l teeth", "human teeth" i m p l i e s " a n i m a l teeth". 3.
A word is defined as a nonempty set of p r o p e r t i e s : for all i, w. c p and w. ~ 0. 1
1
A w o r d must be defined as a set of p r o p e r t i e s to deal with the obvious but important fact that many words a r e ambiguous,
F o r example, a club is both a weapon and a social organization.
If a one-to-one mapping existed between words and p r o p e r t i e s then club would imply " c o n c r e t e " and not imply " c o n c r e t e " , which is a contradiction. Definition of Relations. R1.
Definition of "All" r e l a t i o n s :
"All w. a r e w." if there exists a PieWi and a p e w .
such that pjCTx(P k) ---> P i e T y ( P k ).
59 T h i s says that if Pi i m p l i e s e v e r y p r o p e r t y that pj i m p l i e s , then the set which contains Pi i m p l i e s the set which contains pj.
F o r example, if a m a m m a l contains the p r o p e r t y " m a m -
malian" which i m p l i e s the p r o p e r t i e s " f u r r y " and " a n i m a l " and dog contains the p r o p e r t y " d o g - l i k e " which i m p l i e s the p r o p e r t i e s " f u r r y " and " a n i m a l " and " b a r k s " , then a dog is a mammal.
A special c a s e of this definition is when Pi implies pj.
It follows f r o m the t r a n -
sitive nature of implication that Pi then i m p l i e s e v e r y p r o p e r t y which pj does.
For exam-
ple, "A c a n a r y is a bird" is true because the p r o p e r t y " c a n n a r y - l i k e " i m p l i e s the p r o p e r t y "bird-like". 1%2. Definition of "No" r e l a t i o n : 'rNo w.~ i s a Wm" if t h e r e e x i s t s PiCWi and PmCWm and Pk such that Pi c T(p')x• and pm E Tx(Pk ) and pi~Ty(Pm)_ and pm~Tz(Pi)._
This says that if w.1 and Wm both have p r o p -
e r t i e s which a r e subordinates on the s a m e concept t r e e , they a r e mutually exclusive.
For
example, a canary can not be a robin, since a bird can not be both a robin and a canary.
A
bird can not be a fish, since an animal can not be both a fish and a bird. R3. ~Definition of "Some" r e l a t i o n : " S o m e w.j a r e w."~ if there exists a PiCWi and pje wj such that p j e Tx(Pk)_ ---~ Pi¢ Tv(Pk}" This is the i n v e r s e of the " a l l " r e l a t i o n (R1). canary.
If a c a n a r y is a bird, then s o m e bird is a
If all pets a r e a n i m a l s , then s o m e animal is a pet. R4.
Extension of " S o m e " r e l a t i o n :
"Some w.1 is a wa" if there exists PiCWi and pnCWn and Pk such that pi e Tx(Pk)__ and Pn c Ty(Pk).
This can best be explained by example.
If s o m e animal is a pet by R3 and a
bird is an animal by R1, then since bird i m p l i e s animal and some animal is a pet, s o m e bird might b e a pet.
The c r u c i a l distinction here is that the p r o p e r t i e s of bird and pet a r e
r e l a t e d to the p r o p e r t y of animal under different concept t r e e s (T x and Ty). part of the s a m e concept t r e e , then R2 would apply. relations.
If they w e r e
Of c o u r s e , R3 and R4 define different
R3 defines the existence of something, while R4 only defines the possibility that
something might exist.
But this distinction is absent in n o r m a l English so it has been
glossed o v e r here. R5.
Definition of "Not a l l " relation:
"Not all w.] a r e Wn" if t h e r e exists PiCWi and PLOW.,J such that "All w i a r e w."j and PnC w n such that "No w. a r e w ". 1
n
This conjunction of R1 and R2 says that if a c a n a r y is a kind
of bird and a c a n a r y can not be a robin, then not all birds a r e robins.
Or, if t h e r e e x i s t s a
bird which is not a pet, then not all a n i m a l s a r e pets. The Acquisition of Semantic P r o p e r t i e s . The r e p r e s e n t a t i o n just d e s c r i b e d is g e n e r a t e d by adding new p r o p e r t i e s to concept trees.
Clearly, by the definitions of the r e l a t i o n s R1--R5, all logical r e l a t i o n s h i p s between
60 the property added and every other property in the network are automatically specified. Previously, the problem arose of how to infer abstract logical implication from the representation of perceptual features. But here logical implication is defined a priori, so the problem never arises. It only remains to be shown that this logical system can have perceptual referents. A simple network labelled with its perceptual referents is shown in Figure 3. Note that the property "elm" implies both the perceptual features which make an
percep/tuilm~tree ~
a ] ~ r e s t °f w°rld ~
fu:atq::e~o elm/ perceptual features unique to tree
perceptu i features common to tree and world
perceptual features unique to world
FIG. 3. Representation of How Logical Network Refers to Perceptual Data. elm unique and the property "tree", which implies the features which make Hence,
the property "elm" implies all the perceptual features of an "elm tree".
representation
could have been generated in the following manner.
which he wanted to distinguish from all other trees. 3) was attached to "tree".
Thus an elm is a tree.
Thus,
This
A person saw a tree
So a property (labelled "elm" in Figure Then an implication was drawn
"elm" to the perceptual features which an "elm" possesses trees.
a tree unique.
from
in addition to those unique to all
logical implication can be related to its normal
use in discriminating
percep-
tual classes. Disambiguation
Sentences.
The theory just described can be used to clear up problems where
a eomponential
represented
analysis has run into difficulty.
in other applications
For example,
Katz and Fodor (1963)
the properties defining the word bachelor by the tree structure shown in Figure 4.
In the tree in Figure i, the nouterminal various meanings
properties at the branching points of the tree, which
of bachelor have in common,
are called markers,
while the descriptions
in brackets at the bottom of the tree, which are unique to each meaning called distinguishers. is that it misses
The basic problem
certain generalizations.
certainly have something The academic titles bestowed
in common,
with ordering
meaning
The meanings
of unmated
upon them,
in a hierarchicha[
are fashion
seal and unmarried
but they are placed at the extreme
degree holder and young knight share in common
of bachelor,
man
ends of the tree.
the fact that they have had
but that fact is lacking from the tree.
But if the markers
were
61
bachelor noun
/
(male)
ho has the f i r s t or lowest academic degree
~
who has never married
young knight s e r v i n g under the standard of another knight
(mate) I young fur seal when without a mate during the breeding time
FIG. 4. : The Representation of the Meaning of B a c h e l o r by Katz and Fodor.
r e w r i t t e n and r e o r d e r e d to capture these g e n e r a l i z a t i o n s , then the ones p r e s e n t l y r e p r e s e n ted would be lost.
Though language meaning is tantalizingly h i e r a r c h i c a l , it is just c r o s s -
r e f e r e n t i a l enough to vitiate an attempt to capture meaning in s t r i c t h i e r a r c h i e s .
The r e p r e -
sentation of bachelor in this relationaUy defined theory is shown in F i g u r e 5. As can be seen
baciel°r3
bachelor
1\
m a l e seal
bachelor 2
/\/
unmated
unmarried
mate
knight
bachelor 4
\
man
noble title
a c a d e m i c title
human
FIG. 5. : The Representation of the Meaning of B a c h e l o r in the Relational Theory.
f r o m the figure, the relational theory is m o r e c r o s s - r e f e r e n t i a l in nature than the h i e r a r c h i c a l theory Katz and Fodor propose.
F u r t h e r , t h e r e is a natural way of dealing with the
disambiguation of a word in the context of a sentence.
Disambiguatiou can take place by
taking the i n t e r s e c t i o n s of the concepts a s s o c i a t e d with the p r o p e r t i e s defining the words in the sentence.
F o r example, for "the barking b a c h e l o r " , an i n t e r s e c t i o n is sought with
another concept t r e e which produces a logical relationship between a p r o p e r t y of the concept " b a r k " and a p r o p e r t y of bachelor.
T h e r e is c e r t a i n l y an i n t e r s e c t i o n between " b a r k " and
62 "seal" since a seal bark is a bark and a barking seal is a seal. "Baehelorl" is related to "seal" by R1 (a bachelor is a seal). Thus, by R4, "bachelor" is related to "barking seal" (a "bachelor" is a "seal" and some "seal" is a "barking seal"). Sincethe concept "bark" does not intersect any other concept tree of bachelor, the meaning is completely disambiguated. Thus, within the axioms of the theory, is a complete model of disambiguation. S.~atactic Form and Semantic Meaning The theory can also be applied to the study of the relationship between syntax and symantics. Sintactic rules can be defined as relations between properties, similar to R1-R5.
A s e n t e n c e would then be a c o l l e c t i o n of s e v e r a l s e t s of p r o p e r t i e s defining w o r d s with s o m e of t h e s e r e l a t i o n s defined between t h e i r p r o p e r t i e s .
The following i s a v e r y s i m p l e e x a m p l e
of this. The d i s t i n c t i o n between a n a l y t i c and s y n t h e t i c is well-known. is one which is t r u e by definition.
An a n a l y t i c s t a t e m e n t
F o r e x a m p l e , "All m e n a r e m a l e " is a n a l y t i c since p a r t
of the definition of being a man is b e i n g m a l e .
In c o n t r a s t , while the s t a t e m e n t "All m e n a r e
o v e r one inch tall" is n o n e t h e l e s s t r u e , it is a s y n t h e t i c s t a t e m e n t s i n c e b e i n g o v e r one inch tall i s not p a r t of the definition of m a n , and the s e n t e n c e can only be v e r i f i e d in r e f e r e n c e to the world.
T h i s s e m a n t i c f o r m of the a n a l y t i c - s y n t h e t i c d i s t i n c t i o n r e s t s not on the s y n t a c -
tic r e l a t i o n s h i p of the w o r d s in the s e n t e n c e but s i m p l y on t h e i r m e a n i n g s . second, s y n t a c t i c f o r m of the a n a l y t i c - s y n t h e t i c d i s t i n c t i o n in E n g l i s h . the s y n t a c t i c r u l e s which g o v e r n the c o n s t r u c t i o n of the s e n t e n c e .
But t h e r e is a
T h i s r e s t s solely on
To s e e this, note t h a t
s e n t e n c e (i) i s s o m e h o w odd, o r p e c u l i a r , o r u n g r a m m a t i c a l . (i) *A r o o m m a t e is a w r i t e r . But s e n t e n c e s (ii) and (iii) a r e p e r f e c t l y a c c e p t a b l e . (ii) My r o o m m a t e is a w r i t e r . (iii) A r o o m m a t e i s a p e r s o n . T h e r e s e e m s to be s o m e t h i n g a b o u t the f o r m of (i) t h a t c o m p e l s i t s i n t e r p r e t a t i o n a s a u n i v e r s a l a s s e r t i o n , i n d e p e n d e n t l y of the m e a n i n g of the w o r d s c o n c a t e n a t e d . so, s o m e r u l e s of s e n t e n c e f o r m a t i o n m u s t be specified.
To s e e why t h i s is
The r u l e s a r e given in e x t r e m e l y
s i m p l i f i e d f o r m , with m o s t of the d e t a i l s left out s i n c e they a r e not r e l e v e n t to this e x a m p l e , S1.
The c o n s i s t e n c y r u l e .
Syntactic r u l e s will be u s e d to define new r e l a t i o n s between p r o p e r t i e s .
But if new i m p l i c a -
tions a r e to be d r a w n between p r o p e r t i e s , they c a n not c o n t r a d i c t the ones t h a t a l r e a d y e x i s t . It follows not only f r o m t h i s but any f o r m a l m o d e l , t h a t a s t r u c t u r e w h i c h l e a d s to a logical c o n t r a d i c t i o n i s ill f o r m e d .
T h i s r u l e m a y be s t a t e d : for all Pi and pj c o n t a i n e d in P, i f
pj i m p l i e s Pi o r pj and Pi i m p l y Pk' then Pi can not i m b l y pj.
63 S2.
"A(n)" operator.
A syntactic rule has two parts to it. One part expresses or which should be generated between two properties, rules specified relations between properties.
a relationship which should exist
the same
way the earlier relation
The other part tells what words
these proper-
ties are associated with by specifying the form of the word and where
in the sentence it may
be found.
Pi -->Pa where
For example,
the "A(Q)" operator defines an implication:
Pa is a
p r o p e r t y of the s e t defining __a(n)and Pi i s a p r o p e r t y of a w o r d w.1 following a(n) in the sentence. $3.
The " I s " o p e r a t o r .
The p r e s e n c e of i s in a s e n t e n c e defines an i m p l i c a t i o n : P i - - > P j w h e r e PiCWi which p r e c e d e s i s and p i - - > p a ,
and p . ¢ w. w h e r e w. defines a w o r d w h i c h follows is.
--
j
}
In t r a d i t i o n a l
J
grammatical terms,
w. will be the noun and a(n) will be the d e t e r m i n e r and the s t r u c t u r e 1 that c o n t a i n s P i - ' > P a will be the noun p h r a s e . T h u s , 83 s a y s that in a s e n t e n c e of this type, the noun p h r a s e i m p l i e s i t s p r e d i c a t e .
We have s e e n f r o m o u r t h r e e e x a m p l e s t h a t t h i s i s
true. By a p p l i c a t i o n of $2 to the p h r a s e a r o o m m a t e , a p p l i c a t i o n of $3 to the p h r a s e a r o o m m a t e i s a w r i t e r , F r o m the definition of R t ,
Proommate---> Pa i s defined.
By
P r o o m m a t ¢ - - > P w r i t e r is defined.
t h i s g i v e s "All r o o m m a t e s a r e w r i t e r s " .
But if the l i s t e n e r h a s
s t o r e d a p r o p e r t y i m p l y i n g P r o o m m a t e and s h a r i n g a c o n c e p t with P w r i t e r so that by R5 the l i s t e n e r knows "Not all r o o m m a t e s a r e w r i t e r s " , then a c o n t r a d i c t i o n e x i s t s and by S1 the s e n t e n c e m u s t be ill f o r m e d . S e m a n t i c T h e o r y and P s y c h o l o g i c a l R e a l i t y The b a s i c e x p e r i m e n t a l p a r a d i g m to w h i c h t h i s t h e o r y will be fitted w a s d e v e l o p e d s i m u l t a n e o u s l y by L a n d a u e r and F r e e d m a n (1968) and Collins and Quillian (1969).
A subject
i s f i r s t p r e s e n t e d with a s e m a n t i c c a t e g o r y , such a s b i r d , a n d then a s e c o n d w o r d w h i c h m a y o r m a y not be an i n s t a n c e of that c a t e g r o y .
The s u b j e c t p r e s s e s a button l a b e l l e d t r u e if the
i n s t a n c e w o r d i s a w o r d like c a n a r y , and a button l a b e l l e d f a l s e if the w o r d is a w o r d like rock.
The d e p e n d e n t m e a s u r e is how long it t a k e s the s u b j e c t to r e s p o n d f r o m the o n s e t of
the i n s t a n c e word.
A s e c o n d way t h i s t a s k i s done i s to p r e s e n t the e n t i r e a s s e r t i o n a s a
s e n t e n c e : "A c a n a r y i s a b i r d " and m e a s u r e r e a c t i o n t i m e f r o m the o n s e t of the a s s e r t i o n . T h e r e have been no i n t e r e s t i n g d i f f e r e n c e s r e p o r t e d f r o m v a r y i n g the f o r m of the t a s k (Kintach, e t al, 1970) but the second v e r s i o n of the t a s k i s s l i g h t l y m o r e g e n e r a l . e x a m p l e , one can a l s o p r e s e n t a s e n t e n c e w h i c h r e a d s "A c a n a r y h a s w i n g s " .
For
However,
though s e n t e n c e s of t h i s type a r e often r e p o r t e d to follow the s a m e g e n e r a l p a t t e r n of e x p e r i m e n t a l r e s u l t s as t h o s e w h e r e the a s s e r t i o n s a r e r e s t r i c t e d to the s e t - i n c l u s i o n type (All A
64 a r e B; Some A a r e B), t h e r e a r e many c o n t r a d i c t i o n s in the data r e p o r t e d for them.
Since
it is not known why the data f o r t h e s e s e n t e n c e s i s u n c l e a r , they will not be d i s c u s s e d . The initial r e s u l t s found d e m o n s t r a t e d that t h e r e w e r e r e l i a b l e d i f f e r e n c e s in the time it took to r e s p o n d t r u e and f a l s e to d i f f e r e n t a s s e r t i o n s .
In g e n e r a l , it w a s found that a s e n -
fence of the type (1) "A c a n a r y is a b i r d " is r e s p o n d e d to f a s t e r than a s e n t e n c e of the type (2) "A c a n a r y i s an o r g a n i s m " .
A t f i r s t this w a s thought to r e f l e c t s o m e t h i n g about the l o g i -
cal f o r m in which the m e a n i n g s of the w o r d s w e r e s t o r e d .
F o r e x a m p l e , it w a s though that
(t) was r e s p o n d e d to f a s t e r than (2) b e c a u s e in o r d e r to r e s p o n d to (2) the s u b j e c t had to make m o r e i n f e r e n c e s . c a n a r y ---> bird.
F o r (l), the s u b j e c t would have s t o r e d d i r e c t l y the i m p l i c a t i o n :
But for (2) the s u b j e c t would have to s e a r c h through a n e t w o r k of i m p l i c a -
t i o n s : canary---> b i r d - - > . . . --->o r g a n i s m .
However, s i n c e then, s u b s t a n t i a l e v i d e n c e has
a c c u m u l a t e d that r e a c t i o n t i m e is not c o r r e l a t e d with any m e a s u r e of a chain of logical i m p l i c a t i o n s , but with the intuitive s e m a n t i c d i s t a n c e between two w o r d s (Rips, Shobea and Smith, 1972). than a ~
F o r e x a m p l e , a p e r s o n is much m o r e likely to think of a dog a s an animal
as a mammal.
Do~ and animal a r e intuitively c l o s e .
If a s u b j e c t is a s k e d to
r e s p o n d to (1) "A dog is an a n i m a l " and (2) "A dog is a m a m m a l " , (1) is f a s t e r than (2) in s p i t e of the fact that the chain of logical i m p l i c a t i o n s g o e s d o g - m a m m a l - a n i m a l .
Semantic
d i s t a n c e s c a l e s can be c o n s t r u c t e d by s i m p l y asking s u b j e c t s to rank what w o r d s a r e i n t u itively c l o s e t o g e t h e r o r ( H e i d e r , 1972) by asking t h e m to r a t e good and bad i n s t a n c e s of a particular category.
R e a c t i o n t i m e c o r r e l a t e s c l o s e l y with intuitive d i s t a n c e on t h e s e s c a l e s
and the s c a l e s c o r r e l a t e c l o s e l y with e a c h o t h e r .
It s e e m s that w h e t h e r the question is how
c l o s e the i n s t a n c e is to the c a t e g o r y , o r how good an e x a m p l e it is of that c a t e g o r y , the s a m e q u e s t i o n is r e a l l y being asked.
When logical d i s t a n c e is held c o n s t a n t (A c a n a r y is a b i r d
vs. A chicken i s a bird) the b e t t e r i n s t a n c e of the c a t e g o r y (canary) i s r e s p o n d e d to f a s t e r . Semantic d i s t a n c e is c o r r e l a t e d with conjoint f r e q u e n c y and a s s o c i a t i v e propability.
Con-
joint f r e q u e n c y i s a w o r d count m e a s u r e of how f r e q u e n t l y two w o r d s a p p e a r t o g e t h e r in the language.
A s s o c i a t i v e p r o p a b i l i t y i s the f r e q u e n c y with which one w o r d e l i c i t s the o t h e r a s
a response.
It is r e a s o n a b l e to find that t h e s e m e a s u r e s a r e c o r r e l a t e d with intuitive d i s -
tance, since if w o r d s a r e found t o g e t h e r one would e x p e c t t h e m to be thought of t o g e t h e r , and if w o r d s a r e thought of t o g e t h e r , then one would e x p e c t e t h e m to be p r o d u c e d t o g e t h e r . However, the p r e c i s e m e c h a n i s m of the c a u s e - e f f e c t r e l a t i o n s h i p between acquisition, i n tuition, and production i s still not known.
But all such m e a s u r e s c o r r e l a t e well with r e -
a c t i o n - t i m e m e a s u r e s , to the d e t r i m e n t of logical d i s t a n c e m o d e l s . s u m m a r i z e d as follows.
The b a s i c data m a y be
F o r s t a t e m e n t s which a r e t r u e , the m o r e intuitively s i m i l a r the
c a t e g o r y and i n s t a n c e a r e , the f a s t e r the r e s p o n s e .
F o r s t a t e m e n t s w h i c h a r e f a l s e , the
65
more
intuitively similar the category and instance are, the slower the response.
example,
"A canary is a dog" is responded
In psychology,
to slower than "A canary is a flower".
there is a peculiar distinction between what might be called empirical
validity versus explanatory adequacy.
For example,
a way of handling the true/false data
just presented would be to postulate that two separate processes whether an instance is a member
of a particular category.
are involved in verifying
One of these processes
tive when there is no overlap between the properties of the words are effective where
there is overlap.
results.
said to have empirical validity, since it predicts experimental
to show,
but when they have been proposed,
in a noncircular manner,
which are labelled "set-relation".
"canary".
and both
Such a model would be
results.
Such models
For example, Then,
have
not the slightest attempt has been made
how the logical relationships between the words
inferred from the properties found.
for a path labelled "set-relation"
in the assertion,
is effec-
It is possible to contrive a multiple, serial, set-
search model that accounts for most of the experimental
been proposed,
For
paths between nodes may
are
be defined
to verify that a canary is a bird, one would look
running from a node labelled "bird" to a node labelled
Coding an English sentence as a labelled diagram
certainly does not constitute an
explanation of it. Since such a theory really does not constitute an explanation of how a logical decision is made,
it lacks explanatory adequacy.
The fact that the theory presented
here has attempted to deal with the question of explanatory adequacy the other semantic reaction time models
in psychology.
distinguishes it from
It will now be shown to be empiri-
cally valid as well. The approach
that will be taken here is that what has been called intuitive distance
or intuitive similarity is the causal factor underlying the differences in reaction-time responding to different assertions.
in
So it will first be shown how the theory can be made
account for semantic distance, and how that manipulation
to
also accounts for reaction-time
results. The person who has most thoroughly discussed semantic distance is Eleanor Heider (1972).
She points out that a word like bird often implies not only the properties of all birds,
but the properties which most birds share.
In other words,
feathered animal, which lays eggs and sings. of instances of birds. much
more
For example,
bird usually refers to a srrmll,
These can all be represented
as properties
the sentence "the birds were flying through the air" is
likely than the sentence "the birds were running across the plain".
ostrich, it is the second sentence which is true.
But for an
It appears that instances intuitively close
to the category bird share most of the properties it subtly connotes.
For example,
sparrow,
robin, wren,
and canary can all be described adequately as small feathered animals which
fly and sing.
A few additional properties,
such as "yellow" for canary,
may
define each
66 b i r d uniquely, but t h e s e p r o p e r t i e s a r e additive, not a l t e r n a t i v e , to the ones m e n t i o n e d . But a bad i n s t a n c e , such a s o s t r i c h , r e q u i r e s m a n y additional p r o p e r t i e s in i t s definition, s i n c e it is l a r g e , does not fly, a n d does not sing. fly, and i s r a i s e d f o r eggs and m e a t .
S i m i l a r l y , a c h i c k e n c a c k l e s , d o e s not
T h u s , the m e a s u r e of i n t u i t i v e s e m a n t i c d i s t a n c e will
b e how m a n y additional p r o p e r t i e s it t a k e s to define the i n s t a n c e o v e r the p r o p e r t y w h i c h defines the c a t e g o r y .
F o r e x a m p l e , if a c a n a r y is defined a s a "yellow" " b i r d " and an
o s t r i c h r e q u i r e s t h r e e additional p r o p e r t i e s ( l a r g e , nonfiying, nonsinging, bird) to c a n a r y ' s one, then c a n a r y is i n t u i t i v e l y c l o s e r to b i r d . Though a word was defined a s a s e t of p r o p e r t i e s , so f a r w i h a s been t r e a t e d a s a o n e - e l e m e n t set.
In o r d e r to explain s e m a n t i c d i s t a n c e , a word will now be t r e a t e d a s a
multi-element set containing all the properties it logically implies.
For example, instead
of mammal being defined by the set containing the property "mammal", mammal will be defined by the set consisting of "furry", "animal", etc., which "mammal" logically implies. These two descriptions can be related by defining the sum of the branches of a property B(Pi) as the set of properties such that pj c B(Pi) if Pi --'>Pj" The "furry" and "animal" are the branches of "mammal". The semantic distance between two words defined by B(w.) and i
B(wj) can now be defined by the following formula: Intuitive Distance (lw., w.) = b+ ac I
]
where a is the number of properties in Va and b is the number of properties in vb and c is the number of properties in v . C
pi wi , (pj wj) Pa~Va
if
Pb Vb
if
PeCVc if
pi Tx(Pa)A (pj ¢ Ty(Pa) ~,,pa ¢ wj )
(pbcB(pi)ApbCB(p?)V(pbcB(p?^pb¢(pi) PieTx(Pc)ApjeTx(Pe )
F o r a word p a i r like c a r d i n a l / b l u e jay, v a c o n s i s t s of all the p r o p e r t i e s which have a t r e e c o n t a i n i n g a p r o p e r t y of w with t h e m .
1
and a n o t h e r t r e e c o n t a i n i n g a p r o p e r t y of w. a s s o c i a t e d j
T h e s e a r e p r o p e r t i e s like " f e a t h e r e d " , " b e a k e d " , " t a l o n e d " , etc.
vb consists
of p r o p e r t i e s not in c o m m o n : such a s the N e t t h a t a c a r d i n a l i s a s p o r t ' s e m b l e m ,
v
conc s i s t s of p r o p e r t i e s w h i c h do d i s c r i m i n a t e the s e t s , s u c h a s the p r o p e r t y " c o l o r e d ' , s i n c e both
" r e d " (for e a r d i n a l ) and " b l u e " (for bluejay) a r e m e m b e r s of the s a m e c o n c e p t t r e e " c o l o r e d " and hence, by R2, m u t u a l l y e x c l u s i v e a l t e r n a t i v e s .
F o r a w o r d p a i r like c a r d i n a l / b i r d the
67
p r o p e r t y " r e d " would be in Vb, s i n c e b i r d does not i m p l y any c o l o r , and v It follows f r o m the definition of " A l l " t h a t w
would be empty. c m u s t be e m p t y w h e n e v e r " A l l " is t r u e . Also,
e if a p r o p e r t y in c a r d i n a l i m p l i e s a p r o p e r t y in b i r d , such a s " b i r d - l i k e " , then " b i r d - l i k e "
g o e s into v . The f o r m u l a for i n t u i t i v e d i s t a n c e : the r a t i o of the n u m b e r of p r o p e r t i e s the a w o r d s h a v e in c o m m o n (v a) to the n u m b e r of p r o p e r t i e s n o t held in c o m m o n (v b) plus the p r o p e r t i e s logically d i s t i n g u i s h i n g the w o r d s (v e) follows c l o s e l y the p r e v i o u s d i s c u s s i o n of what the u n d e r l y i n g b a s i s of intuitive d i s t a n c e could be, The r e l e v e n t p a r a m e t e r f o r p r e d i c t i n g r e a c t i o n t i m e is the p e r c e n t a g e of p r o p e r t i e s in the union of the s e t s B(w i) and B(wj) on which a d e c i s i o n about the r e l a t i o n s h i p between the two w o r d s can be b a s e d .
T h i s s t e m s f r o m the a s s u m p t i o n t h a t the union of the w o r d - s e t s
i s s e a r c h e d for a p r o p e r t y on which to m a k e a d e c i s i o n a b o u t t h e i r r e l a t i o n s h i p .
W h e r e the
p e r c e n t a g e of such p r o p e r t i e s i s l a r g e r , the p r o b a b i l i t y of finding one is g r e a t e r and the a v e r a g e t i m e to make a d e c i s i o n should be s h o r t e r . c a r d i n a l i s a b i r d " , Vc will be empty.
In the c a s e of a t r u e s t a t e m e n t like "A
The l a r g e r v a i s , the f a s t e r a d e c i s i o n will be
r e a c h e d ; but by definition, the l a r g e r v a i s the c l o s e r the two w o r d s will be i n t u i t i v e l y . Thus, for t r u e s t a t e m e n t s , the c l o s e r the w o r d s a r e i n t u i t i v e l y , the f a s t e r the r e a c t i o n - t i m e . F o r f a l s e s t a t e m e n t s , like "A c a r d i n a l i s a b l u e j a y " , the r e a c t i o n t i m e will depend on the p e r c e n t a g e of p r o p e r t i e s in v . But w h e r e w o r d s a r e i n t u i t i v e l y c l o s e , and v is l a r g e , v c a e will be p r o p o r t i o n a l l y s m a l l . T h u s , the c l o s e r the intuitive d i s t a n c e for f a l s e s t a t e m e n t s , the s l o w e r the r e a c t i o n t i m e . It would be nice to have s o m e m o r e e v i d e n c e for the e x i s t e n c e of v a, v b, and v cSuch e v i d e n c e e x i s t s .
M e y e r (1970) had s u b j e c t s r e s p e n d t r u e o r f a l s e to s u c h s t a t e m e n t s a s
(1) "All s t o n e s a r e r u b i e s " , (2) "All m o t h e r s a r e w r i t e r s " ,
(3) "All typhoons a r e w h e a t s "
and (4) "Some s t o n e s a r e r u b i e s " , (5) "Some m o t h e r s a r e w r i t e r s " and (6) "Some typhoons a r e wheat". (2) and (3).
T h e r e a r e two p o s s i b l e ways a d e c i s i o n to r e s p o n d f a l s e could be r e a c h e d in (1), One way is t h a t , f o r e x a m p l e , a Pb • B ( r u b i e s ) could be found so t h a t the r e -
q u i r e m e n t of the "All" r e l a t i o n t h a t all Pk i m p l i e d by P r u b i e s a r e i m p l i e d by Pstone could not be s a t i s f i e d .
T h e o t h e r way is t h a t , f o r e x a m p l e , a Pc f o r a n i n s t a n c e of m o t h e r and
w r i t e r could be found so t h a t the r e q u i r e m e n t s of the "Not a l l " r e l a t i o n sould be s a t i s f i e d . It does not m a t t e r w h i c h p r o p e r t y is found s i n c e p r e d i c t i o n s about the s p e e d of r e a c t i o n t i m e a r e b a s e d on the r a t i o b + c / a and a r e thus i n d e p e n d e n t of w h e t h e r the falsifying p r o p e r t y is a Pb o r Pc'
C l e a r l y , s t o n e / r u b y a r e i n t u i t i v e l y c l o s e r t h a n m o t h e r / w r i t e r , which a r e i n -
t u i t i v e l y c l o s e r t h a n t y p h o o n / w h e a t : so (t) should be s l o w e r t h a n (2) w h i c h should be s l o w e r t h a n (3).
The actual a v e r a g e r e a c t i o n t i m e s w e r e 1339 m i l l i s e c o n d s ( m s e c . ) f o r (1), 1263
m s e c . for (2) and 1154 m s e c . f o r (3), w h i c h a r e highly s i g n i f i c a n t d i f f e r e n c e s (p < . 001). T h e r e s u l t s a r e exactly as predicted.
68
For "Some ve,
typhoons are wheats"
the false decision still rests on a property in v b or
so the reaction time for it should not differ from
time of 1115 msee.
the "All 'T statement°
does not differ significantly (p . i0) from
stones are rubies" and (5) "Some
mothers
finding a pa o Clearly,
is greater than b'+c'/a',
than b+c/a.
Thus
if b+c/a
are writers"
1154 rose.
properties
then a'/b'+c'
the theory predicts that (4) will now be faster than (5).
in common
and for (5), 1108 msec., msec.
Thus, 1350
But for (4) "Some
a decision will be arrived at by
tively sound to expect that (5) will be faster than (2), since mother/human more
The measured
than not in common.
will be greater It is also intuiintuitively has
The reaction times are, for (4), 1017 msee.
which is significantly faster (p .01) than the time for (2), 1263
once again all predictions
are confirmed
(see Fig. 6).
ALL
1300 1250 1200
1150 1 i00 1050 i000
/
J SOME
950
stone/ruby
mother/writer
typhoon/wheat
FIG. 6. R e a c t i o n T i m e R e s u l t s f r o m M e y e r (1970). T h e r e is a l s o e v i d e n c e that the "All" and " S o m e " r e l a t i o n s a r e c o r r e c t l y defined. An "All" d e c i s i o n c a n depend on finding s e v e r a l p r o p e r t i e s in Pa' but for a " S o m e " d e c i s i o n only a single p r o p e r t y n e e d be found in Pa" s l o w e r than "Some c h a i r s a r e f u r n i t u r e " .
T h u s , "All c h a i r s a r e f u r n i t u r e " should be T h e i r t i m e s a r e 1182 m s e c . and 99o m s e e .
(p < . 001), r e s p e c t i v e l y . Conclusion T r a d i t i o n a l c o m p o n e n t i a l a n a l y s i s h a s b e e n applied to a wide r a n g e of s e m a n t i c problems,
In t h i s p a p e r logical i m p l i c a t i o n h a s b e e n p r o p o s e d a s a n a l t e r n a t i v e to s e t -
i n c l u s i o n a s the p r i m i t i v e out of which m e a n i n g is built,
t t h a s b e e n shown to apply to a n
equally wide r a n g e of p r o b l e m s a s t h e o r i e s utilizing s e t - i n c l u s i o n , but with the p o s s i b i l i t y of e v e n g r e a t e r s u c c e s s .
The t h e o r y i s potentially a p p l i c a b l e to the study of logic and syntax,
of language a c q u i s i t i o n , and the i n v e s t i g a t i o n of the o r g a n i z a t i o n of m e m o r y t h r o u g h r e a c t i o n time.
69 Acknowledgment Part of the work on this paper was supported by a grant from the Undergraduate Research Council of the State University of New York at Buffalo. I would like to thank John Anderson and Gordon Bower for their time and effort in reading and commenting on the many versions of this paper. References Bollinger, D. (1965), "The Atomization of Meaning", Language, 41__, 555-573. Collins, A.M. and M.R. QuiUtan (1969), "Retrieval Time from Semantic Memory", Journal of Verbal Learning and Verbal Behavior, , 24-248. Heider, E.R. (in press), "On the Internal Structure of Perceptual and Semantic Categories", in T.M. Moore (ed.) Cognitive Development and Acquisition of Language, New York, Academic P r e s s . Katz, J . J . and J. Fodor (1963), "The Structure of a Semantic Theory", Language, 39__,, 170-210. Kintsch, W., E. Crothers and L. Berman (1970), "The Effects of Some Semantic and Syntactic Properties of Simple Sentences upon the Latency of Judgements of Semantic Acceptability", in Studies in Mathematical Learning Theory and Psycholingnistics, The University of Colorado. Landauer, T.K. and J . L . Freedman (1968), "Information Retrieval from Long-Term Memory: Category Size and Recognition Time", Journal of Verbal Learning and Verbal Behavior, 7, 29t-295. Lyons, J. (1969), Introduction to Theoretical Linguistics, London, Cambridge University Press. Meyer, D.E. (1970), "On the Representation and Retrieval of Stored Semantic Information", Cognitive Psychology, 1_, 242-300. Rips, L . J . , E.J. Shoben and E . E . Smith (1972), "Semantic Distance and the Verification of Semantic Relations", unpublished manuscript. Schaeffer, B. and R. Wallace (1970), "The Comparison of Word Meanings", Journal of Experimental Psychology, _~6, 144-152. Schank, R.C. (1971), "Intention, Memory, and Computer Understanding", Stanford Artificial Project Memo AIM-140. Wilkins, A.T. (197t), "Conjoint Frequency, Categroy Size, and Categorization Time", Journal of Verbal Learning and Verbal Behavior, 10__,382-385. Winograd, T. (1972), "Understanding Natural Language", Cognitive Psychology, 3, 1-191.
STRUCTURED-STORAGE AFA Armen Gabrielian Seymour G i u s b u r g U n i v e r s i t y of Southern C a l i f o r n i a Abstract Among the v a r i o u s types of s t r u c t u r e s u s e d f o r s t o r i n g and r e p r e s e n t i n g data t h e r e a r e n u m e r o u s n o n l i n e a r types, e . g . ,
t r e e s (Samuel, 1959), linked s t r u c t u r e s (Newell and
Simon, 1956), and n - d i m e n s i o n a l a r r a y s ( H a r t m a n i s and S t e a r n s , 1965); n e v e r t h e l e s s , m o s t t h e o r e t i c a l m o d e l s of c o m p u t a t i o n a d h e r e to the tape m o d e l of s t o r a g e s t r u c t u r e , e . g . , T u r i u g a c c e p t o r s , pushdown a c c e p t o r s , and s t a c k a c e e p t o r s .
C u s t o m a r i l y , the n o n l i n e a r
types a r e l i n e a r i z e d so that the i n f o r m a t i o n s t o r e d in t h e m is r e p r e s e n t e d on a tape. e v e r , t h i s r e p r e s e n t a t i o n m a y be u n s u i t a b l e for v a r i o u s r e a s o n s .
How-
F o r e x a m p l e , in the c a s e
of l i n e a r coding of o r g a n i c c o m p o u n d s , d i f f i c u l t i e s a r i s e with unique r e p r e s e n t a b i l i t y and decodability.
P e r h a p s the m o s t s e r i o u s d i s a d v a n t a g e of l i u e a r i z a t i o n is that it c o m p l i c a t e s
the p e r f o r m a n c e of n a t u r a l l y and s t r u c t u r a l l y d e p e n d e n t o p e r a t i o n s on t h e data. In r e c e n t y e a r s m u c h of the r e s e a r c h on f o r m a l l a n g u a g e s h a s b e e n within the f r a m e w o r k of A F L t h e o r y ( G i u s b u r g and G r e i b a c h , 1969), i . e . , of f o r m a l l a n g u a g e s c a l l e d " A F L " ,
the study of c e r t a i n f a m i l i e s
S i m i l a r l y , m u c h of the r e s e a r c h on a c c e p t i n g d e v i c e s
h a s been m a d e within the f r a m e w o r k of AFA t h e o r y ( G i n s b u r g and G r e i b a e h , 1969), i . e . , the study of c e r t a i n f a m i l i e s of one-way, n o n d e t e r m i n i s t i c , single s t o r a g e tape a e c e p t o r s called "AFA".
It is known that, in a n a t u r a l m a n n e r , AFA give r i s e to A F L and, c o n v e r s e l y ,
A F L give r i s e to AFA.
In A F A theory, s e v e r a l a l t e r n a t i v e s to the single tape s t o r a g e s t r u c -
t u r e of a e c e p t o r s have been c o n s i d e r e d .
F o r e x a m p l e , n e s t e d m u l t i t a p e A F A a r e u s e d in
G r e i b a e h and G i n s b u r g (1972) to give a d e v i c e - c h a r a c t e r i z a t i o n of the s u b s t i t u t i o n of one full A F L into a n o t h e r .
A g e n e r a l i z a t i o n of this is u s e d in G r e i b a c h (1970) to give a
c h a r a c t e r i z a t i o n of n e s t e d i t e r a t e d s u b s t i t u t i o n - c l o s e d full A F L .
In e a c h of t h e s e c a s e s ,
l i n e a r i z a t i o u is e m p l o y e d to show t h a t c e r t a i n f a m i l i e s of a c c e p t o r s with g e n e r a l i z e d s t o r a g e s t r u c t u r e a r e equivalent, f r o m the point of view of l a n g u a g e s defined, to o r d i n a r y AFA, i . e . , the s i n g l e - t a p e kind.
The p u r p o s e of t h i s p a p e r is to i n t r o d u c e a c l a s s of f a m i l i e s of a c c e p -
t o t s c a l l e d " s t r u c t u r e d - s t o r a g e A F A " ( a b b r e v i a t e d "SS-AFA") in w h i c h the a u x i l i a r y s t o r age is of a v e r y g e n e r a l f o r m .
Indeed, the a u x i l i a r y s t o r a g e is g e n e r a l enough so that SS-
AFA include o r d i n a r y AFA a n d d i f f e r e n t kinds of m u l t i t a p e AFA, a s well a s v a r i o u s f a m i l i e s of a c c e p t o r s obtained by i m p o s i n g g e o m e t r i c a l o r g r a p h i c a l s t r u c t u r e s on the a u x i l i a r y
storage.
71 The p a p e r i s divided into t h r e e s e c t i o n s .
Section 1 i n t r o d u c e s SS-AFA.
Section 2
e s t a b l i s h e s the b a s i c fact that, f r o m the point of view of defining l a n g u a g e s , SS-AFA a r e e q u i v a l e n t to o r d i n a r y AFA.
( T h i s r e s u l t e l i m i n a t e s the need to e m p l o y the l i n e a r i z a t i o n
p r o c e s s e a c h t i m e a new SS-AFA is e n c o u n t e r e d . )
Section 2 a l s o e x a m i n e s the effect of
i m p o s i n g c e r t a i n f i n i t e n e s s conditions on the s t o r a g e s t r u c t u r e of SS-AFA.
Section 3 c o n -
s i d e r s a v a r i e t y of d i f f e r e n t e x a m p l e s of SS-AFA, e a c h of w h i c h e i t h e r is a l r e a d y in the l i t e r a t u r e o r p o s s e s s e s a s t o r a g e s t r u c t u r e of s o m e p r o m i n e n c e .
In view of the n u m e r o u s
s p e c i a l i n s t a n c e s of SS-AFA, it i s r e a s o n a b l e to e x p e c t t h a t the placing of a p p r o p r i a t e r e s t r i c t i o n s on SS-AFA will be a useful tool in o b t a i n i n g new c l a s s e s of a c c e p t o r s and c o r r e sponding families of l a n g u a g e s . References G i n s b u r g , S. a n d S.A. G r e i b a c h (1969), " A b s t r a c t F a m i l i e s of L a n g u a g e s " , in Studies in A b s t r a c t F a m i l i e s of L a n g u a g e s , M e m o i r s of the A m e r i c a n M a t h e m a t i c a l Society, No. 87, 1-32. G r e i b a c h , S.A. (1970), " F u l l A F L ' s a n d N e s t e d I t e r a t e d S u b s t i t u t i o n " , I n f o r m a t i o n and and Control, 16, 7-35. G r e i b a c h , S.A. and S. G i n s b u r g (1972),"Multitape A F A " , J. A . C . M . ,
to a p p e a r .
H a r t m a n i s , J. and R . E . S t e a r n s (1965), "On the Computational Complexity of A l g o r i t h m s " , T r a n s . A m e r . Math. Soc., 117, 205-306. Newell, A . P . and H.A. Simon (1956), " T h e Logic T h e o r y M a c h i n e : A Complex I n f o r m a t i o n P r o c e s s i n g S y s t e m " , IRE T r a n s . IT ~ 2, 61-79. Samuel, A . L . (1959), " S o m e Studies in M a c h i n e L e a r n i n g U s i n g the G a m e of C h e c k e r s " , IBM J. R e s e a r c h and D e v e l p p m e n t , 3_, 211-229.
PREDICATE
CALCULUS
FEATURE
GENERATION
David Rothenberg University College, Rutgers University Abstract A summary description (proof of theorems omitted): An adaptive pattern representation and recognition strategy for application to mechanized interpretation of (sampled) pictorial data (other applications are appropriate) is described. The system generates its own features which are formulae in a subset of the weak (in the sense that only quantification over finite sets is permitted) second order predicate calculus. The models of such formulae define the "objects" in a description of the data, which is hierarchical both with respect to features and extensions. The hierarchy is automatically constructed, thereby implementing changes in "problem representation". Relations between the syntax and semantics of formulae in the weak second order predicate calculus a r e derived (by extending the syntax of the calculus) and utilized. Minimal use is made of the finiteness of the input data by the methods employed. That is, in pictorial pattern recognitions, the adaptive feature generation (i. e . , "learning") algorithms are independent of the fineness of grain of the sampling of an input picture. Because this approach is used, many difficult problems of a purely mathematical nature acquire practical importance. Computation is reduced through the use of topological methods and the system is at present in a stage of development appropriate for programming and use in a variety of practical applications. 1. The Problem The problem definition used conforms well to the theoretical framework proposed by Banerji (1969, 9-16, 103-175): From a finite set of points, U ={Xl,X 2 . . . . . Xn}, two 1 disjoint collections, K and ~, of given subsets of U (called regions) are given, That is, if "R." denotes a region in ~ and 1
"R." a region in C, ]
={RI'R2 ..... Rn} = (%+i'%+2 ..... %+M} ViVj(R i ~ Rj) when R i e K and R.] ¢ ff . A formula in the weak second order predicate calculus, P(A), {called an ideal predicate or ideal feature) must be generated which is strongly satisfied by all regions and by no region R. in ~. J
R. in K, 1
We define a formula as being strongly satisfied by a region if it
i s satisfied by that region and is a proper subset of no other region which satisfies the 1
This, of course, is easily generalized to n distinct collections.
73 formula.
(Of c o u r s e , a f e a t u r e can have only one f r e e (set) v a r i a b l e ("A" in "P(A)"~ )
T r i v i a l l y , s u c h a f o r m u l a c a n be a d i s j u n c t i o n of the c o n j u n c t i o n of (the m e m b e r s h i p s of) all points of U (i. e . , " c o n s t a n t s " ) in e a c h s e t in the c o l l e c t i o n K.
However, the " s h o r t e s t "
p o s s i b l e f o r m u l a ( B a n e r j i , 1969, 103-104) is sought: F r o m a g i v e n s e t of u n a r y and n - a r y a t o m i c p r e d i c a t e s (all s e q u e n c e s of points in U which s a t i s f y e a c h of t h e s e is given), a f o r m u l a is c o n s t r u c t e d in which both the n u m b e r of c o n s t a n t s (specific points i n U) and v a r i a b l e s (both point and set) a r e m i n i m a l .
(This definition of " s h o r t n e s s " s o m e w h a t
r e s e m b l e s t h o s e developed by K o l m o g o r e v and d i s c u s s e d in L~fgren (1967) and Solmonoff (1964a, b). ) Such a s u b s t i t u t i o n of a f o r m u l a c o n t a i n i n g quantified v a r i a b l e s for a f o r m u l a c o n t a i n i n g a l a r g e n u m b e r of c o n s t a n t s (but no quantified v a r i a b l e s ) t e n d s to r e s u l t in the s a t i s f a c t i o n of the f o r m u l a by s e t s not in any of the c o l l e c t i o n s .
Hence, t h e f o r m u l a is
s a i d to g e n e r a l i z e ( B a n e r j i , 1969; 104, 157, 168, 1 7 5 ) t h e e x p r e s s i o n w h i c h i n v o l v e s only constants. A c t u a l l y t h e p r e d i c a t e c a l c u l u s syntax is h e r e e x t e n d e d to include a modified q u a n tifier,
Q(k)x, which m a y b e r e a d ,
"for k . 100 p e r c e n t of x " ( s i m i l a r l y to the r e a d i n g of
"Vx" a s "for all x " and "]Ix" as " t h e r e e x i s t s an x").
The d e g r e e of s a t i s f a c t i o n of a s e t ,
R C U, with r e s p e c t to a f e a t u r e P(A), when R does not s a t i s f y P(A), i s defined as -1 n n p l u s the l a r g e s t value of k in all r e p l a c e m e n t s of a s i n g l e u n i v e r s a l q u a n t i f i e r by ~(k) in P(A) s u c h that R
s a t i s f i e s P(A) a s a c o n s e q u e n c e of t h a t r e p l a c e m e n t . When R n n s a t i s f i e s P(A), the d e g r e e of s a t i s f a c t i o n is the l a r g e s t value of k in all r e p l a c e m e n t s of a single e x i s t e n t i a l q u a n t i f i e r by ~(k) s u c h t h a t R space,
s t i l l s a t i s f i e s P(A). A topology for the n U, is developed so t h a t U c a n be m a p p e d into a s p a c e , S, with f e w e r points, in
s u c h f a s h i o n that the r e s u l t i n g l o s s of r e l e v a n t i n f o r m a t i o n i s bounded.
T h a t is, e a c h
R C U h a s a " d e g r a d e d i m a g e " , R c S, s u c h t h a t if R s a t i s f i e s P(A), t h e n t h e d e g r e e of n n n s a t i s f a c t i o n of R with r e s p e c t to P(A) d i f f e r s f r o m t h a t of R by a bounded (and r e l a n n t i v e l y s m a l l ) amount. If P(A) is an ideal f e a t u r e and all R. c K s a t i s f y P(A) with a d e g r e e 1 of s a t i s f a c t i o n g r e a t e r t h a n z e r o and all R. c C fail to s a t i s f y P(A) (i. e . , have d e g r e e of J
s a t i s f a c t i o n with r e s p e c t to P(A) of l e s s t h a n z e r o ) , t h e n U m a y be m a p p e d into a s p a c e S w h o s e s i z e ( n u m b e r of e l e m e n t s ) i s a function of the d i f f e r e n c e of s u c h d e g r e e s of s a t i s faction.
No loss of r e c o g n i t i o n ability r e s u l t s while t h e c o n s e q u e n t e c o n o m y in c o m p u t a -
t i o n is evident. A l s o , a h i e r a r c h i c a l t h e o r y of l e v e l s is u t i l i z e d so t h a t the p r o b l e m of g e n e r a t i n g a n ideal f e a t u r e c a n be s o l v e d by s t a g e s , w h e r e i n at e a c h e x c e p t the final level, f e a t u r e s w h i c h r e d u c e t h e magnitude of the p r o b l e m a r e g e n e r a t e d .
H e r e the " p o i n t s " at e a c h s u c -
c e s s i v e l e v e l r e p r e s e n t the s e t s w h i c h s a t i s f y the f e a t u r e s at the p r e v i o u s level and t h e " a t o m i c p r e d i c a t e s " at e a c h s u c c e s s i v e l e v e l d e r i v e f r o m the f e a t u r e s at the p r e v i o u s
74 level.
The method a u t o m a t i c a l l y advances one l e v e l w h e n e v e r the n u m b e r of e l e m e n t s in the
regions to be partitioned (or the n u m b e r of such regions) is t h e r e b y reduced. at each l e v e l is iden~cal.
The p r o c e d u r e
A different " r e p r e s e n t a t i o n of the p r o b l e m " is thereby e x p r e s s e d
at each s u c c e s s i v e l e v e l (Nilssen, 1971; A m a r e l , 1962; Sherma~ and E r n s t , 1969). The m a t e r i a l in this paper is o r g a n i z e d as follows. d e s c r i b e d in the f i r s t portion.
The syntactic methods a r e
F e a t u r e s a r e defined as f o r m u l a s in the weak second o r d e r
p r e d i c a t e calculus with one f r e e set variable.
The atomic predicates used specify the p o s i -
tions of, d a r k n e s s of, and other m e a s u r e m e n t s at, and relations between, each of a set of data points called the data space. These atomic p r e d i c a t e s a r e called data independent if they do not v a r y for different inputs (e. g . , the r e l a t i v e positions of the data points at which a photograph is sampled by a flying spot scanner a r e data independent, i . e . , data independent predicates do not depend upon darkness, color, e t c . ) .
Atomic predicates a r e called data dependent if and only if they
v a r y for different inputs (e. g . , the d a r k n e s s of a given d a t a point on a p a r t i c u l a r photograph). Set m e m b e r s h i p is, of course, also a p r i m i t i v e of the s y s t e m .
Objects a r e regions (subsets)
of the data space which satisfy a feature and which a r e m a x i m a l (i. e . , they have no p r o p e r s u p e r s e t s which also satisfy the f e a t u r e - - t h i s is called strongly satisfying the feature).
A
d e s c r i p t i v e basis consists of all f e a t u r e s g e n e r a t e d by the learning algorithm at a p a r t i c u l a r stage in the operation of the s y s t e m .
A description consists of the d e s c r i p t i v e basis and a
list of the e l e m e n t s in e a c h of the objects, All f o r m u l a e a r e put in distributive f o r m .
In such f o r m u l a e the scope of each quan-
t i f i e r is as s m a l l as possible, negations o c c u r only b e f o r e atomic f o r m u l a e , all subformulae within the scope of a u n i v e r s a l quantifier a r e connected by disjunctions, and subformulae within the scope of an existential quantifier a r e connected by conjunctions.
Such a f o r m u l a
is conveniently r e p r e s e n t e d by a d i r e c t e d graph which is a rooted t r e e w h e r e i n the s u b t r e e below each node r e p r e s e n t s a subformula consisting of a quantifier (if such is present) and all subformulae within its scope. Each branch f r o m t h a t node r e p r e s e n t s a subformula within the scope of that quantifier, and unquantified atomic f o r m u l a e a r e the bottom nodes o f t h e t r e e . All f e a t u r e s have c o r r e s p o n d i n g c o m p u t e r p r o g r a m s which evaluate that f o r m u l a in a given space.
If a f o r m u l a is placed in prenex f o r m and adjacent like quantifiers exchanged
to g e n e r a t e additional prenex f o r m s , distinct distributive f o r m s may be obtained by o p e r a t ing on each such prenex form.
This set of distributive f o r m s is called the distributive set
of the f o r m u l a and each such distributive f o r m c o r r e s p o n d s to a c o m p u t e r p r o g r a m which can evaluate the f o r m u l a . A p r o c e d u r e called abstraction c o m p a r e s the distributive set of one feature with that of another to d e t e r m i n e common subformulae (i. e . , c o m m o n s u b p r o g r a m s ) .
Such sub-
75 formulae are extracted, a r e called derived predicates and are retained by the system.
In
effect, abstraction extracts identical subtrees from the graphs of distinct features. Another procedure called relation abstraction exposes the similarity of structure of pairs of t r e e s with different atomic formulae.
That is, formulae which differ only in that
they contain different subformulae which, nonetheless, have the same free variables in corresponding positions, a r e extracted and retained.
In these retained formulae, called
predicate forms, differing subformulae in the features originally compared correspond to a dummy predicate, wherein the free variables a r e shown, but the identity of the predicate is not.
Such dummy predicates will later be replaced by a derived predicate with the same
free variables. Derived predicates and predicate forms are compared to produce two hierarchies of formulae, the predicate hierarchy and the relational hierarchy, later used in the "learning algorithm". P a r t 12 discusses relations between the syntax of the formulae and certain p r o p e r ties of those regions (i. e . , subsets) of the data space which satisfy the formulae.
Such
regions are called models of the formula they satisfy and a r e compared both with those given regions we desire to be models of the formula, K = R1,R 2 .
. . . .
RN, and those given regions
we desire not to be models of the formula, ~ = RN+ 1. . . . . RN+M. Measures of the " s i m i larity" bet~veen the actual models of a formula and the given regions determine various m e a sures of success of the features generated,
The ease of satisfaction of a formula is defined
as the probability that a raadomty selected element from the product space over which its free variables range satisfies the formula.
The overlap of a pair of formulae is the ease of
satisfaction of the conjunction of two formulae.
Mutations map from one feature to another.
Relaxing mutations take a formula into another which is implied by it (ease of satisfaction is increased), r e s t r i c t i n g mutations do the r e v e r s e and neutral mutations connect formulae which are not related by implication. Restricting and relaxing mutations (with the exception of quantifier exchanges) correspond to the addition or elimination of subtrees from some node on the graph of the formula, whereas neutral mutations correspond to replacements of subtrees of the graph,
All are related to the abstraction procedure.
Each feature, P., is broken into a con]unction of two formulae, P! and pD wherein 1
1
P! contains only data independent atomic predicates.
1
I
We now examine the models of Pi in
1
two sets of spaces, one consisting of each of the given regions in K and ~ (some of which we wish to be models of P. and some which we do not) and the other consisting of each model of 1 pD (in the data space). Because models of a feature must strongly satisfy it, we do not, in 1
this case, have the law of the excluded middle (L e.
'
a region which is not a model of P.
1
need not be a model of "~P~), and we are able to derive results relating the ease of satisfac-
?6 tion of a given f e a t u r e with that of m~ ideal f e a t u r e which we a r e a t t e m p t i n g to g e n e r a t e (i. e . , one of which e a c h e l e m e n t of K i s a m o d e l and none in ~ is).
That i s , by e x a m i n i n g the
s i z e and n u m b e r of m o d e l s of a f e a t u r e in the above s p a c e s we a r e able to d e t e r m i n e what kind of mutation i s r e q u i r e d . One r e s u l t of the above i s : if a given r e g i o n R. is an e l e m e n t of K (those we w i s h to 1
be m o d e l s of P.), then (a) when R. d o e s not s a t i s f y P. a r e l a x i n g m u t a t i o n on P. is r e 1
I
1
1
quired, and (b) when
R. satisfies P. but not strongly, a restricting mutation is required. I I When R. is an element of ~ (those regions which must not be models of P.) and R. strongly 1 1 1 satisfies P., any mutation may suffice (although a restricting mutation is usually preferred). 1 These criteria are, of course, averaged for all elements of K and ~. Now
a more
powerful technique of mutation selection is exposed:
Suppose
R. c K and I
R. does not satisfy P.. In P. each of the nested universal quantifiers ("¥x") is replaced i 1 1 one at a t2me, starting with the outermost (highest on the tree) by the modified quantifier "Q(k)x" which computes
the fraction of substitutions of x (as constrained by any set member-
ship atomic predicates,
"c(x, A)" within the scope of that quantifier) for which the entire
formula holds true (such fractions may fier, "Yx",
is weakened
to mean
equal zero).
That is, the original universal quanti-
"for k. i00 percent of x" instead of "for all x".
This is
done independently for each universal quantifier, and such a fractional value is thereby computed for each nested subformula
within the scope of each universal quantifier (i. e., subtree
on the graph).
Clearly this value indicates the amount
the subformula
must be increased if the given set is to be a model of the resulting (modified)
feature.
The "degree of undersatisfaetion"
as the largest of the above computed completed,
by which the "ease of satisfaction" of
of a feature with respect to a given set is defined
fractional values minus one.
After this procedure
is
each of the existential quantifiers ("'~x") are, one at a time, replaced by "Q(k)x"
and again a fractional value is computed existential quantifier.
Now
for each nested subformula
within the scope of an
each (except the bottom) node on the tree representing
a formula
has such a fractional value associated with it. Now
suppose
procedures
R. ~K and satisfies P. but not strongly. Only the latter of the above 1 I is utilized: each existential quantifier is replaced by "Q(k)x" and a set of cor-
responding
values is determined.
"degree of oversatisfaction"
The smallest of these fractional values is defined as the
of the feature with respect to the given set.
It is then shown that modified calculations of the above fractional values can be used to estimate the ease of satisfaction of the various subformulae is evaluated.
Thus,
of a feature when that feature
since derived predicates are extracted from features, their ease of
satisfaction is automatically estimated and stored,
These
derived predicates are the sub-
trees which are added and eliminated from the graph of a feature when mutation formed.
Note that the replacement
is
per-
of universal quantifiers by a "Q(k)x" quantifier is equi-
77 valent to a relaxing mutation of the f o r m u l a consisting of the u n i v e r s a l quantifier and the subformulae within its scope.
Such an a l t e r a t i o n of the quantifier is then r e p a i r e d by p e r -
f o r m i n g an actual relaxing mutation on the subformula and r e s t o r i n g the u n i v e r s a l quantifier. This mutation usually involves the addition or elimination of a subformula (i. e . , d e r i v e d predicate) whose choice is d e t e r m i n e d by its e a s e of satisfaction, which should be as c o n s i s tent as possible with the fractional value of the modified quantifier.
Neutral mutations can
be s i m i l a r l y (but m o r e weakly) d e t e r m i n e d by the e s t i m a t e d " o v e r l a p " (defined previously) of the r e q u i r e d mutations. The above p r o c e d u r e s v,~ill not always uniquely s e l e c t a mutation.
When a feature is
g e n e r a t e d its m e a s u r e s of s u c c e s s a r e evaluated and a c o r r e s p o n d i n g number called a r e i n f o r c e m e n t , which is assigned to that feature together with all d e r i v e d p r e d i c a t e s and p r e d i cate f o r m s contained within it, is a p p r o p r i a t e l y altered.
Thus all f o r m u l a e and predicate
f o r m s have such corresponding indications of t h e i r " s u c c e s s " thus far.
When s e v e r a l m u t a -
tions of a feature a r e equally appropriate, a selection is made as follows: The feature is c o m p a r e d with each f o r m u l a in the list of predicate f o r m s .
That predicate f o r m with the
highest r e i n f o r c e m e n t which will r e s u l t f r o m a mutation of the r e q u i r e d type and amount (of change) is chosen. The dummy predicates are e i t h e r r e p l a c e d by subformulae of the f e a t u r e s being mutated, o r if not t h e r e i n present, such r e p l a c e m e n t s a r e chosen f r o m the list of d e r i v e d p r e d i c a t e s (with the r e q u i r e d numbers of f r e e point and set variables). predicates with the highest r e i n f o r c e m e n t s a r e chosen f i r s t .
The d e r i v e d
Hence the e n o r m o u s p o s s i b i l i -
t i e s of mutation choice a r e reduced first, by the s e m a n t i c c r i t e r i a r e s u l t i n g f r o m e x a m i n a tion of the models of the feature being a l t e r e d , and second, by the syntactic c r i t e r i a which f a v o r those predicate f o r m s and d e r i v e d p r e d i c a t e s which c h a r a c t e r i z e previously s u c c e s s f u l features. The next part of the paper deals with the technique for reducing the size of the p r o b lem by means of the use of a h i e r a r c h i c a l level theory (discussed e a r l i e r ) for r e p r e s e n t i n g the given data by the f o r m u l a e in the d e s c r i p t i v e basis.
That is, a technique is given for
automatically r e p r e s e n t i n g objects as collections of subobjects which a r e collections of s u b subobjects, e t c . , all such objects being defined by f e a t u r e s which a r e s i m i l a r l y h i e r a r c h ically organized. The following part of the paper t r e a t s the topology of the data space as r e l a t e d to techniques for computation reduction.
The m o s t i m p o r t a n t notion is a set of reduction m a p -
pings which s u c c e s s i v e l y r e d u c e the s i z e of the data space (e. g . , the fineness of g r a i n of sampling).
T h e s e mappings a r e r e l a t e d to i n v a r i a n c e s with r e s p e c t to t r a n s f o r m a t i o n s which
dilate o r contract the space.
The way in which r e s u l t i n g e r r o r s depend upon such a reduction
mapping t o g e t h e r with a p a r t i c u l a r feature a r e studied.
78 The final part d e s c r i b e s the o v e r a l l s y s t e m a t i c p r o c e d u r e for solving the problem. Various important open questions, whose solution would vastly affect the power of the s y s t e m a r e discussed.
F o r an i l l u s t r a t i v e example which s i m u l a t e s the procedures d e s c r i b e d ,
see Rothenberg (t973b). It is important to note thgtthe techniques developed p e r m i t a human t r a i n e r to simply and d i r e c t l y i n s e r t any knowledge o r intuitions he has about the p r o b l e m at any t i m e during o r p r i o r to computation.
Hence the task of problem solution begins at the limits of the
trainerTs knowledge and is l e s s difficult than would o t h e r w i s e be expected. The description below is g e a r e d for pictorial pattern interpretation.
It is e a s i l y seen,
however, that other pattern i n t e r p r e t a t i o n problems fit the s a m e f o r m a l i s m 2.
Syntax The s e n s o r y data or input consists of a digitized sampling of a (single) picture.
This
sampling is at a finite o r d e r e d a r r a y of points called the data space, u = { x l , x 2 . . . . . Xn} , which is the s a m e for all pictures.
F o r each such data point (which c o r r e s ponds to a r e c e p -
tor in the s e n s o r y field such as a photosensitive cell), t h e r e is a v e c t o r of k values called its value vector, each element of which is called a data value.
F o r i l l u s t r a t i v e p u r p o s e s we
will h e r e a s s u m e that the f i r s t two positions in the vector specify the row and column p o s i tions, r e s p e c t i v e l y , (in the o r d e r e d array) of the data points.
The r e m a i n d e r of the data
values specify different quantized qualities (i. e . , color or darkness) of that data point on a p a r t i c u l a r picture (these c o r r e s p o n d to a quantized e n e r g y density at a p a r t i c u l a r receptor). The a r r a y of all value v e c t o r s (at all data points) is called the value m a t r i x and is denoted IIv(k)ll where k indexes the positions in the value vector (the p a r t i c u l a r fineness of g r a i n of sampling is specified by the s i z e of the matrix).
F o r convenience, each lower c a s e l e t t e r ,
J~xWr, ,yWt etc. will be used to designate a data point (i. e. the f i r s t two positions of the value vector), the kth data value of point x will be denoted "v(k)(x) ~,, and upper c a s e l e t t e r s ,
"A",
'~Bt~, etc. will denote sets of data points. Data values d e t e r m i n e the truth values of a set of p r i m i t i v e s (also called atomic predicates1),
c~(k' i)(x), which a r e " t r u e " if v(k)(x) -- i and ~fatse ~ otherwise.
Those for
which k > 2 a r e called data dependent p r i m i t i v e s because they differ for different input photographs. Also given are a set of n place p r i m i t i v e s , ~(Xl,X 2 . . . . . x n) which specify ( g e o m e t r i c a d r e l a t i o n s between data points. t
A typical example would be " < ( x , y , z , w ) " which h e r e
Conventionally, the t e r m ~Iprimitive" r e f e r s to a specified '~property" or r e l a t i o n (e. g . , ~E~ for ~'set m e m b e r s h i p " ) , w h e r e a s "atomic p r e d i c a t e " r e f e r s to the p r i m i t i v e together with a specific s e t of arguments (e. g. ~ "~(x, A)"). The usage of t h e s e t e r m s h e r e ignores the distinction,
79 means that data point x is c l o s e r to y than z is to w.
This is usually d e r i v e d f r o m a given
m e t r i c on the data space (which is a function of v(1)(x) and v(2)(x) -- the row and column positions of point x).
Such atomic p r e d i c a t e s (as well as ~(l"i}(x)" and a(2'i)(x) ) a r e called
data independent because they a r e the s a m e for all input photographs. The following definitions c l a r i f y the syntax used. (1) Region:
Any subset of a data space.
(2) F e a t u r e : A formula, P(A), in the second o r d e r predicate calculus with equality such that: (a)
point v a r i a b l e s (x, y . . . . ) range o v e r data points, set v a r i a b l e s
(A, B . . . . )
range o v e r r e g i o n s , and all constants (c 1, c 2 . . . . ) are data points, or sets of data points (C 1, C 2 . . . . ). (b)
all v a r i a b l e s a r e bound except one set variable, A (hence only regions can satisfy a feature).
(c) p r i m i t i v e s include "e (x,A)" (conventionally " x e A " ) and "= (x,y}" (conventionally "x = y") 1, both of which a r e called e l e m e n t a r y p r i m i t i v e s . "card(A) < card(B)" may also be used. (Also included a r e the data independent and dapendent p r i m i t i v e s (atomic predicates) defined
above. ) Certain other conditions may be included which guard against formulae with t r i v i a l interpretations in the pictorial c a s e .
T h e s e will not be d i s c u s s e d here ( s e e Rothenberg,
1973b). (3) Object: A region which s a t i s f i e s a feature (i. e . , is defined as a r e a l i z a t i o n of that feature} and is not a proper subset of any other r e a l i z a t i o n of that s a m e feature (i. e . , it is "maximal}.
Note that while a r e g i o n may be an object because
strongly s a t i s f i e s one f e a t u r e , it may satisfy another f e a t u r e , but not strongly. When an object s a t i s f i e s a feature and is m a x i m a l with r e s p e c t to that p a r t i c u l a r feature it will be said to strongly satisfy that feature and is called a model of the feature (as distinct f r o m a realization}.
When a set of points satisfies a feature
but is not maximal for that feature, it will be said to satisfy that feature, but not strongly. (4) D e s c r i p t i v e Basis:
All f e a t u r e s which have been generated by the learning
algorith m (which g e n e r a t e s s u c h features) at a p a r t i c u l a r stage in the adaptive development of the analysis. 1
Note that although " = (x, y)" will be used as an atomic predicate, it can, in the c a s e of our illustrations, be defined as follows:
y) =- Vz (~,< i x , z , y, z )A "" < (Y, z ,x,z }) (see " = (x, y, z, w)"definition in Example 1).
80 (5) Description:
This contains all features in the d e s c r i p t i v e basis and lists all
objects strongly satisfying each of these f e a t u r e s and the e l e m e n t s of each such object.
The satisfaction of a feature by an object which does not strongly satisfy
that p a r t i c u l a r feature is also indicated. (6) Object Class: A set of objects all of which strongly satisfy the s a m e feature (also defined as the extension of the f o r m u l a for that feature). (7) Derived P r e d i c a t e s : A subformula of a feature which is r e c u r s i v e l y built up f r o m p r i m i t i v e s by the use of logical connectives and quantiflers. denoted "Di~Xl' . . ,x . n, . . A .1 , .
They a r e
, A m ) " where i indexes the predicate and its
argument includes all f r e e v a r i a b l e s . (8) Data Dependent P r e d i c a t e s : All f o r m u l a e (including features) which contain data dependent p r i m i t i v e s .
All other f o r m u l a e a r e called data independent.
Example 1 Let the primitives that the distance between and w.
be "~ (X, A)",
"= (x, y)",
and "< (x, y, z,w)",
which specifies
data point x and data point y is less than the distance between
Also let "D(x)" indicate that a data value (the third position in the value vector at
data point x) is equal to I. the former predicates
Here we assume
that these data values are either one or zero,
indicating a "dark" and the latter a "light" point. 1 are:
= (x,y,z,w) -N<(x,y,z,w)A
~" < ( z , w , x , y )
Ob~,y,~)- ~ =[x,y]^~--[x,~]^
Then
examples
of derived
("ix, y) equals (z,w)") 2
~ =[y,~]^ Vw(= [y~w] v <[x,y,x,w] v <[~,y,z,w]) (y is "between"
D ( A , B ) ~ V x ( e [ x A]'--> ~ [ x , B ] ) c
("A is contained in B")
D (A,B) ~ D ( A , B ) A D (B,A) s e c
("A is the s a m e as B")
o~)
("A is the null set")
-= ~ 3x (~ Ix, A])
x and z)
Dd~) = Vx (~ Ix, A]-->5 Ix])
("all points in A are 'dark' points"~
Dae, y) = ~ =Ix,Y] Aw (~Db[X, z,y])
("x is adjacent to y,,)3
1
z
Note that throughout this description square brackets " [ ] " and round b r a c k e t s " 0 " are used interchangeably and a r e alternated for c l a r i t y only. 2 Note that " = " when h e r e used as a four place predicate, " = (x, y, z , w) ", has a different meaning than when used as a two place p r i m i t i v e , " = (x,y)" (meaning that x and y a r e the s a m e point). 3 F o r use with a c o a r s e (sampling) grid it is suggested that Da(X, y) = ~'~ = (x, y) A Vz (= Ix, z] V ~
81
Note that all y which satisfy Db(X,y,z) when x and z are fixed will approximate a "straight line" even if the grain of the data space is c o a r s e .
As the g r a i n becomes infinitely
fine these points approximate a "true" straight line. F o r convenience 1 we will use the symbol "~/:(xt,x2,. .. ' x n )" to denote that all variables within brackets denote distinct objects: ~b(x,y,z) = ~-- = ( x , y ) A ~ = (x,z)A "~ = (y,z) . Hence Db ('~etweenness") in Example t may be r e w r i t t e n
D
y, z) :
y, z)^ vw(= [y, wl v < Cx, y,x, w]
w])
Note that a large n u m b e r of formulae are tautologous u n l e s s their free v a r i a b l e s a r e distinct. 3.
Semantics (Illustrations) The t r a n s l a t i o n from formulae to p r o g r a m s is simple.
Quantifiers become "do
loops" whose ranges are determined by the m e m b e r s h i p predicates, e.g. VX 1 [,~e{Xl,A)VP(Xl . . . . .
Xn}] = X lA( A P(Xl ' " .. ,Xn)
3x I [p(xI.....Xn)-] _- X l ~VU P(Xl'. . . . x n) 3Xl[e{xl, A) AP(x 1. . . . . Xn)"] K
V P(Xl, ,Xn) XleA '..
( ' U " is the u n i v e r s a l set - - i. e . , the data space) 2 Logical connectives are functions computed by subroutines, and atomic and derived p r e d i cates become subroutines in the main program e x p r e s s i n g the feature. Some exaznples of the r i c h n e s s of the s y s t e m when the primitives of Example 1
are used are: P I(A) =- VxVyVz (e[x, A~ h e [z,A~ A Db[X, y, z]-'> e~¢, A~) is data independent and satisfied by convex sets only (Db[X,y,z ] is defined as in Example 1). tf PI(A) is a feature, then, by definition, only maximal regions are objects. Let Dd(A)~- Vx (e [x, A~--> [) [x] )
(asbefore) .
Then P2(A) :- Dd(A) A PI(A) is a data dependent v e r s i o n of PI(A) 1 Although not described here, the language has been generalized to include "syntactic v a r i a b l e s " which range over formulae and p r i m i t i v e s which specify whether these v a r i a b l e s a r e bound o r free. P r o g r a m m i n g is thereby facilitated when the language is used as a c o m p i l e r (see P e s s i n and Rothenberg reference). 2 Quantified set variables range over regions. Methods of reducing sueh computation a r e d i s c u s s e d in Rothenberg (1973b)
82
defines a "starlike" region, point, x, of a region,
A,
and a derived predicate which is satisfied only by a boundary is given by
A feature satisfied only by a connected I region can be constructed
in two steps:
is a d e r i v e d predicate which a s s e r t s that A is " c l o s e d " in B under the r e l a t i o n of adjacency and P6(A) = V B ~ c [B,A] A"~D~ [B]A D2 [B,A~---> Ds [B,A~ ) is satisfied iff A is a connected region ("for all B, if B c A B = A " - - s e e Example 1).
and B ~ ~ and D 2 [B,A] then
Other equivalent f o r m u l a e exist.
Simple connectedness ('A has no holes") is e a s i l y e x p r e s s e d by stating that the boundaries of both A and its complement are connected sets.
Other illustrations are easily
constructed. An example of a data dependent feature is P7(A) which a c c o m m o d a t e s "noisy" dark areas:
4.
Distributive Normal F o r m s This is a f o r m in which all quantifiers a r e distributed as much as possible along
the t e r m s of a f o r m u l a so that a minimum number of t e r m s lie within the scope of each quantifier.
(It is the opposite of a prenex n o r m a l f o r m and is s i m i l a r to, but not identical to,
the " m i n i - s c o p e " of Wang (1960) and somewhat r e s e m b l e s the "distributive normal f o r m s " of Hintikka (1953).
This is accomplished on a f o r m u l a by r e v e r s i n g the techniques for
placing f o r m u l a e in prenex normal f o r m (Rothenberg, 1971).
A formula is in distributive
n o r m a l (henceforth abbreviated "distributive") f o r m iff: (1) Negations appear only before atomic subformulae. (2) All subformulae 2 within the scope of a u n i v e r s a l quantifier are connected by disjunctions. t
By "connected r e g i o n " is intuitively meant that for e a c h pair of s t ~ - r e g i o n s which partition the r e g i o n t h e r e exists a pair of adjacent points, one of which is contained in each subregion. 2 A subformula is either an atomic predicate or d e r i v e d predicate or either of these quantified a n d / o r negated.
83
(3) All s u b f o r m u l a e within the s c o p e of a n e x i s t e n t i a l q u a n t i f i e r a r e c o n n e c t e d by conjunctions. (4) All s u b f o r m u l a e within the s c o p e of a q u a n t i f i e r a r e e i t h e r p r e f i x e d by a q u a n t i tier or are atomic formulae or their negations.
(That is, they a r e not b r a c k e t -
e n c l o s e d conjunctions o r d i s j u n c t i o n s of o t h e r s u b f o r m u l a e . ) (5) All s u b f o r m u l a e within the s c o p e of a q u a n t i f i e r c o n t a i n t h e quantified v a r i a b l e as an argument. (6) If two o r m o r e like q u a n t i f i e r s a r e a d j a c e n t (e. g . ,
"VxVy" o r " ~ x ~ y ' ) , all
s u b f o r m u l a e within t h e i r scope m u s t c o n t a i n s u c h a d j a c e n t l y quantified v a r i a b l e s as arguments. E x a m p l e 2. Consider the formula
It h a s only one d i s t r i b u t i v e f o r m (in g e n e r a l , t h i s is not t h e c a s e - - t o be d i s c u s s e d } :
F o r c o n v e n i e n c e we h e r e i n s i s t t h a t , within the s c o p e of e a c h q u a n t i f i e r , Ca} s e t m e m b e r ship p r i m i t i v e s p r e c e d e all o t h e r s , (b) quantified t e r m s be p l a c e d l a s t , and (c) t h o s e t e r m s w h o s e a r g u m e n t s contain f e w e r v a r i a b l e s p r e c e d e t e r m s w h o s e a r g u m e n t s c o n t a i n a g r e a t e r n u m b e r of v a r i a b l e s . Note t h a t , b e c a u s e of (1) - (6) above, a d i s t r i b u t i v e f o r m u l a m a y be e a s i l y r e p r e s e n t e d by d i r e c t e d g r a p h w h i c h is a r o o t e d t r e e w h e r e i n t h e s u b t r e e below e a c h node r e p r e s e n t s a s u b f o r m u l a c o n s i s t i n g of a q u a n t i f i e r (if s u c h is present} and all s u b f o r m u l a e within 1 i t s scope. E a c h b r a n c h f r o m t h a t node r e p r e s e n t s a s u b f o r m u l a within the s c o p e of t h a t q u a n t i f i e r , and unquantified a t o m i c f o r m u l a e (and t h e i r negations} a r e the nodes with no successors.
The n u m b e r of d i s t i n c t bound v a r i a b l e s in the f o r m u l a c o r t e s ponds to the n u m -
b e r of e d g e s i n t h e l o n g e s t path down the t r e e .
C i r c u l a r n o d e s will be u s e d to c o n t a i n q u a n -
t i f i e r s of point v a r i a b l e s , two c o n c e n t r i c c i r c l e s will contain q u a n t i f i e r s of s e t v a r i a b l e s , and r e c t a n g u l a r nodes will c o n t a i n a t o m i c p r e d i c a t e s .
T h e point v a r i a b l e s will be indexed
in t h e i r o r d e r of q u a n t i f i c a t i o n ( s t a r t i n g at the top of the tree} by i n t e g e r s . will be s i m i l a r l y indexed by R o m a n n u m e r a l s . 1
Set v a r i a b l e s
F r e e point v a r i a b l e s will be denoted by
When the e n t i r e f o r m u l a i s not quantified but i s i n s t e a d a logical c o m b i n a t i o n of f o r m u l a e which a r e quantified, the top nodes of the g r a p h c o r r e s p o n d to c o n n e c t i v e s " ^ " o r " V " i n s t e a d of q u a n t i f i e r s . T h e g r a p h r e p r e s e n t a t i o n is h e r e u s e d only for i l l u s t r a t i v e p u r p o s e s .
84
01' 02 . . . .
etc. in t h e o r d e r in which they a p p e a r in a t o m i c p r e d i c a t e s .
will be s i m i l a r l y denoted ~1' ~2 . . . .
etc.
Free set variables
When a f e a t u r e is r e p r e s e n t e d , t h e r e will be only
one f r e e v a r i a b l e , the s e t v a r i a b l e , ~. H e r e an example of a f i r s t o r d e r f e a t u r e is shown (for a g r a p h of a s e c o n d o r d e r f e a t u r e s e e E x a m p l e 5): Example 3 C o n s i d e r t h e f e a t u r e P10(A), w h i c h is s a t i s f i e d by r e g i o n s w h i c h a r e " s t r a i g h t l i n e s " c o n t a i n i n g m o r e than two points: A d i s t r i b u t i v e n o r m a l f o r m of t h i s f o r m u l a is
A g r a p h c o r r e s p o n d i n g to t h i s f o r m u l a is
T h u s , to decode the above g r a p h , we m a y b e g i n at t h e bottom q u a n t i f i e r : Let A = ~, x = x 1, y = x 2, z = x 3, w = x 4 and we obtain the following s u b f o r m u l a which we will denote "Dl(x,y,z,A)"(where
x,y,z
and A a r e the f r e e v a r i a b l e s ) :
v w ( : ~, w] v < <x, %x, w ~ , < ~, ~, y, w]) _ , l ~ , y, %,) If we include t h e note above i t we obtain Vz ( - ~ , c ~ , A ? VD 1 ~ , y , z , A ~ )
_ D2(x,y,A).
Now if we w r i t e the f o r m u l a c o r r e s p o n d i n g to the node to the left of the one j u s t considered:
Moving up a node on t h e g r a p h we obtain:
~y(e~z,A] ~ D2[X, y , A ]
AD3[x,y,A])
P r o c e e d i n g to the top node we o b t a i n
= D4(x,A)
.
85
3x (o
^,4
-- P10(A}
which is an alphabetic variant of the f o r m u l a for P10(A) given at the beginning of the e x a m pie. Note that this graph r e p r e s e n t a t i o n e n s u r e s that the m i n i m u m n u m b e r of variables is used (Rothenberg, 1973b). In general, prenex n o r m a l formulae have m o r e than one d i s tributive form; also, a given distributive formula will often have m o r e than one p r e n e x form (e. g., when two branches from the same node contain differing quantifiers and all branches from that node contain quantifiers, e . g . ,
Vx (Vy[p(x,Y)3~/3y[q(x,y)]) c a n b e put
into the prenex n o r m a l forms YxVy3z(p[x, y ] v q [ x , z3) and Vx3zVy(p~x, y~Vq[x, z~). ) Note that the prenex form has m o r e quantified variables than the distributive form. This is often the case. We are here interested in the distributive set of a formula.
This is defined as the
set of formulae obtained by placing a distributive f o r m u l a in prenex form, obtaining all p r e nex forms resulting from the exchange of adjacent like quantifiers, placing each of such prenex forms in distributive form and eliminating the identical formulae which r e s u l t (more economical methods exist}. Another form of i n t e r e s t here is called the nested form.
This corresponds to a
prenex formula wherein the quantifiers are imported 1 into the subformulae within this scope until a further such importation would r e s u l t in a c i r c u l a r node on the graph having two c i r cular nodes as immediate sueees s o t s .
Such a f o r m u l a is obtained from a distributive f o r -
mula,whenever two or more quantified subformulae (i. e. s u b t r e e s on the graph) a r e the immediate s u c c e s s ors of a quantifier (circular node on graph), by consolidating such s u b formulae within the scope of a single quantifier. A nested form of the formula in Example 3 is (also see Example 5): P10(A) = ~ x ~ e ( x , A ) A 3 z ~ [ z , A ~
AVy E~e(y,A)
~/(~b ~ , y , z 3 AVw C: (y,w) v < ( x , y , x , w ) v < (z, y, z, w)])J ) J . Note that both conjunctions and disjunctions now appear within the scope of the quantifier of the variable, y.
Nested formulae may be graphed by including nodes containing
disjunctions and conjunctions. The graph of the nested form of P10(A) above is
1
That is, prefixed to a s u b f o r m u l a r a t h e r than the e n t i r e formula initially within its scope.
86
) 0
Although formulae, if in distributive normal form, translate into efficient computer programs (nested form is more efficient when it contains no more variables than the distributive form), the principal motivation here is the necessity for such form in the procedures which follow. Also, the "learning" algorithm generates and operates upon formulae in their distributive form. 5. Compression and Decompression of Formulae In order to compare different formulae which contain identical derived predicates as subformulae, it is necessary to guarantee that such will not be concealed by their form. Hence, we define the following procedures: Compression: This operation consists of replacing an innermost phrase of a formula by an abbreviation, Di(X1. . ,Xn, A 1. .. Am), with the number of arguments equal to the number of free variables (see Example 3 ~ o v e ) .
If this predicate is a derived predi-
cate which has already been generated (to be discussed), substitute its abbreviation. This is called one layer of compression. This may be repeated until all phrases are exhausted. When
formulae for {all) existing derived predicates are replaced by their abbreviations,
the resulting formula is defined as in collapsed form.
Example 4
PI(A) -: Yx (,re ~ , A] V~y ~(y,A) VDl(X,y, A)])
_
(~e
)
Decompression: This is the revers e of compression where each derived predicate has its definition substituted for its abbrevation in the formula being decompressed. is done one derived predicate at a time, working from the innermost outward.
This
Each such
87 substitution is defined as one l a y e r of d e c o m p r e s s i o n .
This may be r e p e a t e d until only
p r i m i t i v e s remain, in which c a s e the resulting f o r m u l a is called the complete formula.
In
Example 3 above, we would substitute -~(x,y,z)V~w(~[y,w~A~<~x,y,x,w~A'x,<~y,z,w,z~) 6.
for ~'Db(X, y, z) .
Simple Abstraction This is a p r o c e d u r e whereby the collapsed (distributive) f o r m s in the distributive
s e t s of P. and P. (or the graph of these forms) a r e mechanically c o m p a r e d and c o m m o n 1 l subformulac of m a x i m u m length (i. e . , with m a x i m u m n u m b e r of contiguous symbols) a r e e x t r a c t e d and f o r m d e r i v e d p r e d i c a t e s ,
Each c o m p a r i s o n proceeds f r o m t e r m s of the i n n e r 1 most phrase outward to the remaining phrases. Whichever of P. o r P. has f e w e r distinct t i collapsed f o r m s is chosen, say P.. E a c h such f o r m is c o m p a r e d with any collapsed f o r m of t P. and ,~P.. The longest subformulae which a r e c o m m o n to both formulae a r e e x t r a c t e d 1 1 and labeled as d e r i v e d predicates. P e r m u t a t i o n of t e r m s within a phrase in o r d e r to e x t r a c t a longer formula is permitted.
Also p e r m i t t e d a r e exchanges of v a r i a b l e s in p r i m i t i v e s
w h e r e the extension r e m a i n s unchanged t h e r e b y ( e . g , ,
<(x,y,z,w) - < (y,x,z,w)).
Sub-
stitutions of equivalent p r i m i t i v e s may also be used, e . g . , ~, < (x, y, x, z) = < (x, z, x, y)V = (x, y, x, z). When c o m p a r i s o n is made, if a distributive (collapsed) f o r m deviates f r o m nested (collapsed) f o r m because a quantifier has been distributed through b r a c k e t s , an additional c o m p a r i s o n is made after nested f o r m has been r e s t o r e d (see Example 5 below). Notice that the use of collapsed f o r m s guards agains e x c e s s i v e computation by not manipulating f o r m u l a e for existing d e r i v e d predicates (which r e s u l t e d from simple a b s t r a c tion previously performed). Example 5 H e r e we c o m p a r e (the distributed f o r m of) a formula, Pg(A), which is strongly s a t i s fied by the union of all boundary points of dark c i r c u l a r regions (i. e . , those regions which
satis~ derived predicate D II{A) --3x:3y3z(~ Ix,y~ A [~ {z,A) <--->< (x,z,x,y}]) with feature, PI0{A}, of Example 4 (forconvenience ~b{x,y,z) and ~(x,y,z) willnot be decompressed):
,,,,y 1
V<[x,y,x,w-j v <
y, z,w])])].
That is, those strings of symbols (starting f r o m the right of the distributive formulae) which a r e c o m m o n to both formulae and which contain no proper substrings common to both a r e extracted.
88 Although P10(A) is in distributed form, it does not conform to nested form
Hence we con-
solidate all terms quantified by "Vy" within the scope of a single such quantifier:
pl0~l- 3x Fc(x A~^ 3z(o Ez,a ^ vy F~~(y A/ V(~ Ix, y, z] h ~¢w [= (y, w) v < (x, ~,x, w~ v < (z, y, z,
~)])])]
P9(A) has another distributive form, but we here take the negation of the above form and l~tbel our variables so that their order in the innermost phrase is similar to the order of variables in the innermost phrase of P10(A):
V ~ y ~ Ix, z, y] A Y w [ : (y,w)V < (x,y,x,w)V < (z,y, z , w ) ] ) ] ) ] A graphical representation of 'vP9(A) is
,,E(1)I)"]
_
Comparison (from right to left) yields the common maximum subformula Db(X,Y,Z) = ~'(x,y,z) hVw [= (y,w) V<(x,y,x,w) g < ( z , y , z , w ) ] and we see that P10(A) - ~ 3 z ( c [ x , A ]
Ac ~,A-] AVy ~(y,A)
-~Db(X,y, z)])
and if we substitute Da(X, y) for ~ (x, y) AVz (~-,Db [,% z, y]) ("x adjacent to y", see Example 1), P9(A) = Vx3B(c Ix, A] -->c [x, B] A D11 [B] ADd []3] h 3y I ' v e ( y , B) A Da(X,y)]) This procedure is more easily seen by extracting common subtrees of the graphs of ,vP9(A) and P10(A) (see previous section). 1 1Although not done in the diagrams above, for mechanized execution of the abstraction procedure, variables should be indexed in reverse order of quantification (i. e. starting from the bottom of the tree).
89 Comparison of even the simplest formulae often yields derived formulae.
See
Rothenberg (1973b) for examples. In e s s e n c e , the f o r m u l a s o b t a i n e d by abstraction (and by "relation abstraction", to be discussed) are those "properties" of features we consider in this system. 7. The Predicate Hierarch~y Complete abstraction is the application of the techniques of simple abstraction to complete formulae (i. e., formulae which have been decompressed
till only primitives re-
main--see Section 5). This is performed at critical points in the flow of the system, as when computer memory
is becoming exhausted.
such
Then all derived predicates retained by
the system are eliminated and complete abstractionis performed with all pairs of features in the descriptive basis (i. e., "properties" of features are found).
The derived predicates
obtained are retained by the system (for purposes to be described in Section 15) and each
pair of these is again compared to extract maximal common subformulae which again become derived predicates (i. e., "properties of properties" of features are found). This procedure continues until only primitives remain, and the resulting set of derived predicates is called the l~redicate hierarchy.
In this manner an economical representation of the
structure of the descriptive basis may be obtained. 8. Relation Abstraction Here we compare two formulae (again in compressed form), which although they possess no common subformulae, have identical logical form (i. e . , identical relations between their subformulae), or which have subformulae of identical logical form.
Consider
P15(A) = VxVy(¢[x,A~ hOa~X,y~"->~ ~,A~) which expresses closure under the relation of adjacency, and PI6(A) ~ ~¢xVy(e~,A] A D5~x,y~--->e~y,A~) which defines a region closed under another binary relation, D5(x, y). When relation abstraction is here performed a dummy predicate K(x, y), replaces the differing binary predicates in the two formulae, and a predicate for m results:
This variable predicate expresses closure with respect to some unspecified binary relation K ~X, y]. The procedure for comparing P. and P. is as follows: 1
3
Compress whichever of P.
1
and P. has the most variables until both formulae contain the same number of variables. 1 l
1See Example 4 for illustration of how the number of variabIes is reduced by compression.
90 If the substitution of k (at first k = 1) distinct dummy predicates in each of P. and P. 1 j makes them identical, such resulting formula is a predicate form and is retained by the system.
(This compression makes use of the negation of a formula and the other devices
used in simple abstraction. ) When such a substitution does not render P and P. identical, 1 J 1 both formulae a r e compressed one layer and the comparison procedure is repeated. Subformulae of a phrase may be permuted or combined, and a dummy predicate may be inserted where no subformula is present if needed to avoid combining two subformulae (one of which is in the other formula) into a single formula.
(That is, if Pi(x, y) A Pj(y, z)
A Pk(y, z, w) is part of a phrase being compared with Pi(x, y) A Pk(y , z, w), the latter b e comes P.(x,y)A K(y, z) A Pk(y, z,w) and K(y, z) is also substituted for P.(y, z) in the f o r 1 ] mer. ) Example 6 Compare a predicate which is strongly satisfied by the boundaries of all dark convex s ets P17(A) =--.Vx ~(x,A)
> "3BE~(x, B)A P2(B) A 3 y (,-,-,((y, B) A Da(X,y))"])
with a predicate which defines the boundaries of all dark sets with property P3(A): P18(A) ~ Vx ~ ( x , A)---~ 3B ~(x, B)A Vx (~(x, B)----> D(x)) AP3(B)/~3y (,vc(y, B)A Da(X, y ) ) ~
Putting both in distributive normal form: PI7(A) - Vx (Ne(x,A)V 3B [,(x, B)A P2(B)A3y (-v~(y, B) A Da(X,y))~ )
P17(A) has fewer variables than P18(A), Hence compression of P18(A) (indicated by bracket under above formula) yields:
The next to innermost phrase above has a g r e a t e r number of subformulae than the next to innermost phrase of P17(A). Hence subformulae of P18(A) are combined (as indicated by bracket under above formula} to yield:
1
The entire procedure may be repeated for k = 1o 2 . . . . . n (n is chosen to conform with the availability of computer memory and to avoid excessive iteration. Hence it will usually be small. )
91 It is now easy to see that comparison of the compressed sets for PIT(A) and PlS(A) above will yield the following predicate form:
which defines the boundaries of all regions, B, which satisfy K(B), Again, the above procedure can be performed by comparison of the graphs of PI7(A) and P I8(A) for similarity of structure (exclusive of the different atomic formulae}. Relation abstraction is executed whenever simple abstraction is, and predicate forms are also retained by the system I (for uses to be described in Section 15). W h e n complete abstraction is performed and the predicate hierarchy is restructured, relation abstraction is also performed on complete formulae and, in similar fashion, a hierarchy of predicate 2 forms (called the relational hierarchy) is constructed, At each successive level in this hierarchy the predicate forms acquire additional d u m m y variables. None are retained with more than n such d u m m y variables, 9. An Elimination Procedure for the Set Membership Primitive For our purposes we define a feature, P(A) as a tautology iff P is satisfied by all subsets of the data space or only by the entire data space. P(A) is defined as a contradiction iff P is satisfied by no subset of the data space.
Now we eliminate vacant predicates and tautologies which derive from the use of the primitive, "e(x, A)".
Contradictions and tautologies a r e c h a r a c t e r i z e d by: YA (P ~k]) and
~3A (P [A]). These formulae have a similar significance: Ca) Vx(e[x,A]) and (b) ~-~x(e ~,A]~ or Vx ( ~ , c ~ , A ] ) .
(We are not i n t e r e s t e d in the empty set or its c o m p l e -
ment. ) Hence the following partial decision procedure: (1) Examine a f o r m u l a and its negation (both in complete and distributive form). label it "T" (or "F").
(2) If a t e r m of form (a) (or (b)) occurs,
(3) If "T" (or "F") occurs as part of a conjunction or a quantified
conjunction, label the conjunction "T" (or " F ' ) .
(4) If the e n t i r e f o r m u l a becomes
labelled "T" (or "F"), the f o r m u l a is tautologous (or contradictory).
The r e v e r s e is true
for its negation. Hence,
PI9(A)-- VxVy3z (c[x,A] A ¢ [y,A] A Db[X,z,Y]~C [z,A]) which has the following distributive form of its negation
is tautologons (examine its last term} and will not be generated by the system. 1 All nested forms of a predicate form are retained by the system. 2 Actually, r e l a t i o n abstraction is also performed on derived predicates as well as features.
92 S i m i l a r procedures can often be developed to eliminate tautologies and and c o n t r a dictions resulting f r o m the use of other p r i m i t i v e s ,
(The above is, in e s s e n c e , the f a m i l i a r
technique of '~quantifier elimination". ) This of c o u r s e , depends upon the choice of p r i m i tives in the particular application. Other p e c u l i a r i t i e s of the s y s t e m p e r m i t the elimination of other u s e l e s s formulae. Notice that because all objects a r e maximal regions, if
is a data independent feature, an object which strongly s a t i s f i e s it also s a t i s f i e s
Notice that if a f o r m u l a for a feature is used as part of auother feature it becomes a d e r i v e d predicate and the m a x i m a l i t y condition no longer applies.
In such c a s e ~'4 >" must
be substituted for "- -->" in the f i r s t of the above f o r m u l a e if its meaning is to r e m a i n unchanged. Consider a feature with a collapsed f o r m (9. 1)
P(A) - Yx ~ ( x , A ) VDF(X)] =_ Vx ~'ve(x,A)--->DF(X)~
where DF(X) contains no nested phrase of the f o r m (9, 2)
Vy E~c(y,A) VDB
the entire data space.
or
~y ~(y, A) h Dc(Y)~
Hence P(A) wii1 be strongly s a t i s f i e d only by
Also, if the eollapsed f o r m of the feature is P(A) -- 3x [-~e(x,A)
ADE(X)~ its negation will be of f o r m (9.1).
Again, DE(X) can be s a t i s f i e d and still x e A .
Hence P(A) will be strongly satisfied only be sets consisting of the entire data space minus one point.
F e a t u r e s which are strongly satisfied only by sets with a fixed number of points
are of no i n t e r e s t h e r e (except when "eard(A)" is a primitive). 1 formulae contain at least one phrase of f o r m (9.2).
Hence we insist that all
Other elimination methods r e l a t e d to this d i s c u s s i o n a r e d e s c r i b e d in Ilothenberg (t973b).
1
The f o r m u l a e in (9, 2) intuitively specify conditions for m e m b e r s h i p of aAooint in A (i. e . , VyL~(Y,A)---->DB(Y) j ) -- or for n o n - m e m b e r s h i p of a point in A (i. e . , Vy ~'DE (Y) --> ~ (y, A)~). The existentially quantified f o r m u l a in (9.2) may r e s u l t f r o m a negation of the u n i v e r s a l l y quantified formula.
93 10. Models A formula is evaluated in the product set over which the vector of its free v a r i a b l e s ranges.
Thus if S CU is t e m p o r a r i l y our u n i v e r s a l set, P(A) is evaluated in 2 S (i. e . , the
power set of s ) l a n d D(x, y, A, B) is evaluated in SXsx2Sx2 S.
(Note that formulae are never
evaluated in the e n t i r e data space, U (to be discussed), ) A model of the formula, Fi, is an element of the product space (denoted by S), over which the vector of its free variables r a n g e s , such that the substitution of each component of that e l e m e n t for the free variable satisfies the formula,
The model is denoted by M!S)(F.) j 1 where j indexes all such models (the o r d e r i n g is a r b i t r a r y ) . When F. functions as a feature, it is denoted by P. and a model of it M!S)(P.) (as distinct from- -a i"realization", M! S)(F.) ) , .1 1 M (S)(F.) J 1 must strongly satisfy P. in S = 2 S (see Section 2). is the set of all models of a '
1
1
f o r m u l a (called its extension) and M(S)(P.) denotes the extension of feature, P.. 1 2 1 The ease of satisfaction, E(F.), of a formula F i, is defined as 1
E(F.) = 1
card(S~
("card" denotes the n u m b e r of elements in the set within brackets). The ease of satisfaction of a feature, P., is always defined in t e r m s of satisfaction, 1
not strong satisfaction ( i . e . , E(P i) = E(F i) ). The overlap of a pair of formulae, F. a~d F., is defined as E(F. A F.). The following applies only to features:
Pi ) is the set of points of S in the j
model of P. in S = 2 8" Let n(P.) = card(~S)(P_.)). 1
Then we define
1
n(Pi) j=l Restating our problem, we are given two disjoint c l a s s e s of regions
{RI'R2 .....
RN}
~ = {RN+I, RN+ 2 . . . . . RN+M} and we wish to generate an ideal feature, ~, such that (see Section 1} Yk3j ([Mf)(P)I = R k ) i
where R k e K
A i n P(A) is allowed to range only over subsets of S. Points not in A and sets other than A, however, m a y l i e anywhere i n U. (In practice, in most cases, for purposes of c o m p u tation reduction, points not in S a r e r e s t r i c t e d to the boundary of the complement of S in U. 2 Note that E(FI) is generally defined with S = U.
94 AND
V~Vj ([MIU)(l~)1 ~ :R~)
Note
where R~e
that K c M(U)(P), When this inclusionis proper, P is said to generalize K.
W e a r b i t r a r i l y a s s i g n the subscripts of each given region, R..
Now the subseripts of all
1
M!U)(P.) are assigned such that j
1
(10.1)
~" ~
is maximum
for all possible indexings
l_l. Measures
]
1
(any such maximal
indexing suffices).
of Success
Let us denote formula R. EC.
jnlM ])/card%
ard
(i0. I) as ~(Pi,~) when
Let R c. be the complement
]
all R. cK
of R. in U and let ~CJbe
and as ~(Pi,~)
when all
set of R.c for all j suchthat :}
J.
Define 7(Pi,K) = ¢(Pi'~) - ~(Pi'~e) and let 0(P.,1 [ ' ~) = T(Pi' ~) -'Y(Pi' ~)
be defined as the m e a s u r e of success in segmentation. approximates an ideal feature.
It will meas ure how well a feature
Another such m e a s u r e , called the su0cess in partitioning is
defined as
eard( (P., K, C) = i
M
card(K)
eard( card(C)
Other m e a s u r e s are also utilized (see Rothenberg, 1973b and Section 18), Whenever feedback occurs, a set of weightsj called r e i n f o r c e m e n t s , on all features and derived predicates are accordingly altered.
These are i n c r e a s e d (or decreased) by
adding, for example, ~(Pi,K, C) - . 5 + [ 0 ( P . , K , ~ ) - . 57/k (where k is an integer) 1 to the i weights which are initially set at zero. This is done for all derived predicates 2 and p r e d i cate forms which t o g e t h e r 4 o r m a successful (or unsuccessful) feature.
Thus derived p r e d i -
cates and predicate forms accumulate r e i n f o r c e m e n t from a variety of features. 12.
Mutations We generate our features by starting with an a r b i t r a r i l y generated feature, P., arid i
successively altering its syntax by means of mutations which are functions which take one i
An optimal choice of k may b e e s t i m a t e d by experimentation with the program. Actually any method of weighting ~(Pi' K, C) more heavily than O(P.,K, C) will suffice. 2 i The negations of derived predicates are also treated as derived predicates and accumulate r e i n f o r c e m e n t accordingly.
95 feature into another (L e . , fn(Pi) = Pk ). We specify three c l a s s e s of mutations: r e s t r i c t i n g (iff M(U)(Pk) ~ M(U)(Pi)), relaxing (iff M(U)(P i) ~ M(U)(P k) ), and neutral (otherwise). Relaxing mutations (said to "relax" a formula) demonstrably include (Rothenberg,
1973b). (1) Adding a subtree (i. e. derived or atomic predicate) below a node (on the graph of a distributive formula) which contains a universal quantifier. (2) Eliminating a subtree below a node containing an e x i s t e n t i a l q u a n t i f i e r . (3) Replacing a universal quantifier by an existential quantifier (and restoring d i s tributive form). (4) Replacing " 3 ~ Y" by '%ry3x" and restoring distributive form. (5) Replacing a subformula (i. e. subtree) by one which it implies. (6) Replacing "Vx (e [x,A~ V D ~ , . . . ~ ) " by "~x(e[x,A~ AD[x . . . . ~ ) "
when the
former formula is not v a c u o u s .
(7) Replacing "Vx(c[x,A7 VD[x . . . . 3) " by "~x ('vE[x,A~ h D [ x . . . . ~ ) " when the former formula is not satisfied by the universal set. The inverses of the above are r e s t r i c t i n g mutations (said to " r e s t r i c t " a formula). A neutral mutation consists of the replacement of a subtree on the graph of the distributive form of a feature by another subtree with the same free variables such that neither subtree r e p r e s e n t s a formula which implies the other. of variables in a subformuta.
This includes permutations and exchanges
Other "composite" mutations a r e discussed in Rothenberg
(19735). 13. Mutation Choice Here, among others, we will develop the following c r i t e r i a : if given region R. is an J element of ~ , then (a) when R. does not satisfy P. a relaxing mutation on P. is required, ]
I
i
and (b) when R. satisfies Pi but not strongly a r e s t r i c t i n g mutation is required.
When R.
J_
J
is an element of C and R. strongly satisfies P., either mutation may suffice (although a ]
I
r e s t r i c t i n g mutation is usually preferred).
These c r i t e r i a are, of course, averaged for all
elements of K and C. Assuming we have a feature, P., we now consider how to choose an appropriate 1
T
11%
mutation: Let P. be decomposed into a conjunction of two formulae, P.~ and P~ such that
i pD I I i P_ is data independent and . contains data-dependent atomic predicates and only those data 1
l
independent predicates which are inseparable from this portion of the formula (Rothenberg, 1973b). (13. i)
p. = p! ^ pD. 1
1
1
96 The f e a t u r e s a r e generated in f o r m (13.1).
Suppose we wish r e g i o n R . ¢ K to be a model of 2Rj. 1 J Now let S. = Let "p [A]" r e p r e s e n t the
P. (note: strong satisfaction is required).
I probability
.....
of event A.
~::J-
It has been proved (Rothenberg,
-
19735) that if R. is a randomly J
s e l e c t e d region, (sj)
(t3.2)
I"
E(P~) ~<E(pI)---> p ~ard (M~I'I(p~)) > card (M " (Pk))17 > t/2 . T~
This holds only because all e l e m e n t s of M J(P.) a r e maximal r e a l i z a t i o n s (i. e . , 1 models) of P.. (13.2) can be intuitively s e e n if we c o n s i d e r the number of m a x i m a l : 2 (1) 1
circles,
(2) convex regions and (3) " s t a r l i k e " regions that may be contained in a r e g i o n c o n -
sisting of the i n t e r i o r of an a r b i t r a r y ink blot.
C l e a r l y the n u m b e r s d e c r e a s e f r o m (1) - (3)
while the e a s e of satisfaction of the f o r m u l a e (features) c o r t e s ponding to (1)-(3) i n c r e a s e .
isJ
.
L et " M r j (P.) , ,, denote the largest (i. e. most points) model of P.I in S.. ~u 1 ] also holds:
1
I
r
is.)
I"
r
(s.)
Then the following
_
(13.3)
.
(s,)
..
.. (s.)
..
(13.4)
E(P~)
(13.5)
E(pI)--=->
(S.)
(s.)
/7
(Sj) 1
I\7
is )
(13.6) The above indicates that if m o r e than one object s a t i s f i e s P. in S. or, if the l a r g e s t t j model of P. in S. is s m a l l e r than d e s i r e d , a r e l a x i n g mutation of P. is appropriate. Now
I
)
M(U) (PD) [I
we c o n s i d e r the c a s e where S. = 2 J .th J D_ all subsets of the j model of P : ) .
i
i I for all j ( i . e . , we evaluate P! in the space of 1
1
Suppose for some j, Mj(U)(PiD) ~ R j
tR..j
Then PDI will be r e l a x e d until, for
all j (13.7)
R.j CJ [M
(U)
D
(Pi)1
and such that 1 For brevity in notation, w h e n m o d e l s of features are denoted, the " - - " o v e r the super script will be omitted (e. g . , M(U)(P.} will be denoted by M(U)(P.) ). 2 1 1 If we o m i t " m a x i m a l " , c l e a r l y the o r d e r i n g is the other way around.
97
n(p.)
i s as s m a l l as p o s s i b l e .
Note that it is possible that
oard(, s_
esj)_ >l ("1"
means
"such that"), i . e . , there are several RiC S.. j Consider P.1 and ~ which is ideal feature. Let
.
( .~ =
j
1
.
=S.
j
l
(note: R . C MU)(P ) ). Then it can be shown that (Rothenl
berg, 1973b): . (s.)
(13.8)
oardCM ' (P~))< card(Hs ~/a s C Sj>---> p[E(P~)> E(~')] > lie
(13.9)
]
iS.}
])~
That is, if there are two many models of pI relax pI. if there are two many points to a i i' given model of Pi'I restrict pIi" (U) D recognition system (intuitively, a system wherein the magnification of a geometric figures does not alter the features wMeh it satisfies or fails to satisfy), the probable proportion of models in a space S. does not alter with the size of the space (after a certain minimal size). J
Hence (13.8) still holds (although (13.9 and (13.10) do not___). Hence a comparison of how well (13.8) and (la. 9) agree, i . e . , the probability that .
(13. 11)
is.)
is.)
card(M J (P~))< card(Ric[~iR£¢ Sj)< > card(iM L] (P~)l) > eard(R.)]
inc~Cates how well IM~U}(~D)I and be too large.
M~U)(pD)] match.
Because of(13.7), E(~ D)
can only
Hence' ' Jfailure a of (13. ll)J ~'indicates that pD should be restricted in accordance 1
with (13.9), but without violating (13.7). When pD is usually composed of one place atomic predicates (i. e., "gray scales"}, 1
mutations of pD are simpler than those on P! and are performed first (when there is such a 1
I
choice}. Other techniques similar to the above exist (Rothenberg, 1973b).
Since probabalistic
considerations are involved in the above, (13.8) and (13.9) should be averaged over all j (there are N such) and normalized; i . e . , examine both 1
Except at the boundary of U.
98
N
(S)
X = N1 Z]_± ~ c a r"~ d~M
](P~'b-card(R~eKtR~ ¢S,~/card(R#eKIR~CSj} )
and
Y=~
N
(S.)
b
j=± \~
and apply the s t r a t e g i e s previously outlined according to whether X and Y a r e g r e a t e r than zero.
We have assumed that each R. is randomly s e l e c t e d f r o m subsets of U. This is J p! ., not reasonable. However, since models of strongly s atisfy pI a region, R., which is 1 3 ¥ not a model of P.~ need not be a model of "vP a.. Hence by applying our c r i t e r i a for mutation 1
- -
1
choice to both formulae, pI. and ,~pI. (both of which for this purpose a r e t r e a t e d as f e a 1
1
tures), it appears that we may a m e l i o r a t e our a s s u m p t i o n of the r a n d o m n e s s of R. (i. e . . ] "models of P! a r e as random as models of ~ p I , , ) ! Thus the d i s a g r e e m e n t of c r i t e r i a when 1 i applied to both p I and ~pI. (e. g . , both P and ~- P. r e q u i r e r e l a x i n g mutations) indicates I
1
1
the need for a neutral mutation. flected in the above.
14.
i
Notice that both measures of success (Section ii) are re-
(Details are discussed in Rothenberg (1973b),)
Modified Quantifiers Now a m o r e powerful technique of mutation s e l e c t i o n is d e s c r i b e d ; note that all
quantified point variables in a f o r m u l a P(A) range o v e r e i t h e r the set A (e. g . , a r e r e s t r i c t e d by " e ( x , A ) " within the scope of the quantifier) o r the complement A e of A, o r range o v e r the u n i v e r s a l set (when no set m e m b e r s h i p atomic predicate r e s t r i c t s its range). Thus when, on the graph of a f o r m u l a , a node containing a quantifier has no branch containing a set m e m b e r s h i p atomic predicate, the quantified v a r i a b l e ranges o v e r the u n i v e r s a l set (or its power set if the variable is a set variable). v
x ranges over
(Note that in subformula
and in subformula
x ranges o v e r A . However, when the subformula is -qx ¢)
V
, A] A F Ix . . . . .
.... J),
x ranges
o v e r A, etc.)~. Let Y denote the set o v e r which variable x ranges (where q indexes the v a r i a b l e s q 3 q in a n e s t e d f o r m -- s e e Section 3). Since we a r e dealing with a size invariant s y s t e m (see 1
That is, if this randomness with r e s p e c t to the above c r i t e r i a does not hold, it appears t h a t in t h e s e c a s e s , we will know it because of opposite indications. 2 Note that xeA etc. 3 r
q
always is one of A, A e, U, 2A, or 2
xeA Ac
.
99 previous section) we h e r e c o n s i d e r the fraction of x
in F (rather than the fraction of x q q q in U) for which a f o r m u l a is satisfied. (Similar r e m a r k s apply to set variables. ) 1 The following notation will be utilized: all distributive f o r m s in the distributive set
will be a r b i t r a r i l y indexed and p!k)" will indicate the kth such distributive f o r m of feature 1 P.. It can be shown that an a r b i t r a r y indexing of quantifiers in a prenex f o r m f r o m which the distributive set d e r i v e s (see Section 4) will ,result in a c o r r e s p o n d i n g o r d e r i n g of the quantifiers in each distributive f o r m .
This follows f r o m the following, which a r e c o n s e -
quences of the definition of the distributive set and of distributive f o r m (see Section 4). (a)
If, in placing a prenex formula into a distributive f o r m , one quantifier in the prenex f o r m u l a b e c o m e s m o r e than one quantifier (using the s a m e variable) in the distributive f o r m (e. g . , Vx --(FI(X) h F2(x) )_ b e c o m e s Vx ~FI(X)) A Vx (F2(x))), this o c c u r s in all distributive f o r m s in the d i s t r i b u t i v e set.
(b)
If two or m o r e quantifiers using distinct v a r i a b l e s in a prenex formula become quantifiers using the s a m e variable in a distributive f o r m (e. g . , Vx'qY(Fl(X) hF2(Y)) b e c o m e s VX(Fl(X)) h ~ x ( F 2 ( x ) ) ) ,
the number of
quantifiers in the prenex f o r m and in the distributive f o r m a r e equal. Note that in (a) above the number of quantifiers i n c r e a s e s and the number of v a r i ables r e m a i n s unchanged.
In (b) the number of variables is reduced, but the number of
quaatifiers r e m a i n s unchanged. Hence all distributive f o r m u l a e in a distributive set have the s a m e number of quantifiers.
All nested f o r m u l a e (see Section 4) have the s a m e number of quantifiers as the
prenex f o r m u l a (but possibly f e w e r variables).
However, each of the distributive f o r m u l a e
may have m o r e quantifiers than the prenex f o r m u l a (even when the distributive f o r m u l a e have f e w e r variables).
Hence two o r m o r e quantifiers using the s a m e v a r i a b l e in a d i s t r i b -
utive f o r m u l a may have the s a m e index (as d e r i v e d f r o m the indexing of quantifiers in the prenex formula).
Such quantifiers will be a b r i t r a r i l y o r d e r e d .
That s a m e o r d e r i n g will be
maintained in all distributive f o r m u l a e , each of whose quantifiers will be subscripted by this ordering. Let Q(k) denote the r th quantifier in (i. e . , the kth distributive f o r m of I).), r (k) (k F (k) 1 x r denote the variable quantified by~,_Qr ) and r denote the subformula within its scope (i. e . , its s u b t r e e s on the graph of P!~)).
Simple methods exist (Rothenberg, 1973b) which
1
guarantee that the subscripting of two o r m o r e quantifiers of the s a m e variable is consistent a c r o s s all distributive f o r m s in the s e n s e that, if t h e y have the s a m e subformula within 1 When f" = U - S.; for computational economy, x is allowed to range only over points in the boundaqry of th~ complement of S. Note that no__.~ormula is e v e r evaluated in U.
100
t h e i r scope, they have the s a m e subscript.
Note that each variable, x "°"
is subscripted
according to its ctuantifier's subscript, and hence the s a m e variable m a y have different subs c r i p t s when s e p a r a t e l y quantified (i, e . , such quantifiers a r e on the s a m e l e v e l of the graph of P!~)." 1 "x tKj = x tKt''. p r
If x (k}'" and x (k}" a r e two such v a r i a b l e s we will denote t h e i r equivalence by p r (Note that such v a r i a b l e s a r e not s e p a r a t e l y quantified in nested formulae. )
[,(k) will denote the set o v e r which x (k}~" ranges (if x (k}'" is not a point variable, F"(k) P p p p denotes the power set o v e r which x
(k) ranges).
P
Let "Q(k)sQ(k)" denote that ~ ( k ) i s a s u c c e s s o r node of Q ~ ) i n p r ~p
p!k) (i.e. , Qr(k) 1
is within the scope of Q(k)),p ,,Qp = V" and "Qp = "4" will indicate, r e s p e c t i v e l y , that Qp is a u n i v e r s a l quantifier and that Qp is an extential quantifier (note that the s u p e r s c r i p t is u n n e c e s s a r y because of the c o r r e s p o n d e n c e in indexing a c r o s s distributive f o r m s of a f o r mula). S u p e r s c r i p t s will henceforth be included only when r e l e v a n t o r useful for purposes of clarity.
Given a region, Rq, and a feature, Pi, notethat "R
t
omitted because i r r e l e v a n t ) denotes strong satisfaction of Pi by R ,
t~ /
1
= IM J(P )]" (subscript is q 1 i ct
(s)
in space, S., and J
"Rq = IiM J(Fi) " denotes satisfaction but not necessarily strong satisfaction, i Now consider a feature, Pi(A), and two given regions R e ~ and Rt c ~. Suppose q 2 we wish to alter P.i so that Rq strongly satisfies the altered feature and Rt does not. From the preceding section we see that, in general: (a)
if R q ~ M(Sj)(Fi )
(i.e. ' R q fails to satisfy P,) 1
a relaxing mutation is r e q u i r e d ,
• (5)"(Fi) but a ÷ M(5) (Pi)l , (h) ISRq=IM q
(i. e . , R
q strongly),
s a t i s f i e s P. but not 1
a r e s t r i c t i n g mutation is r e q u i r e d ,
either aneutra~ mutation or a mutation according to (a) o r (b) above is required.
(Note:
R t c ~ -- to be discussed in Section 15. ) Combinations of (a), (b) and (c) will also be d i s c u s s e d in the next section. 1
Note that it is h e r e a s s u m e d that R C S.. q J
2 As in the previous section, explicitly noted here).
P! 1
1
a r e independently mutated (although tMs is not
101
F i r s t we consider c a s e (a) above Note that A is no longer a free variable
Let A = R
(A is the free set variable in P.(A)}. q p(k) 1 A r b i t r a r i l y s e l e c t a distributive form, . .
Now s e l e c t p such that Qp = V and such that for all quantifiers Q{~)r (in all distributive forms,
P!~) 1
(14. 1)
~....~c a: \r d ( Q
^(~) .J./.~. r (~) = V ) ) r(D (Qr(~) SUp
is maximum (i. e . , such that Qp has the maximum number of s u c c e s s o r nodes which are u n i v e r s a l quantifiers, summed over all distributive f o r m s in the distributive set of P.). If 1 two or more quantifiers are equivalent in this r e s p e c t , an a r b i t r a r y selection will suffice. Now r e p l a c e Qp by the modified quantifier, Qp, which computes Kp, which is d e fined as that fraction of the x ' s p
in [~ for which P. is true (i. e. for which R = p 1 ' q
(sj)
IM " (Fi)l; which definition is elarified by the following procedure- e v e r y subformula of P.
1
is of the form Q1Xp(Fp[-X _
. . . . .
xo-])~ (although it may be n e c e s s a r y to r e - i n d e x the quantifiers
in the distributive form so that only v a r i a b l e s in F element of I~
n
be denoted by e ( k ) where k n
n
n
are indexed--see Section 4). Let each P indexes all elements of r . Define n
K(Cl(kl), c2(k 2) . . . . . Cp_l(kp.1) ) as the fraction of substitutions of the elements of f~p for Xp f o r w h i e h Pi(A) i s t r u e w h e n
(x . . . . . x
1) = ( c ~ ( k 1) . . . . .
Qn = V, define K(Cl(k 1) . . . . . Cn_l(kn_ 1
n~in
e
l(k
1)).
Forall n
if
l(kl) . . . . . Cn(kn)); ff Qn = 3, then
K (Cl(k1) . . . . . Cn_l(kn_l) ) = max(K~ct(kt) . . . . n, Cn(kn)>. W h e n n = 1, by definition, K = K . kn \ \ 1 1 p 1 However, if when K = 0, R still fails to satisfy Pi' by definition K = -co. Note that P q P is the same r e g a r d l e s s of the distributive f o r m chosen (i. e . , r e g a r d l e s s of the k in F(k)). P P Actually, the procedure will operate on a prenex form (it is best to use nested form). ~(k) Now we r e p l a c e Qp by another modified quantifier, which simi l a r t y computes ~(k), which is defined as that fraction of the x(k)'s in F'(k) Up p for ,which F (k) is satisfied, p
p
averaged over all substitutions of eonstants for the free v a r i a b l e s in
rl
Note that
the distributive form chosen is now relevant and the s u p e r s c r i p t of K (k~) is n e c e s s a r y . P ~(k) (Superscripts of x (k) if(k) and ~(k) a r e included for clarity. ) That is, is such that p ' p P feature P. is s a t i s f i e d and ~(k) is t~psuch that subformula F (k) is satisfied. Hence always p P ~(k) >_ 0. o
Note that Q and Qp,_ {k) are c o r r e c t l y i. n t e r p r e t e d as , for Kp. 100 percent of x " and as p P "for ~ k , 100 percent of x ~K)", r e s p e c t i v e l y (strictly speaking, the s u p e r s c r i p t of x (k) is unP P (k P^ (k) n e c e s s a r y . Also note that E(P.) and E(F ~ a r e i n c r e a s e d by r e p l a c i n g Q by Q and Q , r e --(k) 1 p P p ^o spectively. Kp and Kp a r e called the values of their r e s p e c t i v e quantiflers (Qp and Qp(k) ). 1
This can be built into the above algorithm.
102
Now define Dp = Kp- 1. Restore the original quantifier, Qp, and select another u n i v e r s a l quantifier, Qr' according to condition (14. 1). Again, replace Qr by Qr and ~(k)r and compute the corresponding values of K , ~(k) and D . Restore Qr and repeat the pro(..r r r cedure till all u n i v e r s a l quantifiers in p:k) a r e exhausted. (This can be done in a single 1
computation (Rothenberg, 1973b). ) Suppose for some p, K = -co. Using Q is defined as equivalent to replacing F (k) P P P by a tautology. A value of "1" in the n u m e r a t o r of K ( i . e . , K = 1/card(W )) is equivalent P P to replacing Qp by an existential quantifier. When K = -oo we may conclude that no r e l a x (Sj) . P'I (Sj) I ing mutation within F (k) will r e s u l t in R = M (Fi) I (or, of course, in R = M (Pi) ). P q q (Hence modifications of (tuantifiers which a r e s u c c e s s o r s of Qp on any tree of any d i s t r i b utive form in the distributive set may be avoided. ) Note, however, that replacing a u n i v e r sal by an existential quantifier may i n c r e a s e E(F (k)) {and hence E(F.) ) l e s s than a mutation p 1 (k) --which adds a subformula within F . Therefore when K = 0 (i. e. -co < K < i/card(f" ) ) P P P i~ l~ P we may
not without some Now
hazard avoid considering the modification of Qr ~(Q~ _ i s Q~ i).
consider case (b) where
R q
satisfies P. but not strongly. i !k)
s i m i l a r : Again, a r b i t r a r i l y select a distributive form, P
1
The procedure
is
and let A = R . Select p such q
that Qp = 3 and such that (similarly to (14.1))
r
r
p
r
is maximum. Replace Qp by Qp, and compute K (k) for all k~ no replacement
of F
which is defined as previously except that if
P by a contradiction
(i. e., a formula which is always false)
P
reqults in Rq ~ M(Fi), 1 by definition, K~p = co.
We then replace Qp by Qp which com-
putes _K wbieh is defined s i m i l a r l y to K exeept that we now i n s i s t that R = M(P.) (strong p p q 1 satisfaction) instead of 1t = M(F.) (satisfaction but not strong satisfaction). Note that K q 1 p and K are not n e e e s s a r i I y equal when K = co. P P "~(k) ^ k( ) . . . . . New we again replace Qp by the modified q u a n t l h e r Q which computes K (k), P (k) (k) p which is defined (as previously) as the fraction of the x s in W for which F is s a t i s fied, averaged as before ~ / c a r d ( f '
)<
< 1 . Note that E(P i) and E(F ) a r e de-
c r e a s e d by replacing Qp by the modified quantifiers (here, always, Kp _> 1/card(Fp) ). Now define Dp = Kp. Restore the original quantifier, Qp, select another existential quantifier according to (14.2) and repeat the above procedure until all existential quantifiers in "Pl.~) a r e exhausted. 1
(Again this can be done in a single computation (Rothen-
berg, 1973b). ) 1 (s.) The superscript of "M ] (El)" is henceforth omitted in such equations when Rq r'Sj.
103
Note that when ~
= co or K = co, we may conclude that no r e s t r i c t i n g mutation P P within F xk'~ for any k (i. e . , any distributive form) will r e s u l t in R = M(F.) or R = M(P.) p q 1 q 1 r e s p e c t i v e l y . When K = 1 (or K = 1 o r ~(k) = 1) we a r e , in effect, r e p l a c i n g an e x i s P P P tential by a u n i v e r s a l quantifier. Note, however, that r e p l a c i n g an existential by a u n i v e r s a l quantifier may d e c r e a s e E(F (k)) l e s s than a mutation which adds a subformula within F (kJ." P P Hence, in this case, we may not without some h a z a r d avoid considering the modification of
Q
r
3~(Qr(~)sQ~p(~)). With appropriate modifications the above procedure will operate on quantifiers of
set (as well as point) variables, We now define the degree of satisfaction, D(Pi, Rq), of Rq with respect to feature, P, ~ as 1
o4. 3)
"(Pi'aq) = m~x(Sp~IA = aq p
("A" is the f r e e variable in "P.(A)'9 1 When D(Pi, R q) is negative (universal quantifiers have been modified because R
~ M(F i) ), D(P i,Rq) is called the degree of undersatisfaction of R with r e s p e c t to P.. q q 1 When D(Pi,R q) is positive (existential quantifiers have been modified because Rq = M(F.)I ) but R ~ M(Pi), D(P i, Rq) is called the degree of o v e r s a t i s f a c t i o n of R with r e s p e c t to p. 1 q ........... q 1
In our p r o b e l m h e r e we w i s h to g e n e r a t e a feature, R k = M(P i) and R i E ~
> R l ~ M(Pi).
P i ' such that R k ~ K -->
Accordingly we define the satisfaction gap, G{Pi),
of P. as 1
(14.4)
°(Pi) = rain_ Rk~K
R~ e C
Note the c o r r e s p o n d e n c e between G(P.) and our " m e a s u r e s of s u c c e s s " (Section 11). 1 Although we are seeking strong satisfaction while G(P i) deals with satisfaction which is not n e c e s s a r i l y strong (appropriate adjustements can be made in the definition of G(P i) ), it will be s e e n that G(P.) has uses other than m e a s u r i n g the s u c c e s s of P. -- s e e Section 17. 1
I~
Note
1
also that modified quantifiers of the F ~kj in P. c o r r e s p o n d to mutations of such subformulas p 1 both in kinds and amounts of alterations in e a s e of satisfaction, E(P.) and E(F(k)). Also, p the alterations in f o r m u l a e due to quantifier modification a r e r e l a t e d by implication just as a r e the alterations due to mutations. t
Note the intuitive i n t e r p r e t a t i o n of t h e s e notions--e, g . , of two concave (i. e . , non-convex) regions (e. g . , ~ and ( ~ ) , one may be " m o r e concave" (the f o r m e r ) than the other. E x perimentation will r e v e a l that the above techniques p r e s e r v e such intuition (e. g . , ~ and ( ~ are approximately "equally e o n c a v e " ) - - s e e Rothenberg (1973b).
104
15.
E a s e of Satisfaction; Mutation
It is e a s i l y seen that, when ~(k) is evaluated for all R. eK and all R.e ~ and I~(k) is 1 P 3 defined as an a v e r a g e of all such ~(KT, P
E(F(:))~
Kp * (k)
(" ~ " means "approximately equal to")
" ~'" is used instead of " = " in the above because the probability of such equivalence depends upon and sharply i n c r e a s e s as the number of regions (R. and R.) for which ~(k) is evaluated 1
and a v e r a g e d i n c r e a s e s .
3
P
All modified quantifiers a r e e s t i m a t e d in a single evaluation of P. by considering all 1
substitutions for the v a r i a b l e s in the f o r m u l a ( r e g a r d l e s s of whether the v a r i a b l e s are quantified by u n i v e r s a l or existential quantifiers).
Such methods (which r e s u l t in e s t i m a t e s for
E ( F p) ) " r a t h e r than exact values) a r e e s s e n t i a l to reduce computation to feasible proportions. Hence, when P i is evaluated, we obtain an e s t i m a t e of the e a s e of satisfaction E ( F ! k)) of all its subformulae, F (k). Some such s u b t r e e s (on the graph of P.) become d e r i v e d predicates p P 1 a f t e r the simple abstraction p r o c e d u r e is applied. When t h e s e d e r i v e d predicates a r e stored, t h e i r available e s t i m a t e d e a s e s of satisfaction a r e also retained. Each of the K
(and ~ ) a r e also a v e r a g e d 1 for all R. e K (in c a s e s (a) and (b)) o r P P _~ ~ for all R. c ~ (in c a s e (e) - to be discussed). Let K (and Kp) r e p r e s e n t such a v e r a g e s , J P Then a mutation of subformula F (k) (of P.) which r e s u l t s in another subformula, ~(k) such "" * p 1 p ' h ~ (k) 2: = t at E(F ) = K (or K , if appropriate) may r e s u l t in t r a n s f o r m i n g P into an ideal f e a P P P (k) i t u r e , No other mutation of F" " can achieve this r e s u l t . Hence mutations a r e chosen so (~(k)) P that new subformulae r e s u l t where, as closely as possible, E ( F (k )) = K F o r this P P p" purpose, the F (k) in P. a r e c o m p a r e d with the d e r i v e d predicates previously e x t r a c t e d and, p 1 in general, in much simplified f o r m , the p r o c e d u r e is: (1) D e t e r m i n e whether a relaxing, r e s t r i c t i n g or neutral mutation (to be discussed) is r e q u i r e d . (2) Examine that distributive i b r m ,
p(k). , of the f e a t u r e , 1
P., to be mutated (usually 1
that feature with the highest m e a s u r e of s u c c e s s ) which, when c o m p a r e d (by the simple a b stration procedure) to other f e a t u r e s previously generated, yields a derived we4caie ~ e f o r m u l a is the longest of those of d e r i v e d predicates so obtained (k is thus chosen to f a c i l i tate the c o m p a r i s o n s (in the abstraction procedures) of the new feature generated with p r e viously generated features).
When we evaluate P. (A) we thereby compute K and ~'(k) for 1 p p all p. At f i r s t the collapsed f o r m (see Section 5) of p(k) is examined. 1 1" M o r e refined methods than averaging a r e used; see next section.
105
(3) C o m p a r e e a c h s u b f o r m u l a ,
F
(k)
, w h i c h would r e s u l t f r o m e a c h m u t a t i o n of the P r e q u i r e d type (on a c o r r e s p o n d i n g subformtfla F (k) of P.) with e a c h of the d e r i v e d p r e d i p 1 c a r e s w h i c h h a s bee~ stored. C o m p a r i s o n p r i o r to m u t a t i o n avoids t h e e x a m i n a t i o n of m a n y p o s s i b i l i t i e s (see R o t h e n b e r g , 1973b).
A l s o , of m u t a t i o n s w h i c h add s u b f o r m u l a e within t h e
s c o p e of a q u a n t i f i e r (see Section t2) only t h e addition of s u c h s u b f o r m u l a e a s a r e i n c l u d e d in the list of d e r i v e d and a t o m i c p r e d i c a t e s a r e c o n s i d e r e d . (4) Select t h a t m u t a t i o n w h i c h p r o d u c e s a s u b f o r m u l a ,
~(k) (of t h e new f e a t u r e ) , P w h i c h is i d e n t i c a l to a d e r i v e d p r e d i c a t e w h o s e e a s e of s a t i s f a c t i o n m o s t c l o s e l y a p p r o x i m a t e s that required
(i. e., E(F~))~Kp;_
see above discussion). 1 If there are several such
m u t a t i o n s , s e l e c t t h a t which r e s u l t s in t h e m o s t r e i n f o r c e d (i. e. with the h i g h e s t ( r e i n f o r c e m e n t " - - s e e Section 11) d e r i v e d p r e d i c a t e . Section 5) "P(k) 1 one l a y e r and t r y again. t r i b u t i v e f o r m of P.. If this also fails:
If r e p e a t e d d e c o m p r e s s i o n f a i l s , t r y a n o t h e r d i s -
(5) C o m p a r e the f e a t u r e s to be a l t e r e d , Section 8) which have b e e n s t o r e d .
If none s u c h e x i s t s , d e c o m p r e s s (see
P i ' with e a c h of the p r e d i c a t e f o r m s (see
Select a p r e d i c a t e f o r m (if such exists} which r e s e m b l e s
P. (or one of its s u b f o r m u l a e ) in the s e n s e t h a t , if c o r r e s p o n d i n g s u b f o r m u l a e of P. (which 1 1 m u s t have the s a m e f r e e v a r i a b l e s as the d u m m y p r e d i c a t e ) r e p l a c e the d u m m y p r e d i c a t e s , a f o r m u l a r e s u l t s which c o n s t i t u t e s a m u t a t i o n of P. of the r e q u i r e d type. If s u c h r e s u l t i n g 1 f o r m u l a contains a n o t h e r d u m m y p r e d i c a t e , r e p l a c e it by the m o s t r e i n f o r c e d d e r i v e d p r e d i c a t e (or a t o m i c predicate} with the s a m e f r e e v a r i a b l e s -- s e e l a s t p a r t of E x a m p l e 6 low.
be-
If a choice of p r e d i c a t e f o r m s e x i s t s , s e l e c t t h a t with the h i g h e s t r e i n f o r c e m e n t .
If
no s u c h p r e d i c a t e f o r m e x i s t s , a m u t a t i o n of the r e q u i r e d type is c h o s e n at r a n d o m and the footnote below a p p l i e s . Criteria restricting such random choice exist.
The d e t a i l s (Rothenberg, t'973b) of
t h e above p r o c e d u r e s a r e c o n s t r u c t e d so t h a t it i s e x t r e m e l y unlikely t h a t r a n d o m choice will b e n e c e s s a r y (except e x t r e m e l y r a r e l y )
a f t e r t h e initial s t a g e s of the l e a r n i n g p r o -
c e d u r e (note t h a t , a l m o s t always, s o m e m u t a t i o n will b e t t e r a p p r o x i m a t e the r e q u i r e d e a s e of s a t i s f a c t i o n of a s u b f o r m u t a of P.1 t h a n a n o t h e r ) . It is, of c o u r s e , p o s s i b l e (in s o m e applications) to c o n s t r u c t (from c e r t a i n a t o m i c p r e d i c a t e s ) i n h e r e n t l y t w o - v a l u e d p r e d i c a t e s (such a s t h a t a s e t m u s t c o n t a i n a n even n u m b e r of points) so that q u a n t i f i e r m o d i f i c a t i o n c a n n o t guide the m u t a t i o n p r o c e d u r e .
In t h i s c a s e , t e c h n i q u e s which develop t h o s e of S e c -
t i o n 13 m u s t be u s e d and m u c h g r e a t e r u s e of r a n d o m choice r e s u l t s .
1
If the e a s e of s a t i s f a c t i o n of a d e r i v e d p r e d i c a t e (or s u b f o r m u l a r e s u l t i n g f r o m a p o s s i b l e mutation) is not a v a i l a b l e by the m e t h o d s d e s c r i b e d , it m a y be e s t i m a t e d by v a l u a t i o n s of the f o r m u l a in r a n d o m l y c h o s e n s m a l l s p a c e s (as well as by o t h e r e s t i m a t i n g t e c h n i q u e s - s e e R o t h e n b e r g , 1973b).
106
Example 6 Suppose we wish to r e l a x P21(A), which a s s e r t s that the region substituted for A has a point which is "between"lno two other points in the region (i. e . , a "cusp" if the region is connected).
Let D14(x,y) ~ Vz E~c(z,A)V ,V Db(Y,X, Z)~ and let Db(Y,X,Z) (previously
defined} be in our list of derived predicates so that the collapsed form of our chosen d i s tributive form of P21(A) is
Q1
Q2
Suppose we wish a relaxing mutation and our list of derived p r e d i c a t e s contains Da(X, y) (previously defined) and
DI5(X) - Vy E,~(y,A) VDa(X,y) VDl4(x,y~ and E(DI5(X ~ ~ K 2 (which corresponds to Q2 in the distributive form above).
Then our
mutation (of type 1 -- Section 12) yields a new feature,
-- 3xE a.A)
wy
[y. v,aCx, y? vD14[x.y?)
Suppose we could not find D15(x) or any other s u i t a b l e derived predicate.
We then d e c o m -
p r e s s P21(A} one l a y e r to obtain
Q1
Q2
Q3
Suppose we now find among our derived p r e d i c a t e s D16(x,y) = Vz E,,¢(z,A)v "~Dbly, x , z ) VDa(X,Z)~ , and E (D16(x,y}) ~ K3" The r e s u l t of our mutation yields:
(which formula has the same significance as that resulting from our f i r s t mutation). If instead of D16(x,y), we found among our derived p r e d i c a t e s
, and E (D17(x)) = K2' our mutation of type 5 -- Section 12 yields 1
Use definition of "betweeness" in Example 1, p. 8 3
107
P(.k} ~ :Ix ~C{X,A)V'Vy(~,,c[y,A-~ VDa[x,y ] VVz [-,~(z,A} VDb(Y,X,Z) VDa(X,Z)]~ . Suppose no suitable derived predicate is found after examining decompressed forms of the various distributive forms of P21' but we discover among our predicate forms S10(A, KI[X,y], K 2 ~ , y ) : "~X ~c(X,A) VVy('v~ ~ , A ] VK 1 Ix, Y3 VK2 ~,y-J)~ We first note that D14(x, y) has the same variables as K2(x, y) and hence D14(x, y) replaces K2(x, y). Then, if Da(X, y) is the most reinforced derived predicate with two free variables, Da(X, y) replaces Kl(X, y) and our mutations will yield the same result as our first mutation in this example.
(The r e a d e r may find it amusing to find models of the formulae resulting
from the above mutations. ) Possible mutation operations a r e too numerous for extensive illustration here -- see Rothenberg (1973b}. We now consider ca se (e) of the previous section where Rt ~ ~ strongly satisfies P"I We wish this not to be the case and hence we consider a relaxing mutation if R ~ K fails to q
satisfy P. and a restricting mutation if R i strategies are unlikely to be successful.
sidered.
satisfies, P..
If R
strongly satisfies P. both
q I q I In all these cases a neutral mutation should be con-
This also applies when both Rq and R t fail to satisfy P..1 When modified quanti-
f i e f s are evaluated, in all cases both universal and existential quantifiers a r e modified and values of K
and ~(k) are computed for all such quantifiers. Suppose a neutral mutation of P P P.i is appropriate, and within the scope of U~(k) (with subformula F p(k)) there is another p quantifier, ~q-(k) (with subformula F (k})q such that Qp is a universal quantifier and Qq is existential (or vice v e r s a ) .
Then, if t~ ~ K , a neutral mutation of F (k} which does not q P P alter the ease of satisfaction of F ~k~ is appropriate. In general, a comparison of K and P P will indicate the degree to which (if any} and in which way a neutral mutation should alter q I1~ the ease of satisfaction of F ~1. The extent to which such a mutation may alter the overlap P v (see Section 10) of P. and the feature, P. (which r e s u l t s from the mutation}, without disturb1
1
ing E(P i} (i. e . , such that E(P.)~I E(P.}}I is indicated by min(Kp, Kq). In place of a neutral mutation, a "composite mutation" which combines a relaxing and r e s t r i c t i n g mutation may be used.
These will not be discussed here (see Rothenberg, 1973b), but some indications
as to the general scheme are shown: The required overlap of P. and P. when a neutral or 1
1
composite mutation is performed may be determined by examination of the measure of s u c cess, 0(Pi,K, ~} (see Section 10}. Neutral mutations are chosen by comparison with p r e d i cate forms (somewhat as in step (5) above}: If the o r d e r of variables in a dummy predicate in a predicate form (which matches P.1 as in step (5)) differs from the order of variables in a corresponding subformula of P~ to be mutated, the o r d e r of variables in that subformula
108
is
a l t e r e d to match those of the dummy predicate.
If a predicate f o r m differs f r o m P
in
the identity of a corresponding subformula (to the subformula to be mutated), the subformula of P. is r e p l a c e d by that of the predicate form, 1
k ^ Note that when, for some feature P . F ( ) - F(k)^ F (k) K will e s t i m a t e E'F(k)A F (k)~j while ¢.(k) .~(k) win.. estimate . 1 • P E(Fq . (k~.) and r E(F . r(k).)p respectively. If F (k) Ix ann~ r~
qF and F(k) r
r q r (k) become derived predicates, we will be able to estimate the "overlap" of F (k) and q r q
" This is useful in selecting neutral mutations of a feature which r e p l a c e a subformula
which is identical to F (k) by one which is identical to F (k) in such fashion as to control the q r overlap between the original and mutated feature. In general, we compute a I~ and K for each quantifier of a feature and for each P P R i e I~ and R.] c ~. Hence these values of modified quantifiers should p r o p e r l y be denoted "Kp(Rq)" and "Kp(Rq)'= where Rq r e p r e s e n t s the region substituted for the f r e e variable, A, in P.(A) when these values a r e computed. 1
When selecting a mutation, p r o p e r t i e s of the
distributions of Kp(R a)_ and K p(Rq) o v e r all Rq ¢ K and o v e r all R q 6 C a r e considered ( a v e r a g e s a r e seldom u s e d - - t h e i r use was mainly for i l l u s t r a t i v e purposes).
importance is min(I~plRqeI~ ) and max(KpIRqeC and min(Kp Rq c C) when Qp is existential. G(P i), as large as possible.
Of p a r t i c u l a r
) when Qp is universal and max(KIRqeK
)
We are attempting to make the satisfaction gap,
Particular properties of the distributions of Kp(Rq) and
Kp(Rq) corollate with our measures
of success and with G(Pi).
Although the detailed strat-
egy for mutation choice is too detailed for description here (see Rothenberg, of its properties can be deduced by the reader from the preceding discussion. Kp(Rq) differs from Kp(Rq) in that it is based upon strou~ satisfaction.
1973b), many Note also that
The condition for
s u c c e s s that Rq< C may satisfy P i ' but not strongly, elucidates the use of I~p(Rq) in the mutation choice procedure.
Also of use in selecting neutral mutations is that it has been
shown (Rothenberg, 1973b) that: (15.1) A neutral mutation on a c o m p r e s s e d formula (for a feature) is m o r e likely to reduce the o v e r l a p between the feature and its mutant then a s i m i l a r mutation on a subformula which is exposed by d e c o m p r e s s i n g the f o r m u l a one (or more) l a y e r s . t6.
Initial F e a t u r e s The s y s t e m r e t a i n s three l i s t s (of fixed m a x i m u m length) of f o r m u l a e : (1) features,
(2) atomic predicates and derived predicates and (3) predicate f o r m s .
Those with lowest
r e i n f o r c e m e n t (except atomic predicates) a r e eliminated when the c o m p u t e r m e m o r y a l l o c a tion for these l i s t s fill (except when complete abstraction is p e r f o r m e d and the predicate h i e r a r c h y is rebuilt).
109
In m o s t applications the t r a i n e r will i n s e r t any p r i o r knowledge he has of the p r o b l e m by the direct insertion Cat any t i m e in the learning procedure) of d e r i v e d p r e d i c a t e s a n d / o r features. When no initial features o r d e r i v e d p r e d i c a t e s a r e provided by the t r a i n e r , the initial f e a t u r e s a r e generated as follows: an atomic predicate is selected Cat random) and is c o m bined with quantifiers and set m e m b e r s h i p p r i m i t i v e s to form formulae which both satisfy the syntactic r e q u i r e m e n t s for features and survive the elimination p r o c e d u r e . itive does not suffice, a conjunction o r disjunction of s e v e r a l is used. possible a r e used. Pt9(A>
If a single p r i m -
As few p r i m i t i v e s as
Using the p r i m i t i v e s for Example 1, such initial features might be:
= 3x~y(~,y-] AVz ~<(z.A)v
P20CA) = ' ~ x 3 y ~ [ x , y - j AYz E~,¢(z,A)v,< ( z , x , z , y ~ )
(strongly satisfied by c i r c u l a r regions) Cstrongly satisfied by l i n e a r partitions of U)
At the s t a r t of the generation procedure g(x, y . . . . ) is used for all v a r i a b l e s (as above) to prevent c o n t r a d i c t o r y or tautologous f e a t u r e s o r f e a t u r e s with e x c e s s i v e l y high o r low e a s e s of satisfaction. Consider features which a r e of the following c o m p r e s s e d f o r m wherein D(x) contains no set m e m b e r s h i p p r i m i t i v e s :
PICA) -- Vx [~(x,A) YD(x)]
(~ Vx [,~c(x,A) --> D(x)])
P2(A) =_ Vx ['~eCx,A) YD(x)]
( - Vx [¢(x,A) --> D(x)])
P3(A) -- 3x [e(x,A)A D(x)] P4(A) -- ~
[-,,e(x, A) A D(X)]
Notice that all realizations of P2(A) also satisfy PI(A) because of the strong satisfaction requirement. 1 PICA), however, is strongly satisfied by all data spaces. (Note that it may no__!tbe satisfied by subsets of a data space and is hence not a tautology. ) P3(A) is strongly satisfied by the null set or an entire data space. P4(A) is strongly satisfied either by the null set or by an entire data space with one point removed. Clearly, at the beginning of the feature generation procedure, we wish to avoid features with very high or low ease of satisfaction (so that some measure of success is probable). Hence we choose features with compressed form similar to P2(A) when generating initialfeatures. More refined restrictions can be deduced (see Rothen berg, 1973b) 1
The above statements may be intuitively illustrated by interpreting "D(x)" as being s a t i s fied only by points on the boundary of the data space.
110
In general, if a mutation r e s u l t s in a tautology, the tautology is r e s t r i c t e d , and if the 1 mutation r e s u l t s in a contradiction, it is r e l a x e d . In initial feature generation, the following o r d e r of mutations is used, which approximates an o r d e r i n g by the probability of d e c r e a s i n g the o v e r l a p of a feature and its mutant: R e s t r i c t i n g Mutations first:
add ~@(xl... x )" to the formula by a conjunction (to the i n n e r m o s t n
phrase) where ~x 1. ..Xn} includes all bound point v a r i a b l e s . second: add"E(xi,A)" or " ~ e ( k i , A)" to a phrase by a conjunction.
This is one
phrase at a time f r o m the i n n e r m o s t phrase outward (i takes on appropriate values) - - s e e example in Section 19. Relaxing Mutations first:
eliminate "~(Xl..._ xn ) f r o m the formula.
second: add " , ~ ¢ { x l . . . __xn)" to the i n n e r m o s t phrase of the formula by a disjunction (where ~ X l . . . X n } includes all bound variables). third:
add "¢(xiA)" or "~-c(.x., A) to a phrase by a disjunction. 1 phrase at a time, f r o m the i n n e r m o s t phrase outward.
This is done one
Neutral Mutations first:
negate a subformula (from the i n n e r m o s t p h r a s e outward).
second: permute the v a r i a b l e s in a subformula. Also, in the initial generation of f e a t u r e s , a selection weighting function assigns s o m e what higher probabilities of selection to mutations which i n c r e a s e the length of formulae. (Such selection weighting function l a t e r p r e v e n t s the generation of f o r m u l a e with an e x c e s s i v e n u m b e r of v a r i a b l e s . )
As soon as m o r e than one successful feature is obtained, simple and
relation abstractions a r e p e r f o r m e d . Note also that initial d e r i v e d predicates of any given number of f r e e v a r i a b l e s (when such a r e r e q u i r e d by the mutation p r o c e d u r e and a r e scarce) may be g e n e r a t e d by combining atomic or derived predicates; e° g., Dl(x, y, z) = Da(X, y) A Da(Y, z); D2(x, y) = 3Z(Db[X, y, z~) 17,
Topology 2 Because of the use of quantified v a r i a b l e s in features ( r a t h e r than constants--i, e . , the
propositional calculus), each feature has many models (i. e . , " g e n e r a l i z a t i o n " -- see Section 1 2
This applies to tautologies and contradictions which s u r v i v e the elimination procedure.
The author wishes to e x p r e s s his gratitude to Prof. John Myhill of Leeds U n i v e r s i t y and to Prof. B e r n a r d Jaulin of the U n i v e r s i t y of P a r i s for their suggestions that a topological approach would prove fruitful.
111
1, o c c u r s ) . other -- i.e.,
F u n c t i o n s m a y e x i s t w h i c h m a p m o d e l s of the s a m e f e a t u r e into e a c h the d a t a s p a c e m u s t c o n t a i n c e r t a i n s y m m e t r i e s .
In g e n e r a l , it i s d e s i r a b l e
t h a t the s a t i s f a c t i o n of f o r m u l a e by r e g i o n s of the d a t a s p a c e be i n v a r i a n t w i t h r e s p e c t to the t r a n s f o r m a t i o n s of s u c h r e g i o n s by " t r a n s l a t i o n " , " r o t a t i o n " , " r e f l e c t i o n " , " d i l a t i o n " (i e . , " m a g n i f i c a t i o n " ) , and " c o n t r a c t i o n " (i. e , " r e d u c t i o n " ) .
It i s c o n v e n i e n t to i m p o s e a
topology on the data s p a c e so that t h i s will hold and so t h a t t h e above t r a n s f o r m a t i o n s m a y be defined.
Also, c o n t r a c t i o n (the i n v e r s e of dilation) c a n v a s t l y r e d u c e the n u m b e r of points in
the data space (and e a c h of i t s r e g i o n s ) .
The a p p l i c a t i o n of o u r t e c h n i q u e s to s u c h a s p a c e
with a r e d u c e d n u m b e r of points c a n r e s u l t in e n o r m o u s c o m p u t a t i o n r e d u c t i o n and, in fact, the p r a c t i c a l i t y of the p r o c e d u r e s developed r e s t upon the u t i l i z a t i o n of s u c h " r e d u c t i o n m a p p i n g s " (to be defined).
The topology d e s c r i b e d below also r e l a t e s the finite s p a c e s h e r e
e m p l o y e d to t h e infinite s p a c e s which they a p p r o x i m a t e and with which model t h e o r y c u s t o m a r i l y d e a l s : (See appendix (p. 124) f o r the m o t i v a t i o n for the definitions below. ) (S, n s) e ff
(i. e . , is a g r a p h i c s p a c e iff
(S = s p a c e ,
ns = neighborhood system)
(a) S is a V - s p a c e (A F r e c h e t s p a c e - s e e S i e r p i n s k y , 1952) w i t h > l element (b) Each point h a s a m i n i m u m n e i g h b o r h o o d (e) E a c h m i n i m u m n e i g h b o r h o o d h a s >2 points (a c o n s e q u e n c e of (d)) (d) ~ and S a r e the only c l o s e d ( o r o p e n ) s e t s ("~" h e r e denotes the null set) Let e a c h point be defined as " a d j a c e n t " to all points in its m i n i m u m neighborhood.
This
induces a m e t r i c ; i . e . , t h e g r a p h m e t r i c d e r i v e d f r o m the n u m b e r of e d g e s in the s h o r t e s t p a t h (along a d j a c e n t points in the graph) f r o m one point to a n o t h e r . for that metric. finite. )
(t7.1)
W r i t e "xy < zw", e t c . G (Note t h a t , b e c a u s e of (d) above, S is c o n n e c t e d , but not n e c e s s a r i l y ^ (S, n s) c I"I
s)
(is a g r a p h a b l e space) iff
np)Cr) ^ (s :
( NINcn p ,, N is
VNi'5 CnP) i 5-- Ni i
minimal?) ^ t
Given a n o r d e r i n g , -~0' of a l l e l e m e n t s of S 2 and of p 2 , we define (where ¢ is a n e l e m e n t of S x S )
1
The l a t t e r condition is added in o r d e r to p r e s e r v e the intuitive notion. Also, p o s s i b l y c o n v e r g e by n e i g h b o r h o o d s r a t h e r t h a n m i n i m u m n e i g h b o r h o o d s is sufficient. Note t h a t " P " h e r e d e n o t e s a s p a c e , not a f e a t u r e ( f e a t u r e s a r e always s u b s c r i p t e d , e . g . " P ~ ") °
112
x y - z w < 0 ¢ - : - x y < 0 z w V 3 v [(xv< 0 zw/k vy <0¢)/x (xy <0ZVAWV < 0 c)~ and so
1 x y - z w >0 c -- xy >0 zWA Vv [(Xv< 0 zw-->vy >0C)A (Xy <0ZV"-~WV >0C)]
(P, np,_< 0) e %(e) (is a uniform graphic space) iff (P, n ) e F
and (Vx, y, z, u e P)
(xy - z u >0 e-->xY>O zu) (and so xy 50 zu ^ xy >0 zu---> 3v [(xv ~0 zu n vy ~0 ¢) v (xy -~0 z v ^ uv -~0 e)~). (P, n ) is a b a s i s for S (given _~0) iff (where 6 is an element of S × S): P , S s a t i s P ties (17. 1) for some ~ and (17, 2)
(3~: S--~ P)(36)(~¢x, y, z , w c S) [(xy- zw >0 6)'-~ (~px,~y) >G (~bz,~v)~
(17.3)
~:S-->PcRd
E
(is an c-reduction map) iff
(a) ~ is continuous where
(i.e., ~b(E')c ~b(E) U(~(E))' for all E c S,
E' is the derived set of E) 2
and onto
(b) Se F (e) U
(c) P i s a b a s i s
for S ( i . e . ,
17. 2 holds with the same ~ and
with 6 = e) (d) Card P < c a r d S The following observations are g e r m a n e in graphable s p a c e s (all r e f e r e n c e s a r e to Sierpinsky, 1952): (a) "x is a limit point of A " means that x is adjacent to an element of A (p. 3). E' (the derived set) consists of add non-isolated points of E, plus the neighboring points of E (p. 3), (b) Only the whole space is closed or open but r e l a t i v e closure and openness makes sense (pp. 4, 6, 15): E C F is closed in F if it is s e p a r a t e d from the r e s t of F by a "white band".
Relative open is the same as relative closed.
(e) "Dense in itself" here means no isolated points (p. 13). 1 Note that our definition can be weakened to include the case where,(xy
Condition (a) may be deducible from the other conditions (see T22, p. 25 of Sierpinsky, 1952, and definition 17, i here).
113
(d) " S c a t t e r e d " means only isolated points (p. 13). (e) Two sets a r e " s e p a r a t e d " if a white band s e p a r a t e s them (p. 16). (f) "Connected in the topological sense coincides with the g e o m e t r i c a l sense (p. 16). (g) " F r o n t i e r " of A is points of A adjacent to points not in A, plus points not in A next to points in A (page 19 - hence c o r o l l a r y on s a m e page). (h) "Continuous mapping" (from one d i s c r e t e space to another) means two neighboring points go into neighbors
or into the s a m e point.
(i) "Bicontinuous map" (bomomorphism) m e a n s " x next to y iff f(x) next to f(y) (p. 28). Reduction maps a r e continuous maps and all p r o p e r t i e s p r e s e r v e d under continuous maps a r e p r e s e r v e d by reduction maps; i . e . ,
connectedness (but not disconnectedness)
boundary connected (i. e . , simply connected), f r o n t i e r s a r e p r e s e r v e d (but boundaries a r e not), etc. The t r a n s f o r m a t i o n s of "contraction" and "dilation a r e defined by reduction maps and their i n v e r s e s .
"Rotation", " t r a n s l a t i o n " and " r e f l e c t i o n " a r e defined (together) by a bi-
unique mapping f r o m a space S to a t r a n s f o r m e d space, P, such that (a), (b) and (c) (but no__~.t(d)) of (17.3) a r e satisfied. ¢~:S ---> P e Rd
C
The following have been proved (Rothenberg, 1973b) where
and where each feature, P., is constructed using the atomic predicate, 1
"<(x, y, z , w ) " : If S
, [M(Pi)] ," • then P = ,I'[M(P::')[- where P~' is the s a m e formula as P. except that 1
1
its degree of satisfaction has been a l t e r e d by a bounded amount which is a function, f, of e and of the syntax of P. - - i . e . , in g e n e r a l (see (14.1): 1
(17.4)
[D(Pi, S) - D(Pi, P)i <-- f(e, I1)
(note again that P is a space, not a feature like P.) 1
where n depends upon the number of o c c u r r e n c e s of the atomic predicate "<(x, y, z , w ) " in the subformula of P'l within the scope of the r e l e v a n t (to D(P l, P)) modified quantifier. Actually f depends upon other factors as well, but m i n i m a l l y so when the p r e i m a g e of each point pic P is chosen so that ¢ is m i n i m a l o v e r all of S4.
(In this c a s e note that if S is
homogeneous, the number of points in the p r e i m a g e of each Pi ¢ p is d e t e r m i n e d by e. ) When S is, for example, a bounded r e c t a n g l e in Euclidean space, and P is finite, f = K / c a r d ( P ) where K is a function of n and specific m e t r i c p r o p e r t i e s of S, and card(P) is a f u n c t i o n o f ¢.
( N o t e : a s e - - - ~ 0, f--~) 0.)
Let S be finite.
Suppose R C S and R is the image of R in P under the above
reduction mapping (henceforth called the d e g r a d e d image of R).
Let each pj ~ ~(R) such
114 that the p r e i m a g e of pj c o n t a i n s e l e m e n t s both of R and S - R denoted by "pj c d(R)".
be c a l l e d a dubious point 1
Then
(17.5)
]D(Pi, R) - D(Pi,~ I < g(E, n, m)
w h e r e m = e a r d ( p j l p j c d ( R ) ) / c a r d ( p j [ P i e R ) and the above r e m a r k s (in (17.4)) about f also apply to g.
L a r g e r v a l u e s of m r e s u l t in l a r g e r v a l u e s of g.
c h o s e n c o n n e c t e d r e g i o n in a u n i f o r m l y s a m p l e d place,
(Note: if R is a r a n d o m l y
m is a p p r o x i m a t e l y equal to its
c i r c u m f e r e n c e divided by its a r e a . ) S i m i l a r (but tighter) bounds e x i s t when ~ e x e c u t e s a t r a n s l a t i o n , r o t a t i o n o r r e f l e c t i o n . Note the s i m i l a r i t y b e t w e e n the teft hand p o r t i o n s of the f o r m u l a e (17.4) and (17.5) and the definition (14. 2) of the s a t i s f a c t i o n gap of P., 1 (17.4) and (17.5) is a p p r o p r i a t e :
G(P.). l
Hence the following a p p l i c a t i o n of
N o r m M l y the data space, U, i s u n i f o r m l y s a m p l e d - along a s q u a r e grid.
2
The
i m a g e of U u n d e r a r e d u c t i o n m a p is s i m i l a r , but i s a s m a l l e r (i. e . , f e w e r points) space. When G ( P . ) > 0 we m a y ( f r o m G(P.)) d e t e r m i n e an 6 (and s e l e c t a c o r r e s p o n d i n g r e d u c t i o n 1 1 mapping) such t h a t the a p p l i c a t i o n of P. to the d e g r a d e d r e g i o n s in P s t i l l r e s u l t s in 1 G(P.) > 0 (see r e m a r k s following (17.4)). F o r s u c h c no i n f o r m a t i o n r e l a t i v e to t h e s a t i s f a e 1 tion of P. is lost by a p p l i c a t i o n of the r e d u c t i o n map. Since f ~ K / e a r d ( P ) (see above) and 1 c a r d ( P ) d e c r e a s e s with ¢, a r e d u c t i o n m a p with a s m a l l E ( i . e . , c a r d ( P ) is large) will r e d u c e G(P i) by a s m a l l amount.
However, the a m o u n t of c a l c u l a t i o n r e d u c t i o n t h a t can
r e s u l t f r o m the a p p l i c a t i o n of s u c h a r e d u c t i o n m a p p i n g i s e n o r m o u s ,
tn addition to u s e s
when G(P.) > 0, the evMuation of P. in a r e l a t i v e l y l a r g e s p a c e m a y be avoided by c o n 1 1 fining such evaluation to r e g i o n s which a r e p r e i m a g e s of r e g i o n s in s m a l l e r s p a c e s w h e r e m o d e l s of P. have b e e n found. ( T h a t is, the e v a l u a t i o n of a f e a t u r e in a s m a l l e r s p a c e p r e l c e d e s t h a t in a l a r g e r , the f o r m e r with a l o w e r r e q u i r e d d e g r e e of s a t i s f a c t i o n than the latter.
E v a l u a t i o n in the l a r g e r s p a c e is then p e r f o r m e d in that r e s t r i c t e d s u b s p a c e w h i c h
i s the p r e i m a g e of a m o d e l found in the s m a l l e r . ) O t h e r m e t h o d s of c o m p u t a t i o n r e d u c t i o n a r e used.
R e g i o n s of the s a m p l e s p a c e which
s a t i s f y (or fail to satisfy) c e r t a i n p r e d i c a t e s have t h e i r i d e n t i f i c a t i o n s s t o r e d t o g e t h e r with the p r e d i c a t e a b b r e v i a t i o n s .
Then "do l o o p s " in f o r m u l a e r a n g e only o v e r t h o s e r e g i o n s
which a r e a l r e a d y known to s a t i s f y s u b p r e d i c a t e s included within t h e s e loops, I
Further
Data d e p e n d e n t u n a r y a t o m i c p r e d i c a t e s (e. g . , D(x) in E x a m p l e t) c a n be included by extending the definition of dubious point to include any P i e ¢,(R) w h o s e p r e i m a g e c o n t a i n s e l e m e n t s with d i f f e r i n g vMues of the u n a r y a t o m i c p r e d i c a t e . Then (17.5) s t i l l a p p l i e s . 2 Note: a s a m p l i n g along the c e n t e r s of a c o v e r i n g of U by r e g u l a r hexagonal t i l e s would permit
a smaller
choice
of c in a reduction
map.
115
computation reduction is accomplished by the use of implications between f e a t u r e s (these a r e known a c r o s s all but neutral mutations -- see Section 12) to avoid u n n e c e s s a r y testing. A s i m i l a r technique may be employed to m i n i m i z e s e a r c h and storage when data independent p r i m i t i v e s a r e r e l a t e d by implication ( " t r e e s e a r c h e s " a r e used). Note also that the data independent portions of features,
p Ii' a r e always evaluated in
~ f
e i t h e r some R. o r M (U) (P~)I (see Section 13), n e v e r in the e n t i r e data space. In this J J ............ m a n n e r e n o r m o u s computations a r e avoided. F u r t h e r m o r e , P. is evaluated in S by utilizing 1 the p r o p e r t i e s of nested n o r m a l f o r m to r e m o v e points f r o m the space until a model (i. e . , strong satisfaction) is encountered. then S = ']M~U). ( N F j ) l. '
Roughly (see Rothenberg, 19735), if S 5~ Iyf (U) (pi)] , j Then points a r e r e m o v e d , one at a time so that ~,P is violated a s
soon as possible. 18. H i e r a r c h y of Representation Here the size of the p r o b l e m is reduced (and its r e p r e s e n t a t i o n changed) by the use of h i e r a r c h i c a l l e v e l s wherein objects a r e e x p r e s s e d as collections of subobjects, e t c . , all such objects being defined by f e a t u r e s which a r e s i m i l a r l y h i e r a r c h i c a l l y organized.
At each
such h i e r a r c h i c a l level the techniques previously d i s c u s s e d a r e independently applied. All "atomic p r e d i c a t e s " ( " p r i m i t i v e s " ) , " f e a t u r e s " , "points", " o b j e c t s " and "object c l a s s e s " (see Section 2) will now have " f i r s t l e v e l " prefixing their names.
Also the following
r e s t r i c t i o n s a r e added to our syntax: (18.1)
(a) F o r each f i r s t l e v e l feature, Pi(A), which begins with a string of existentially quantified set v a r i a b l e s (e, g.,
" 3 B 3 C 3I) . . . . "), those set
v a r i a b l e s (B, C, D . . . . ) a r e allowed to range only o v e r models of other features already in the d e s c r i p t i v e basis (i. e . , objects). (b) F u r t h e r m o r e , the above set v a r i a b l e s may no_itbe identical with the region that is substituted for the free v a r i a b l e , A, in Pi(A) above (i. e . , " ( A = B)A ~, (A = C)A ~ (A = D) . . . ) . Also the following, called the identification m e a s u r e is added to our m e a s u r e s of s u c c e s s (see Section 10) and is used when generating f e a t u r e s : (18,2)
0(P.,I~,C) = T ( P . , K ) + 7 ( P . , C ) 1 1 1
•
Whenever we advance a l e v e l (from the n th to the n+ 1st level) the following operations a r e p e r f o r m e d (see Example 7 which follows). (18.3)
(a) All n th level objects, A 1 , A 2 . . . A r become n+ 1st l e v e l points, P l ' P2 . . . . ' Pr"
116
(b) All nth level f e a t u r e s ,
PI(A), P2(A) . . . .
P
(A) become n + l s t level m p r i m i t i v e s (atomic predicates), V:l(X), ~2(x) . . . . ~m(x). (c) If P.(A) is an n th level feature which begins with a s t r i n g of ~ i s t e n t i a l l y 1
quantified set variables, i . e . , P.(A)I = 3B1 . . . . . 3 B j ~'(A,B 1. . . . . ~)'] then F(A,B 1
. . . . .
(all sets of n
th
Bj) becomes an n + l s t level primitive f?(Xl,X2 . . . . xj+l). .... level objects which together satisfy F a r e known (see (18.1).
Hence the values ("true" or "false") of F(A, B 1. . . .
B.) are known for all 3 arguments and so a r e the corresponding values of fi(x 1,x 2 . . . . X]+l). This procedure p e r m i t s both relations between objects and between other relations
to be expressed at successive l e v e l s - - s e e i l l u s t r a t i o n below and Rothenberg (1973b). Whenever the following conditions a r e met we advance a level:
(18.4)
F o r each R.t ~ ~ there exists a set of objects (A1,A2 . . . .
A n~ such that
n
(a)
U A k = R.1 k=l
and
(b)
n < card (R.)
and
(e) There does not exist an R. c ~ and a set of objects ( B 1, B 2 . . . . that both
1
Bm} such
~1 B£ = Rj and all such B~ (and n-tuples of B~) satisfy the s a m e
next level latomic predicates as the A k (and n-tuples of A k) in an (A1,A 2....
A n } satisfying condition (a) of (18, 4) above.
(Note that these
atomic predicates include all features and relations constructed therefrom (as in (18.3) above) at the level being examined.
This condition a s s u r e s that there
exist no R. cK and R. E ~ which consist of unions of indistinguishable (by 1 j features satisfied) objects.) The above r e s u l t s in a change of h i e r a r c h i c a l level whenever a reduction in the data to be r e p r e s e n t e d r e s u l t s thereby.
(Provis ion for t r a i n e r intervention when d e s i r e d exists
(see Rothenberg, 1973b). ) The definition of descriptive basis r e m a i n s the same as in Section 2 except that all features at all levels are included,
nth level objects a r e defined in t e r m s of n th level
points ( i . e . , n - 1st level objects) exactly as f i r s t level objects are defined in t e r m s of th first level points, Similarly, the definitions of n leve ! f e a t u r e s , object c l a s s e s , e t c . , a r e defined as before in t e r m s of the other entities at the s a m e level.
However, the defi-
nition of nth level primitive is extended to include the atomic predicate, c (i" k)(x(k-i),x(k))
117
wherein k and i are fixed (of course, k - i < k < n + 1 where n is the h i e r a r c h i c a l level). This atomic predicate is called r e c u r s ive containment and a s s e r t s that there exists a sequence of points at the indicated (by the superscript} levels such that, if we put x (k - i ) in front of the sequence and x (k) at the end, each t e r m in the sequence is an element of the next (there are i + 1 t e r m s in such a sequence}.
(This allows statements about first level
points in second level (and higher level features -- see Rothenberg, 1973b). ) Example 7 Suppose we wish to distinguish photographs (from above} of any n u m b e r of stacked discs from pictures of any other objects.
Then the R.1 ~K will consist of d i a g r a m s of the
following form:
.
(1)
- ~ f
~
(5) (4) (3)
etc.
R. c ~ will consist of d i a g r a m s which cannot be formed by a finite n u m b e r of overlapping ] c i r c l e s . Suppose we have thus far generated feature Pi(A) = Dd(A) A "4x ~rVz (c (x, A)--> ~, < Ix, y, x, z~) (see Example 1, Section 1) which is strongly satisfied by dark c i r c u l a r regions and P2(A) = ~ B 3 x ( ~ x , A ] h c [ x , B ] ) .
(Note that 0 ( P 2 , ~ , ~ } ,
(18.2), is very high in this example. ) Note that conditions (18.4) are now met.
see
Hence we
advance a level as follows (see (18.3)): (1) All first level objects become second level points, P l ' P2 . . . . . Pr"
(2) PI(A) becomes a second level atomic predicate, al(X) ( s a t i s -
fied by second level points}.
(3) P2(A) is t r a n s f o r m e d by eliminating the existential quan-
tifier at the left of its f o r m u l a to form F(A, B) = 3x(¢ Ix, A] A ~ Ix, B']) which is satisfied by pairs of f i r s t level objects (which are now second level points}. F(A, B) now becomes a second level atomic predicate, ~(x, y). We may now, at the second level, construct the
secondlevel feature, P~IZ)(A)='"Vx(~ _ [x,A~-'-~I(x)
h Vy [e(y, A) --'-> ~ (x, y ) ] )
fied only by R. c K. 1
The following definitions should also be added to those of Section 2:
whichiss a t i s -
118
Description: This is a list of all features at all levels used in the descriptive basis and lists all objects strongly satisfying and satisfying each of these features and the elements 1 of each such object, Filter predicates: (optional) These are formulas of the same form as f i r s t level features which must be satisfied by regions of the sample space called infra-objects in o r d e r for sub-regions of these infra-objects to be candidates for becoming first level objects. These may be used to substantially reduce the number of regions which must be considered for satisfaction of a firs t level feature.
Typically, filter predicates might include formulae
whose satisfaction guarantees that all points in a region have data values in a specified range and/or that such regions form connected sets.
Filter predicates might also be used
to eliminate the need for data dependent features. 19,
Flow of the System Features are evaluated by whether they axe strongly satisfied by the required
regions and perform required partitions (as initially specified).
Other c r i t e r i a (in the
absence of initially specified objects and partitions) include feedback to a human trainer wherein objects in the description of a picture are projected on a cathode ray tube for comparison with the input picture.
Also, at the request of the trainer, such objects may be
successively replaced by other regions which satisfy (and strongly satisfy) the same features as these objects, so that the t r a i n e r may " s e e " what the system does not " s e e " as well as what it does " s e e " (see Rothenberg, 1973b)).
The trainer thereby evaluates the p e r -
formanee of the system, which he indicates by means of positive or negative "feedback". Note again that the t r a i n e r may include any prior knowledge he has of the problem by direct insertion at any time of derived predicates and/or features.
The system, in a practical
application, thereby begins "learning" as if it had already "learned" all such knowledge provided by the trainer. When feedback is absent and success has been obtained, the system nonetheless continues to compute, attempting to increase the satisfaction gap, to lessen the number of quantifiers in features and to eliminate unnecessary objects; i . e . , those which are in the description, but which are not relevant to the problem as defined initially (i, e . , do not correspond to any element of K or ~) or by the t r a i n e r , Very briefly, the overall flow of the system is as follows (trainer interaction is not included): 1
In some applications, if two features at the same level have identical data dependent e x pressions in their formulae and if an object which strongly satisfies one feature is properly contained in an object which strongly satisfied the other feature, the former object is omitted from the description.
119
(1) Initial f e a t u r e s a r e either provided by the t r a i n e r or constructed as in Sec. 16. (2) Simple and r e l a t i o n abstraction a r e p e r f o r m e d .
(When the lists a r e full
r e c o n s t r u c t the predicate h i e r a r c h y and if t h e r e a r e still too many f o r m u l a e , those with lowest r e i n f o r c e m e n t a r e eliminated. ) (3) That feature with the highest r e i n f o r c e m e n t is selected for mutation. (4) The mutation is chosen as in previous sections.
If the new feature m e e t s
syntactic r e q u i r e m e n t s and has a higher m e a s u r e of s u c c e s s than s o m e o t h e r feature (and if the l i s t of f e a t u r e s is full), the feature with the lowest m e a s u r e of s u c c e s s is e l i m i n a t e d and the new feature is added to the list of f e a t u r e s . (5) Is the satisfaction gap > 0 ?
If so, p e r f o r m reduction map.
(6) Is the satisfaction gap large enough to stop computing? ( P r e d e t e r m i n e d by trainer. ) (7) A r e the c r i t e r i a for advancing a h i e r a r c h i c a l l e v e l m e t ? level, and r e t u r n to step (1) at the next level.
If so, advance the
If not,
(8) R e t u r n to step (2). The p r o c e d u r e is, of course, e x t r e m e l y complex in detail 1 - s e e Rothenberg (1973b). 20.
Open Questions A c h a r a c t e r i z a t i o n of the s e m a n t i c s of predicates which can be constructed f r o m a
given set of p r i m i t i v e s is r e q u i r e d .
A partial d e c i s i o n procedure which, as far as possible,
avoids the generation of features which a r e conjunctions of c o n t r a d i c t o r y p r e d i c a t e s is needed.
The generation of p r e d i c a t e s with no finite model should also be avoided where pus-
sible (e. g . ,
P. (A) =- Vx~Vy-qz(~ [x, A] A ~ [x, A]---> [e (z, A) h D b (x, z, y)]) is satisfied by c o n 1
vex sets only when the
data
space is infinite--otherwise it is vacant).
It would be useful
to know in advance the m i n i m u m depth (of nested phrases) of the m o s t deeply nested p r e d i 2 cate needed to s e p a r a t e all inequivalent regions in a data space. The r e p l a c e m e n t of " e a s e of satisfaction" by an e a s i l y computed m e a s u r e which r e t a i n s the o r d e r i n g of formulae, F. (by E(F.)), when the data space is infinite is d e s i r a b l e . 1
1
Also significant is the question of whether the f r a c t i o n consisting of the n u m b e r of logically inequivalent f o r m u l a e with n quantifiers divided by the total number of f o r m u l a e with n quantifiers i n c r e a s e s o r d e c r e a s e s as n i n c r e a s e s .
This indicates the chances of
skipping and not r e c a p t u r i n g a needed feature in a "depth f i r s t " s e a r c h on the t r e e of f e a 1 2
Actually, a s t a t e - s p a c e s e a r c h p r o c e d u r e is u s e d - - s e e Nilsson (1971).
Note that a m a x i m u m m u s t be set on the number of quantifiers in a feature, both for p r a c t i c a l r e a s o n s and to avoid t r i v i a l solutions (L e , , no generalization).
120 t u r e s and is r e l e v a n t to s e a r c h s t r a t e g y (i. e . , to proceed by l a r g e mutations which change the number of quantifiers o r by s m a l l mutations ?), The extension of the s y s t e m to include r e s t r i c t e d t h i r d - o r d e r quantification Cover f e a t u r e s in lower levels of the d e s c r i p t i v e basis) is being considered.
This would enable
the s y s t e m to d e s c r i b e objects of s i m i l a r kind (with r e s p e c t to the d e s c r i p t i v e basis) and would i m p a r t to it s o m e s e l f - d e s c r i p t i v e power of a limited sort.
However, p r o b l e m s
a r i s e due to the potential incompleteness of the d e s c r i p t i v e basis at any t i m e during the learning procedure. Another important p r a c t i c a l problem is the elimination of f e a t u r e s which a r e t r i v i a l (as well as vacant o r tantologous) in a p a r t i c u l a r application (e. g . , -~x3y ( e ~ , ~ ] A c [ y , A ] A ~ [ x , y ] ) } .
P. (A) = 1 Such methods have already been derived, but the need
for additional techniques will doubtless become evident during e a r l y e x p e r i m e n t s using the program. It appears that for cerLain choices of p r i m i t i v e s (such as those in the e x a m p l e s here) the use of Skolem functions (Schoenfeld, 1967) may r e s u l t in eliminating existential quantifiers and hence in substantial simplification of the mutation procedure. Of g r e a t e s t significance, however, is that it appears that the e n t i r e s y s t e m can be reduced to a f i r s t o r d e r s y s t e m by (a) an appropriate choice of p r i m i t i v e s (see Beth defineability t h e o r e m (Schoenfeld, 1967; Reyes, 1969)) o r (b) the development of a p r o c e d u r e for r e p l a c i n g some second o r d e r f o r m u l a e by first o r d e r p r e d i c a t e s r e c u r s i v e l y defined.
For
example, a second o r d e r f o r m u l a a s s e r t i n g that "point x is connected to point y " may be r e p l a c e d by
F(x,y) = ~Da(X,Z) A Da(z,y) ] V [Da(X,Z) h F(z,y)]) The computation expended in the p r o c e d u r e s d e s c r i b e d i n c r e a s e s rapidly as the number of p r i m i t i v e s used i n c r e a s e s .
Hence, the choice of the s m a l l e s t number of p r i m i -
t i v e s which a r e appropriate to a problem and which r e s u l t in a sufficiently e x p r e s s i v e l a n guage is of central importance. <(x,y,z,w),
(Note that the p r i m i t i v e used in our i l l u s t r a t i o n s ,
is e x t r e m e l y p o w e r f u l - - e . g . ,
" x y + y z = x z " can be defined by using Db(X,y,z)
and then we may define "xy +vw = pq" by using x y + y z = xz, = (V, z, v, w) and = {x, z, p, q), etc. ) When many p r i m i t i v e s a r e r e q u i r e d , it is advisable to distribute them among the h i e r a r c h i c a l levels used--i, e . , so that p r i m i t i v e s not at the f i r s t level a r e introduced at the second ( e . g . ,
"card(A)"), etc,
F u r t h e r work on this question could prove to be of c o n s i -
derable p r a c t i c a l value. It should also be noted that although this has not been done in the e x a m p l e s here, it appears profitable to e x p e r i m e n t by using the s y s t e m with p r i m i t i v e s specifying (optical) s p e c t r a l o r holographic information.
121
Appendix to Section 17 (Topology) All r e f e r e n c e s a r e to S i e r p i n s k i ' s (1952) G e n e r a l Topology, f i r s t c h a p t e r (which d e a l s with F r e c h e t s p a c e s ) .
F i r s t we d r a w a t t e n t i o n to s e v e r a l r a t h e r p e c u l i a r p r o p e r t i e s of
F r e c h e t s p a c e s which a r e no___t.topological t spaces: (1) In a F r e c e t s p a c e t h e s u m of two c l o s e d s e t s n e e d not be c l o s e d (p. 11, bottom). the i n t e r s e c t i o n of open s e t s n e e d no_._~tbe open.
Hence
[Note t h a t c l o s e d s e t s a r e s e t s w h i c h c o n t a i n
all t h e i r l i m i t e l e m e n t s , p. 6 (4.), and open s e t is defined a s the c o m p l e m e n t of a c l o s e d s e t , p. 11 (6.), and t h a t t h e condition " e v e r y n e i g h b o u r h o o d of a n e l e m e n t of K is a n open s e t " i s n e c e s s a r i l y t r u e only when K s a t i s f i e s the definition of a t o p o l o g i c a l s p a c e (p. 38, middle). See also p. 12, T h e o r e m 5.~ (2) In a F r e c h e t s p a c e the condition t h a t " a n e i g h b o u r h o o d c o n t a i n s a n e l e m e n t " does not i m p l y t h a t "the n e i g h b o u r h o o d is a n e i g h b o u r h o o d of t h a t e l e m e n t " .
That is, a neighborhood
m a y c o n t a i n an e l e m e n t of which it i s not a n e i g h b o r h o o d (i. e . , x c N(y) and N(y) ~ N(x) ). Note also that conditions ~/ and 5 on p. 38 m a y not b e s a t i s f i e d - - i , e . , d i s t i n c t e l e m e n t s m a y not b e c o n t a i n e d in d i s t i n c t n e i g h b o r h o o d s . B e f o r e a t t e m p t i n g to e x p l a i n the m o t i v a t i o n for t h e a x i o m " T h e only c l o s e d s e t s a r e the null s e t and the u n i v e r s a l s e t " , we p r e s e n t an i l l u s t r a t i o n of s u c h a F r e c h e t s p a c e : Let our space,
S, c o n s i s t of nine points: S = { a , b , c , d , e , f , g , h , i } .
F o r convenienc
convenience, picture them in a rectangular array: a.b.c. d.e.f. g.h.i. L e t the m i n i m u m n e i g h b o r h o o d of a point c o n s i s t of a l l points w h i c h a r e n e a r e s t to it in the above r e c t a n g u l a r a r r a y (i. e . , t h o s e points d i r e c t l y above, below, to t h e r i g h t of and to the left of it (if t h e r e a r e s u c h
points) ). Thus o u r m i n i m a l n e i g h b o r h o o d s a r e : :
{a,b,d}
N(b) = {a,b,c,e}
N(C) = { b , c , f } N(d) = {a, d, e, g}
N(e) = {b, d, e, f,h} N(f) = {c.e,f,i} N(g) = {d, g, h}
= {e,g,h,i} N(i) = {f.h.i} F o r a l l x, the n e i g h b o r h o o d s of x will be all s e t s which include N(x).
122
Note t h a t all s e t s e x c e p t the null and the e n t i r e s e t a r e n e i t h e r open n o r closed. e x a m p l e , let E = ( c , e, f, g, h , i } . limit elements). setof
F.
Let E ' denote i t s d e r i v e d set of E (i. e . , the s e t of a l l i t s
Let F denote the c o m p l e m e n t of E ( i . e . ,
Then F = {a,b,d t.
Note a l s o t h a t e e F ' .
Hence
For
Notethat beE'. ~(F'c
Hence
S - E ) and F ' denote the d e r i v e d ~(E'aE)
and E i s not elosed.
F) and F is not c l o s e d .
The s a m e a r g u m e n t c a n be r e p e a t e d f o r a l l o t h e r n o n - e m p t y p r o p e r s u b s e t s of S. A l s o , S c a n be m a d e a s l a r g e a s we w i s h by adding- e l e m e n t s to the r o w s and c o l u m n s of t h e rectangular array.
If the m i n i m a l n e i g h b o r h o o d s a r e s i m i l a r l y defined a l l n o n - e m p t y p r o p e r
s u b s e t s of S will be n e i t h e r open n o r c l o s e d . O b s e r v e t h a t the i m p o s i t i o n of the conditions on a F r e c h e t s p a c e t h a t " e a c h point h a v e a m i n i m a l n e i g h b o u r h o o d and t h a t ~ and S be the only c l o s e d s e t s " is c o n s i s t e n t with t h e definition of a m i n i m a l n e i g h b o r h o o d of a point in a finite s p a c e as c o n t a i n i n g t h o s e points " a d j a c e n t " to t h a t point.
The conditions also p r e v e n t the s h r i n k a g e of n e i g h b o r h o o d s to the
t r i v i a l c a s e w h e r e they contain only one point.
T h i s is as m i g h t i n t u i t i v e l y be e x p e c t e d in a
finite s p a c e , and the p r e s e r v a t i o n of conditions (a) - (i) of Section 17 is also in a c c o r d a n c e with intuition.
Also, we a r e a s s u m e d t h a t o u r e n t i r e s p a c e is c o n n e c t e d so t h a t a g r a p h
m e t r i c m a y be i m p o s e d (Section 17). Note that condition 5 on page 38 of S i e r p i n s k i does not hold, and h e n c e we do not h a v e a topological s p a c e . A n o t h e r a l t e r n a t i v e was a v a i l a b l e : The condition t h a t "~ and S a r e the only c l o s e d s e t s " m i g h t h a v e b e e n r e p l a c e d by 'tall s e t s a r e both open and c l o s e d " .
T h e n we could h a v e
defined the m i n i m a l n e i g h b o r h o o d s in o u r above i l l u s t r a t i o n n o n - u n i q u e l y : Nl(a) = { a , b }
N2(a) = ( a , d }
Nl(b) = ( b , a }
N2(b) = { b , e }
NI(C) = ( c , b }
N2(c) = ( c , f }
Nl(d) = ~ d , a }
N2(d) = ~ d , e }
N3(d) = { d , g }
Nl(e) =<e,b}
N3(e) : 6, d}
N3(e) : 6'ft
N3(b) = ~ b , e }
N4Ie) : {e,h}
etc. Now e a c h e l e m e n t would be the i n t e r s e c t i o n of all its n e i g h b o r h o o d s .
But now condition
fl on page 38 of S i e r p i n s k i would not be s a t i s f i e d and we s t i l l would not h a v e a topological space.
Of g r e a t e r i m p o r t a n c e , how would we now obtain the n e c e s s a r y
graph metric.
Also, the i m p o s i t i o n of t h i s type of n e i g h b o r h o o d s y s t e m on a finite s p a c e s e e m s counterintuitive.
123
In s u m m a r y , t h e p u r p o s e of the t o p o l o g i c a l s e c t i o n of t h i s p a p e r i s to i m p o s e conditions on the given s t r u c t u r e <S, ..
so t h a t it could b e m a p p e d into a s t r u c t u r e with 1 fewer elements, , s u c h t h a t if a s u b s e t , A, of S i s a m o d e l of a f o r m u l a , F, the image,
f(A), of A (in P) is a m o d e l of a f o r m u l a ,
~, which is obtained f r o m F by the s u b -
s t i t u t i o n of a modified q u a n t i f i e r f o r a u n i v e r s a l o r e x i s t e n t i a l q u a n t i f i e r in F.
1
xy~
zw = x y - z w ~
c
124
BIBLIOGRAPHY Addison, J. , L. Henkin and A. Tarski (1965), "Theory of Models", Proc. 1963 International Symposium at Berkeley, North Holland Co., Amsterdam. Amaret, S. (1962), "An Approach to Automatic Theory Formation", in Principles of Self Organization, Trans. Illinois Symposium on Self Organization (Von Foerster and Zopf, Eds.), Pergamon Press, New York. Banerji, R. (1969), Theory of Problem Solving, An Approach to Artificial Intelligence, American Elsevier Publishing Co., Inc., New York. Bell, J. and A. Slomson (1969), Models and Ultraproducts, North Holland Co., Amsterdam. Block, H.D., N.J. Nilsson and R.O. Duda (1964), "Determination and Detection of Features in Patterns", Computer and Information Sciences (J. T. Tou and R. Wilcox, Eds), Spartan Books, 75-110, Bongard, M. (1970), Pattern Recognition (J.H. tIawkins, Ed.), Spartan Books Bremermana, H.J. (unpublished manuscript), "Artificial Intelligence and Neurobiology", Department of Mathematics~ University of California, Berkeley. Etlentuck, E. (1971), "Direct Products of Relational Systems", U.S. Air Systems Command Final Report, Jan. 1, 1971 (see Rothenberg, 1971) Fogel, L . J . , A.J. Owens, and M.J. Walsh (1966), Artificial Intelligence through Simulated Evolution, John Wiley and Sons, New York. Friedberg, R.M. (1958), "A Learning Machine, Part I", IBM J. Research and Development, 2, 2-13. Frtedberg, R.M., B. Dunham and J.H. North (1959), "A Learning Machine, Part II", IBM J_~..Research and Development,, 3, 282-287. Hawkins, J. (1970), "Textual Properties of Pattern Recognition", Picture Processes and and Psyehopictorics (B. S. Lipkin and A. Rosenfeld, Eds), Academic Press, New York, 347-370. Hintikka, J. (t953), "Distributive Normal Forms in the Calculus of Predicates", Acta Philos~ Fennics, 6, 1. Julesz, B. (1969), "Cluster Formation at Various Perceptual Levels", in Methodology of Pattern Recognition (S. Watanabe, Ed.), Academic P r e s s , New York, 297-315. Kolers, P.A. (1970), "The Role of Shape and Geometry in Picture Recognition", Picture Processing and Psychoptetories (B, S. Lipkin and A. Rosenfeld, Eds.), Academic Press, New York, 181-202. Lee, R. C.R. (1967), "A Completeness Theorem and a Computer Program for Finding Theorems Derivable from Given Axioms", Ph.D. Dissertation, University of California at Berkeley. Lofgren, L. (1967), "Recognition of Order and Evolutionary Systems", in Computer and Information Science - II (J. Tot~ Ed.), Academic Press, New York, 165-175. Minsky, M. (1963), "Steps Toward Artificial Intelligence", in Computers and Thought (E.A. Feigenbaum and J. Feldman, Eds.), McGraw-Hill Book Co., Inc., New York, 406 -450. Muerle, J.L. (1970), "Some Thoughts on Texture Discrimination by Computer", Picture Processing and Psychopictories (B. S. Lipkin and A. Rosenfetd, Eds. ), Academic Press, New York, 371-380.
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Nilss~a, N.J. (1971), Problem-solving Methods in Artificial Intelligence, McGraw-Hill Book Co., Inc., New York. Pessin, A. and D. Rothenberg (in preparation), The Weak Second Order Predicate Calculus as a Programming Language. Reyes, G.E. (t969), "Local Defineability Theory", Annals of Math. Logic, _1, 101. Rosenfeld, A. and B. Lipkin (1970), "Texture Analysis", in Picture Processing and Psychopictorics (B. L i p k i n a n d A . Rosenfeld, Eds.), Academic Press, New York, 309-347. Rothenberg, D. (1971), "An Adaptive Linguistic Model for Pattern Representation and Recognition", Final Report, Air Force Systems Command, USAF, Directorate of Mathematical and Informational Sciences, January t, 1971. Rothenberg, D. (1973a), "A Pattern Recognition Model Applied to the Perception of Pitch", AFOSR Technical Report, January 1969, revised version to appear as series of articles in Mathematical Systems Theory. See also this volume. Rothenberg, D. (1973b), "A Theory of Feature Generation", a series of four articles to be submitted to Mathematical Systems Theory (Copies available from Mathematics Department, Rutgers University, University College, New Brunswick, New Jersey}. Schoenfeld, J. (1967), Mathematical Logic, Addison Wesley, Reading, Massachusetts. Selfridge, O.G. and U. Neisser (1963), "Pattern Recognition by Machine", i n Computers and Thought (E.A. Feigenbaum and J. Feldman, Eds.), McGraw-Hill Book Co., Inc., New York, 237-250. Sherman, R. and G.W. Ernst (1969), "Learning Patterns in Terms of Other Patterns", in Pattern Recognition, Pergamon P r e s s , New York, (Vol. 1), 301-313. Sierpinsky,
(1952), General Topology, University of Toronto Press, Toronto.
Single, J . , C. Chang and R. Lee (1969), "Completeness Theorems for Semantic Resolution in Consequence Finding", Proc. International Joint Conference on Artificial Intelligence, May 7-9, 1969, Washington, D.C. (D.E. Walker and J.M. Norton, Eds.), 281-285. Solomonoff, R . J . (1964), "A Formal Theory of Inductive Inference - Part r ' , Information and Control, 7, i-22. Solomonoff, R . J . (1964), "A Formal Theory of Inductive Inference - Part H", Information and Control, ~ 224-254. Uhr, L. and C. Vossler (1963), "A Pattern Recognition Program that Generates, Evaluates, and Adjusts its Own Operators", in Computers and Thought, (E.A. Feigenbaum and J. Feldman, Eds.), McGraw-Hill Book Co., Inc., New York, 251-268. Wang, H. (1960), "Toward Mechanical Mathematics", IBM J. Research and Development, 4, 2-22.
A MATHEMATICAL APPLIED TO
THE
MODEL FOR PERCEPTION
PERCEPTION OF PITCH
David R o t h e n b e r g Inductive I n f e r e n c e , Inc.
Abstract A m a t h e m a t i c a l model for p e r c e p t i o n which d e r i v e s f r o m a t h e o r y of e f f i c i e n t data r e p r e s e n t a t i o n in the c e n t r a l n e r v o u s s y s t e m is d e s c r i b e d . space,
A p o s s i b l y infinite and continuous
S ( o v e r which s e n s o r y s t i m u l i r a n g e ) , is mapped into a finite s p a c e of d i s c r e t e points,
C (the i n d i c e s on the c l a s s i f i c a t i o n of such stimuli).
An o r d e r i n g of p a i r s of points in S × S
t o g e t h e r with e i t h e r the n u m b e r of c l a s s i f i c a t i o n s r e q u i r e d o r a m a x i m u m t o l e r a b l e e r r o r i s given and a s s u m e d to d e r i v e f r o m f e e d b a c k ( e x p e r i e n c e ) . i s t i c " finite s u b s e t , P ,
The model c h o o s e s a " c h a r a c t e r -
of the s t i m u l u s space, which defines a function and a r a n g e about e a c h
e l e m e n t of the s u b s e t s u c h t h a t the union of all such r a n g e s is m a x i m a l and that the function p r o v i d e s a m e t r i c (on C) which p r e s e r v e s the given o r d e r i n g .
R e s t r i c t i o n s on the c h o i c e of
P d e r i v e f r o m l i m i t a t i o n s in the i n f o r m a t i o n c a r r y i n g c a p a c i t y of the r e s u l t i n g c l a s s i f i c a t i o n system.
A c o n t e x t - d e p e n d e n t " G e s t a l t " d e s c r i p t i o n of p e r c e p t i o n r e s u l t s in which e x t r e m e l y
c o m p l e x and v a r i e d p h e n o m e n a can be p e r c e i v e d without p r o p o r t i o n a t e l y l a r g e h u m a n m e m o r y . F o r e a c h a p p l i c a t i o n s p e c i f i c d i s t o r t i o n s of p e r c e p t i o n in s p e c i f i e d c o n t e x t s a r e p r e d i c t e d . The s y s t e m , in some a s p e c t s , r e s e m b l e s an h i e r a r c h i c a l c l u s t e r i n g s c h e m e .
It is hence u s e -
ful for r e p r e s e n t i n g d i f f e r e n t p a t t e r n s of s a t i s f a c t i o n of s e v e r a l " f e a t u r e s " in a p a t t e r n r e c o g nition s c h e m e by a single s e t of i n t e g e r s with m e t r i c p r o p e r t i e s w h i c h r e f l e c t r e l e v a n c e to the t a s k (i. e . , 1.
a n n - v a l u e d logic r e p l a c e a t w o - v a l u e d logic).
P r o p e r Mappings 1 C o n s i d e r S to be the r e t i n a of the eye.
L e t the p a i r s of points (S x S)
by the endpoints of the p r o j e c t i o n of a r i g i d r o d on the r e t i n a of the eye. would a l t e r with r o t a t i o n of the rod.
be o r d e r e d
Such p r o j e c t i o n
Since we know the r o d is r i g i d , all s u c h p r o j e c t i o n s
would be c l a s s i f i e d a s e q u i v a l e n t with r e s p e c t to s i z e .
The p r o j e c t i o n s of a n o t h e r rod, w h i c h
e x p e r i e n c e has taught us is l o n g e r , would f o r m a d i f f e r e n t c l a s s f i c a t i o n and an o r d e r i n g r e l a tion would e x i s t between the two c l a s s i f i c a t i o n s .
Of c o u r s e , in b i n o c u l a r v i s i o n e a c h point in
S would be a duple ( e a c h e l e m e n t of a duple f r o m one endpoint of the p r o j e c t i o n on e a c h retina) a n d S ~,S would b e a p a i r of s u c h duples. A r e l a t i o n , <_, c a l l e d the initial o r d e r i n g , i.e.
t r a n s i t i v e , c o n n e c t e d and r e f l e x i v e .
is defined on S X S which is a p r e o r d e r ,
Define (x,y)~,~(z,w) to m e a n (x,y) <_(z,w)A (z,w)
<_(x,y) and (x,y) < (z,w) to m e a n (x,y) < _ ( z , w ) ^ ,v ~ z , w ) ~ ( x , y ) ~ .
R e q u i r e that always
(x, y),,J (y, x). 1
See Section 10 for t h e o r e t i c a l b a s i s (which m a y be r e a d before Section 1).
127
It is a s s u m e d that S x S is m a p p e d into the code, C, by s o m e function, w h i c h is dependent only upon the p r e o r d e r i n g of S x S (i. e. c o n t a i n s no s t a t i s t i c a l w e i g h t s o r o t h e r arbitrary parameters).
Since S i s v e r y l a r g e ( p o s s i b l e infinite in the m a t h e m a t i c a l model),
it is a s s u m e d t h a t h u m a n m e m o r y c a n n o t contain a p r e o r d e r i n g of S × S. code m u s t t h e r e f o r e depend upon much l e s s s t o r e d i n f o r m a t i o n .
The m a p p i n g into the
The f i r s t step in th~ s t r a t e g y
of the model i s to find a finite s u b s e t P of S and a function, f, s u c h t h a t f is dependent only on the p r e o r d e r i n g of p x P and f: PxP--->C.
{Consider C to the the set, {1 . . . . . n}, of i n t e g e r s . )
f i s specified in the following m a n n e r : l e t ab denote the p a i r Ca, b) c P x P. choose s o m e p a i r , ~ P defined a s a d j a c e n t .
xP,
We
c a l l e d the link s i z e , such that iff ab <_~y', a and b a r e
The choice of xy i s d e t e r m i n e d by the d e s i r e d c a r d i n a l i t y of C o r by
a given t o l e r a n c e - - t o be d e s c r i b e d .
A s e q u e n c e of e l e m e n t s in P, ( a , b , e , d . . . ) such that a
is a d j a c e n t to b, b to c, c to d, e t c . , i s c a l l e d a chain,
f{a,b); a, b c P ,
i s n o w defined
a s the c a r d i n a l i t y of {the n u m b e r of e l e m e n t s in) the s m a l l e s t c h a i n c o n n e c t i n g a a n d b, m i n u s one ( i . e . , the n u m b e r of edges).
When P i s c h o s e n i t is thus n e c e s s a r y f o r the model
to know { r e m e m b e r ) only which e l e m e n t s in P a r e
a d j a c e n t in o r d e r to m a p P x p into C.
(It m a y s o m e t i m e s be n e c e s s a r y to add " i d e a l " points to P which a r e not in S in o r d e r t h a t P × P be connected.
T h i s is a n a l a g o u s to the b r a i n ' s " f i l l i n g in" i m a g e s in locally d a m a g e d
p o r t i o n s of the r e t i n a . ) Different p o s s i b l e c h o i c e s of PC" S and link size a r e c l a s s i f i e d a c c o r d i n g to w h i c h of the following s t a t e d p r o p e r t i e s a r e s a t i s f i e d : (a)
ab > cd --> f(a, b) >_ fne, d)
(b}
ab Ned --> f{a, b) = f{c, d) .
If (a) i s s a t i s f i e d both f and P a r e defined a s p r o p e r ; if both (a) and (b) a r e s a t i s fied f and P a r e c a l l e d s t r i c t l y p r o p e r i if f and P a r e not p r o p e r they a r e c a l l e d i m p r o p e r . a b i s c a l l e d a n a m b i g u o u s p a i r iff t h e r e e x i s t s a p a i r cd c P x P s u c h t h a t ab ,-,cd and f{a, b) f(c,d).
B o t h a b and cd a r e c a l l e d c o n t r a d i c t o r y iff ab < cd and f(a,b) > f(c,d). F o r r e a s o n s which b e c o m e obvious
that proper sets,
if (a) and {b) a r e e x a m i n e d ~ it is h y p o t h e s i z e d
P, c o r r e s p o n d to " G e s t a l t s " o r " r e f e r e n c e f r a m e s " .
It is a l s o e a s i l y s e e n
that when P i s s t r i c t l y p r o p e r f p r o v i d e s a m e t r i c on P f ( a , b ) + f ( b , c ) >_ f{a,c); f(a,b) = f(b,a); f(a,a) = 0 wMeh p r e s e r v e s the p r e o r d e r i n g on P x P. 2.
Mapping f r o m P X S into C L e t P be p r o p e r .
1 2
We define a p r o p e r m o d i f i c a t i o n of P a s a n a s s i g n m e n t to e a c h
A c t u a l l y , t h i s defiMtion i s w e a k e r in m o s t a p p l i c a t i o n s - - s e e Sections 3 and 10. See Section 10.
128
Pi c P of a " n e i g h b o r h o o d " , i . e . ,
s e t R.¢S1 such t h a t , if we define g(pi,x) = f(Pi'- ri(x)~+
w h e r e ( x e R i) <==> --(rib¢) = _P~)' g i s a p r o p e r mapping.
(The definition of g m a k e s s e n s e
only i f the R. a r e disjoint; t h i s i s so in a l l i n t e r e s t i n g c a s e s (see R o t h e n b e r g , 1969), ) A x
m a x i m u m p r o p e r m o d i f i c a t i o n is a p r o p e r modification, R = .v R. which is p r o p e r l y 1
con-
1
t a l n e d in no o t h e r p r o p e r modification. C l e a r l y , we m a y b e g i n with, say P2' and choose R 2 s u c h t h a t it is m a x i m a l , t h e n do the s a m e for P5' then Pl" etc.
That is, the R. m a y be s u c c e s s i v e l y m a x i m i z e d f o r
ele-
1
merits of a n y p e r m u t a t i o n of the e l e m e n t s of P.
E a c h s u c h m a x i m a l R. will c o n s t r a i n t h e r e 1 m a i n i n g Ri, and we m a y obtain one or m o r e d i f f e r e n t p r o p e r m o d i f i c a t i o n s , R = .u1 R.1 for 1 e a c h p e r m u t a t i o n , ~ = i, j, k, ~ . . . . . of the e l e m e n t s of P. F o r s i m p l i c i t y , c o n s i d e r the c a s e
w h e r e e a c h p r o p e r modification, R {a), i s unique f o r t h a t a. We define the r a n g e , R., of p. a s the m a x i m a l R. when Y(j fi i)R~ ={p.). Let 1
x
,,
I
j
j
= .U ~.. It is easily shown that R = %1 R t~2. Similarly, we define R = :~ R %
= la
(~)
such that
i(°:)"(where i indexes the different R.I obtained in the order specified by ~). _R is
called the blu__._~r of Pl and for all i, _~g~RiGR i. R and _Rare obviously more easily computed
usually maximal). Note t h a t often, R ~ S. 3 a n d 10).
Actually, we a d j u s t o u r m e t h o d s so t h a t R = S (see Sections
F o r t e c h n i q u e s for m a p p i n g f r o m S ×S to C, s e e R o t h e n b e r g (1969).
Note t h a t the s p a c e in which P i s e m b e d d e d n e e d not be E u c l i d e a n and m a y h a v e d i f f e r ing local " d i m e n s i o n a t i t y " at e a c h point ( r e l a t e d to the n u m b e r of c h a i n s p a s s i n g t h r o u g h t h a t point). 3.
Tolerance Notice t h a t the c a r d i n a l i t y of P i s s p e c i f i e d by a r e a l p a t t e r n r e c o g n i t i o n t a s k in one
of two w a y s : e i t h e r the size of the code ( a l p h a b e t o r n u m b e r of c l a s s i f i c a t i o n s r e q u i r e d ) is specified ( t h i s c o r r e s p o n d s to the l e n g t h of the m a x i m u m chain), o r a m a x i m a l t o l e r a b l e c o n fusion i s s p e c i f i e d t o g e t h e r with a r a n g e o v e r w h i c h s u c h l i m i t a t i o n a p p l i e s . l e t S be i n t e r p r e t e d a s c e l l s in the r e t i n a of the eye.
For example,
Suppose that in o r d e r to p e r f o r m a p a r -
t i c u l a r t a s k , d i s c r i m i n a t i o n of s t r a i g h t l i n e s which d i f f e r in l e n g t h by outy one c e n t i m e t e r and which a r e o b s e r v e d a t a fixed d i s t a n c e f r o m the eye, i s r e q u i r e d .
Then any p a i r of points on
the r e t i n a which c o r r e s p o n d to a p r o j e c t e d d i s t a n c e of m o r e than one c e n t i m e t e r c a n n o t lie in the s a m e R. (or s o m e n e c e s s a r y d i s c r i m i n a t i o n s would fail).
Such a t o l e r a n c e , ~, m a y be
1
i n c o r p o r a t e d into the s y s t e m by w e a k e n i n g 2(a) and 2(b) (above) to a c c o m o d a t e r e v e r s a l s of 2 o r d e r i n g l e s s than ¢. Then the R. will u s u a l l y o v e r l a p and m a y be c o n s i d e r e d "fuzzy s e t s " . Z
1
F o r m e t h o d s of c o n s t r u c t i n g p r o p e r m o d i f i c a t i o n s , s e e R o t h e n b e r g (196 9). 2 R e l a t i o n s b e t w e e n • (which is a n e l e m e n t of SXS) and t h e link size e x i s t so t h a t S can be c o v e r e d by R. See R o t h e n b e r g (1969).
'~29
4.
Sufficient Sets Since e a c h P m a y be a " G e s t a l t " or a " p h o n e m i c a l p h a b e t " , p r o b l e m s a r i s e when
m o r e than one " G e s t a l t " m a y be u s e d (as in vision), o r when a l i s t e n e r s p e a k s m o r e than one language.
M i n i m a l c u e s for the i d e n t i f i c a t i o n of the a p p r o p r i a t e set, P (the " G e s t a l t " o r
"alphabet"), m u s t be d e r i v e d .
T h e s e m i n i m a l c u e s a r e s u b s e t s of S w h i c h allow a unique
i d e n t i f i c a t i o n of a p a r t i c u l a r P C S f r o m all p o s s i b l e P ' s a v a i l a b l e to the l i s t e n e r . t h e s e a s y s t e m of mapping P ( a s well a s P × P) into an " a l p h a b e t " i s d e r i v e d .
From
(In v i s i o n t h i s
a p p l i e s to the fixing of the position of an o b j e c t in the v i s u a l field. ) C o n s i d e r a s e t { P v } (the s e t of l e a r n e d " G e s t a l t s " ) , w h e r e v i n d e x e s d i f f e r e n t P c S. ( H e r e we a s s u m e t h a t v t h e r e e x i s t no PX, and Pw in ~ - ~ P v~ s u c h t h a t P x c Pw. ) We define a sufficient s e t f o r P v ' u s a subset, Q, of P
such t h a t f o r all w, if w ~ v , Q i s not a s u b s e t of P . v w C o n s i d e r a language whose p h o n e m i c a l p h a b e t (code) c o n t a i n s n d i s t i n c t e l e m e n t s
("phonemes" or "letters"). bet?
How m a n y d i s t i n c t n - l e t t e r w o r d s can be f o r m e d u s i n g t h i s a l p h a -
Of c o u r s e , c e r t a i n r e s t r i c t i o n s e x i s t which l i m i t the s e q u e n c e s of l e t t e r s which can
o c c u r (e. g . , no m o r e than two c o n s o n a n t s in a row o r , a s in C h i n e s e , all w o r d s have only one syllable).
The m o r e d i s t i n c t w o r d s t h a t c a n be f o r m e d w h o s e length i s l e s s than o r equal to
s o m e m a x i m a l n, the m o r e " e f f i c i e n t " the a l p h a b e t (code) m a y be s a i d to be. C o n s i d e r all n o n - r e p e a t i n g s e q u e n c e s of all points ( s a y n) in P . T h e r e a r e n ' s u c h v s e q u e n c e s . L e t s. be the n u m b e r of e l e m e n t s in e a c h s e q u e n c e which m u s t a p p e a r b e f o r e a 1 sufficient s e t i s e n c o u n t e r e d . T h e n , F ( P ) i s defined a s the a v e r a g e , v F ( P v) = ~ s . / n } i=t * F ( P v) m a y be i n t e r p r e t e d as the a v e r a g e n u m b e r of e l e m e n t s in a n o n - r e p e a t i n g s e q u e n c e of the n e l e m e n t s of P v r e q u i r e d to uniquely d e t e r m i n e v.
Efficiency, E(Pv), is defined as
F ( P v ) / n and r e d u n d a n c y , R ( P v ) ' a s 1 - E ( P ) . E(P v) m a y be i n t e r p r e t e d a s a m e a s u r e of the a s s y m m e t r y of P
v
with r e s p e c t to all
r o t a t i o n s and t r a n s l a t i o n s of itself.
5. The Directed Graph Suppose t h e r e e x i s t P x and Pw in -. ~ P v~ s u c h t h a t P x C P w .
All e l e m e n t s of ,,~'Pv~
m a y be a r r a n g e d in a d i r e c t e d g r a p h , G, in which a c o n n e c t i o n f r o m P
to P i n d i c a t e s t h a t x y with the f e w e s t e l e m e n t s a r e a t the b o t t o m of G. We now define a
P c P and w h e r e the P x y w g r a p h sufficient s e t for g r a p h node, HoP
Pv' as a subset, >p
w
cp v
w
H, of P v s u c h t h a t for all w
130
If we utilize the h y p o t h e s i s t h a t c l a s s i f i c a t i o n p r o c e d u r e s a r e a s efficient a s p o s s i b l e (i. e . , w h e n e v e r s e v e r a l p o s s i b i l i t i e s e x i s t , the l o w e s t node on the g r a p h will be used), we define a node sufficient s e t for P
HC
V
as a s u b s e t H, of P
P
>p W
c p V
V
such that for all w
or P W
--
C P W
V
We now define g r a p h efficiency, E G, and node efficiency, E N, as b e f o r e , u s i n g g r a p h o r node sufficient s e t s i n s t e a d of sufficient s e t s . To e a c h set, H, which is a s u b s e t of a n e l e m e n t of { P v }
there corresponds a subset
of {Pv} , V(H) = { P v l H C Pv } Two sets, H i and H 2 are called graph equivalentiff V(H I) = V(H2). To each H let there correspond a number I (H) = card{Pv} - card V(H)
I(H) is the number of graph nodes of which H is not a subset, and is called the information value of H with r e s p e c t to graph, G. We now define the i m a g e d i s t a n c e ,
I'(H1,H2), between two s e t s ,
H 1 and H 2 as
card @ ( H I)n V(H2)) T(H 1, H 2) = 1-
6.
card (V(H i)U V(H2))
Application to Spoken Speech C o n s i d e r the r e c o g n i t i o n of r a n d o m vowel sounds by a monolinguat s p e a k e r of a n a t u r a l
language, s a y F r e n c h .
The s e t of r a n d o m vowel sounds would c o r r e s p o n d to S and the s e t of
F r e n c h vowel sounds to P (note: P c S). We now obtain our initial o r d e r i n g on S × S by n o i s e modulating the vowel sounds in S and noting the r e l a t i v e confusions of p a i r s of vowels. l
hypothesis
--
s t a t e s t h a t t h e r e should e x i s t a link s i z e , x y ,
Our
such that P is p r o p e r (as defined)
and such that the r a n g e , R i, of e a c h P i e P c o r r e s p o n d s to the s e t of r a n d o m vowels confused with Pi" ( P would c o r r e s p o n d to points in the "vowel q u a d r i l a t e r a l " u s e d by l i n g u i s t s . ) Suppose we have a m u l t i l i n g u a l s p e a k e r a n d we a r e p r e s e n t i n g h i m with r a n d o m s y l l a b l e s which include t h o s e in all of the l a n g u a g e s he s p e a k s ,
{ P v }" Now o u r g r a p h s u f f i -
c i e n t s e t s should be the m i n i m a l c u e s n e c e s s a r y f o r h i m to choose a p a r t i c u l a r language, P x ' a s a r e f e r e n c e f r a m e o r " G e s t a l t " so that s u b s e q u e n t c o n f u s i o n s a r e a s p r e d i c t e d f o r P above).
X
(as
The a m o u n t of i n f o r m a t i o n supplied by a s u b s e t of vowels ih a p a r t i c u l a r language is
given by the i n f o r m a t i o n v a l u e of t h a t s u b s e t and d e t e r m i n e s the p r o b a b i l i t y of his choosing tha~ 1
Or a n a p p r o p r i a t e n e i g h b o r n o o d s y s t e m - - s e e footnote, Sections 2 and 10.
131
language as a "Gestalt". for different sequences 7.
Hence,
we here have different predictions generated by the model
of stimuli (i. e., it is "context dependentL').
Visual Illusions C o n s i d e r the f a m i l i a r optical illusion w h e r e b y the moon a p p e a r s l a r g e r on the h o r i z o n
than when high in the sky.
Note that buildings, t r e e s and a i r p l a n e s b e c o m e s m a l l e r as they
a p p r o a c h the horizon a c c o r d i n g to the laws of p e r s p e c t i v e . us, the moon does not. due to p e r s p e c t i v e . distant objects.
B e c a u s e of its g r e a t d i s t a n c e f r o m
Hence our n o r m a l P d e r i v e s f r o m a m e t r i c involving c o n t r a c t i o n s
T h i s P , h o w e v e r , i s i n a p p r o p r i a t e f o r judging r e l a t i v e s i z e s of v a s t l y
In s i m i l a r fashion, f a m i l i a r illusions due to p e r s p e c t i v e , such as the " r a i l -
r o a d tie" illusion:
a r e e a s i l y explained. 1 When applied to c o l o r vision, if the initial o r d e r i n g i s obtained by m e a s u r i n g c o n f u s i o n s of c o l o r s on a p a r t i c u l a r photograph, if a s e p a r a t e P i s c o m p u t e d for each of t h r e e p r i m a r y c o l o r s and we then a s s u m e c o l o r mixing a c c o r d i n g to the values of C obtained by our m a p ping, we have a quantitative v e r s i o n of Edwin L a n d ' s " T h e o r y of the R e t i n e x " . Note a l s o that if a chain (as h e r e defined) i s i n t e r p r e t e d a s a s e r i e s of c o n n e c t e d n e u r o n s , and if a s e t of n e u r o n s , all of which a r e c o n n e c t e d to e a c h o t h e r , i s i n t e r p r e t e d a s a range, our function, g, which counts e d g e s in a s h o r t e s t chain, c o r r e s p o n d s roughly to the length of t i m e r e q u i r e d for a s t i m u l u s to p a s s f r o m one n e u r o n c l u m p (range) to a neuron end which i s not in that clump. 8.
Analogies to "shadow e n e r v a t i o n " on the r e t i n a a l s o can be m a d e .
Application to the P e r c e p t i o n of P i t c h H e r e we a s s u m e a s i m p l e o r d e r i n g on S (pitch o r frequency} t o g e t h e r with the p r e -
o r d e r i n g on S ~ S ( " m u s i c a l i n t e r v a l s " ) , and the following a x i o m l i m i t s d i m e n s t o n a l i t y : (a) x < y < z
>xy, yz<xz
F o r W e s t e r n m u s i c we may also a s s u m e that (b) x y < z w
> ' q u ( x y ~ , z u ) and z < u < w
or w
(c5 (x < y < z) ^ (u < v < w) a (xy'~uv) ^ ( y z ~ ' v w )
> xz '~uw
(c) s p e c i f i e s that equal m u s i c a l i n t e r v a l s "added" to equal i n t e r v a l s a r e equal.
(b) p o s t u l a t e s
the ability to i n t e r p o l a t e a pitch between two o t h e r p i t c h e s , and p e r m i t s us to obtain the p r e o r d e r i n g on S x S a s follows: to c o m p a r e zw with xy add u on the s a m e side of z a s w so 1 See Stratton (18975, T h o u l e s s (19315 and yon Senden (19325 p e r t a i n i n g to t h e a b s e n s e of a stable m e t r i c in the visual c o r t e x .
132
that xy,-, zu and d e t e r m i n e if u is internal to zw. zw < xy; otherwise xy--,zw.
If it is, zw > xy; if u is e x t e r n a l ,
The u at which zu is p e r c e i v e d as equivalent to xy has been
e x p e r i m e n t a l l y shown to depend upon the t i m b r e of the tones, x, y, z and u, as does the 1 entire p r e o r d e r obtained in this manner. Axiom (a) will accommodate all music (except "klangfarbenmusik", for which the m o r e g e n e r a l model must be used. ) Now, let i in Pi range f r o m -co to +co and index Pie P such that the simple o r d e r i n g on P c
S is obeyed.
Then f(Pi'Pj) = l i - j l
and if
we
define 6ij = Pi+~Pi '~
-i6 = mhj 6..~j and 5i = maxj 6..13 it is e a s i l y shown that
(d) Vi(g i < ~+l) is a n e c e s s a r y and sufficient condition for a s t r i c t l y proper mapping, and if "_<" r e p l a c e s "<" in the formula, for a proper mapping.
Otherwise P is i m p r o p e r .
P has period n if for all i, j, piPk ~ Pi+nPk+n and if n is the l e a s t positive integer satisfying the condition.
In this c a s e ( e . g . , octave equivalence), the positive i n t e g e r s suf-
fice to rank the o r d e r of all pairs 6.. e P • P according to the initial ordering. U o r d e r for 5ij = Pi+~,p~. is called --~i' and it is now possible to define a m a t r i x ,
This rank [~i~
$
called the reduced m a t r i x of P: 0~
i, i
.
.
.
.
OL
l,n
~(n-l), I" " ~(n-l),n Now, if we define a . = r r m a . , and ~ = r n ~ . . , we may r e p l a c e ( d ) (above) by -% 1,j i j 1,j "Vi(~i- < --i+Yv~ }" which is applied only to e l e m e n t s of the reduced matrix. It is now e a s i l y proved that, if P is proper and periodic, R i (range) is an i n t e r v a l about Pi ( i . e . ,
ri(x) is proper), all R i and Ri+ 1 i n t e r s e c t at one point at most (all others
a r e disjoint), if e v e r y two consecutive ranges i n t e r s e c t , then
U R. = S. Simple methods for i 1 computing proper modifications, r a n g e s , blurs, and a method for generating all p r o p e r P such that P a S (S is finite) have been developed.
Efficiency, E, as well as all other quan-
tities and sets defined a r e e a s i l y computed or generated by operations on the reduced matrix. Whenever t h e r e exists an ai, j = ~i+1, k we have an ambiguous pair (musical intervM) 1
See Shouten (1962), E v e t t s (1958), L i c k l i d e r (1959), P r a t t (1928) and M u n s t e r b e r g (1892) for interactions between pitch and t i m b r e .
133
and if qij > a i + l , k we h a v e c o n t r a d i c t o r y p a i r s .
In o u r i n t e r p r e t a t i o n P i s a " m u s i c a l s c a l e "
(and a l l i t s " m o d e s " ) o r a " c h o r d " (and all i t s " i n v e r s i o n s " ) ,
The reduced matrix where P
i s the " m a j o r s c a l e " (as tuned in 12-tone equal t e m p e r a m e n t ) i s shown below and a m b i g u o u s ~.. 1j a r e e n c i r c l e d : --2
2
2
1
2
2
~-
4
4
3
3
4
3
3
Q
5
5
5
5
5
5
7
7
7
Q
7
7
7
9
9
8
8
9
9
8
ii
i0
I0
i0
it
i0
I0
[aij] =
p l P 4 and p4Pl a r e a m b i g u o u s
Note t h a t the f i r s t column c o n t a i n s all m u s i c a l i n t e r v a l s l e s s t h a n an octave which h a v e the "fourth d e g r e e " of the m a j o r s c a l e as an end point.
H e n c e we note t h a t the two t r i -
t o n e s b e t w e e n the f o u r t h and s e v e n t h d e g r e e s and s e v e n t h and f o u r t h d e g r e e s a r e a m b i g u o u s pairs.
We also define s t a b i l i t y ,
the m a j o r s c a l e § =
S, as the p r o p o r t i o n of u n a m b i g u o u s p a i r s in P x P
For
9524.
Below a r e two d i a g r a m s which show the r a n g e s , R i ' and b l u r s , -~i' of e a c h Pi in one octave of the "C m a j o r s c a l e " (the f i r s t d i a g r a m ) and in a "C m a j o r t r i a d " (the second d i a gram).
E a c h I~. is shown by the b r a c k e t s e n c l o s e d on top and b o t t o m , and e a c h R. by a 1
d a r k e n e d rod.
"--1
The n u m b e r s u n d e r the d i a g r a m index the " s e m i t o n e s " in the octave s t a r t i n g
with C = 0:
(lst (2nd (3rd (4th (hth (6th (Tth (8th degree) degree) degree) degree) degree) degree) degree) degree) (c) (~,) (E) (F') (~) (^) (B) (c)
(L., A-,, J, L, ,L,, J, , J ,L, I
0
I
\
I
2
%
3
(c')
0
/
~
,~
~
5
6
CE)
1
2
3
4
p..
7
8
i
9
S
10
.I \ 11
(c4
5
6
7
12
(c')
8
9
10
11
12
l
134
Since ambiguous pairs cannot be unambiguously classified except in reference to adjacent elements, it is an obvious rule of musical usage that an ambiguous musical interval (except when it occurs in a "chord" in which it is not ambiguous) must be approached or left by a "step" (i°e., it must "resolve" (move) to a tone adjacent to one of its component tones which, together with its other component tone, no longer forms an ambiguous pair (musical interval)).
It is also predicted that a tone, x c S - P which is in the range R. of some p. c P I
i
must either replace or "resolve" (move) to that Pi" Hence the explanations of "auxilliary" and "altered" tones. "Root" or "tonality" corresponds to an element of P which temporarily is an endpoint of all pairs measured by f (or g). In cases where a drone or "ostinato" is used, it is not significant that P is proper, and improper P ("scales") are often used (to be
discussed). Note that a given P m a y have m a n y P s c a l e , those P
V
are "chords".
a P which a r e p r o p e r . When P i s the m a j o r v T h o s e p r o p e r P which a r e s u b s e t s of both the " m a j o r " V
and " m i n o r " s c a l e s and t h e i r " a l t e r a t i o n s " c o m p r i s e the " f i g u r e d b a s s " s y s t e m of W e s t e r n classical music.
C o n s i d e r a s t r i n g of p r o p e r s e t s ,
P I ' P2" P3 . . . . .
P m such that for all
k < m, t h e r e e x i s t s a p r o p e r modification of P k ' R(Pk) such that Pk ¢ R(Pk-1)" analogous to an h i e r a r c h i c a l c l u s t e r i n g . )
(This is
If c l a s s i f i c a t i o n is always p e r f o r m e d by the s m a l -
l e s t p r o p e r set, the r a n g e s of the points in t h a t s e t will include the e l e m e n t s of the l a r g e r s e t next in the above s t r i n g . " G e s t a l t " (i. e . ,
Hence, if we t e m p o r a r i l y use the "C m a j o r t r i a d " as our
P) when u s i n g o t h e r tones of the "C m a j o r s c a l e " , its r a n g e s d e t e r m i n e the
t r a d i t i o n a l " r u l e s of voice l e a d i n g " ; i . e . , to G" ( s e e d i a g r a m above).
"B l e a d s to C, D and F lead to E and A l e a d s
S i m i l a r t r a d i t i o n a l r e s u l t s obtain f r o m a p p l i c a t i o n to the m i n o r
s c a l e s and to p r o p e r s u b s e t s of i m p r o p e r s c a l e s w h o s e r a n g e s include all t o n e s in t h a t i m proper scale.
In the l a t t e r c a s e a p a r t i t i o n of s c a l e tones into " p r i n c i p a l " a n d " a u x i l i a r y "
tones results.
S i m i l a r a n a l y s i s e l u c i d a t e s c h r o m a t i c u s a g e within the m a j o r - m i n o r s y s t e m 1
It i s also p r e d i c t e d that t h o s e p r o p e r s c a l e s will o c c u r in the m u s i c of d i f f e r e n t c u l t u r e s which a r e m a x i m u m both in s t a b i l i t y , S, and efficiency, E, a s c o m p u t e d when all keys
(i. e.
t r a n s p o s i t i o n s ) of the s c a l e a r e in ~Pv}'- A p p r o p r i a t e c o m p u t a t i o n s have 2 been m a d e , and t h i s h a s been shown to be the c a s e (note: the i n i t i a l o r d e r i n g i s d e t e r m i n e d by the t i m b r e of the i n s t r u m e n t s u s e d in e a c h c u l t u r e ) .
R e s t r i c t i o n s on the tuning of s c a l e s
such t h a t the initial o r d e r i n g i s r e t a i n e d have b e e n computed.
T h e s e , in g e n e r a l , a r e m o r e
s t r i n g e n t in i m p r o p e r s c a l e s than p r o p e r s c a l e s , and a r e c o n s i s t e n t with c r o s s - c u l t u r a l o b servations.
In J a v a t h e r e e x i s t two s c a l e s y s t e m s , " S l e n d r o " and " P e l o g " , e a c h containing
a v a r i e t y of " s c a l e s " .
It h a s been o b s e r v e d t h a t all s c a l e s in the " S l e n d r o " c l a s s a r e s t r i c t -
ly p r o p e r and that all in the " P e l o g " c l a s s a r e i m p r o p e r . 1,2
See R o t h e n b e r g (1969).
In a study conducted with the
135
a s s i s t a n c e of M r . Surya B r a t a of the M i n i s t r y of E d u c a t i o n and C u l t u r e , J a k a r t a , t h e u s e s 1 of t h e s e s c a l e s y s t e m s w e r e o b s e r v e d to b e i n a c c o r d with the p r e d i c t i o n s of t h i s model. Notice t h a t in a p r o p e r s c a l e with h i g h s t a b i l i t y , t h e m u s i c a l i n t e r v a l s (i. e . , p a i r s in P x P) a r e , for the m o s t p a r t , u n a m b i g u o u s l y c l a s s i f i e d ( m e a s u r e d ) .
Hence s e q u e n c e s of
s i m i l a r i n t e r v a l p a t t e r n s (i. e . , " m o d a l m o t i v i c s e q u e n c e s " ) a r e e a s i l y a p p r e h e n d e d a s s i m i l a r , a n d t h e i r u s e would be anticipated.
When e f f i c i e n c y is also high, the e l e m e n t s of P h a v e
t h e i r p o s i t i o n s (absolute pitches) quickly fixed ( r e l a t i v e to t o n e s p r e v i o u s l y h e a r d ) and we would a l s o expect the u s e of " t o n i c s " , i . e . ,
"tonal m u s i c " .
When e f f i c i e n c y i s low, it would
not be e x p e c t e d t h a t t o n a l i t y would be u s e d (as m u s i c a l m a t e r i a l ) . the c a s e .
T h i s indeed a p p e a r s to be
Motivic s e q u e n c e s a r e u s e d in p r o p e r s c a l e s , but t o n a l i t y is avoided (or i r r e l e v a n t )
in t h o s e with low efficiency (high redundancy} s u c h a s the " t w e l v e tone s c a l e " (note the m o t i vie p r o p e r t i e s of '%one r o w " use), t h e "whole tone s c a l e " , etc.
When t h e m a j o r and m i n o r
s c a l e s as well a s the whole tone s c a l e s a r e i n c l u d e d in the d i r e c t e d g r a p h ,
G, and the g r a p h
efficiency, E G ( p ), is c o m p u t e d w h e r e P i s the whole tone s c a l e , t h i s e f f i c i e n c y b e c o m e s v v T h i s i s c o n s i s t e n t with the s u c c e s s f u l e x t e n s i v e t o n a l u s e of the whole tone s c a l e only
high.
i n conjunction with t h e m a j o r and m i n o r s c a l e s by D e b u s s y and R a v e l . When a s c a l e i s i m p r o p e r ( o r of low stability) m a n y m o t i v i c ( i n t e r v a l l i c ) s i m i l a r i t i e s a r e not a p p a r e n t ( e a s i l y p e r c e i v e d ) .
T h e r e f o r e , it i s i m p o r t a n t t h a t the t o n e s in the s c a l e be
quickly fixed, i. e., the s c a l e " d e g r e e s " be identified) so that the p a r t i t i o n between " p r i n c i p a l " and " a u x i l l i a r y " tones be c l e a r ( w i t n e s s the c u s t o m a r y u s e of a d r o n e in Indian m u s i c ) . Hence high r e d u n d a n c y (low efficiency) i s r e q u i r e d .
T h i s a p p e a r s to c h a r a c t e r i z e a l l i m -
proper scales observed. Note t h a t in tonal m u s i c a " c a d e n c e " m u s t fix both the s c a l e , e l e m e n t of P . )
P, and i t s " t o n i c " (an
Hence it would be e x p e c t e d t h a t it would contain a sufficient set, the tonic
(and p r o b a b l y a tone a "fifth" above it so t h a t i t i s r e i n f o r c e d by the r e s u l t i n g d i f f e r e n c e tone)f The c a d e n c e would c o n t a i n a s m a n y p r o p e r s u b s e t s of P a s p o s s i b l e , so a s to r e v e a l i t s h a r monic s u b s t r u c t u r e .
All t r a d i t i o n a l c a d e n c e s a r e a c c o u n t e d for in t h i s m a n n e r , a s well a s
c a d e n c e s in " f r e e twelve tone m u s i c " (when all p r o p e r s u b s e t s of the " c h r o m a t i c s c a l e " a r e included on the d i r e c t e d g r a p h , G). When all s u c h p r o p e r s u b s e t s of the c h r o m a t i c s c a l e a r e i n c l u d e d on the g r a p h , i n f o r m a t i o n v a l u e s and i m a g e d i s t a n c e s a s s u m e s i g n i f i c a n c e .
The p r o p e r s u b s e t s with high
s t a b i l i t y and with six o r m o r e e l e m e n t s (those with f e w e r usually function a s " c h o r d s " ) w h i c h a p p e a r on the g r a p h a r e all s c a l e s which can be f o r m e d by c h o o s i n g an e l e m e n t of P f r o m S and s e l e c t i n g the r e m a i n i n g t o n e s of P f r o m S in c l o c k w i s e fashion in a c c o r d a n c e with the 1 2
See R o t h e n b e r g (1969), r e v i s e d v e r s i o n , a n d a l s o Kunst (1949) and Hood (1954, t966). See H e l m h o l t z (1948).
136
following vectors tone",
etc.).
(where "I" represents
Each
a distance of one "semitone"
and "2" of a "whole
vector denotes a set of "keys" of a scale"
(l,l,l,l,l, Ijl, l,l,l,l,l)
("twelve tone" scale)
(2,1,1,2,1;1,2,1,1)
(of b o r d e r l i n e stability - used by Olivier M e s s i a e n - - s e e M e s s i a e n (1944))
(2,1,2,1,2,1,2,1)
( " s t r i n g of pearls")
(2,2,2,2,1,2,1)
("melodic minor")
(3,1,3,1,3,1) (2,2,2,2,2)
("whole tone" scale)
The information value of a given group of tones indicates the e a s e with which a P is chosen f r o m the graph for its c l a s s i f i c a t i o n (by a l i s t e n e r ) .
The three tone sets ("triads")
of maximal information value a r e given by the v e c t o r s , (10, 1, 1) ( e . g . , (e.g.,
C, Ab , B), ( 8 , 1 , 3 ) ( e . g . ,
C, F #, G).
C, G# , A), ( 6 , 5 , 1 ) ( e . g . ,
C, A # , B), (8,3, 1)
C, F #, B) and ( 6 , 1 , 5 ) ( e . g . ,
The frequent use of these " c h o r d s " in twentieth century music (and e s p e c i a l l y in
"cadences") is well known; hence " m i n j o r ''1 chords and chords in fourths. ample is Anion W e b e r n ' s "Piano V a r i a t i o n s " (opus 27).
A startling e x -
The e n t i r e composition consists of
a s u c c e s s i o n of graph sufficient sets for the "twelve tone s c a l e " , although the " s e r i a l " t e c h nique of composition does not guarantee this. I m a g e distance is an e x t r e m e l y s e n s i t i v e indicator of apparent differences between " c h o r d s " in twentieth century w e s t e r n music.
In addition to obvious s i m i l a r i t i e s due to the
p r e s e n c e of common tones, the following subtle differences a r e r e v e a l e d (the image d i s tances a r e below the p r o g r e s s i o n s ) : ,v ,,
,
~
[~l/
"
,
~
~k'-)
/~,
"!
B
i = 2/5
Y
,'
I~
kl]
C-!
~]/
! '
I,.--r
P
A
,~
A
C
i = 6/7
A
I
,
O
"|
v
.I '.....
B
z = 2/3
A
C
i = 6/v
(A) = (8, 1,3); on C
(A) = (8, 3, 1); on G b
(B) = ( 6 , 5 , 1); on C
(B) = (8, 1,3); on C
(C) = ( 6 , 5 , 1 ) ;
(C) = (8,1,3); on B
on D b
The model is far m o r e detailed than h e r e p r e s e n t e d and m o r e complete in its musical application. 1 That is, a combination of a m i n o r and m a j o r triad (e, g . , C, E b, E~., G).
137
10.
The D e v e l o p m e n t of New M a t e r i a l s In addition to e t h n o - m u s i c o l o g i c a l t e s t i n g , which h a s thus f a r s t r o n g l y s u p p o r t e d the
theory, e x p e r i m e n t s and e q u i p m e n t have been d e s i g n e d for the t e s t i n g of the application to the p e r c e p t i o n of pitch.
T h e s e e x p e r i m e n t s in p a r t r e s e m b l e those d e s i g n e d for t e s t i n g the
p e r c e p t i o n of vowels in a n a t u r a l language,
R e l a t e d to t h e s e i s the u s e of the model for the
g e n e r a t i o n of new m u s i c a l m a t e r i a l s I The e x p e r i m e n t a l e q u i p m e n t i s a p p r o p r i a t e for t h e i r m u s i c a l exploitation.
Many of the new m u s i c a l m a t e r i a l s g e n e r a t e d a r e c o n s i s t e n t with
Western musical tradition. Of p a r t i c u l a r i n t e r e s t i s the g e n e r a t i o n of m a t e r i a l s for "tone c o l o r " m u s i c ( K l a n g f a r b e n m u s i k " ) , which r e s e m b l e s the a p p l i c a t i o n to spoken vowels: The c o m p o s e r i s a s k e d to supply a s e t of sounds he l i k e s , S. be.
He i s then a s k e d how l a r g e he w a n t s h i s a l p h a b e t , P, to
The i n i t i a l o r d e r i n g on S × S i s obtained by d i r e c t i n q u i r y a s to p e r c e i v e d s i m i l a r i t y o r
by m e a n s of confusion in the p r e s e n c e of n o i s e (as in the c a s e of v o w e l s - - n o o r d e r i n g on S exists here).
All p r o p e r P c S and the r a n g e s of e a c h of t h e i r e l e m e n t s a r e g e n e r a t e d .
T h e s e P ' s a r e c h o s e n such t h a t they have a p r o p e r modification, R = S. then c h o o s e s one of t h e s e P ' s .
The c o m p o s e r
The p r o p e r t i e s of the p e r c e p t i o n of the e l e m e n t s of S a r e
d e t e r m i n e d a s in the c a s e of pitch and t h e i r m u s i c a l usage i s s i m i l a r l y c i r c u m s c r i b e d .
Note t h a t , while the t e c h n i q u e known a s " c l u s t e r a n a l y s i s " h a s b e e n s u c c e s s f u l l y applied to the study of the p e r c e p t i o n of p h o n e m e s of n a t u r a l l a n g u a g e s , t h i s tool i s not a d e quate f o r o u r purpose h e r e (i. e . , to g e n e r a t e a m u s i c a l " p h o n e m i c a l p h a b e t " - - a s y n t h e t i c alphabet of m u s i c a l m a t e r i a l s } .
The p r i n c i p a l r e a s o n is t h a t , i a e x p e r i m e n t s involving the
p h o n e m e s of a n a t u r a l language, deviant p h o n e m e s tend to " c l u s t e r " about n o r m a t i v e phon e m e s i n t h e language as a c o n s e q u e n c e of the context of t h a t language in the m i n d of the s u b j e c t p r o v i d i n g e x p e r i m e n t a l data.
However, in a language not known to the s u b j e c t (e. g . ,
a " m u s i c a l a l p h a b e t " of n o v e l sounds), no s u c h effect c a n b e expected.
Experimental data
p r o d u c e d by a s u b j e c t b e f o r e he i s f a m i l i a r with a language will a l m o s t c e r t a i n l y d i f f e r f r o m d a t a produced a f t e r he i s fluent in the language, therefore produce different results.
A c l u s t e r a n a l y s i s of e a c h s u c h s e t
will
By c o n t r a s t , t h e q u e s t i o n we h e r e c o n s i d e r i s , 'rhow
c a n we s e l e c t o u r " a l p h a b e t " of m a t e r i a l s (P) in s u c h f a s h i o n t h a t t h e e x p e r i m e n t a l l y p r o duced o r d e r i n g ( i. e . , t h e " i n i t i a l o r d e r i n g " ) b e f o r e t h e s u b j e c t h a s l e a r n e d the " a l p h a b e t " will b e m i n i m a l l y a l t e r e d in e x p e r i m e n t s p e r f o r m e d a f t e r he h a s l e a r n e d t h a t " a l p h a b e t " ? 1
In p a r t i c u l a r , m a n y novel " m u s i c a l scales*' e m p l o y i n g " m i c r o t o n e s " h a v e b e e n g e n e r a t e d . See R o t h e n b e r g (1969).
138
It c a n b e s h o w n that this question is logically equivalent (if we a s s u m e that the hypotheses of this paper a r e valid) to the problem of s e l e c t i n g a " p r o p e r " set (P) f r o m the domain (of stimuli),
S, utilized in the experiments.
A s i m i l a r application to the perception of patterns (in p a r t i c u l a r , textures) used in a b s t r a c t animated films is planned. 11. T h e o r e t i c a l Basks:
1
Let L be a (possibly infinite) set whose e l e m e n t s c o r r e s p o n d to given stimuli o r (alternatively) s e n s o r y r e c e p t o r s .
Assume a set of atomic predicates which specify an o r -
dering of pairs of elements of S (e. g , , the o r d e r i n g specifying the r e l a t i v e s i m i l a r i t i e s of p a i r s of stimuli; i . e . , y than z is to w).
"xy < zw" would indicate that x is " m o r e s i m i l a r " (or " c l o s e r " ) to
Suppose we weaken the o r d e r i n g " < " by the introduction of s o m e c
which is an e l e m e n t of S~ S; i . e . , we say that "xy e< zw" if and only ff xy exceeds zw by at %
least ~ (precisely, xy e> zw - xy > zw a V v ((xv _< zw --> vy > e) ,qxy < zv ---> wv > <)) ). We
considermapping the structure, <S, >> intoa substructure, where P is a finite d i s c r e t e space (intended to c o r r e s p o n d to a " G e s t a l t " , " r e f e r e n c e f r a m e " or classification of stimuli) such that the following conditions hold (and define ~ ): P is a F r e s c h e t space (see Sierpinski, 1952, Chapter 1) wherein points a r e assigned to neighborhoods such that each point has a m i n i m u m neighborhood (i. e . , no other neighborhood is p r o p e r l y contai ned therein) and such that P and the null set a r e the only c l o s e d sets, 2 We define points in the s a m e m i n i m a l neighborhood as " a d j a c e n t " and define a m e t r i c , f(x, y), on P as the number of edges in the s h o r t e s t path along adjacent points f r o m x to y.
If f(x, y) is g r e a t e r than f(z, w) we denote this by "xy ~ zw".
When @ is
a continuousmapping from <S, >,> onto such thatif x,y,z,wcS, x y >e zw --> @(x)@(y) ~ @(z)@(w), P is called a '%asis" for S and ~ is called a "reduction mapping",
Such a mapping f r o m
<S, >G) onto a s u b s t r u c t u r e
G) is of i n t e r e s t because
it can be shown that any subset A of S which s a t i s f i e s a f o r m u l a of the second o r d e r p r e d i cate calculus (i. e . , a " p r o p e r t y " of the set, A, of stimuli) has an i m a g e @(A) in P which satisfies a modified v e r s i o n of that s a m e formula.
This modification is accomplished by
replacing u n i v e r s a l quantifiers of the f o r m u l a by " n u m e r i c a l quantifiers",
Q(k)x, which may
be r e a d "for k 100 percent of x" ( s i m i l a r l y to the reading of "Vx" as " f o r all x"), in the following m a n n e r : 1
A b r i e f m a t h e m a t i c a l t r e a t m e n t of this section can be found in another paper in this volume entitled " P r e d i c a t e Calculus F e a t u r e Generation", Sections 14 and 17. 2 This guarantees the eonnectedness of the space (Rothenberg, 1974).
139
Suppose we a r e given a formula, F, with one free set variable, and a set B c S 1 which satisfies F. When~B) does not satisfy F, we consider the replacemen£ of a single universal quantifier by a numerical quantifier, Q(k), where k is maximal such that ~(B) satisfies the modified formula, F'.
This is done for all universal quantifiers, and unity
subtracted from the largest value of k thus obtained is defined as the "degree of satisfaction oft(B) with respect to F " , I ~ ( t ~ , ~ .
When ~b(B) does satisfy F, we define
I~(B),F)
as the largest value of k over all replacements of a single existential quantifier by Q(k) such that ~(B) still satisfies the modified formula, F ' .
Note that when B is finite, the latter
definition, if applied to B instead of ~(B), defines the degree of satisfaction of B with respect to F, D(B,F). Intuitively (dealing with geometrical figures), if F is satisfied only by starlike 2 sets and both B and C a r e starlike, "D(B, F) > D(C, F)" says that "B is more starlike (i. e . , more nearly convex) than C".
If neither B nor C is starlike, the inequality is intuitively
interpreted as "C is less starlike (i. e . , more "hollow" at its "center") than B. Hence we define D ( B , F) - D(~(B), F) as the "degradation of B with respect to F by ~", and ~(B) is called the "degraded image" of B. 3 A bound on such degradation may be computed by e x a m iuation of the syntax of F and the smallest c (in " < " ) such that ~ is a reduction mapping. E Similarly, a hound, b(F,~), on the maximum degradation o v e r all B such that B CS and B
satisfies F may be computed (i e.,
(D(B, F)-
F)[ B
S^ B satisfies F ) )
This latter quantity exposes the effect that ~ has on the satisfaction of F; i. e, it is a measure of the degree to which properties of subsets of our set of stimuli, S, a r e p r e served by their images in our "reference f r a m e " , P. 4 If P is a musical scale" we hypothesize that those properties which enjoy minimal degradations (as defined above) are those which will define the relations used in the construction of musical forms.
We also propose
1
Note that the discussion applies if B is not a set, but a sequence of substitutions of e l e ments (and/or subsets) of S for the free variables in a formula with many free variables. 2 That is, there exists a point in B such that for e v e r y other point of B all elements between the two points are contained in B. 3 Metaphorically, performing a "reduction mapping" is analogous to tearing a hologram into s e v e r a l parts. One of these parts corresponds to a ' ~ a s i s " of the mapping. The image produced from that part would correspond to the "degraded image" of the image produced by the entire hologram. A formula with one free set variable (as above) would be a property of an image which makes it recognizable (e. g . , "it is spherical" or "it consists of a cube adjacent to a convex ball"). Such a property might be less (or possibly more) pronounced (e. g . , "less spherical") in the image produced by a portion of the hologram than in the image produced by the entire hologram. The degree to which such a property (of a particular image) is pronounced is analogous to the "degree of satisfaction" of that property by the image, intuitively a degraded image may be thought of as a "fimzy" image of the original, much as a picture of a object on a television screen is a "filzzy" image of that object. 4 That is, it is a m e a s u r e of specific kinds of information loss resulting from the utilization of ~.
140
that the "rules of voice-leading" and "rules of harmony" of various cultures are chosen so as to r e s t r i c t the degradation of the properties on which these relations are based.
Although
psychological experiments for direct verification of the predictions of the model have been designed, results are not yet available.
However, the musical predictions of the rm del are
in accord with Western musical practices and those of various Asian musical cultures which
have been exaznined (l~othenberg.
1969,
revised version).
For brevity, we hereleliminate e from consideration, call the weakened reduction mappings which result "proper mappings" and confine our discussion to Western music. " P r o p e r modifications" are preimages of points in the '%asis" of a "proper mapping" and "contractory (and ambiguous) pairs" a r e those pairs of musical intervals (i. e . , musical intervals are pairs of elements) which account for degradations by virtue of the inversion (or collapsing) of their order as a consequence of the mapping,"Stability" isarough measure of the degradations of that class of formulae whose "models" (i. e . , sets of substitutions for the free variables such that the formula is satisfied) a r e invariant with respect to t r a n s lation 2 (and various other transformations as well) under the particular proper mapping (or corresponding basis, P) chosen.
"Efficiency" similarly ranks the bases of various map-
pings according to the degradation of formulae whose models are no__~tinvariant with respect to translation.
These various bases are considered as "musical alphabets" and models of
negligibly degraded formulae a r e hypothesized to be the units of "musical f o r m " (e, g . , "motifs", etc. ) when such alphabets a r e used in a musical composition. References Evetts, J . E . (1958), "The Subjective Pitch of a Complex Inharmonic Residue", unpublished report, Pembroke College, England. Helmhottz, H. L . F . (1948), On the Sensations of Tone as a PhFsiotogical Basis for the Theory of Music (Translated by A . J . Ellis~ 1885), New York; P e t e r Smith. Hood, M. (1954), The Nuclear Theme as a Determinant of Patet in Javanese Music Groningen, D]akarta: J. B. Wolters. Hood, M. (1966), "Slendro and Pelog Redefined", Selected Peports, Institute of Ethnomusicology, University of California at Los Angeles. Kunst, J. (1949), Music in Java, The Hague; Martinus Nijhoff. Licklider, J. C.R. (1959), "Three Auditory Theories", in S. Koch (ed.), Psychology: A Study of a Science, New York: McGraw-Hill, pp, 41-144. Messiaen, O. (t944), Technique de mon langage musical, P a r i s : Alphonse Leduc.
1
For a treatment of c, see Rothenberg (1969, 1974).
2 That is, still satisfiy the same formula after translation.
See Rothenberg (1974).
141
Munsterburg, H. (1892), "Vergleichen der Tondistanzen", Beitrage zur experimentelle Psychology, 4, 147-177. Pratt, C.C. (1928), "Comparison of Tonal Distance", and "Bisection of Tonal Intervals Larger than an Octave", J. Experimental Psychology, 1__~1,77-87 and 17-36. Rothenberg, D. (1969), "A Pattern Recognition Model Applied to the Perception of Pitch", Air Force Office of Scientific Research Technical Report, Dept. of Information Sciences. Revised version to appear as series of articles in Mathematical Systems Theory Rothenberg, D. (1974), "Predicate Calculus Feature Generation", this volume. Sehouten, J. F . , R.J. Ritsma and B. L. Cardozo (1962), "Pitch of the Residue", J. Acoustical Society of America, 3.~4, 1418-1424. Sierpinski,
(1952), General Topology, Toronto: University of Toronto Press.
Stratton, G.M. (1897), "Vision Without Inversion of the Retinal Image", Psych. Rev., 4, 341-360; 463-481. Thouless, R.H. (1931), "Phenomenal Regression to the Real Object", British J. Psychology, 21__, 339-359. von Senden, M. (1932), Raum- und Gestaltauffassung beioperierten Blindgeborenen vor und naoh der Operation, Leipzig: Barth.
MODELS OF SPEECH PRODUCTION Chin-W. K i m U n i v e r s i t y of Illinois i.
Introduction T h e m o s t d o m i n a n t i s s u e a m o n g p h o n e t i c i a n s in r e c e n t y e a r s h a s b e e n the q u e s t i o n of
the model of s p e e c h production, in p a r t i c u l a r the s i z e o r the unit of s p e e c h e n c o d i n g (cf. F r o m k i n , 1965, 1966, 1968, 1971; Kim, 1971b; Kozhevnikov and C h i s t o v i c h , 1965; Ladefoged, 1967; MacKay, 1970; M a c N e i l a g e , 1970; Ohala, ±970; Ohman, 1967; T a t h a m , 1971; T a t h a m and Morton, 1969; W h i t a k e r , 1970; and W i e k e l g r e n , 1969). E v e r s i n c e Sapir (1933) a d v a n c e d the notion of the " p s y c h o l o g i c a l r e a l i t y of phon e m e s " , the c o n c e p t of a phoneme h a s been a c c e p t e d a s s o m e t h i n g r e a l w h i c h h a s a n i n v a r lent c o r r e l a t e a t s o m e l e v e l .
And n e e d l e s s to say, the c o n c e p t of the p h o n e m e w a s the
c o r n e r s t o n e of s t r u c t u r a l l i n g u i s t i c s .
The notion of p h o n e m i c r e a l i t y w a s s t r e n g t h e n e d by
the supposedly s u p e r i o r a l p h a b e t i z e d w r i t i n g s y s t e m s of the W e s t e r n world.
If the language
u s e r w r i t e s two d i f f e r e n t sounds with one and the s a m e symbol, he m u s t r e g a r d t h e m a s m e n t a l l y the s a m e , so goes the a r g u m e n t . it m i g h t s e e m .
Actually, t h i s notion i s not so u n e h a l l e n g e a b l e a s
F o r one thing, I a m not a t all s u r e if a w a r e n e s s of this s a m e n e s s i s equally
s t r o n g in s p e a k e r s of l a n g u a g e s w h i c h do not have w r i t i n g s y s t e m s .
Secondly, the a l p h a -
b e t i c s y s t e m i s not the only w r i t i n g s y s t e m t h a t the h u m a n c i v i l i z a t i o n h a s known.
As a
ma t t e r of fact, the e a r l i e s t w r i t i n g s y s t e m s w e r e s y l l a b i c , and m a n y l a n g u a g e s like A r a b i c , C h i n e s e , and J a p a n e s e s t i l l have s y l l a b i c w r i t i n g s y s t e m s , and even though they have b e e n looked upon as s o m e t h i n g l e s s e c o n o m i c a l and l e s s o p t i m u m , I think they should be given m o r e m e r i t for the r e a s o n s that will b e c o m e c l e a r s h o r t l y . It is sufficient at this point to cite one i n t e r e s t i n g e x p e r i m e n t .
In a r e c e n t a r t i c l e in
Science, R o z e n et at (1971) r e p o r t e d an e x p e r i m e n t w h i c h showed t h a t A m e r i c a n c h i l d r e n with r e a d i n g p r o b l e m s chould e a s i l y l e a r n to r e a d E n g l i s h when r e p r e s e n t e d by C h i n e s e characters. deficiency.
E i g h t to nine y e a r old e x p e r i m e n t a l c h i l d r e n w e r e c h o s e n for t h e i r r e a d i n g " T h e y had difficulty in identifying w o r d s by i n i t i a l o r final sounds a n d in c o m -
bining a s e q u e n c e of l e t t e r s into a known E n g l i s h w o r d " , and " w e r e unable to r e a d r e l i a b l y a s e t of r h y m i n g w o r d s (cat, fat, mat, sat) a f t e r b e i n g given the p r o n u n c i a t i o n of a t . " (p. 1264) Yet a f t e r a few h o u r s of t r a i n i n g in which they l e a r n e d C h i n e s e c h a r a c t e r s r e p r e s e n t ing s o m e E n g l i s h w o r d s , they could r e a d quite e a s i l y s e n t e n c e s m a d e of t h e s e c h a r a c t e r s
(e. g.,
"father bought a black ear").
The authors hypothesize that reading d i s -
/
ability in the c a s e of s e n t e n c e s w r i t t e n in E n g l i s h a l p h a b e t i s l a r g e l y a c c o u n t a b l e in t e r m s of the highly a b s t r a c t n a t u r e of the p h o n e m e , while f a c i l i t y in the c a s e of s e n t e n c e s r e p r e -
~Part of this paper appeared in: "A Survey of Linguistic Science", (1971),
16 - 128.
143 s e n t e d with C h i n e s e c h a r a c t e r s i s on the a c c o u n t of the fact that C h i n e s e c h a r a c t e r s m a p into s p e e c h a t the level of w o r d s r a t h e r than of p h o n e m e s o r a l p h a b e t i c l e t t e r s a s i s the c a s e in E n g l i s h , a n d p r o p o s e the unit of s y l l a b l e a s the m o s t s u i t a b l e v e h i c l e for t e a c h i n g r e a d i n g . The m e r i t of s y l l a b l e will b e c o m e m o r e a p p a r e n t in the c o u r s e of t h i s p a p e r . 2.
U n i t s in P r o d u c t i o n of Speech W h a t is the unit of s p e e c h p r o d u c t i o n ?
It would be i d e a l l y e c o n o m i c a l if o u r
b r a i n s t o r e d s e p a r a t e i n s t r u c t i o n s f o r e a c h p h o n e m e and g e n e r a t e d t h e m in the o r d e r of s e q u e n c e in a p h o n e m i c s t r i n g .
Note, for e x a m p l e , the following t r a d i t i o n a l a c c o u n t of the
p r o c e s s of s p e e c h production: We s h a l l a s s u m e that the s p e a k e r h a s s t o r e d in h i s m e m o r y a t a b l e of all the p h o n e m e s and t h e i r a c t u a l i z a t i o n s . T h i s t a b l e l i s t s the d i f f e r e n t ~ocal t r a c t c o n f i g u r a t i o n s o r m o t o r g e s t u r e s that a r e a s s o c i a t e d with e a c h phoneme and the conditions u n d e r w h i c h e a c h is to be used. In producing the u t t e r a n c e the s p e a k e r looks up, a s it w e r e , in the table the individual p h o n e m e s and then i n s t r u c t s h i s vocal t r a c t to a s s u m e in s u c c e s s i o n the c o n f i g u r a t i o n s o r g e s t u r e s c o r r e s p o n d i n g to the p h o n e m e s . (Halle and Stevens, 1964, p. 605) T h i s s i m p l e view, h o w e v e r , r u n s into m a n y difficulties. iant a c o u s t i c c o r r e l a t e s .
S p e c t r o g r a m s do not show i n v a r -
A t t e m p t s to c o n s t r u c t an a u t o m a t i c s p e e c h r e c o g n i t i o n d e v i c e
(e. g . , a m a c h i n e that will take s p e e c h a s input and produce a p r i n t - o u t of i t s p h o n e m i c i n s t a n t i a t i o n a s output) h a s failed despite y e a r s of e f f o r t s .
Speech s y n t h e s i z e r s that "look up"
p h o n e m i c t a b l e s produce all but n a t u r a l speech. The s e a r c h for phonemic i n v a r i a n c e a t the m o t o r level t h r o u g h e l e c t r o m y o g r a p h i c studies (e.g.,
F r o m k i n , 1965; H a r r i s e t al, 1965; L u b k e r and P a r r i s ,
1970; M a c N e i l a g e ,
1963; MacNeilage and Sholes, 1964, e t c . ) h a s not p r e s e n t e d c a u s e for m o r e o p t i m i s m .
One
then began to s p e c u l a t e that the unit of s p e e c h e n c o d i n g m a y have a d i f f e r e n t m a g n i t u d e than the s i z e of a p h o n e m e , and t h e r e a r e s o m e i n d i c a t i o n s t h a t it i s p e r h a p s s o m e t h i n g l a r g e r than the p h o n e m e s i z e . T h i s p a p e r is a r e v i e w of m o d e l s of s p e e c h production.
F o r e a s e of e x p o s i t i o n , I
will p o l a r i z e the two a l t e r n a t i v e m o d e l s , p e r h a p s m o r e than it i s w a r r a n t e d .
L a t e r I will
e x a m i n e s o m e p h e n o m e n a in s p e e c h t h a t s e e m to f a v o r one model o v e r the o t h e r . T h e two opposing m o d e l s m a y be given the d e s c r i p t i v e n a m e s : the " s a w - t o o t h " m o d e l and the " c o m b " model ( s e e d i a g r a m below).
The s a w - t o o t h m o d e l a s s u m e s t h a t
s p e e c h g e n e r a t i o n is nothing but a c o n c a t e n a t i o n of a s t r i n g of p h o n e m e s , e a c h p h o n e m e r e c e i v i n g a s e p a r a t e i n s t r u c t i o n f r o m the b r a i n , and functioning a s a s t i m u l u s for e l i c i t i n g the next phoneme in the s t r i n g , in a m a n n e r that p s y c h o l o g i s t s c a l l a c h a i n - a s s o c i a t i o n .
In
this model, the chain f o r m s a c l o s e d - l o o p , a s a p h o n e m e i s i n t e g r a l l y linked to the next segm e n t a s the f e e d b a c k to the l a t t e r ' s r e a l i z a t i o n , m u c h a s a guided m i s s i l e d e t e r m i n e s i t s flight path upon r e c e i v i n g i n f o r m a t i o n a b o u t i t s t a r g e t ' s d i r e c t i o n , speed, etc.
On the o t h e r
144
]and, the c o m b m o d e l of s p e e c h p r o d u c t i o n a s s u m e s t h a t s o m e t h i n g l a r g e r t h a n a p h o n e m e , s a y a s y l l a b l e o r a w o r d , f o r m s an i n s t r u c t i o n a l a n d o p e r a t i o n a l unit a t the n e u r o - m u s c u l a r level.
T h a t i s , it i s the s y l l a b l e o r w o r d b l o c k s , not individual p h o n e m e s , t h a t r e c e i v e
spearate brain instructions.
In t h i s view, s p e e c h is a s e r i a l b e h a v i o r in the s e n s e of L a s h -
t e y (1951), like s k i l l e d t y p i n g o r piano p l a y i n g in w h i c h the p e r f o r m e r ' s h a n d s r e c e i v e p r e p r o g r a m m e d i n s t r u c t i o n s of the s i z e of a w o r d o r a m e a s u r e f r o m the b r a i n .
This model
then i s o p e n - l o o p e d , m u c h like a t r a f f i c light s y s t e m t h a t d o e s not take the a c t u a l t r a f f i c v o l u m e into a c c o u n t in c h a n g i n g s i g n a l s o n c e the t i m e p a t t e r n i s built into the s y s t e m . One can d i a g r a m the two a l t e r n a t i v e v i e w s in the following w a y : The "Sawtooth" Model
Y a b c x y z
-
V v
b e g i n n i n g of a n e u r a l c o m m a n d b e g i n n i n g of the m o v e m e n t m o m e n t of a r r i v a l of a f f e r e n t i m p u l s a t i o n about b t i m e i n t e r v a l for c o n d u c t i o n of n e u r a l c o m m a n d t i m e i n t e r v a l for c o n d u c t i o n of f e e d b a c k s i g n a l l a t e n t p e r i o d until the n e x t n e u r a l c o m m a n d ( a d a p t e d f r o m K o z h e v n i k o v and C h i s t o v i c h , i966, p. 94)
T h e " C o m b " Model A
11111
a
b
c
d
e
A - simultaneous instruction a-e - sequential realization ( a d a p t e d f r o m Ohala, 1970, p. 12t) M a j o r a r g u m e n t s for f a v o r i n g the c o m b m o d e l a r e : (1)
In r u n n i n g s p e e c h , p h o n e m e s a r e not s e g m e n t a b l e into d i s c r e t e u n i t s but o v e r l a p in t h e i r a r t i c u l a t i o n .
(2)
T h e r e i s n o t e n o u g h t i m e for a r o u n d - t r i p t r a n s i t of n e r v e i m p u l s e s . m o t o r r e s p o n s e s to s e n s o r y s t i m u l i t a k e a s long a s 1/8 s e c o n d .
(3)
T h e r e a r e d i f f e r e n c e s in n e r v e i m p u l s e c o n d u c t i o n t i m e s , a m o n g a r t i c u l a t o r m u s c l e s , of up to 30 m s e c ( L e n n e b e r g , 1967, p. 96). A s i n g l e s t i m u l u s c a n n o t t r i g g e r a c o m p l e x of c o m m a n d s w h i c h h a v e to be t e m p o r a l l y s t a g g e r e d .
Human
145
P r o p o n e n t s of the phoneme t h e o r y c o u n t e r a r g u e that the lack of i n v a r i a n t s is due to such f a c t o r s as the m e c h a n i c a l c o n s t r a i n t s i n h e r e n t in the p e r i p h e r a l vocal a p p a r a t u s , l i m i t a t i o n s in the r e s p o n s e c a p a b i l i t i e s of the m u s c u l a r s y s t e m , and o v e r l a p p i n g in t i m e of the e f f e c t s of s u c c e s s i v e p h o n e m e c o m m a n d s .
A s f o r (2} above, Ohala (t970) a r g u e d , a la F a i r -
b a n k s (I954), t h a t it is not n e c e s s a r y t h a t a g e s t u r e be 100 p e r c e n t c o m p l e t e d b e f o r e a f f e r e n t i n f o r m a t i o n on the p r o g r e s s of the g e s t u r e is r e p o r t e d to the b r a i n .
Afferent signals are
p r o b a b l y c o n t i n u o u s l y s e n t to the b r a i n which i s c a p a b l e of p r e d i c t i n g on the b a s i s of initial preliminary information.
T h i s i s p l a u s i b l e , Ohala c o n t i n u e s , b e c a u s e a s s p e e c h m o v e m e n t s
b e c o m e m o r e a u t o m a t i c and m o r e r e m o v e d f r o m the c o n s c i o u s c o n t r o l , the a r t i c u l a t i o n b e comes more reflexive, which requires shorter latencies.
Ohala a l s o c o u n t e r a r g u e d the
point (3) above by saying that l a r y n g e a l s y n c h r o n i z a t i o n i s not p r e c i s e in voice o n s e t t i m e in a s p i r a t i o n a n d voicing,
F u r t h e r m o r e , O h a l a ' s r e c o n s t r u c t i o n of conduction t i m e d i f f e r e n c e s
between o r o - f a e i a l n e r v e s and l a r y n g e a l n e r v e s showed only a d i f f e r e n c e of 9 m s e c . w h i c h Ohala doubts will p r e s e n t any g r e a t p r o b l e m for the kind of p r e c i s e c o o r d i n a t i o n b e l i e v e d to be r e q u i r e d by L e n n e b e r g . Strong s u p p o r t for the c l o s e d - l o o p model c a m e r e c e n t l y f r o m W i c k e l g r e n (1969) who a r g u e d that L a s h l e y ' s r e j e c t i o n of the a s s o c i a t i v e - c h a i n m o d e l w a s p r e m a t u r e in t h a t one n e e d s to d i s t i n g u i s h n o n - c r e a t i v e s e r i a l b e h a v i o r f r o m c r e a t i v e s e r i a l b e h a v i o r and that L a s h l e y ' s objection a p p l i e s only to the l a t t e r , but not to the f o r m e r to which w o r d p r o d u c t i o n belongs.
W i c k e l g r e n a t t e m p t s to r e s c u e the a s s o c i a t i v e - c h a i n h y p o t h e s i s by p r o p o s i n g " c o n -
t e x t - s e n s i t i v e " a s s o c i a t i v e - c h a i n t h e o r y which a s s u m e s t h a t s e r i a l o r d e r i s encoded by m e a n s of a s s o c i a t i o n s between " c o n t e x t - s e n s i t i v e e l e m e n t a r y m o t o r r e s p o n s e s " (EM-R). In speech, this m e a n s t h a t a w o r d s u c h a s stop is a s s u m e d to be coded a l l o p h o n i c a l l y a s [4~st] ( Is] in the context of ~ _ _ [ t ] ) ,
[sto] , [ t ° p ] '
and [op~¢] . In this model the b a s i c units of production
would be c o n t e x t - s e n s i t i v e a l l o p h o n e s , a n d s e r i a l o r d e r i n g of t h e s e E M i l ' s would be a c h i e v e d , a s W h i t a k e r (1970, p. 3) put it, by i n t e r l o c k i n g the p r e c e d i n g and following c o n t e x t s in puzzle fashion.
Since s t r e s s , pitch, and d u r a t i o n a l d i f f e r e n c e s in o t h e r w i s e i d e n t i c a l c o n t e x t s c o n -
stitute i n d e p e n d e n t E M R ' s , it t u r n s out t h a t one n e e d s to s t o r e about 104 to 106 E M R ' s for s p e e c h production.
W i c k e l g r e n c l a i m s t h a t this is not a p r o b l e m b e c a u s e the h u m a n c o r t e x
c o n t a i n s about l010 n e u r o n s .
R e g a r d l e s s of the c o u n t e r - i n t u i t i v e n a t u r e of t h i s n e c e s s i t y
for such an e n o r m o u s c e r e b r a l s t o r a g e and r e t r i e v a l c a p a c i t y , I think the n u m e r o l o g y h e r e i s quite m e a n i n g l e s s . T h e r e is no e v i d e n c e w h a t s o e v e r that e a c h n e u r o n is a s s o c i a t e d with a specific allophone o r an EMR.
A v a i l a b l e e v i d e n c e is r a t h e r that a c o m p l e x n e t w o r k of n e u r -
ons b e h a v e s in a m u l t i p l y i n t e g r a t e d way, and u n l e s s s o m e s p e c i f i c r e l a t i o n s h i p s between a l l o p h o n e s and n e u r o n s a r e o b s e r v e d , a c l a i m t h a t the n u m b e r of a l l o p h o n e s does not begin to e x h a u s t the n u m b e r of n e u r o n s in the b r a i n i s p o i n t l e s s ,
146
W i e k e l g r e n ' s paper occasioned a c r i t i c i s m by MacKay (1970) who, citing e x a m p l e s f r o m German and English s p o o n e r i s m s , argued that W i c k e l g r e n ' s model would fail.
Speci-
fically, MacKay a s s u m e d that since association in the chain model is unidirectional, it would predict that r e v e r s e d s e g m e n t s in s p o o n e r i s m s would o c c u r much m o r e frequently following r e p e a t e d segments, i . e . , ABCBDA ---> ABDBCA, e . g . , C a v a l e r i e --> C a l a v e r i e ,
and that
positions of r e v e r s a l s would be random as long as the r e v e r s e d s e g m e n t s share s i m i l a r contexts in the chain.
But MacKay found that r e p e a t e d s e g m e n t s followed the r e v e r s e d s e g -
ments as often as they p r e c e d e d them, e . g . , W a s s e r f l a s c h e ---> F t a s s e r w a s c h e , and that, as others have also o b s e r v e d (Boomer and L a v e r , 1968; F r o m k i n , 1971), r e v e r s e d s e g ments occur only in the s a m e position within the syllable, i . e . , the switching occurs only between two syllable-initial segments, between two syllable nuclei, o r between two s y l l a b l e final segment. In a pilot study, Tatham and Morton (1970) tested W i c k e l g r e n ' s model by trying to detect any EMIR differences between two [ k ] ' s in [aku],
[uka] and ~iku~,
~ki].
They
r e a s o n e d that if W i c k e l g r e n was right, EIVIR for [k] in the f o r m e r set should be different f r o m that in the latter set; in p a r t i c u l a r , the muscle contraction for lip-rounding in the former
[k] would be g r e a t e r and longer than in the l a t t e r c a s e because
[a] is neutral to
lip-rounding, while [i] r e q u i r e s l i p - s p r e a d i n g which i s antagonistic to lip-rounding.
Since
EM-R's a r e c o n t e x t - s e n s i t i v e , the two [ k ] ' s would constitute two different E M R ' s and t h e r e fore would show different EMG patterns.
But the r e s u l t was negative, i . e . , EMG activity
in [aku ] was not different from that in [iku ].
Since there are nonetheless differences in
spatial configurations of the lips during the two [ k ] ' s ,
Tatham and M o r t o n ' s conclusion was
that "['~aku-] and [iku ] exist a f t e r coarticulation, but only [k] exists before coarticulation" (pp. 121-122). M a c K a y ' s c r i t i c i s m of Wickelgren was soon rebutted by Whitaker (1970) who a s s u m e d that Wickelgren could have argued that MacKay's hypotheses about r e v e r s a l s in s p o o n e r i s m s a r e not r e l e v a n t to a s s o c i a t i v e - c h a i n theory because it is after the e r r o r has been made that the c o r r e c t group of E M R ' s is a s s e m b l e d .
Thus, all e x a m p l e s that MacKay cited agains
Wickelgren could no longer be damaging, since E M R ' s a r e activated after t h e i r e r r o n e o u s o r d e r i n g has been established.
This assumption, Whitaker says, is borne out well by the
fact that r e g a r d l e s s of the cause of e r r o r , the e r r o r output is still an a d m i s s i b l e sequence in the language obeying all the phonological constraints.
(I have noted, however, the follow-
ing exceptions: comb---> nogwp r~owp (Hockett, 1967, p. 915, fn. l l ) s t i c k s h i f t - - > shtick sift (Fromkin, 1971, p. 32).) Whitaker gives a hypothetical example of a s p o o n e r i s m which would c r u c i a l l y bear on the issue.
That is, if a loud spanking spoonerized into a poud stank-
ing, the model would predict that the attophone of p would be a r t i c u l a t e d as a w o r d - i n i t i a l
147
h a s p i r a t e d [p ], not a s a post-s_ u n a s p i r a t e d [p].
If a n y r e s i d u e s of the o r i g i n a l l y i n t e n d e d
context r e m a i n e d in the r e a l i z e d output, the m o d e l would be d i s e o n f i r m e d . MacKay, Ohala, and W h i t a k e r p r o p o s e a m o d e l of language p r o g r a m m i n g in which an e n t i r e p h r a s e i s s i m u l t a n e o u s l y d i s p l a y e d in a " b u f f e r " zone, then i s r e a d off o r s c a n n e d in a u n i d r e c t i o n a l fashion.
T h a t i s , s p e e c h u n i t s a r e put into s o m e kind of a " h o p p e r " (Ohala,
1970, p. 133) w h e r e they a r e e n t e r e d in a p a r t i c u l a r o r d e r - - t h e o r d e r in w h i c h they will be "fed" to the a p p r o p r i a t e m u s c l e s .
In W h i t a k e r ' s w o r d s , " t h e phonological output of the
g r a m m e r i s d i s p l a y e d in a b u f f e r a n d the
tracking
m e c h a n i s m t r a c k s t h i s d i s p l a y by a c t i -
vating the r e q u i s i t e v o c a l t r a c t c o m m a n d s e t s " . Kozhevnikov and C h i s t o v i c h ' s (1965) view is s o m e w h a t e c l e c t i c , e c l e c t i c in the s e n s e that t h e i r model u s e s both the o p e n - l o o p m e c h a n i s m s .
They h y p o t h e s i z e that what m a k e s
the s t r i n g of s e g m e n t s in a s y l l a b l e be r e a l i z e d in a s e r i a l o r d e r i s not a s e p a r a t e and d i r e c t i n s t r u c t i o n for e a c h s e g m e n t f r o m the s p e e c h c e n t e r , n o r a p r o p r i o c e p t i v e i m p u l s a t i o n ( s e n sation of t e n s i o n in m u s c l e s ) o c c u r r i n g upon the m o v e m e n t s such t h a t s t i m u l a t i o n for the following m o v e m e n t is s u b l i m i n a l , which b e c o m e s above the t h r e s h o l d value to c a u s e an e x t e r n a l effect when g i v e n a n additional push c r e a t e d by the i m p u l s a t i o n o c c u r r i n g upon a r t i c ulation of the p r e c e d i n g s e g m e n t . MaeNeilage (1970) is a bit m o r e s p e c i f i c about the r e f l e x m e c h a n i s m .
He b o r r o w s
H e b b ' s (1949) notion of m o t o r e q u i v a l e n c e which is to say that the m o t o r s y s t e m is c o n t r o l l e d by i n t e r n a l s p e c i f i c a t i o n s of c e r t a i n s p a t i a l t a r g e t s so a s to a c h i e v e a single r e s u l t , e . g . , r e a c h i n g for a d o o r - k n o b , a t e n n i s p l a y e r r e a c h i n g f o r the ball h i t by the opponent, etc.
In
M a c N e i l a g e ' s model, the open-loop c o m p o n e n t e m i t s a c o n t e x t - i n d e p e n d e n t c o m m a n d for an a r t i c u l a t o r to r e a c h a c e r t a i n position and c l o s e d - l o o p c o n t r o l c i r c u i t s c o n s t a n t l y s a m p l e the m e c h a n i c a l s t a t e of the a r t i c u l a t o r and a d j u s t the c o m m a n d a c c o r d i n g l y ( s p e a k i n g w i t h a c i g a r e t t e o r a pipe in o n e ' s mouth p r o v i d e s a good e x a m p l e of s u c h a d j u s t m e n t ) .
MacNeil-
age s p e c u l a t e s that t h i s feedback loop is c o n t r o l l e d by the g a m m a m o t o r s y s t e m . 3.
Syllable a s a C a n d i d a t e Having so f a r d i s c u s s e d in s o m e d e t a i l a few m o d e l s of s p e e c h p r o d u c t i o n , I will now
t u r n to an e x a m i n a t i o n of s o m e p h e n o m e n a in s p e e c h t h a t s e e m to i n d i c a t e the s y l l a b l e a s a unit of linguistic p e r f o r m a n c e .
While the e x a m p l e s do not c o n s t i t u t e d i r e c t n e u r o p h y s i o l o g i -
cal e v i d e n c e , I think t h a t t h e i r i m p l i c a t i o n s a r e f a i r l y and a t t r a c t i v e l y strong.
The b a s i s of
t h i s o p t i m i s m i s the a s s u m p t i o n that, while i t i s not a b s o l u t e l y r e q u i r e d t h a t n e u r o p h y s i o logical m a t r i x be m a p p a b t e into b e h a v i o r a l m a t r i x in a s t r i c t o n e - t o - o n e fashion, t h e r e i s enough p r o j e c t i o n onto b e h a v i o r so t h a t i t s e x a m i n a t i o n can lead to r e a s o n a b l e a s s u m p t i o n s about the u n d e r l y i n g m e c h a n i s m s .
The r a t i o n a l e i s h a r d l y needed.
A physicist's assumption
of a t o m i c s t r u c t u r e s , E i n s t e i n ' s s p e c i a l t h e o r y of r e l a t i v i t y , P e r c i v a l L o w e l l ' s p r e d i c t i o n
148
of the e x i s t e n c e of Pluto on the b a s i s of his o b s e r v a t i o n of the a n o m a l o u s m o t i o n of U r a n u s , i n t e r n a l r e c o n s t r u c t i o n in h i s t o r i c a l l i n g u i s t i c s , etc. all b e a r t e s t i m o n y to r e a s o n a b l e n e s s of the a s s u m p t i o n . E a r l i e r , I e x p r e s s e d a doubt of p s y c h o l o g i c a l r e a l i t y of p h o n e m e s in n a i v e s u b j e c t s . Since S a p i r ' s 1933 p a p e r h a s been a powerful a n d i n f l u e n t i a l one, it i s w o r t h r e e x a m i n i n g s o m e of his e x a m p l e s that p u r p o r t to show the p s y c h o l o g i c a l r e a l i t y of p h o n e m e s .
His f i r s t
e x a m p l e had to do with his Southern P a i u t e i n t e r p r e t e r n a m e d Tony w r i t i n g his n a t i v e w o r d ~pa ./3a'~.
Tony w r o t e , " p a . ,
pause, pal"
T h i s " a s t o n i s h e d " Sapir, b e c a u s e Tony was not
" h e a r i n g " p h o n e t i c a l l y but p h o n e m i c a l l y , for in Southern P a i u t e , p r e v o c a l i c / p , s p i r a n t i z e d , b e c o m i n g a l l o p h o n i c a l l y ~/3, r , ~ ] r e s p e c t i v e l y .
t, k/ a r e
Since Tony did not w r i t e i n t e r -
vocalic / p / a s [t3] a s it w a s p r o n o u n c e d but a s [p], this c o n s t i t u t e d to Sapir the e v i d e n c e of a psychological r e a l i t y of p h o n e m e s .
But what i s " a s t o n i s h i n g " is not so m u c h the f a c t
that Tony w r o t e the s e c o n d s y l l a b l e a s p_~ a s the fact that he b r o k e the p h r a s e into two s y l l a b l e s : pa -, pause, pa'.
Given the fact t h a t t h e r e was a pause b e f o r e the s e c o n d s y l l a b l e
and given the f a c t t h a t a s p i r a n t does not o c c u r i n i t i a l l y in S o u t h e r n P a i u t e , Tony had n o choice but to w r i t e p._~. I will e x a m i n e a n o t h e r e x a m p l e f r o m Sapir in a l a t e r s e c t i o n . An i n d i c a t i o n t h a t people a r e not a w a r e of individual s e g m e n t s a s m u c h a s the a l p h a b e t i c w r i t i n g s y s t e m i m p l i e s c o m e s f r o m an e x p e r i m e n t with r e v e r s a l of w o r d s .
In an
i n f o r m a l t e s t , I once p r e p a r e d a l i s t of p o l y s y l l a b i c s e q u e n c e s which c o n t a i n e d a c t u a l E n g l i s h and Swahili w o r d s , and a s k e d the s u b j e c t s to r e v e r s e the w o r d s (by speaking, not by w r i t i n g ) . In n e a r l y all c a s e s of s u c c e s s f u l a t t e m p t s , the w o r d s w e r e r e v e r s e d in t e r m s of s y l l a b l e s e q u e n c e s , not in t e r m s of s e g m e n t a l s e q u e n c e s ( t h i s a f t e r p r a c t i c i n g with a few m o n o s y l labic w o r d s such a s cat, T i m , bat, e t c . )
F o r e x a m p l e , h o s p i t a l would c o m e out as ~ ! l - p i -
ho_._~s(not latipsoh), T i m b u k t u a s t u - b u k - t i m , a s k i - m a - s a (not a s i k a m a s ) ,
etc.
kikapu, " b a s k e t " , a s p u - k a - k i ,
samaki, "fish",
T h i s shows t h a t people a r e not a s m u c h a w a r e of s e g -
m e n t a l s e q u e n c e s as s y l l a b l e s e q u e n c e s . A s i m i l a r r e s u l t has been o b t a i n e d f r o m p e r c e p t u a l e x p e r i m e n t s which have shown t h a t people could not identify the t e m p o r a l o r d e r i n g of s e t m e n t s (Ladefoged, 1967; T h o m a s et at, t970).
F o r e x a m p l e , when u n f a m i l i a r n o n s e n s e s e g m e n t s w e r e p r e s e n t e d , l i s t e n e r s
could e a s i l y d i f f e r e n t i a t e between c o m p l e x s t i m u l i w h i c h d i f f e r e d in the o r d e r of t h e i r c o m ponents.
But they d i f f e r e n t i a t e d the s t i m u l i a s wholes and could not tell the r e l a t i v e t i m e s of
a r r i v a l of the c o m p o n e n t p a r t s .
T h a t i s , the s t i m u l i w h i c h d i f f e r e d only in the o r d e r of t h e i r
c o m p o n e n t s w e r e p e r c e i v e d as b e i n g d i f f e r e n t s i m p l y in o v e r a l l quality.
An e x p e r i m e n t with
l o c a l i z a t i o n of a c l i c k showed the s a m e n e g a t i v e r e s u l t s , s u g g e s t i n g t h a t l i s t e n e r s have d i f fictflty in d e t e r m i n i n g the p h y s i c a l o r d e r of a r r i v a l of individual i t e m s , and t h e r e f o r e t h a t i m m e d i a t e p e r c e p t i o n is p e r h a p s in t e r m s of l a r g e r units than p h o n e m e .
149
A n o t h e r indication t h a t the s y l l a b l e is the m i n i m a l unit of p r o d u c t i o n is found in c h i l d r e n ' s b a b b l i n g s which a r e e m i t t e d always in t e r m s of the CV type, e . g . , Ema, m a , ma~, J~da, da, d a ] , o r [ka, ka,
ka-]. " T h e y
ma, ma, ma,
do not b a b b l e in a n y way t h a t i n d i -
c a t e s a w a r e n e s s o r c o n t r o l of t~iits s m a l i e r than a s y l l a b l e , t~ (Ladefoged, 1967, p. t48) T h u s , even when they have m a s t e r e d the above p h r a s e s , s u c h i n d e p e n d e n t and a r b i t r a r y sequences as
[madaka~ ,.
[~kadam1,
[amkad),
etc. do not o c c u r .
A n o t h e r i m p r e s s i o n i s t i c e v i d e n c e of s y l l a b l e is s t u t t e r i n g and f a l s e s t a r t s .
It h a s
been o b s e r v e d t h a t s t u t t e r e r s s t u t t e r in t e r m s of s y l l a b l e s ( T a y l o r , 1966), and that in f a l s e s t a r t s one cannot c o r r e c t o n e s e l f b e f o r e the c o m p l e t i o n of at l e a s t the f i r s t s y l l a b l e of an u t t e r a n c e , even though the s p e a k e r h a s c o m e to be a w a r e of the m i s t a k e b e f o r e o r d u r i n g the f i r s t s e g m e n t of the u t t e r a n c e .
(cf. "Such e r r o r s a r e p r a c t i c a l l y n e v e r c o r r e c t e d until a
whole s y l l a b l e a t l e a s t h a s been e m i t t e d . "
F r y , 1964, p. 219)
An a r g u m e n t can be m a d e to
the effect t h a t if the n e u r a l c o m m a n d s w e r e s e n t out p h o n e m e b y p h o n e m e , one should be able to stop a t any s e g m e n t in the u t t e r a n c e , but if the unit of n e u r a l c o m m a n d s i s s y l l a b l e , then one would have to c o m p l e t e the m o t o r e x e c u t i o n of t h a t s y l l a b l e once the c o m m a n d h a s b e e n s e t out, i . e . ,
t h e r e would be no way to r e t r i e v e i t in the m i d d l e of the s y l l a b l e .
phenomenon m a y be likened to a p i t c h e r ' s t h r o w o r a g o l f e r ' s swing.
The
D u r i n g the final w i n d -
up b e f o r e the ball l e a v e s the p i t c h e r ' s hand, the p i t c h e r m a y r e a l i z e that the ball is going to be a wild pitch, and the g o l f e r m a y r e a l i z e d u r i n g his final swing b e f o r e his club hits the ball that it i s going to be s h o r t of the g r e e n .
Despite this r e a l i z a t i o n , h o w e v e r , he i s unable
to change the p a t t e r n of the swing, a s n e u r o n a l c o m m a n d s f o r that final swing have a l r e a d y been f i r e d , and once f i r e d , they m u s t be executed.
Any any one who h a s typed is p r o b a b l y
f a m i l i a r with the s e n s a t i o n that he h a s often e x p e r i e n c e d when he " h a d to" hit a w r o n g key even though the r e a l i z a t i o n t h a t it w a s a w r o n g one had c o m e p r i o r to the d o w n w a r d m o v e ment. A m o r e r e l e v a n t p h e n o m e n o n i s a t e m p o r a l o v e r l a p between a r t i c u l a t i o n s of two o r more phonemes.
In s u c h w o r d s a s two.__, who, coo, one c a n e a s i l y d e t e c t the l i p - r o u n d i n g
beginning with o r b e f o r e the i n i t i a l c o n s o n a n t , not a f t e r ( c o m p a r e t h e s e w o r d s with te.__a.a, he.._, key).
Daniloff and Moll (1968) found that c o a r t i c u l a t i o n of r o u n d i n g can begin a s e a r l y a s
four c o n s o n a n t s b e f o r e it is s e g m e n t a l l y due (e. g . , in s u c h w o r d s a s s i n c e t r u e and c o n s t r u e , the l i p - r o u n d i n g for u was o b s e r v e d to s t a r t a t n_). One can a r g u e that if t h e r e w e r e a s e p a r a t e and i n d e p e n d e n t n e u r a l c o m m a n d for e a c h p h o n e m e and if t h e s e c o m m a n d s w e r e g e n e r a t e d sequentially, how i s it p o s s i b l e t h a t a p a r t of a r t i c u l a t i o n of the following p h o n e m e i s e x e c u t e d d u r i n g the a r t i c u l a t i o n of the p r e c e d i n g p h o n e m e ? D o e s n ' t this s u g g e s t that our e f f e c t o r o r g a n s a l r e a d y p o s s e s s i n f o r m a t i o n c o n c e r n i n g the s e c o n d s e g m e n t a t the s a m e t i m e a s the a r t i c u l a t i o n of the f i r s t s e g m e n t i s b e i n g a c c o m p l i s h e d ?
It w a s in t h i s vein t h a t
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K o z h e v n i k o v and C h i s t o v i e h (1965) h y p o t h e s i z e d t h a t the m i n i m a l unit of a m o t o r c o m m a n d i s a s y l l a b l e , not a p h o n e m e , and a c c o r d i n g l y , t h a t s e g m e n t s within a s y l t a b l e r e c e i v e a s i m u l t a n e o u s package of i n s t r u c t i o n s for a r t i c u l a t i o n . One c a n of c o u r s e a r g u e that a s e a r l y a s 1951, L a s h l e y t h e o r i z e d t h a t c e r t a i n b e h a v i o r s involving r a p i d s e r i a l m o v e m e n t s , e . g . ,
piano playing, galloping, speaking, etc.
m u s t be c e n t r a l l y p r o g r a m m e d r a t h e r than p e r i p h e r a l l y o r g a n i z e d .
But L a s h l e y was n e v e r
specific about the m e c h a n i s m in s p e e c h production, and p h o n e m i c r e a l i s m in s t r u c t u r a l l i n g u i s t i c s was so s t r o n g t h a t L a s h l e y ' s h y p o t h e s i s went [ a r g l y i g n o r e d and u n t e s t e d .
Thus,
m o s t e a r l y EMG studies w e r e p r i m a r i l y occupied with finding m o t o r i n v a r i a n t s of p h o n e m e s , and the s o - c a l l e d a s s o c i a t i v e - c h a i n model d o m i n a t e d the thinking of b e h a v i o r a l and p h y s i o logical p s y c h o l o g i s t s . A n o t h e r i m p o r t a n t finding of Kozhevnikov a n d C h i s t o v i c h w a s that in d i f f e r e n t r a t e s of speech, s y l l a b l e was the s m a l l e s t unit t h a t was c o m p r e s s e d o r l e n g t h e n e d in equal p r o portion to the total length of t i m e of the u t t e r a n c e .
C o m p r e s s i o n r a t i o of individual s e g -
m e n t s w a s not in the s a m e p r o p o r t i o n to t h a t of the l a r g e r u n i t s .
Since it i s r e a s o n a b l e to
a s s u m e t h a t the d u r a t i o n a l r e l a t i o n s h i p in the c a s e of the v a r y i n g tempo of s p e e c h i s such that the s u m of the v a r i a n c e s of the c o m p o n e n t p a r t s should be equal to the v a r i a n c e of the total u t t e r a n c e , the s m a l l e s t c o m p o n e n t that shows j u s t t h i s kind of r e l a t i o n s h i p would b e c o n s i d e r e d to be a p r o g r a m m i n g unit in t e m p o r a l o r g a n i z a t i o n of the u t t e r a n c e .
On the o t h e r
hand, one would e x p e c t a d u r a t i o n a l i n t e r a c t i o n a m o n g s u b p a r t s of the unit in s u c h a way a s to c o m p e n s a t e o r c o m p l e m e n t e a c h o t h e r in o r d e r to m a i n t a i n the s p e c i f i e d a v e r a g e value. T h i s i s what Kozhevnikov and C h i s t o v i c h found with r e s p e c t to the s y l l a b l e ( s e e a l s o L i n d blom, 1968). L e h i s t e (1970) o b t a i n e d a s i m i l a r r e s u l t by e x a m i n i n g t e m p o r a l o r g a n i z a t i o n of s e v e r a l E n g l i s h w o r d s s u c h a s s t e a d , s t a y , stayed, steady, etc.
She found t h a t t h e r e w a s
i n d e e d a t e m p o r a l c o m p e n s a t i o n between c o m p o n e n t s e g m e n t s in t h e s e w o r d s .
For example,
in v a r i o u s p r o n u n c i a t i o n s of the w o r d steady, t h e r e w a s a n e g a t i v e c o r r e l a t i o n between the two vowels in such a way that if one vowel was longer, the o t h e r w a s p r o p o r t i o n a t e l y s h o r t e r , thus all tokens m a i n t a i n i n g the s a m e a v e r a g e d u r a t i o n ,
W h a t i s m o r e s u r p r i s i n g w a s the
fact that the w o r d s s t e a d and s t a i d had the s a m e total d u r a t i o n , although the vowel in s t e a d is i n t r i n s i c a l l y s h o r t e r than that in staid. d u r a t i o n s of c o n s o n a n t s .
A t e m p o r a l c o m p e n s a t i o n was m a d e by l o n g e r
T h i s study shows t h a t the unit of a r t i c u l a t o r y p r o g r a m m i n g i s c e r -
tainly l a r g e r than the s i z e of a s e g m e n t , and m a k e s it difficult to b e l i e v e that a r t i c u l a t i o n is m e r e l y a s e r i e s of t a c k i n g - o n of s e g m e n t s (if t h i s w e r e the c a s e , a w o r d in i s o l a t i o n and the s a m e w o r d a p p e a r i n g a s a s t e m in a d e r i v a t i o n a l f o r m would h a v e the s a m e d u r a t i o n , and the l a t t e r word would be l o n g e r than the f o r m e r j u s t by the a m o u n t of the n u m b e r of s e g m e n t s
151
in the suffix), and p r o v i d e s a r a t h e r s t r o n g a r g u m e n t a g a i n s t the view t h a t s p e e c h p r o d u c t i o n is a phoneme-by-phoneme Markovian process. T h e r h y t h m in s p e e c h a l s o a p p e a r s to b e h a v e in t e r m s of u n i t s of s y l l a b l e s i z e . T h u s , i a m b i c p e n t a m e t e r m e a n s t h a t t h e r e a r e five s e t s of p a i r e d s y l l a b l e s in a line, e . g . , And A L L the AIR a SOLemn S T I L L n e s s HOLDS. ( F r o m T h o m a s G r a y ' s Eiigy.
Capital l e t t e r s i n d i c a t e s t r e s s e d s y l l a b l e s . )
Although the
r o l e of r h y t h m in p r o s e language is not c o m p a r a b l e to t h a t in poetry, it should not be m i n i mized, s i n c e , although not always r e g u l a r , it definitely h a s tendency to m a i n t a i n i a m b i c rhythmicity.
Note the s t r e s s c o n t o u r in such w o r d s a s A R t i F I c i A L i T Y , DIFfeRENtiAtion,
O P p o r T U n i T Y , etc.
A good e x a m p l e of the a l t e r n a t i n g s t r e s s p a t t e r n is found in F i n n i s h
w h e r e the s t r e s s fails r e g u l a r l y on odd n u m b e r s y l l a b t e s , e . g . , " a b o u t my i g n o r a n c e " .
TIEtt~lVl~tt6MYYdesTAni
The u n d e r l y i n g c u r r e n t of r h y t h m i s so s t r o n g t h a t when r h y t h m i s
g r o s s l y v i o l a t e d l a n g u a g e s s e e m to have r u l e s w h o s e sole function i s to r e a d j u s t the s t r e s s p a t t e r n so a s to m a i n t a i n the r h y t h m i c i t y .
T h u s , E n g l i s h h a s r u l e s like S t r e s s A d j u s t m e n t
and A u x i l i a r y Reduction (cf. C h o m s k y and Halle, 1968, C h a p t e r 3) t h a t p r o d u c e r h y t h m i c contours.
F o r e x a m p l e , to d e r i v e s o l i d i t y f r o m SOLid+ity, the o r i g i n a l p r i m a r y s t r e s s
m u s t be w e a k e n e d to t e r t i a r y a s the new p r i m a r y s t r e s s i s a s s i g n e d on the next s y l l a b l e in the noun f o r m .
To give a n o t h e r e x a m p l e , in T u n i c a (an A m e r i n d i a n language in Louisiana)
TAkoMEli " a s p e c i e s of t r e e " i s d e r i v e d f r o m t a " d e t e r m i n e r " + KO " t r e e " + MEli " b l a c k " , and the u n d e r l y i n g f o r m of T I y a h P A n i " s h e was hungry, it i s s a i d " As t i + Y A H p a + An__!i. It h a s a l s o been o b s e r v e d that in E n g l i s h a l e x i c a l s t r e s s c a n shift to c o n f o r m with the r h y t h m i c stress, e.g.,
JUST f o u r T E E N vs. FOURteen SHILLings; QUITE . unKNOWN vs. an UNknown
LAND ( J o n e s , 1959, p. 253). It h a r d l y n e e d s a m e n t i o n that the d o m a i n of s t r e s s is a l s o a s y l l a b l e , not j u s t a vowel o r a s y l l a b l e n u c l e u s .
S t r e s s j u s t does not s h i f t to an i m m e d i a t e l y n e i g h b o r i n g s e g -
m e n t but to the p r e c e d i n g o r the n e x t s y l l a b l e , and not only does the vowel m a n i f e s t the s t r e s s , but s o do c o n s o n a n t s in the s a m e s y l l a b l e by being a s p i r a t e d , by having a l o n g e r d u r ation, by n e v e r b e c o m i n g voiced in a n o t h e r w i s e v o i e e a b l e e n v i r o n m e n t , etc.
The utility
of s y l l a b l e in s t r e s s r u l e s i s well e x e m p l i f i e d in s u c h t r a d i a t i o n a l s t a t e m e n t s a s : In Czech, s t r e s s falls r e g u l a r l y on the f i r s t s y l l a b l e ; In P o l i s h , s t r e s s falls r e g u l a r l y on the p e n u l t i m a t e syllable; In F r e n c h , s t r e s s falls r e g u l a r l y on the final s y l l a b l e . The s a m e can be said of t o n e s .
Typically, the d o m a i n of a tone i s a s y l l a b l e , and
again, vowels a r e not the only s e g m e n t s t h a t c a r r y t o n e s .
A tone s a n d h i m a y a p p e a r in
such c o m p l e x f o r m s a s inducing n a s a l i z a t i o n o r voicing of the initial c o n s o n a n t in low tone
152
s y l l a b l e s a s in Kpelle (e. g . , p¢r~ " " a h o u s e " , " pcrcx ~ " [ b c r ¢ i ] " t h e h o u s e " ; 1S_~ " l i e down", 1_~_~[~a~3 " l a y it down".
Weimers,
1962), or high tone inducing devoicing o r a s p i r a t i o n a s
happens in s o m e C h i n e s e d i a l e c t s . 4.
Linguistic Implications I will now e x a m i n e s o m e l i n g u i s t i c i m p l i c a t i o n s of the s y l l a b l e model.
Firstly,
difficulty in speaking a f o r e i g h language having the s a m e s e t of sounds (e. g . , E n g l i s h and Swahili) o r in p r o n o u n c i n g n o n s e n s e f o r m s v i o l a t i n g the n a t i v e m o r p h e m e s t r u c t u r e c o n d i tions (e. g , , t r y
m p s t g l i ; i t i s m e r e l y a r e a r r a n g e m e n t of the word g l i m p s e d by shifting
gH. to the end) can be e x p l a i n e d in t e r m s of this model.
T h a t i s , it would be n a t u r a l l y diffi-
cult to r e - p r o g r a m the s e r i a l b e h a v i o r . Furthermore,
if we a c c e p t the a s s u m p t i o n t h a t the m i n i m a l unit of s p e e c h p r o d u c t i o n
i s a s y l l a b l e , then t h i s a s s u m p t i o n d i c t a t e s t h a t e o a r t i c u f a t i o n and m e t a t h e s i s between s e g m e n t s a c r o s s a s y l l a b l e b o u n d a r y should s e l d o m o c c u r , while they would o c c u r m a x i m a l l y within a syllable.
T h a t is, in the s t r i n g abc____#xyz w h e r e ~ i s a s y l l a b l e b o u n d a r y , it should
be r a r e for b and y, o r a_ and z to be c o a r t i c u l a t e d o r to c h a n g e s p l a c e s , while it p r e d i c t s t h a t c o a r t i c u l a t i o n and m e t a t h e s i s between s e g m e n t s within a s y l l a b l e would be m o r e c o m mon.
T h i s follows s i n c e our p e r i p h e r a l o r g a n s would p r e s u m a b l y have the i n f o r m a t i o n a b o u t
the whole s y l l a b l e at the beginning of a r t i c u l a t i o n of the s y l l a b l e , and one can e x p e c t c a s e s w h e r e c o m p a t i b l e a r t i c u l a t i o n s f o r d i f f e r e n t s e g m e n t s within a s y l l a b l e a r e s i m u l t a n e o u s l y m a d e o r w h e r e a s e q u e n c e of s e g m e n t s a r e r e a l i z e d in a r e v e r s e d o r d e r p e r h a p s due to excessively simultaneous coarticulation. metathesis, e.g.,
horse (from hros),
give one m o r e s p e c i f i c e x a m p l e h e r e .
H i s t o r y and c h i l d r e n a r e abound with e x a m p l e s of
b i r d ( f r o m brid), a s k - ax, l i s p - lips~ etc.
I will
In F r e n c h , n a z a l i z a t i o n of vowels o c c u r s only b e f o r e
a t a u t o s y l l a b i c n a s a l , but not b e f o r e a n a s a l t h a t c o n s t i t u t e s the initial s e g m e n t of the n e x t syllable, e.g., so?
langage ~i~agaJ'] " l a n g u a g e " but l a n i c e
lanis
"woolen".
Why should t h i s be
The l o w e r i n g of the v e l u m in a n t i c i p a t i o n of the following n a s a l should o c c u r i r r e s p e c -
tive of the position of the s y l l a b l e b r e a k .
The s y l l a b l e t h e o r y would explain a n a s a l c o a r t i c -
ulation in a t a u t o s y l l a b i c VN and i t s a b s e n c e a c r o s s a s y l l a b l e boundary in a n a t u r a l way. C o n s i d e r now the a s s u m p t i o n which h a s been f a v o r e d h e r e in c o n n e c t i o n with a f a m i l i a r phonological r u l e of the f o r m : X --~ Y~
Z
If one c h e r i s h e s a c h a i n - a s s o c i a t i o n model (a m o d e l that e x p l a i n s a s e r i a l b e h a v i o r ABC a s A e l i c i t i n g B r e s p o n s e which then b e c o m e s a s t i m u l u s for C r e s p o n s e , and so on), then it s e e m s to m e t h a t he c a n n o t w r i t e such a r u l e , f o r how can he explain t h a t what follows (Z) t r i g g e r s a change in what p r e c e d e s (X to Y), when a t the s a m e t i m e one is about to
153
a c c o m p l i s h the r e q u i r e d a d j u s t m e n t , one h a s no i n f o r m a t i o n a t all a b o u t Z a s Z h a s to w a i t for the c o m p l e t i o n of X ?
In view of the f a c t t h a t t h e r e a r e r a r e l y c o n t e x t - f r e e r u l e s , but
that m a n y phonological r u l e s a r e c o n t e x t - r e s t r i c t e d , like the above, n e c e s s i t a t i n g the s p e c i fication of such units a s s y l l a b l e s o r s t r e s s g r o u p s a s the d o m a i n in w h i c h a change i s t r i g g e r e d , one can only w o n d e r what the phonological r u l e s would look like and how m a n y " e x c e p t i o n s " and " e n v i r o n m e n t s " one could d i s p e n s e with if r u l e s w e r e w r i t t e n in t e r m s of the m o t o r u n i t s .
It i s known, f o r e x a m p l e , t h a t V e r n e r ' s Law and c o n s o n a n t c l u s t e r s a r e
e x c e p t i o n s to G r i m m ' s Law.
A s Ladefoged (1967, p. 172) s p e c u l a t e s , if all t ' s w e r e the
s a m e in the s e n s e t h a t all t ' s had the s a m e s t o r a g e p a t t e r n and the s a m e n e u r a l coding in the b r a i n , then s h o u l d n ' t all t ' s have changed in the s a m e way without e x c e p t i o n ? But once we a c c e p t the a s s u m p t i o n t h a t e a c h t had a d i f f e r e n t n e u r a l r e p r e s e n t a t i o n depending upon the s y l l a b i c s t r u c t u r e of w h i c h i t i s a p a r t , t h e n e x c e p t i o n s m a y c e a s e to be e x c e p t i o n s . T h i s g i v e s m o r e m a g n i t u d e to the s t a t u s of s y l l a b l e a s a l i n g u i s t i c unit.
Indeed, in
phonological r u l e s t h e r e a r e s o m e c o n s t r a i n t s t h a t s e e m to be due to the n a t u r e of the s y l lable a s a unit.
F o r e x a m p l e , one c a n i m a g i n e r u l e s t h a t s t a t e : " s h i f t the s t r e s s to the
next syllable", " l e n g t h e n a vowel in e v e r y second s y l l a b l e " (e. g . , in T~ibatulabal, a U t o A z t e c a n in Cali fornia, ta.-,vogimanada "go along c a u s i n g h i m to s e e " ) , o r " d e s t r e s s in f r o n t of a s t r e s s e d s y l l a b l e " , e t c . , but h a r d l y s u c h r u l e s a s " s h i f t the s t r e s s to the n e x t s e g m e n t " "lengthen every second segment", etc. 5.
Syllabic P r o p e r t y Having deposed s e g m e n t for s y l l a b l e , one m i g h t a s k what the e x a c t n a t u r e of s y l l a b l e
is, as I have b e e n u s i n g the t e r m in a r a t h e r loose way.
U n f o r t u n a t e l y , d e s p i t e the fact t h a t
m o s t l a y m e n t can i n t u i t i v e l y d e t e r m i n e the c o r r e c t n u m b e r of s y l l a b l e s within a w o r d , we d o n ' t know yet e x a c t l y w h a t a s y l l a b l e i s .
S t e t s o n ' s (1951) t h e o r y t h a t e a c h s y l l a b l e i s a c -
c o m p a n i e d b y a " b a l l i s t i c c h e s t p u l s e " h a s b e e n r e f u t e d by Ladefoged (1962), and p h o n e t i c i a n s a r e s t i l l unable to find s i m p l y physiological c o r r e l a t e s of s y l l a b l e s . s y l l a b l e a s a unit m u s t have s o m e m i n i m u m r e q u i r e m e n t s .
It s e e m s t h a t a
T h i s r e q u i r e m e n t m a y be in
t e r m s of a c e r t a i n n u m b e r of s e g m e n t s , m o r a e , o r a c e r t a i n a m o u n t of m u s c l e w o r k . E n g l i s h m o n o s y l l a b i c w o r d s , for e x a m p l e , m u s t contain a t e n s e vowel, e . g . , m a y , boyp e t c . , o r m u s t end in a c o n s o n a n t if the vowel i s lax, e . g . ,
~
~
how,
apt m e t ,,
be_~d etc. It i s a p p r o p r i a t e a t this point to e x a m i n e a n o t h e r e x a m p l e t h a t S a p i r (1933) g i v e s a s e v i d e n c e of the p s y c h o l o g i c a l r e a l i t y of p h o n e m e s .
S a p i r n o t i c e d t h a t in h i s p h o n e t i c s
c l a s s e s h i s s t u d e n t s had an i l l u s i o n of h e a r i n g a final glottal stop in s u c h d i c t a t e d n o n s e n s e •
/
1
j
w o r d s a s s m c , or plla, but not in the c a s e of pila o r pila:.
Noting, a s we did in the p r e -
ceding p a r a g r a p h , t h a t a n a c c e n t e d final s y l l a b l e in E n g l i s h m u s t end in a t e n s e (long) vowel
154 with o r without a following c o n s o n a n t , o r in a s h o r t vowel t h a t m u s t be followed by a e o n s o n e n t o r a c o n s o n a n t c l u s t e r , Sapir w r i t e s t h a t "the i l l u s i o n of the final glottal stop i s e s s e n t i a l l y the i l l u s i o n of a g e n e r a l i z e d final c o n s o n a n t n e e d e d to c l a s s i f y the d i c t a t e d w o r d s into a known c a t e g o r y " .
In w h a t s e n s e , then, does the i l l u s i o n of h e a r i n g the w o r d - f i n a l ?
c o n s t i t u t e an e v i d e n c e of the p s y c h o l o g i c a l r e a l i t y of p h o n e m e s ?
? i s not e v e n a n E n g l i s h
p h o n e m e , n o r i s it a n allophone of a p a r t i c u l a r c o n s o n a n t in the w o r d - f i n a l position (it m a y o c c u r as an allophone of t in such w o r d s as c a p t a i n [ k a e p ? m ] , bottle [ b o ? l ] ,
etc.).
mountain [mann?n],
I s n ' t it m o r e r e a s o n a b l e and p l a u s i b l e to say t h a t the i l l u s i o n is due to
the p s y c h o l o g i c a l r e a l i t y of s y l l a b l e s whose c o n s t i t u e n t s m u s t m e e t s o m e m i n i m u m r e q u i r e m e n t s in E n g l i s h ? When one c o n s i d e r s the fact t h a t the i l l u s i o n of ? did not happen in the c a s e of s y U a b l e s t h a t m e t such r e q u i r e m e n t s , i t b e c o m e s c l e a r t h a t the ? i l l u s i o n i s not to be explained on a s e g m e n t a l level but within the f r a m e of a s y l l a b l e . The s o - c a l l e d " c o m p e n s a t o r y l e n g t h e n i n g " of a vowel in the e n v i r o n m e n t of a c o n s o n e n t a l l o s s ( e . g . , Late OE nyht -->ME u~t " n i g h t " , PIE n i s d o s - - > L a t , n~dus " n e s t " , P G m c . guns -->OE ~
" g o o s e " ) , o r a c o n s o n a n t g e m i n a t i o n e n s u i n g a vowel l o s s o r s h o r t e n i n g
( e . g . , Lat. v i d e r e " s e e " : P P . v i s u s , but s e d e r e " s i t " : P P . s e s s u s ) i s a l s o , I feel, f u n c t i o n a l l y r e l a t e d to the e f f o r t to p r e s e r v e the unity of a s y l l a b l e . Kozhevnikov and C h i s t o v i c h have a d v a n c e d the notion t h a t any n u m b e r of p r e v o e a l i c c o n s o n a n t s f o r m s a s y l l a b l e - u n i t w i t h the following vowel (the final p o s t v o c a l i c c o n s o n a n t is said to c o n s t i t u t e a s y l l a b l e of i t s own).
L a t e r , they s u g g e s t t h a t the s i m p l e s t and the m o s t
b a s i c a r t i c u l a t o r y unit i s the c o n s o n a n t - v o w e l s y l l a b l e c o m p l e x , and t h a t m o r e c o m p l e x c c o m b i n a t i o n s of the type CCV a r e nothing but the g r o u p s of t h e s e s i m p l e e o m p l e x s o r g a n ized in s u c h a way t h a t the s e c o n d c o m p l e x i s a c c o m p l i s h e d p a r t i a l l y in p a r a l l e l with the a c c o m p l i s h m e n t of the f i r s t , s u p p o r t i n g F a i r b a n k s and G u t t m a n ' s (1958) c o n c l u s i o n t h a t m u l tiple r e p e t i t i o n s of a c o m p l e x of m o v e m e n t s which f o r m a l l y coincide with a CV s y l l a b l e a r e v e r y typical f o r a r t i f i c i a l s t u t t e r i n g . Can we s a y that CCV = C X + C V w h e r e X i s a vowel r e f l e x ? to t h i n k t h a t a l l s y l l a b l e s a r e of a s i m p l e CV type. m i g h t be a p r o t o - t y p e .
It i s c e r t a i n l y t e m p t i n g
T h e r e a r e c e r t a i n i n d i c a t i o n s t h a t t~is
Many l a n g u a g e s have e i m p l e CVCV s e q u e n c e s (e. g . , J a p a n e s e ,
Swahili, Yoruba), and l i n g u i s t i c c h a n g e s s e e m to f a v o r t h o s e w h o s e output is c l o s e r to the o p t i m u m s e q u e n c e CVCV (it i s a c h i e v e d , for e x a m p l e , by b r e a k i n g up c o n s o n a n t c l u s t e r s , by e p e n t h e s z i n g vowels, e t c . ) e.g.,
CVCV s e q u e n c e s a r e a l s o c h i l d r e n ' s f a v o r i t e v o c a b u l a r y ,
m o m m y , daddy, doggie, e h i e k i e , e t c .
F o r m s like room, dad, dog, etc. a r e c e r -
tainly later developments. The d i s t i n c t i o n between a " w e a k " s y l l a b l e and a " s t r o n g " s y l l a b l e in E n g l i s h s t r e s s r u l e s ( C h o m s k y and Halle, 1968, p. 29) is i n t e r e s t i n g to e x a m i n e in t h i s r e s p e c t .
For
155
example, both w o r d s PERson and DIAlect have the s t r e s s in the initial syllable, but when the affix -al is added, the s t r e s s r e m a i n s on the f i r s t syllable in P E R s o n a l but shifts to the next syllable in DIAlect to give d i a L E C T a l .
The r e a s o n is of c o u r s e given as being due to
the different syllable s t r u c t u r e s of the penultimate syllable, -so__~n- being a "weak" syllable, while - l e q t - being a " s t r o n g " syllable.
But if we a s s u m e that the consonant c l u s t e r in l e c t
has an intervening vowel r e f l e x (which is deleted a f t e r the s t r e s s rule), both f o r m s would have the s t r e s s on the antepenultimate syllable.
Admittedly, this is a wild speculation, "but
I think it is worth having a f r e s h look at all phonological behavior, e s p e c i a l l y the prosody and exceptions, with the t h e o r y of sullable integrated into the c u r r e n t theory. Recently, MacKay (1971) proposed that the encoding of the syllable of the type CVC be made in two stages: 1.
S(yllable) - - > C P + V P
2.
VP - - ~ V ( o w e l ) + C P
CP = consonant part, VP = vocalic part
(I cannot help but notice the p a r a l l e l i s m between the above and Chomsky's well-known g e n e r a t i v e r u l e s : S --->NP+VP, VP --->V+NPI).
As evidence for justifying the p r i m a r y
break between C and VC r a t h e r than between CV and C, MacKay cites r h y m e in poetry (e. g . , h/old, t / o l d , g/old), P / t g L / a t i n ( i g - p - a y a t i n - l - a y ) , and s p o o n e r i s m s (e. g . , shell = sh(out)+(y)ell).
(I may add h e r e a l l i t e r a t i o n , e . g . , w / i l d and w / o o l y , as another s u p -
porting phenomenon.)
In the light of evidence cited, I find MacKay's postulation of s y l -
labic p r o p e r t y and encoding p r o c e d u r e v e r y appealing and interesting, 6.
Concluding R e m a r k s In this paper, I have d i s c u s s e d a few specific models of speech production.
I have
also examined some speech phenomena that s e e m to indicate the utility of the syllable as a model of linguistic p e r f o r m a n c e .
While such indications a r e quite a few in n u m b e r , as we
have seen, one must guard oneself against undue o p t i m i s m in the absence of any c o r t i c a l evidence, and against unwarranted impudence, the s o r t that shrouded the phoneme for d e c ades.
Chronologically, the syllable theory may have been adopted, as the phoneme began to
l o s e its explanatory power, simply a s a cautious m e a s u r e , as the i m m e d i a t e l y next phonological unit l a r g e r than a phoneme is a syllable.
This is to say that what s e e m s to be c e r -
tain is not so much that the syllable is the (minimal) unit of speech production as that the phoneme is too s m a l l a unit. other unit than a syllable. unit.
Thus, there is no r e a s o n why one may not speculate on some
Indeed, t h e r e a r e a few signs that point to the word a s a plausible
F o r example, a t e m p o r a l compensation in Slovak and Estonian o p e r a t e s i n t e r s y l l a b i c -
ally as well as i n t r a s y l l a b i e a l l y (i. e . , the s t r u c t u r e of a preceding syllable d e t e r m i n e s the vowel length of the following syllable, el. Lehiste, 1965).
Also, the e x a m p l e s f r o m Lehiste
(1970) that I cited e a r l i e r fit a word model better than a syllable model.
And the domain of
156
m o s t phonological r u l e s , e . g . , vowel harmony, s t r e s s , etc. is the word, that is, they operate only within the word boundary (hence the t e r m , w o r d - l e v e l phonology).
Spoonerism
is another example where association jumps a c r o s s a syllable boundary, s o m e t i m e s a word boundary, although t h e r e is a constraint in such a way that a switching takes place only b e tween segments that occupy the s a m e position in syllabic s t r u c t u r e , i . e . , only between s y l lable initials, syllabic nuclei, or between the post-nuclei s e g m e n t s in two syllables (el. F r o m k i n , 1971; MacKay, 1970). Even if the syllable theory p r o v e s to be c o r r e c t , there still r e m a i n s a m a j o r task of defining the syllable at a n e u r o - m u s e u l a r level and of specifying its exact role as an explana t o r y tool in phonetic and phonological d e s c r i p t i o n s .
Linguists, who a r e e s s e n t i a l l y non-
e x p e r i m e n t a l i s t s , like to d e s c r i b e syllables in t e r m s of language-specific surface constraints on phonemic strings (the s o - c a l l e d m o r p h e m e s t r u c t u r e conditions).
Since these constraints
differ f r o m language to language, does this mean that each language has differently sized units of neural commands ? P e r h a p s so, although one would like to think that n e u r a l m e c h a n i s m s a r e basically the s a m e for all men.
Also, in the c a s e of speech rhythm, what does
it mean to say that a language is s y l l a b l e - t i m e d , while another is s t r e s s - t i m e d ?
Exactly,
how does the relationship between rhythm and syllable differ in these two types of languages ? Another p r o b l e m is the question of the syllable at two leve
of description: phono-
logical (deep, linguistics) and phonetic (surface physiological),
Is the syllable definable at
both l e v e l s o r only at one of the two? If the latter, which o n e ?
Could it be, for example,
that syllables a r e units only at the physiological level, while the units at the linguistic level are segments?
If so, just what is the relationship between the two? (cf. Tatham, 1971.)
N e e d l e s s to say, resolution of these questions will have to wait further study and neurophysiological evidence.
As the new wave of phonetics s a i l s f r o m the tongue to the
brain (cf. Kim, 1971a), it should find such evidence hovering offshore, if not in the offing. Meanwhile, I suppose one is allowed to play the r o l e of e i t h e r a defense lawyer o r a p r o s e cuting attorney disputing about c i r c u m s t a n t i a l evidence.
P e r s o n a l l y , I find it difficult to
believe that the command sequence for the word yes is just the r e v e r s e of that for the word say, s e y .
Somehow, I feel that their storage and r e t r i e v a l in the brain is unitary, indepen-
dent, and f r e e of segmental association. Reference s B o o m e r , D.S. and J. D.M. L a v e r (1968), 'Slips of the Tongue", B r i t i s h Journal of D i s o r d e r s of Communication, 3__, 1-12. Chomsky, N. and M. Halle (1968), The Sound P a t t e r n of E n g t i s h , H a r p e r and Row, New York. Daniloff, R. and K. Moll (i968), "Coarticulation in Lip Rounding", J o u r n a l of Speech and He_.._aring R e s e a r c h , 1__tl, 707-721.
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Fairbanks, G. (±954), "A Theory of the Speech Mechanism as a Servo-System", Journal of Speech and Hearing Disorders, 19__, 133-139. Fairbanks, G. and N. Guttman (1958), "Effects of Delayed Auditory Feedback upon A r t i c u lation", Journal of Speech and Hearing Research, 1_, 12-22. Fromkin, Victoria (1965), "Some Phonetic Specifications of Linguistic Units: An E l e c t r o myographic Investigation", UCLA W0rking Papers in Phoneties~ No. 3. Fromkin, Victoria (1966), "Neuromuscular Specification of Linguistic Units", Language and Speech, 9_ 170-199. Fromkin, Victoria (1968), "Speculations on Performance Models", Journal of Linguistics,
i, 47-68. Fromkin , Victoria (1971), "The Non-Anomalous Nature of Anomalous Utterances", Language, 4__7.27-52. Fry, D.B. (1964), "The Functions of the Syllable", Zeitschrift fur Phonetik, 1_7, 215-237. Halle, M. and K.N. Stevens (1964), "Speech Recognition: A Model and a P r o g r a m for Research", in Fodor and Katz (eds.), The Structure of Language, Prentice-Hall, 604-612. Harris, Katherine, S., G.F. Lysaught, and M.M. Schvey(1965), "Some Aspects of the Production of Oral and Nasal Labial Stops", Language and Speech, 8_, 135-147. Hebb, D.O. (1949), "The Orga~zation of Behavior, Wiley, New York. Hockett, C.F. (1967), "Where the Tongue Slips, There Slip r ' , in To Honor Roman Jakobson, Vol. It, Mouton, The Hague, 910-936. Jones, D. (1959), An Outline of English Phonetics, 8th ed., E.P. Dutton, New York. Kim, C-W (1971a), "A New Direction in Phonetics", Language Sciences, 16_, 35-40. Kim, C-W (1971b), "Experimental Phonetics", in W.O. Dingwal(ed.), A Surveyof Linguistic Science, University of Maryland Press, 16-128. Kozhevaikov, V.Ao and L.A. Chistovich (1965), Rech: Artikulatsia i Vosprovatiye, MoskvaLeningrad, English Translation by Joint Publications Research Service, U.S. Department of Commerce, Speech: Articulation and Perception (1966). Ladefoged, P. (1962), "Sub-glottal Activity During Speech", Proceedings of the IVth I n t e r national Congress of Phonetic Sciences, Mouton, The Hague, 73-91. Ladefoged, P. (1967), "Units in the Perception and Production of Speech", in his Three A r e a s of Experimental Phonetics, 143-172. Lashley, K.S. (1951), "The Problem of Serial Order in Behavior", in L. A. Jeffress (ed.), Cerebral Mechanisms in Behavior, Wiley, 112-136. Reprinted in Saporta (ed.), Psycholinguistics, 180-198. Lehiste, Ilse (1965), "The Function of Quantity in Finnish and Estonian", Language, 41__, 447 -456. Lehiste, Ilse (1970), "Temporal Organization of Spoken Language", Ohio State University Working P a p e r s in Linguistics, 4_., 95-114. Lenneberg, E.H. (1967),
Biological Foundations of Language, Wiley, New York.
Lindblom, B. (1968), "Temporal Organization of Syllable Production", Quarterly P r o g r e s s and Status Report, 1968.2/3:1-5, Speech Transmission Laboratory, Royal Institute of Technology, Stockholm.
158
Lubker, J . P . and Pamela J. P a r r i s (1970), "Simultaneous Measurement of Intraoral Pressure, Force of Labial Contact, and Labial Electromyographic Activity During Production of the Stop Consonant Cognates / p / and / b / " , Journal of Acoustic Society of America, 47, 725-633. MacKay, D.G. (1970),"Spoonerisms: The Structure of E r r o r s in the Serial Order of Speech", Neuropsychologia, 8_, 323-350. MacKay, D.G. (1971), "The Structure of Words and Syllables: Evidence from E r r o r s in Speech, MS. MacNeilage, P . F . (1963), "Electromyographic and Acoustic Study of the Production of Certain Final Clusters", Journal of Acoustic Society of America, 35.__, 461-463. MacNeitage, P . F . (1970), "Motor Control of Serial Ordering of Speech", Psychological Review, 77, 182-196. MacNeitage, P . F . and G.N. Shotes (1964), "An Electrom ographic Study of the Tongue During Vowel Production", Journal of Speech and Hearing Research, 7, 209-232, Ohala, J. (1970), "Aspects of the Control and Production of Speech", UCLA Working Papers in Phonetics, No. 15. Ohman, S. (1967), "Peripheral Motor Commands in Labial Articulation", Quarterly Progress and Status Report, 1967.4:30-63, Speech Transmission Laboratory, Royal Institute of Technology, Stockholm. Rozin, P . , Susan Poritsky, and Raina Sotsky (1971), "American Children with Reading Problems can Easily Learn to Read English Represented by Chinese Characters", Science, 171, 1264-1267. Sapir, E. (1933), "La realite psychologique des phonemes", Journal de Psych01ogie, 30___, 247-265. (English translation in ~D. G. Mandelbaum (ed.), Selected Writing of Edward Sapir ~ University of California P r e s s (t963), 46-60, Stetson, R.H. (1951), Motor Phonetics, North Holland Publishing Co., Amsterdam. Tatham, M.A.A. (1971) ,"Motor Building in Phonetic Theory", Language Sciences, 14___,16-19. Tatham, M.A.A. and Katherine Morton (1969), "Some Electromyography Data Towards a Model of Speech Production", Language and Speech, 1_~2, 39-53. Tatham, M.A.A. and Katherine Morton (±970), "Explaining Some Apparently ContextSensitive Effects in Speech", University of Essex Language Center Occasional Papers, 9_, 116-122.
Taylor, I.K. (1966), "What Words are Stuttered?" Psychological Bulletin, 65_._,236-242. Thomas,
I.B. and P.B. Hill, F.S. Carrol, and B. Garcia (1970), "Temporal Order in the Perception of Vowels", Journal of Acoustic Society of America, 4_88, I010-I013~
Welmers,
W.E.
(1962), "The Phonology of Kpelle", Journal of African Languages,
i., 69-93.
Whitaker, H.A. (1970), "Some Constraints on Speech Production Models", University of Essex Language Occasional papers, 9_, 1-13. Wickelgren, W.A. (1969), "Context-Sensitive Coding, Associative Memory, in (Speech) Behavior", Psychological Review, 76___,1-15.
and Serial Order
TOWARDS A THEORY OF LINGUISTIC MEMORY T e r r e n c e J. K e e n e y U n i v e r s i t y of C a l i f o r n i a , R i v e r s i d e E v e r y day we a r e exposed to a v a s t a m o u n t of l i n g u i s t i c m a t e r i a l .
We h e a r the s p e e c h
of o t h e r s ; we r e a d s i g n s , n e w s p a p e r s , books, e t c . ; and we h e a r our own speech.
Although
we do not a t t e n d to and fully p r o c e s s all of t h i s l i n g u i s t i c m a t e r i a l , t h e r e r e m a i n s a s u b s t a n t i a l s u m of it which we do pay a t t e n t i o n to, p r o c e s s , and c o m p r e h e n d .
Yet, of all t h i s
c o m p r e h e n d e d m a t e r i a l , we know f r o m o u r own p e r s o n a l e x p e r i e n c e t h a t we can r e c a l l v e r b a t i m only a tiny portion.
When a c c u r a t e v e r b a t i m r e c a l l b e c o m e s i m p o r t a n t , f o r e x a m p l e ,
in c o u r t r o o m t e s t i m o n y , the fallibi, lity of e v e n t h i s tiny portion b e c o m e s woefully a p p a r e n t . R a r e l y can we a c c u r a t e l y r e c a l l even what we o u r s e l v e s have said p r e v i o u s l y . It i s t h i s m e m o r y f o r what we have h e a r d o r r e a d t h a t I a m r e f e r r i n g to b y the t e r m "linguistic memory."
L i n g u i s t i c m e m o r y m u s t not be c o n f u s e d with s e m a n t i c m e m o r y , the
topic of the p a p e r s by Loftus and G l a s s in t h i s c o n f e r e n c e .
Semantic m e m o r y r e f e r s to the
" p e r m a n e n t " s t o r a g e of m o r p h e m e s o r c o n c e p t s in the h u m a n o r a r t i f i c i a l b r a i n .
Linguistic
m e m o r y , on the o t h e r hand, r e f e r s to the r e p r o d u c t i o n o r r e c o g n i t i o n of p r i o r l i n g u i s t i c i n puts.
A s we s h a l l see, l i n g u i s t i c m e m o r y d r a w s upon s e m a n t i c m e m o r y , but it i n c l u d e s , in
addition, i n f o r m a t i o n about specific, t i m e - d e p e n d e n t , p r i o r e v e n t s . By analogy to m e c h a n i c a l d e v i c e s that have b e e n d e s i g n e d f o r p r e s e r v i n g s p e e c h and w r i t i n g , we m i g h t c o n s i d e r l i n g u i s t i c m e m o r y a s a p r o c e s s w h i c h p r e s e r v e s the p h y s i c a l f e a t u r e s of the l i n g u i s t i c input.
J u s t a s a photocopy o r a tape r e c o r d i n g encode the e l e c t r o -
m a g n e t i c a n d a c o u s t i c f e a t u r e s of w r i t i n g and s p e e c h , p r e s e r v e t h e m for s o m e t i m e , and then r e p r o d u c e o r play b a c k the o r i g i n a l p h y s i c a l f e a t u r e s with s o m e d e g r e e of a p p r o x i m a t i o n to the o r i g i n a l , so the h u m a n s e n s o r y and c e n t r a l n e r v o u s p r o c e s s e s would encode the p h y s i c a l f e a t u r e s of the input, s t o r e t h e s e f e a t u r e s , and then r e p r o d u c e t h e m a t s o m e l a t e r t i m e t h r o u g h the a p p r o p r i a t e n e u r o - m u s c u l a r c o m m a n d s . In o r d e r to a c c o u n t for o u r v e r y l i m i t e d ability to a c c u r a t e l y r e c a l l the p r i o r inputs, we would n e e d to i n t r o d u c e i n e f f i c i e n t encoding and d e c o d i n g d e v i c e s and a r a p i d l y d e t e r i o r ating s t o r a g e o r a s t o r a g e which is e a s i l y d i s r u p t e d by i n t e r f e r e n c e f r o m p r i o r and s u b s e quent inputs.
P h y s i c a l a n a l o g i e s to t h e s e p r o c e s s e s could be c o n s t r u c t e d by the u s e of v e r y
g r a i n y e m u l s i o n s for a photocpy o r l o w - f i d e l i t y m a g n e t i c tape for a sound r e c o r d i n g , c o p i e s that fade o r r e c o r d i n g tape that d i s i n t e g r a t e s o v e r t i m e , the s u p e r i m p o s i t i o n of photos o r t a p e s , and i n e f f i c i e n t r e a d - o u t o r playback e q u i p m e n t . The c r i t i c a l f e a t u r e of all s u c h d e v i c e s , d e s p i t e the d i f f e r e n t i a l a c c u r a c y of t h e i r r e p r o d u c t i o n s , i s the fact t h a t they o p e r a t e on the p h y s i c a l f e a t u r e s of the input.
Any l o s s
160
of i n f o r m a t i o n o r e r r o r s in r e c a l l m u s t t h e r e f o r e be r e l a t e d to the p h y s i c a l f e a t u r e s of the input.
F o r h e a r d m e s s a g e s , p h o n e m e s that sound alike would be s u b s t i t u t e d for one a n o t h e r
and p a u s e s and i n t o n a t i o n s would be l o s t o r d i s t o r t e d .
F o r r e a d m e s s a g e s , the s a m e types
of e r r o r s would o c c u r and, in addition, l e t t e r s t h a t looked a l i k e m i g h t be s u b s t i t u t e d f o r one a n o t h e r and punctuation m a r k s and s p a c i n g s would be l o s t o r d i s t o r t e d .
F r o m s t u d i e s of
a u d i t o r y and v i s u a l p e r c e p t i o n and s h o r t - t e r m m e m o r y for i s o l a t e d p h o n e m e s and l e t t e r s ( W i c k l e g r e n , 1966; Gibson, e t al, 1962), we know enough a b o u t the r e l e v a n t v i s u a l and a u d i t o r y a n a l y z e r s to be able to p r e d i c t the n a t u r e of t h e s e types of e r r o r s in l i n g u i s t i c m e m o r y . However, when we i n v e s t i g a t e the n a t u r e of the e r r o r s m a d e in both the r e c a l l and r e c o g n i t i o n of c o h e r e n t l i n g u i s t i c m a t e r i a l , we find that the model d e s c r i b e d above w h i c h is b a s e d on the p h y s i c a l f e a t u r e s of the input f a i l s .
B a r t t e t t (1932) studied the r e c a l l of s h o r t
s t o r i e s and found r e g u l a r i t i e s within the e r r o r s c o m m i t t e d ; but t h e s e e r r o r s had to do w i t h the " s h a r p e n i n g " and " l e v e l i n g " of t h e m e s and i d e a s , not with the confusion of s i m i l a r sounds or letters.
F i l l e n b a u m { 1966) i n v e s t i g a t e d m a m o r y f o r n e g a t i v e c o n s t r u c t i o n s and found a c -
c u r a t e m e m o r y of the " g i s t " of what had been r e a d , but not of the specific w o r d s o r s y n t a c t i c constructions. Sachs (t967a) studied m e m o r y f o r c o n n e c t e d d i s c o u r s e by i n t r o d u c i n g r e c o g n i t i o n - t e s t s e n t e n c e s a t the c o n c l u s i o n of s h o r t s t o r i e s .
When t h e s e t e s t s e n t e n c e s w e r e a c t i v e - t o -
p a s s i v e o r " f o r m a l " t r a n s f o r m s of o r i g i n a l s t o r y s e n t e n c e s , h e r s u b j e c t s gave tittle e v i d e n c e t h a t they could d e t e c t any d i f f e r e n c e between the o r i g i n a l and t e s t s e n t e n c e s , even though they w e r e e x p l i c i t l y i n s t r u c t e d to t r y and d e t e c t c h a n g e s in f o r m as well a s in m e a n i n g .
On the
o t h e r hand, t r a n s f o r m s of an o r i g i n a l s t o r y s e n t e n c e which r e v e r s e the a c t o r - o b j e c t r e l a t i o n or in s o m e o t h e r way d r a s t i c a l l y a l t e r e d the m e a n i n g of a s t o r y s e n t e n c e w e r e r e a d i l y d e t e c t ed a s b e i n g different.
T h e s e m e a n i n g c h a n g e s w e r e d e t e c t e d d e s p i t e the f a c t t h a t they did not
a l t e r the word o r d e r o r actual f o r m of the s e n t e n c e any m o r e than did the u n d e t e c t a b l e a c t i v e t o - p a s s i v e and " f o r m a l " t r a n s f o r m s . In a s u b s e q u e n t study (Sachs, 1967b), she found t h a t change even in the a c t u a l w o r d s of a s e n t e n c e i n t r o d u c e d by the s u b s t i t u t i o n of s y n o n y m s in the r e c o g n i t i o n t e s t w a s g e n e r a l l y not detected.
F u r t h e r m o r e , the a c t i v e - t o - p a s s i v e , " f o r m a l " , and s y n o n y m - s u b s t i t u t i o n
t r a n s f o r m s w e r e u n d e t e c t a b l e when t h e r e was a s l i t t l e a s 7 . 5 s e c o n d s of filled (with f u r t h e r s t o r y m a t e r i a l ) t i m e e l a p s i n g between the s e n t e n c e a n d i t s r e c o g n i t i o n t e s t . Mathewson and K e e n e y (1971) modified the S a c h s ' p r o c e d u r e so that m e a n i n g and f o r m could be v a r i e d c o m p l e t e l y independently of one a n o t h e r .
By the use of d e e p - s t r u c t u r e a m b i g -
uous s e n t e n c e s such a s , "He was in fact the one to h e a r b e f o r e l e a v i n g " , we w e r e able to change the m e a n i n g of a s e n t e n c e without i n t r o d u c i n g any c h a n g e in the f o r m of the s e n t e n c e itself.
This was a c c o m p l i s h e d by v a r y i n g the l i n g u i s t i c context in which the s e n t e n c e was
161
embedded.
In the e x a m p l e below, the m e a n i n g of the s e n t e n c e c h a n g e s f r o m the s t o r y to the
t e s t s i m p l y a s a r e s u l t of i t s context. Story The s p e a k e r f r o m New York was by f a r the b e s t . He spoke a b o u t the p r o b l e m s of d e a f education. He b r o u g h t f o r t h s o m e v e r y i n t e r e s t i n g i d e a s . The i d e a s c o n c e r n e d the s o c i a l and p e r s o n a l p r o b l e m s in e d u c a t i n g deaf people. None of the o t h e r s p e a k e r s i m p r e s s e d m e as much. The one I m e n t i o n e d w a s the only one w o r t h l i s t e n i n g to. He was, in fact, the one to h e a r b e f o r e leaving. Test happy.
Of all the people who u n d e r w e n t s u r g e r y for d e a f n e s s , only Bill left the h o s p i t a l He was~ in fact, the one to h e a r b e f o r e leaving. The findings of t h i s study c o n f i r m e d and e x t e n d e d t h o s e of S a c h s ' .
Meaning changes,
even when they involved no change in the f o r m of the s e n t e n c e , w e r e r e a d i l y d e t e c t e d . c h a n g e s such as a r t i c l e o r a d v e r b m o v e m e n t t r a n s f o r m s , e . g . ,
Form
"He was, in fact, the one to
h e a r b e f o r e l e a v i n g . / I n fact, he was the one to h e a r b e f o r e l e a v i n g " , which p r e s e r v e d the m e a n i n g , w e r e not d e t e c t e d even when a s few a s 25 s y l l a b l e s of m a t e r i a l i n t e r v e n e d between the s e n t e n c e s , as in the e x a m p l e above.
In this study, a s well as in t h o s e of Sachs, t h e s e
r e s u l t s c a n n o t be a s c r i b e d to the s u b j e c t s ' lack of a w a r e n e s s of the types of c h a n g e s to be detected.
P r i o r to the e x p e r i m e n t the s u b j e c t s w e r e i n s t r u c t e d to d e t e c t both f o r m a l c h a n g e
and m e a n i n g change.
In addition, they w e r e shown e x a m p l e s of e a c h type of change.
In o r -
d e r to produce such r a p i d f o r g e t t i n g of the p u r e l y f o r m a l f e a t u r e s of l i n g u i s t i c m a t e r i a l , i t i s n e c e s s a r y only t h a t the s u b j e c t be l i s t e n i n g o r r e a d i n g n o r m a l l y and c o m p r e h e n d i n g the m a t e r i a l r a t h e r than m e m o r i z i n g o r r e h e a r s i n g the s e n t e n c e s .
Since it is i m p o s s i b l e f o r
m o s t people to m e m o r i z e s e n t e n c e s a s r a p i d l y a s they a r e h e a r d in n o r m a l speech, such r a p i d f o r g e t t i n g i s the c o m m o n e x p e r i e n c e .
In the e x p e r i m e n t s c i t e d above, the s u b j e c t s did
not know which of the s e n t e n c e s in the s t o r y they would s u b s e q u e n t l y be p r e s e n t e d in the r e c ognition t e s t .
Although i n s t r u c t e d to r e m e m b e r f o r m , they w e r e unable to do so.
A r e c e n t e x p e r i m e n t ( J a r v e l l a and H e r m a n , 1972) s u g g e s t s that the p r o c e s s of c o m p r e h e n s i o n may, i t s e l f , c o n t r i b u t e to, o r a t l e a s t allow for, the d e t e r i o r a t i o n of v e r b a t i m memory.
They i n v e s t i g a t e d the r u n n i n g m e m o r y s p a n for c o n n e c t e d d i s c o u r s e .
When r e c a l l
w a s c a l l e d f o r at the c o n c l u s i o n of a c o m p l e x s e n t e n c e , v e r b a t i m r e c a l l of the f i r s t c l a u s e in the s e n t e n c e was b e t t e r for s e n t e n c e s of s u b o r d i n a t e c l a u s e - m a i n c l a u s e c o n s t r u c t i o n than for their converse.
If we m a k e the r e a s o n a b l e a s s u m p t i o n t h a t a s u b o r d i n a t e c l a u s e i s not fully
p r o c e s s e d and thus not fully c o m p r e h e n d e d until its m a i n c l a u s e h a s been h e a r d and c o m p r e hended, we s e e that in the s e n t e n c e in which the s u b o r d i n a t e c l a u s e p r e c e e d s the m a i n c l a u s e the s u b o r d i n a t e c l a u s e m u s t be held in s o m e p a r t i a l l y p r o c e s s e d s t a t e u n t i l the m a i n c l a u s e i s understood.
A m a i n c l a u s e can, of c o u r s e , be c o m p r e h e n d e d in i s o l a t i o n .
T h u s , we s e e that
162
a t the c o n c l u s i o n of the e n t i r e s e n t e n c e the f i r s t c l a u s e in a s u b o r d i n a t e - m a i n s e n t e n c e is s t i l l only p a r t i a l l y c o m p r e h e n d e d , while the f i r s t c l a u s e in a m a i n - s u b o r d i n a t e s e n t e n c e h a s b e e n fully c o m p r e h e n d e d f o r some t i m e .
T h e r e f o r e , the p o o r e r v e r b a t i m r e c a l l in the m a i n -
s u b o r d i n a t e s e n t e n c e s u g g e s t s t h a t c o m p r e h e n s i o n of a c l a u s e a c t u a l l y d i m i n i s h e s the l i s t e n e r s ' ability to recall that clause verbatim. When
exact verbatim
etc., in which the form devices of rhyme
recall is required,
of expression
and/or rhythm
as in remembering
is as important
are almost
songs,
as the meaning,
poems,
invariably present in the material itself.
devices provide a definite formal
structure into which the meaning
may
a given idea, yet only one way which possesses
be many
ways
priate meter
of expression
and rhyme.
on the form by the rhyme glnal message meter
Thus memory and rhythm
for the meaning
and rhyme
in the myths
combined
restfit in accurate verbatim
without any actual verbatim
rituals,
the special mnemonic
must
be fitted.
These
There the appro-
with the restrictions reconstruction
storage of that message.
put
of the ori-
The pervasiveness
of
and unwritten histories of preliterate cultures belies the fact
that v e r b a t i m r e c a l l without t h e s e s t r u c t u r a l r e s t r i c t i o n s i s difficult, if not i m p o s s i b l e . A m o d e l of l i n g u i s t i c m e m o r y w h i c h c o n s i s t s of a copy of the s u r f a c e p h y s i c a l f e a t u r e s of the l i n g u i s t i c input is c l e a r l y i n a d e q u a t e .
The individual sounds, m o r p h e m e s , and s y n t a c -
tic s t r u c t u r e s a r e not d i r e c t l y s t o r e d in m e m o r y ; n o r does the input s i m p l y excite c e r t a i n p r e - e x i s t i n g m o r p h e m e " n o d e s " in a p e r m a n e n t m e m o r y s t o r a g e s y s t e m .
Rather, linguistic
m e m o r y depends on an a n a l y s i s of the m e a n i n g of the input. T h a t m e a n i n g and not f o r m is the p r i m a r y i n f o r m a t i o n r e t a i n e d in the m e m o r y of w o r d s e m b e d d e d within s e n t e n c e s was d e m o n s t r a t e d by Bobrow (1970).
He s e l e c t e d a n u m b e r
of a m b i g u o u s nouns and p r e s e n t e d t h e m a s s u b j e c t and o b j e c t n o u n s of s e n t e n c e s .
The m e a n -
ings of t h e s e a m b i g u o u s nouns w e r e d e t e r m i n e d by the context e s t a b l i s h e d by the s e n t e n c e in which they w e r e e m b e d d e d .
An e x a m p l e of t~vo s e n t e n c e s in which the nouns take on d i f f e r e n t
m e a n i n g s is "The pine b o a r d s h o r e d the r i v e r b a n k . / T h e s e c u r i t i e s b o a r d c l o s e d the s h a k y bank." A list of sentences pair was presented of three ways:
to the subjects to study for later recall.
twice in the course of this list.
I) exact sentence repetition,
tained the same meanings.
was presented
noun meanings,
III) change
These
If) change
in the sentence
in the sentence
On the recall test the subject-noun
noun pairs were
Each noun
repeated
in one
context which main-
context which changed
the noun
was given as a cue for the recall of the object-
noun. If the actual physical features of the input were three conditions of repetition should have been equal, in all conditions,
However,
if the meaning
remembered,
then the recall in all
since the nouns were
of the input was remembered,
presented
twice
then recall in
163
Condition I and tI should have b e e n s u p e r i o r to r e c a l l in Condition HI.
The m e a n i n g of the
nouns changed f r o m the f i r s t to the s e c o n d p r e s e n t a t i o n of the s e n t e n c e in Condition tII; so, in effect, the w o r d m e a n i n g s w e r e not r e p e a t e d .
Thus, by virtue of t h e i r r e p e t i t i o n within
the l i s t , the w o r d m e a n i n g s in Conditions I and II should have been r e c a l l e d with a g r e a t e r f r e q u e n c y than the o n c e - p r e s e n t e d m e a n i n g s of Condition III. B o b r o w ' s r e s u l t s c o n f i r m e d the m e a n i n g - b a s e d m e m o r y m o d e l and w e r e o p p o s e d to the f o r m - b a s e d m e m o r y model. call in Condition III.
R e c a l l in Conditions I and II was equal and s u p e r i o r to r e -
Changing the meaning of a w o r d when it i s r e p e a t e d does not i n c r e a s e
the probability of its r e c a l l a s much a s if it w e r e r e p e a t e d with the s a m e meaning. Not only a r e the f o r m s of m o r p h e m e s , w o r d s , and s e n t e n c e s not r e t a i n e d e x a c t l y in m e m o r y , even the division between s e n t e n c e s is not r e t a i n e d .
B r a n s f o r d and F r a n k s (1971)
p r e s e n t e d s e n t e n c e s that contained one, two, o r t h r e e of a total of four r e l a t e d " l i n g u i s t i c ideas".
In a s u b s e q u e n t r e c o g n i t i o n t e s t in which the s u b j e c t s w e r e a s k e d to indicate w h e t h e r
o r not they had actually h e a r d the e x a c t s e n t e n c e b e f o r e , the s e n t e n c e s with all four of the r e l a t e d i d e a s w e r e m o s t often " r e c o g n i z e d " a s having b e e n h e a r d b e f o r e , even though, in r e a l i t y , they had not been p r e v i o u s l y p r e s e n t e d .
In linguistic m e m o r y , then, the s e p a r a t i o n
between s e n t e n c e s is not r e t a i n e d and i d e a s which "go t o g e t h e r " , but which a r e h e a r d in s e p a r a t e s e n t e n c e s , a r e combined. The e v i d e n c e that we have r e v i e w e d up to this point is c o n s o n a n t with a model of m e m o r y in which not the f o r m of the linguistic input, but s o m e o t h e r a s p e c t of it, which we have c a l l e d " m e a n i n g " , is encoded, s t o r e d , and r e t r i e v e d .
H o w e v e r , even this m e a n i n g -
b a s e d model is inadequate, for we " r e m e m b e r " m e a n i n g s that we have n e v e r b e f o r e h e a r d explicitly s t a t e d .
T h e r e is evidence to s u g g e s t that we cannot d i s t i n g u i s h our i n f e r e n c e s f r o m
our p e r c e p t u a l inputs. B r a n s f o r d et a1.(t972) have d e m o n s t r a t e d f a l s e r e c o g n i t i o n of s e n t e n c e s w h i c h contain i n f o r m a t i o n that was not explicitly s t a t e d in the initial input s e n t e n c e s .
Rather, these falsely
r e c o g n i z e d s e n t e n c e s e x p r e s s e d m e a n i n g s which could be i n f e r r e d f r o m a combination of the previous linguistic input and the l i s t e n e r s ' knowledge of t h r e e - d i m e n s i o n a l s p a t i a l r e l a t i o n s . F o r e x a m p l e , the s e n t e n c e " T h r e e t u r t l e s r e s t e d on the floating log and a f i s h s w a m beneath t h e m . ", along with spatial knowledge, l e a d s to the i n f e r e n c e that the s e n t e n c e " T h r e e t u r t l e s r e s t e d on the floating log and a fish s w a m beneath i t . " is a l s o t r u e . In a s e r i e s of e x p e r i m e n t s , B r a n s f o r d e t al. p r e s e n t e d s e n t e n c e s such a s the one with " t h e m " above and found that in a s u b s e q u e n t r e c o g n i t i o n t e s t , in w h i c h the s u b j e c t s w e r e to indicate w h i c h s e n t e n c e s they actually had h e a r d b e f o r e , f a l s e r e c o g n i t i o n of the i n f e r r e d s e n t e n c e s , such a s the one with "it" above, was j u s t a s f r e q u e n t a s the c o r r e c t r e c o g n i t i o n of
164
the s e n t e n c e s actually h e a r d .
The s u b j e c t s r e s p o n d e d e x a c t l y a s if they had h e a r d the s e n -
t e n c e s that w e r e only i n f e r e n c e s f r o m what they had heard.'
Control s e n t e n c e s that w e r e only
o n e - w o r d d i f f e r e n t f r o m the p r e s e n t e d s e n t e n c e s , but w e r e not i n f e r e n c e s f r o m t h e m , w e r e not r e c o g n i z e d a s having been h e a r d b e f o r e ; so the above r e s u l t s a r e not due to the s m a l l f o r m a l d i f f e r e n c e bet~veen the p r e s e n t e d and the i n f e r r e d s e n t e n c e s . t e s t s w e r e p a r a l l e l to t h o s e of the r e c o g n i t i o n t e s t ,
The r e s u l t s of r e c a l l
The s u b j e c t s " r e c a l l e d " s e n t e n c e s which
they had n e v e r h e a r d b e f o r e , but which w e r e i n f e r e n c e s f r o m t h e s e p r e v i o u s l y h e a r d s e n tences. So, linguistic m e m o r y d o e s not involve s i m p l y an encoding o r e x t r a c t i o n of s o m e information, i . e . ,
m e a n i n g , which i s d i f f e r e n t f r o m the p h y s i c a l f o r m of the input.
Rather,
linguistic m e m o r y i s b a s e d on an a c t i v e , c o n s t r u c t i v e p r o c e s s which is t r i g g e r e d by s o m e linguistic input, but is not r e s t r i c t e d to that input. p r o c e s s u n d e r s t a n d i n g o r thinking.
F o r lack of a b e t t e r t e r m , I will call t h i s
The r e c a l l i n g o r r e - t h i n k i n g of t h e s e initially t r i g g e r e d
thoughts c o n s t i t u t e s the phenomenon we call linguistic m e m o r y .
When a s s e s s i n g the a c c u -
r a c y of m e m o r y we should s e e k i s o m o r p h i s m not between the p e r c e p t u a l input at t i m e T O and the s u b s e q u e n t o r g a n i s m i c output at t i m e T1, but between the thought p r o c e s s e s at t i m e s T O and T 1 Any combination of linguistic and situational e v e n t s which p r o d u c e s the s a m e thoughts o r l e a d s to the s a m e u n d e r s t a n d i n g should be i n d i s t i n g u i s h a b l e in m e m o r y .
For example,
we would e x p e c t that all five of the following would t r i g g e r b a s i c a l l y the s a m e thoughts and would t h e r e f o r e all be c o n f u s e d one with the o t h e r in a r e c o g n i t i o n t e s t . 1.
"John had a f a v o r i t e toy.
He l o s t it.
He w a s s a d . "
2.
" J o h n ' s f a v o r i t e toy was l o s t , so he was s a d . "
3.
"John had a f a v o r i t e toy.
He was sad b e c a u s e he l o s t i t . "
4.
"John had a f a v o r i t e toy.
It w a s l o s t .
5.
"John l o s t his f a v o r i t e toy. ", h e a r d while looking a t a p i c t u r e of a boy with a sad
That m a d e John s a d . "
e x p r e s s i o n oa his face. What combination of e v e n t s a c t i v a t e s a given thought is not r e m e m b e r e d , r a t h e r , the thought p r o c e s s i t s e l f is r e p e a t e d at the t i m e of r e m e m b e r i n g .
Not being able to r e m e m b e r w h e t h e r
we have r e a d s o m e t h i n g or h e a r d it is a c o m m o n e x p e r i e n c e , and often we eventually d i s c o v e r that in fact we had n e i t h e r r e a d it n o r h e a r d it, but only thought it o u r s e l v e s . If the f o r m of a linguistic input is v e r y unusual, for e x a m p l e , if it i s u n g r a m m a t i c a l , poetic, e m p l o y s s t r a n g e m e t a p h o r s , contains v e r y l o w - f r e q u e n c y w o r d s , is spoken in an u n f a m i l i a r a c c e n t , e t c . , then c e r t a i n thoughts m a y be a c t i v a t e d by the f o r m i t s e l f .
Since
thoughts a r e the " s t u f f " of m e m o r y , t h e s e thoughts about the f o r m a r e capable of being r e thought, i . e . , r e m e m b e r e d , at s o m e l a t e r t i m e giving the illusion that the f o r m of the input
165
h a s been r e t a i n e d ,
H o w e v e r , the thoughts about the u n u s u a l f o r m a r e v e r y r a r e l y , if e v e r ,
i s o m o r p h i c with the f o r m of the input i t s e l f .
F o r e x a m p l e , upon h e a r i n g an u n g r a m m a t i c a l
s e n t e n c e , the m e a n i n g s a c t i v a t e d by the s e n t e n c e and the fact that it i s u n g r a m m a t i c a l m a y be thought.
Then, in a l a t e r r e c o g n i t i o n t e s t , a g r a m m a t i c a l s e n t e n c e t h a t a c t i v a t e d the
s a m e thoughts about m e a n i n g would not be r e c o g n i z e d b e c a u s e it did not a c t i v a t e the f a c t of u n g r a m m a t i e a l i t y a s did the i n i t i a l input.
However, any n u m b e r of u n g r a m m a t i c a l inputs
which a c t i v a t e d the a p p r o p r i a t e thoughts would be c o n f u s e d with one a n o t h e r and all " r e c o g nized'.' a s baying been h e a r d b e f o r e . about it, i . e . ,
The f o r m i t s e l f i s not r e m e m b e r e d , only what i s thought
t h a t it is u n g r a m m a t i c a l .
Once the focus i s r e m o v e d f r o m the c o m p a r i s o n of the p e r c e p t u a l input to the output a s a m e a s u r e of the a c c u r a c y of m e m o r y and p l a c e d i n s t e a d on the c o m p a r i s o n of the thought p r o c e s s e s a t t i m e s T O and T 1, m u c h of the o b j e c t i v i t y of r e s e a r c h into m e m o r y p h e n o m e n a i s s e m m i n g l y lost.
The thoughts which a r e a c t i v a t e d by l i n g u i s t i c and o t h e r e v e n t s a r e not
a c c e s s i b l e to d i r e c t o b s e r v a t i o n a s a r e the e v e n t s t h e m s e l v e s .
H o w e v e r , i n f o r m a t i o n about
the n a t u r e of the r e l e v a n t thought p r o c e s s e s c a n be obtained by a c o m p a r i s o n of t h o s e u t t e r a n c e s w h i c h a r e all r e c o g n i z e d a s being what was h e a r d p r e v i o u s l y .
T h a t i s , the n a t u r e of
the thoughts a c t i v a t e d by a n e v e n t a r e r e v e a l e d by what is c o n f u s e d in a r e c o g n i t i o n t e s t . Thus, if all five of the e v e n t s d e s c r i b e d above a r e " r e c o g n i z e d " a s b e i n g w h a t p r e v i o u s l y o c c u r r e d , then we know t h a t the i n f o r m a t i o n which d i s t i n g u i s h e s t h e s e e v e n t s one f r o m a n o t h e r is not p a r t of the thought p r o c e s s e s a c t i v a t e d by t h e i r input. It is r ~ a s o n a b l e to suppose t h a t t h e r e will not be p e r f e c t i s o m o r p h i s m between the i n i t i a l thought and the r e - t h o u g h t at the t i m e of r e m e m b e r i n g .
R e c a l l o r r e c o g n i t i o n will not
always be p e r f e c t .
The e v e n t that a c t i v a t e s the r e - t h o u g h t will, itself, influence the n a t u r e
of t h i s r e - t h o u g h t .
F u r t h e r m o r e , i n t e r f e r e n c e f r o m o t h e r thoughts and, p e r h a p s , s o m e
p u r e l y t i m e - d e p e n d e n t d e c a y p r o c e s s will a l t e r the n a t u r e of the r e m e m b e r i n g .
Assessments
of the e o n f u s a b i l i t y o r s u b s t i t u t a b i l i t y of a n u m b e r of inputs at v a r i o u s t i m e s a f t e r the i n i t i a l input will give an indication of t h e s e c h a n g e s in m e m o r y . One such change is a l o s s of d i f f e r e n t i a t i o n o r l o s s of s p e c i f i c i t y o v e r t i m e .
Gary
Olson (1971) i n v e s t i g a t e d m e m o r y for p r e n o m i n a l a d j e c t i v e s h e a r d in i s o l a t e d s e n t e n c e s .
In
r e c o g n i t i o n t e s t s he found f a l s e r e c o g n i t i o n for a d j e c t i v e s that w e r e of the s a m e g e n e r a l c l a s s a s the a d j e c t i v e s a c t u a l l y h e a r d in the initial s e n t e n c e s .
F o r e x a m p l e , with the initial s e n -
t e n c e , "A stone t o w e r stood a l o n g s i d e the old building. ", t h e i n c o r r e c t a d j e c t i v e "wooden", was r e c o g n i z e d m o r e f r e q u e n t l y than the i n c o r r e c t a d j e c t i v e s , " r o u n d " and " s q u a r e " .
The
o r i g i n a l u n d e r s t a n d i n g of the a d j e c t i v e , " s t o n e " , i n c l u d e d the g e n e r a l notion of "type of m a t e r ial".
T h i s g e n e r a l u n d e r s t a n d i n g o r thought r e m a i n e d when the m o r e specific i n f o r m a t i o n of
the e x a c t type of m a t e r i a l was lost.
It is in d e s c r i b i n g the n a t u r e of t h i s u n d e r s t a n d i n g o r
166
thought p r o c e s s , not the n a t u r e of s e m a n t i c m e m o r y , p e r se, that I b e l i e v e the c u r r e n t w o r k of Loftus and G l a s s in t h i s c o n f e r e n c e will prove valuable. The thoughts which a r e a c t i v a t e d upon h e a r i n g a given l i n g u i s t i c input a r e r e l a t e d to the e n t i r e context in which they a r e h e a r d ,
tn fact, meaning i t s e l f d e p e n d s on this context,
a s we have seen in the c a s e of the ambiguous w o r d o r s e n t e n c e .
As David Olson (1970) has
pointed out, the w o r d s of a m e s s a g e s e r v e to d i f f e r e n t i a t e s o m e event f r o m s o m e s e t of a l ternatives.
It is this d i f f e r e n t i a t i o n p r o c e s s which is the meaning of the word.
I s u g g e s t that we r e m e m b e r only the d e g r e e of d i f f e r e n t i a t i o n r e q u i r e d by the context in which a m e s s a g e i s h e a r d .
Take, for e x a m p l e , the s e n t e n c e , "He ate the a p p l e . " When
h e a r d in the context of a d e s c r i p t i o n of a man at a s m o r g a s b o r d c h o o s i n g and eating v a r i o u s foods the m e a n i n g of t h i s s e n t e n c e will be r e m e m b e r e d a c c u r a t e l y .
However, this v e r y s a m e
s e n t e n c e , when h e a r d in the c o n t e x t of a s t o r y about a s t a r v i n g man who b r e a k s into a house to get s o m e t h i n g to e a t will be r e m e m b e r e d at s o m e t i m e , T 1, a s "He ate the f r u i t " , and at s o m e l a t e r t i m e , T 2, a s "He ate the f o o d . " Despite this much l o s s of s p e c i f i c i t y o v e r t i m e , it will not continue the p r o c e s s and take on only the meaning, "He did s o m e t h i n g " ,
because
the context r e q u i r e s p r e s e r v a t i o n of the i n f o r m a t i o n that the s t a r v i n g man ate s o m e t h i n g n o u r ishing,
The context of the s m o r g a s b o r d s t o r y r e q u i r e s s p e c i f i c a t i o n of the type of fruit; the
context of the s t a r v i n g - m a n s t o r y r e q u i r e s s p e c i f i c a t i o n only at the l e v e l of food.
The though
thoughts activated upon h e a r i n g any given w o r d a r e fitted into the c o n t e x t of the e n t i r e m e s s a g e and, o v e r t i m e , t h e s e thoughts b e c o m e only a s s p e c i f i c a s r e q u i r e d by that context. Thus d e t a i l s often s e e m to "fade" in m e m o r y . If it is the thoughts of the l i s t e n e r r a t h e r than the linguistic input i t s e l f which is the "content" of m e m o r y , then e v e r y t h i n g that a f f e c t s thought a f f e c t s m e m o r y .
A whole host of
o r g a n i s m i c v a r i a b l e s which a r e i g n o r e d in the i n p u t - b a s e d m e m o r y model b e c o m e v e r y i m p o r tant in the t h o u g h t - b a s e d model.
The l i s t e n e r a c t i v e l y c o n s t r u c t s thoughts, and the e x t e r n a l
input i s only one of the many d e t e r m i n a n t s of t h e s e thoughts. a c t i v a t e two v e r y d i f f e r e n t thoughts in two d i f f e r e n t people. what they have h e a r d o r s e e n will be v e r y d i f f e r e n t .
E x a c t l y the s a m e input m a y welt Consequently, t h e i r m e m o r i e s of
Although a model which i n c o r p o r a t e s the
n e c e s s a r y o r g a n i s m i c v a r i a b l e s will n e c e s s a r i l y be much m o r e c o m p l e x than the s t i m u l u s b a s e d model, the a t t r a c t i o n of the s i m p l e r model should not blind us to its i n a d e q u a c i e s . We have a r g u e d that, in s o m e s e n s e , linguistic m e m o r y does not e x i s t .
Rather,
things h e a r d o r r e a d contribute to the c o n s t r u c t i o n of i n t e r n a l thoughts in the l i s t e n e r . thoughts a r e r e - t h o u g h t , with m o r e o r l e s s fidelity, at the t i m e of r e m e m b e r i n g . thoughts, not the linguistic input, which a r e r e m e m b e r e d .
These
But it is the
167
References Bartlett, F.C. (1932), Remembering, Bobrow,
Cambridge,
England: Cambridge
University Press.
S.A. (1970), "Memory for Words in Sentences", Journal of Verba ! Learning and Verbal Behavior, 9, 363-372.
Bransford, J.D. and J.J. Franks (1971), "The abstraction of Linguistic Ideas", Cognitive Psychology, 2_, 331-350. Bransford, J.D., J.R. Barclay and J.J. Franks (1972), "Sentence Memory: versus Interpretive Approach", Co~itive Psychology, 3, 193-209. Fillenbaum, S. (1966), "Memory 9, 217-227.
Constructive
for Gist: Some Relevant Variables", Language and Speech,
Gibson, E.P., J.J. Gibson, A.D. Pick and H. Osser (1962), "A Developmental Study of the Discrimination of Letter-like Forms", Journal of Comparative and physiological Psychology, 55__, 497-906. Jarvella, R.J. and S.J. Herman (1972), "Clause Structure of Sentences and Speech Processing", P,erc,eption and Psychophysics, t ~__,381-384. Mathewson, G.C. and T . J . Keeney (t971), "Memory for F o r m and Meaning of Sentences Embedded in Paragraphs", Paper presented at the Western Psychological Association, San Francisco, April 22, 1971. Olson, D.R. (1970), "Language and Thought: Aspects of a Cognitive Theory of Semantics", Psychological Review, 77__, 257-273. Olson, G.M. (1971), "Memory for Prenominal Adjectives in Ordinary English Sentences", Cognitive Psychology, 2, 300-312. Sachs, J.S. (1967a), "Recognition Memory for Syntactic and Semantic Aspects of Connected Discourse", Perception and Psychophysics, 2, 437-442. Sachs, J.S. (1967b), "Recognition of Semantic, Syntactic, and Lexical Changes in Sentences", Paper presented at Psychonomic Society, October, 1967. Wicklegren, W.A. (1966), "Distinctive Features and E r r o r s in Short-term Memory for English Consonants," Journal of the A qoustical Society of America, 39, 388-398.
THE
GRAMMAR
OF RELATIVE
ADJECTIVES
AND
COMPARISON
Renate Bartsch FU B e r l i n and U n i v e r s i t y of C a l i f o r n i a , Los A n g e l e s Theo V e n n e m a n n g e n a n n t N i e r f e l d U n i v e r s i t y of C a l i f o r n i a , L o s A n g e l e s 1.
E a r l i e r A p p r o a c h e s to the P r o b l e m of R e l a t i v e A d j e c t i v e s and C o m p a r i s o n . S p e a k e r s of E n g l i s h know that the following s e n t e n c e s have s o m e t h i n g in c o m m o n . (0} J o h n is 5 feet tall. (i) John is tall. (i,) M a r y is tall. (2) John is t a l l e r than Mary. (3) J o h n i s as tall a s M a r y . (4) John is the t a l l e s t of P e t e r ' s sons. (5) John i s s h o r t . (6) John is s h o r t e r than M a r y . (7) John i s a s s h o r t as Mary. (8) John is the shortest of Peter's sons. Contemporary
have assumed
syntaeticians have tried to account for this knowledge.
that sentences (i) and (I') are somehow
involved in the derivation of sentences
(2) - (4), viz. as part of their deep structures from which their more tures are derived by means arise accidentally. these sentences,
Most of them
of syntactic transformations.
complex
This procedure
It is suggested by the relative complexity
surface struc-
did not, of course,
of the surface structures of
and is thus a direct result of the preoccupation
of contemporary
syntac-
t i c i a n s with s u r f a c e - s y n t a c t i c p r o p e r t i e s of l a n g u a g e s . The f a i l u r e of m o s t c o n t e m p o r a r y s y n t a c t i c i a n s to a n a l y z e and f o r m u l a t e the p r o p e r t i e s of r e l a t i v e a d j e c t i v e s and c o m p a r i s o n p r o p e r l y is, of c o u r s e , by no m e a n s novel.
On
the c o n t r a r y , they p e r p e t u a t e (or renew) a v e n e r a b l e t r a d i t i o n which s t a r t e d two and a half thousand y e a r s ago with P l a t o ' s T h e a e t e t u s , and m a y thus c o n s i d e r t h e m s e l v e s in e x c e l l e n t company.
B e r t r a n d R u s s e l l (±945, p. 159) w r i t e s ,
T h i s p a p e r a p p e a r s in L i n g u i s t i s c h e B e r i c h t e , 21. It is a s h o r t v e r s i o n of an a r t i c l e , " R e l a t i v e A d j e c t i v e s and C o m p a r i s o n " , UCLA P a p e r s in Syntax, 2, edited by P a u l S c h a c h t e r and G e o r g e B e d e l l . The t h e o r y of g r a m m a r on which this a r t i c l e is b a s e d , as well as a p p l i c a t i o n s of this t h e o r y to o t h e r p r o b l e m s of g r a m m a t i c a l a n a l y s i s , i s p r e s e n t e d i n o u r b o o k , Semantic Structures: A Study in the Relation Between Semantics anti Syntax, Frankfurt: Athenaum-Verlag 1972.
169
" T h e r e a r e , at this point, s o m e p u z z l e s of a v e r y e l e m e n t a r y c h a r a c t e r . We a r e told that s i n c e 6 is g r e a t e r than 4 but l e s s than 12, 6 i s both g r e a t and s m a l l , w h i c h i s a c o n t r a d i c t i o n , Again, S o c r a t e s i s now t a l l e r than T h e a e t e t u s , who is a youth not yet full grown; but in a few y e a r s S o c r a t e s will be s h o r t e r than T h e a e t e t u s . T h e r e f o r e S o c r a t e s is both tall a n d s h o r t . T h e idea of a r e l a t i o n a l p r o p o s i t i o n s e e m s to have puzzled P l a t o , a s i t did m o s t of the g r e a t p h i l o s o p h e r s down to Hegel (inclusiveL" L o g i c i a n s a n d s e m a n t i c i s t s e m p h a s i z e that (2), (3), and (4) do not i m p l y (1) and (1'). John m a y be a v e r y s h o r t boy, which m e a n s that (1) is false; yet (2), (3), and (4) m a y be true.
N e v e r t h e l e s s , e v e n though t h e r e e x i s t s no i m p l i c a t i o n a l r e l a t i o n between s e n t e n c e s
(2) - (4) on the one hand and (1) and (1') on the o t h e r , t h e r e c l e a r l y e x i s t s a s e m a n t i c r e l a t i o n s h i p w h i c h a t h e o r y of g r a m m a r m u s t a c c o u n t for. T r a n s f o r m a t i o n a l g r a m m a r i a n s have p r o p o s e d deep s t r u c t u r e s of the following kind in w h i c h (1) a n d / o r (19 a p p e a r a s c o n s t i t u e n t s of (2) 1. (A)
S John is Mod tall
m o r et h a / ~ n
S
M a r y i s tall
(B)
S NP
I S John is tail
VP V
I
more
NP
I
S
M a r y i s tall The s y n t a c t i c p r o p e r t i e s in t h e s e a p p r o a c h e s r u n c o u n t e r to the s e m a n t i c s of s e n t e n c e s like (2): they p r e s e n t deep s t r u c t u r e s of c o m p a r a t i v e s e n t e n c e s w h i c h contain deep s t r u c t u r e s of the c o r r e s p o n d i n g s e n t e n c e s with p o s i t i v e s a s c o n s t i t u e n t s .
T h i s is i n c o r r e c t , b e c a u s e the
m e a n i n g s of s e n t e n c e s with p o s i t i v e s a r e not conceptual c o n s t i t u e n t s of the m e a n i n g s of the c o r r e s p o n d i n g c o m p a r a t i v e s e n t e n c e s : one does not have to i n t e r p r e t the positive s e n t e n c e s in o r d e r to i n t e r p r e t the c o m p a r a t i v e s e n t e n c e s .
In (B), the c o n s t i t u e n t
s u g g e s t s t h a t t e r m s a r e the a r g u m e n t s of the r e l a t i o n m o r e . what kind of n o m i n a l i z a t i o n of S is intended.
I
[[John i s tall] SJNP
But a s it s t a n d s we do not know
It i s e v e n w r o n g b e c a u s e it s u g g e s t s that the
F o r d e t a i l s and r e f e r e n c e s , cf. the c o m p l e t e a r t i c l e ( B a r t s c h a n d V e n n e m a n n , 1972).
170
nominalization of the positive sentences John is tall and M a r y is tall is intended,
This would
be something like John's being tall and Mary[s being tall, o r J o h n ' s t a l l n e s s and M a r y ' s t a l l n e s s where t a l l n e s s is the p r o p e r t y of being tall, even though sentence (2) does not imply that e i t h e r John o r Mary have the p r o p e r t y of being tall.
Note further that with analyses of
type (A) and (B), sentences like Bill b e l i e v e s he is t a l l e r than he is a r e not r e p r e s e n t a b l e , as o b s e r v e d by Ross and P e r l m u t t e r (1970). Several semantic approaches to the problem of r e l a t i v e a d j e c t i v e s and c o m p a r i s o n have been proposed.
It would be too t i m e - c o n s u m i n g to d e m o n s t r a t e in this lecture that
those made by Reiehenbaeh (1947), Seuren (1970), and W i e r z b i c k a (1972) a r e inadequate. The only authors who have shown an understanding of the problem a r e Sapir (1944), F i l l m o r e (1965), Wunderlieh (1970), B i e r w i s e h (1971) and Dik (1971).Of t h e s e , on t h e o n e h a n d , and W u n d e r l i c h on t h e o t h e r , r e p r e s e n t are similar
to o u r p r o p o s a l .
Sapir-Fillmore
Sapir-Fillmore
contributions
that
s u g g e s t that t h e c o m p a r a t i v e
is
not b a s e d on t h e p o s i t i v e s e m a n t i c a l l y ,
despite the morphology.
And Wunder-
l i c h i n t r o d u c e s t h e c o n c e p t of m e a s u r e
f u n c t i o n s r e l a t e d to r e l a t i v e a d j e c t i v e s . 1 Here, rather than pointing these
Both proposals have several inadequacies. out, w e p r e f e r to p r e s e n t 2.
o u r o w n a n a l y s i s of r e l a t i v e a d j e c t i v e s and c o m p a r i s o n s .
Semantic R e p r e s e n t a t i o n s of (0) - (8). In this section, we use ~T as an abbreviation for the m e a s u r e function as applied to
the dimension Height (Tallness).
We will explain this abbreviation in Section 3. NT, y
r e p r e s e n t s the a v e r a g e of the heights of the objects in the r e f e r e n c e set Y within which x is compared.
The angular brackets a r e used to indicate presuppositions. (0a)
f~T(X) = 5 feet
(la)
fT(X)>NT,y
(2a)
f~T(X) > ~TIT(y)
(3a)
~TI(x)=f~T(y)
M
(and 2 <~T(Y)>NT, y }
John = (lx [Soo(x (5a)
(On) I
)
(y (Son(y Pe)
x
¢(y@
~TT(X)< Nw, y
/and
The semantic approaches a r e analyzed in the complete a r t i c l e ( B a r t s c h and Veanemann, 1972). 2 Sentence (3) has a second i n t e r p r e t a t i o n with > r a t h e r than =. Similarly, (7) with < for = .
171
(Ta)
¢(y) and
N ,y>
(Sa) John = (Ix)FSon(x,Pe) ~ (y)(Son(y,Pe) & x ~ y ~ fTM(x)<~T(y))3 3.
The G e n e r a l M e a s u r e F u n c t i o n fM. The g e n e r a l m e a s u r e function fM is a 2 - p l a c e o p e r a t o r w h e r e one a r g u m e n t i s a
d i m e n s i o n D, and the o t h e r , an o b j e c t m e a s u r e d in t h i s d i m e n s i o n , fM(x, D). We u s e two abbreviations:
For example,
•DD
(x) = def t~I(x' D)
if D i s c o n s t a n t ;
~xx(D) = def flvI(x" D)
if x is constant.
D is c o n s t a n t in John is t a l l e r than M a r y , a n d x i s c o n s t a n t in T h i s house i s
b r o a d e r than high. The g e n e r a l m e a s u r e function fM m a p s p a i r s of o b j e c t s x and d i m e n s i o n s D on e q u i v a l e n c e c l a s s e s of o b j e c t s r e l a t i v e to D.
The s e t of e q u i v a l e n c e c l a s s e s
r e l a t i v e to D is l i n e a r l y o r d e r e d . In Section 2, D w a s e x e m p l i f i e d with T ( T a l l n e s s ) .
O t h e r d i m e n s i o n s for which
both E n g l i s h and G e r m a n have r e l a t i v e a d j e c t i v e s a r e : B r e a d t h , T e m p e r a t u r e , Speed; I n t e l ligence, Beauty, L i v e l i n e s s . a s s o c i a t e d with it.
Note that a d i m e n s i o n m a y or m a y not have a quantified s c a l e
Of the above, the f o r m e r t h r e e do, the l a t t e r do not (but a t t e m p t s have
been m a d e to quantify intelligence).
Many, m o r e , m o s t , a n d t h e i r a n t o n y m s few, fewer,
fewest, m i s t a k e n by s o m e l i n g u i s t s as q u a n t i f i e r s like al..~la n d s o m e , a r e a l s o r e l a t i v e a d j e c t i v e s , s p e c i a l in that t h e i r d i m e n s i o n (D = P, for P o w e r , the n u m b e r of e l e m e n t s in a set) a p p l i e s to s e t s r a t h e r than individual o b j e c t s .
Much ( m g r e , most) and its a n t o n y m little
( l e s s , least) a r e r e l a t i v e a d j e c t i v e s r e f e r r i n g to the d i m e n s i o n Quantity a p p l i c a b l e to m a s s nouns.
4.
On the Syntax of R e l a t i v e A d j e c t i v e s . We now t u r n to the s y n t a c t i c d e r i v a t i o n of c o m p a r a t i v e s e n t e n c e s .
The m a i n t y p e s
of r u l e s we u s e a r e l e x i c a l i z a t i o n r u l e s , c o n s t i t u e n t s t r u c t u r e r u l e s , and s e r i a l i z a t i o n r u l e s . By l e x i c a l i z a t i o n we m e a n the c h o i c e of l e x i e a l i t e m s a p p r o p r i a t e to c o n v e r t s e m a n t i c r e p r e s e n t a t i o n s into p h o n e t i c a l l y i n t e r p r e t a b l e s t r u c t u r e s . verbalize a given semantic representation.
T h e r e a r e usually s e v e r a l ways to
F o r e x a m p l e , the s e m a n t i c r e p r e s e n t a t i o n
m a y be e x p r e s s e d a s in (2 0 ) and (2). (2 0)
"Direct" verbalization: T h e h e i g h t of John i s g r e a t e r than the height of M a r y .
172
(2)
"Better" verbalization: John is taller than Mary.
Of these sentences,(20) seems to verbalize (2a) more directly. As a consequence it is longer than (2).
The difference between a "direct" verbalization and a "better" one is even
more obvious in cases like (la)
~TIT(xj) > NT, Y
(I 0)
John's height is greater than the average of the heights of boys.
(I)
Johnis tall.
and ~I) used in a ¢~ontextwhich makes clear that John is compared to other boys.J
The function of syntax is not simply to verbalize semantic
representation
i. e., to provide shortcuts which allow the speaker to express complex simple surficial structures.
but to do it "well",
meanings
in short and
Thus,(2) is better than (2 0) and (I) is better than (I0), where
"better" here stands for "expressing
the same
semantic
representation with fewer words
and a simpler surface phrase structure". From
this point of view, the "invention" of the relative adjective (just like that of the
relative verb) was a stroke of genius of the human
linguistic mind.
A dimension
is inherently
a nominal concept, and so is the value associated in it with an individual by the measure function fM. compared
The relations
>, <, = are the verbal elements
values are the arguments
hidden in the argument
of the measure
present the second-degree face arguments.
of the relations.
arguments
function.
The names
sentences.
The
of the objects compared
are
Relative adjectives permit the speaker to
of the semantic representations
Consider the semantic and surface representations
(2), (in), and (I): John is a second-degree argument
in comparison
argument
in (2) and (I) (see Figure i). Furthermore,
be inferred from the context allows semantic
as first-degree
sur-
of the sentences
(2a),
in (2a) and (In), but a first-degree the deletion of an average which can
representations
of the kind (In) to assume
face structures like (~) which look like the most primitive kind of predications: This rose is red,
Mary
is human
is optimal, while all semantic
(Figure 2).
The shortcut aspect of relative adjectives becomes
completely recoverable.
most apparent from the fact that
only the most frequently referred to dimensions
have a corresponding
tive.
"liveliness",
"being prepared for the contest" or "being more
f(x), as in
The gain in surface shortness and simplicity
properties of the sentence remain
English has lexical units for "largeness",
sur-
lexical relative adjec-
and "weight",
balanced racially".
but not for
(Perhaps
the deverbal
adjective integrated will develop into a special relative adjective for the dimension balance" of a given set of people, as this dimension
becomes
more
important
of "racial
socially and is
173
P
Pre~Arg
I
Pre~Arg
I
>
Com!d
C2
C1 Op/~Arg
i
!
Op~Arg
I i
I FIGURE 1
P
P
Pre
/A
Arg
I
L
>
I
/~ Op
i
¢
Pred
ND'y Arg
I
x
FIGURE 2
Arg
1 Y
174
more
frequently referred to.) Even certain simple geometrical
adjectives in English,
e. g,, the (maximal)
diameter
dimensions
and circumference
a rectangle or circle, while the length and width of appropriate tence ;'.,Allcircles are more
circumferenced
proper rectangles are longer (or more of dimensions
becomes
than diametered
long) than wide.
e,g., in Igbo with Tallness:
In such eases the nominal character
"John (sur-)passes
Mary
is greater than the diathan in diameter.
languages even with more
John ka Mary
ogologo,
(in) tallness", (examples
common
due to Marianne
ness, Largeness,
Bigness
and Greatness
are interesting universals of dimensional plausible to us that there may
dimension.
Celce-Mureia,
dimensions syncretism
German
literally UCLA).
into one adjective uses gross for the Tall-
separated in English. to be discovered here.
Certainly there It seems
quite
exist languages lacking relative adjectives altogether in which
nouns (weight, duration) or verbs (weigh,
last) are used instead.
long in addition to the nouns and verbs. ) However, tire and the verb shortcuts seems less likely.
(English has heavy and
that a language renounces
Note also that while many
both the adjeo-
of the relative nouns
lacking related relative adjectives are rather long, relative adjectives themselves short, with many
The
dimensions,
'~John is taller than Mary",
Often a language neutralizes several semantically similar dimensions English uses long for a spatial and a temporal
There is no sen-
as there is the sentence All
meter for all circles, or All circles are greater in circumference to arise in some
of a figure such as
figures do.
apparent on the surface: The circumference
analogous situation seems
do not have
of the most basic ones being monosyllabic
tend to be
in English: long, high, wide,
broad, far, tal___l,large, big, great, smart (together with sharp and bright preferred to the
unwieldy intelligent), late, good, bad, etc. A relative adjective is r e p r e s e n t e d in the lexicon with no other information than the dimension(s) to which it r e f e r s , and the information Adjective (in addition to its phonetic shape, and perhaps e e r t a i n i d i o s y n e r a t l c properties it may have, such as supptetiveness). Let d be a basic ("unmarked") relative adjective with the phonetic shape d and the d i m e n sion D. The lexical r e p r e s e n t a t i o n of d is Ed, D. AdjectiveJ.
The ("marked") antonym
of d is r e p r e s e n t e d with a "mark", for which we may use < itself: L~, D, <. AdjectiveJ. We will now give the typical lexieaiizations for sentences containing positives, c o m paratives, superlatives, and a s . . . a s constructions. We use Y to designate the r e f e r e n c e set.
This r e f e r e n c e set is needed for the superlative, where the "superlative" element has to
to be a m e m b e r of it, and in the positive, where the average "different meanings" of ~
ND, y
is based on it; cf. the
in This mouse is big and This elephant is big, which follows from
the fact that NBig ' {x: mouse(x)} ¢ NBig, {x: elephant(x)}"
175
(9)
(a) fM(x)> ND, Y ~
x(Posd)
-~D(X)
x(Posd)
fDM(x)> ¢ ( y )
x(Compd) y
(b)
~
¢(x) <¢(y) (e)
(d)
x(0omp)y
(ix)~y)~yeY
& x#y)
D fDM(X)> fDM(y) & < x ~ Y ) ] ' - " - ~ ( x ) ( x ( s u p d ) Y )
(lx) E y ) ( ( y e Y
& x#y)
~ fMD(x) ~ - - ~ , < x)(x(sup'd)Y)
as.,.
as constructions
¢ ( x ) = ¢ ( y ) " ~ x(Equal d)y ¢ ( x ) = fM(y)
&
< ¢ ( y ) < ND, y ) ~"-~ x(Equal ~) y
These texicatization r u l e s r e q u i r e a n u m b e r of clarifying r e m a r k s .
F i r s t , we note
that no other aspects leading to the lexiealization of these semantic r e p r e s e n t a t i o n s have been dealt with than those leading to the introduction of the r e l a t i v e adjectives themselve s. Other r u l e s a r e needed to lexicalize the constituents Pos (Positive) [usually lexicalized as zeroS, Comp (Comparative), Sup (Superlative), and Equal, and to introduce further m o r phological m a r k e r s r e q u i r e d for a given language. We will show in some detail what these r u l e s are for English. Second, we must point out that (9) is a simplified r e p r e s e n t a t i o n which we have introduced in the i n t e r e s t of expository clarity.
A false i m p r e s s i o n one may get from (9) is
that the constituents of the forms left and right of the arrow a r e n e c e s s a r i l y a r r a n g e d in a l i n e a r sequence.
It is apr!or.i, the case that the logical forms to the left of the arrow a r e
logically devoid of any spacial or temporal sequentiality. The fact that they are ordered on paper is based on the desire to save additional notational apparatus.
To make explicit that
the l i n e a r a r r a n g e m e n t of logical forms is accidental (i, e . , has only pragmatic but no logical reasons), we must introduce a notation which makes explicit what the relationship of each a r g u m e n t to its relation is.
Thus, we label the two a r g u m e n t s o c c u r r i n g in the definition of
> in some a r b i t r a r y fashion, e . g . , by means of n u m b e r s 1, 2 (but an a s t e r i s k and a heart would serve the same purpose).
Let ! m a r k that place in the relation > which in the c o n -
ventional notation a > b is occupied by a; let 2 m a r k the place of b. We write
~
indicate that we use this particular notation for the relation " g r e a t e r (more) than".
to (Note
that this procedure is not c i r c u l a r because the relationships of the two a r g u m e n t s to the relation > follows directly from the meaning of the relation.) Now all possible l i n e a r
176 a r r a n g e m e n t s of the constituents a, >,
(iO)
b of a
>
b are
equivalent:
Q ~ 2 ,±a2b)
(b~
ba ~
'
In fact, v e r t i c a l a r r a n g e m e n t s would likewise be equivalent. We enclose constituents whose order is i r r e l e v a n t in b r a c e s , with an a r b i t r a r y a r r a n g e m e n t of the constituents
1
2
1,2
More explicitly, since ~
and its a r g u m e n t a r e likewise unordered, 1
we
write:
2
Note that the operation fM and its two a r g u m e n t s can likewise be r e p r e s e n t e d as an u n o r deredset, e.g.,
{fMxl~};
but we use the abbreviatory notation, a s l e s s c o m p l e x .
We suggest that the output of the lexicalization rules (9) is likewise unordered.
Thus,
we intend our rules in (9) to be interpreted on the analogy of (9b'), which is a formal r e p r e sentation of (9b) in the model just outlined:
2
Our basis for this suggestion is that different languages s e r i a l i z e (linearize) the constituents in comparison sentences differently, and that some languages s e r i a l i z e the same constituents
differently indifferent contexts. Thus, English serializes {~Compd}l 2} as ~{Comp d} 21, Hindi as ~i 2{Comp d}I , Sanskrit in any order" , German serializes {Comp d } as [d Comp], Spanish as [Comp d~, English as [d Comp] if Comp is l e x i c a l ized as - e r , otherwise as LComp dj.
On the other hand, s e r i a l i z a t i o n depends on t e x i c a l -
ization: Whether a relational concept is lexicalized as a relative adjective, or a verb, or a noun, can have incisive consequences for the s e r i a l a r r a n g e m e n t of the whole sentence.
177
We will now show with a s a m p l e d e r i v a t i o n how r u l e s (9')
E= (9) r e i n t e r p r e t e d along
the l i n e s of (9b')J work, and what o t h e r l e x i c a l i z a t i o n and s e r i a l i z a t i o n r u l e s m a y be n e e d e d in a given language.
We u s e an E n g l i s h s e n t e n c e , (2) J o h n i s t a l l e r than M a r y . Its s e m a n t i c
r e p r e s e n t a t i o n is (14a), w h e r e x j and x M a r e i n d i c e s for the two i n d i v i d u a l s which a r e l e x i e a l i z e d a s J o h n and M a r y , r e s p e c t i v e l y ( r a t h e r than as, say, S m i t h ' s son and M i l l e r ' s daughter).
T h i s l e x i e a l i z a t i o n , with all the s y n t a c t i c and s e m a n t i c p r o p e r t i e s of the l e x i c a l
i t e m s involved, has taken place in (14b).
One m u s t r e c a l l the n o t a t i o n a l d i f f e r e n c e between
d and d in o r d e r to i n t e r p r e t l e x i c a l r e p r e s e n t a t i o n s p r o p e r l y : 1
2
(14) 1
2
Application of (9b') yields (15): (15)
omp tall
John
Mary
Lexicalization of Comp follows the well-known rule according to which a certain class of common monosyllabic and disyllabic words (with certain phonological conditions imposed on disyllables) take -er, while all others take more: taller, redder (or more red), happier, commoner (or more common) vs. more serene, more beautiful, more jaune, more dead. Let us designate the class of adjectives taking -er be ER; i . e . , let us consider all such adjectives as marked ER in the lexicon. The rule can then be formulated as (16), where both -er and more are, of course, marked lexically as containing the feature Comp. (16)
Equivalently,
(17)
(a)
Comp
~-er/ER
(b)
Comp
~
more
.
we can write (17):
Comp ~
I }-er/ER more
but we do not intend any order of application to be involved here. The theory of grammar provides conventions which prohibit the general case from applying if a special mark r e quires the application of a restricted rule. The need for at least two more lexicalization rules of a morphological nature is obvious. The first introduces than as part of the second argument of a comparative:
lZ8
Note that the information Comp is still present even after the lexicalization of the constituent Comp, because it has to be part of the lexical representation of both - e r and more in order for rules (16a, b) to find them in the lexicon as suitable lexicalizations of Comp. Note further that than is introduced in such a way that it builds a new constituent together with the old constituent 2. The second morphological rule is that which introduces the copula in verbless subjectpredicate sentences. We formulate only that part of the rule which applies to adjectives as predicates, because only that part is needed in our fragment. We do not formalize certain restrictions related to the modality of the sentence, or to such information as "predicative use of the adjective", Note that different from than, be is not adjoined to any one of the constituents of the sentence. The reason for this is that we want this rule to be neutral to the modality of the sentence, e . g . , to the information "declarative", "interrogative", "impera-
tive": (19)
be/{--
~ ~
{1} {Adjective X } Y } .
The remaining rules are familiar enough. The verb of an English sentence takes the number and person of the first argument:
(20)
(a) (b)
,
> E~ Number] /
{~ Nmnber
~
{/3 Pe:Lrs°n}}
Person] / { I ~ ]
These are feature spreading rules, or agreement transformations.
'
The verb (be is char-
acterized in the lexicon as a verb) likewise adopts properties from the modality of the sentence. We have not given any part of the modality here. In a complete grammer, the situation must be presented in such a way that the verb can adopt the features equivalent to Present and Indicative. The lexicalization rules for the verbal categories can now operate. If the verb is regular, they add -s__, given the verbal categories of our sentences.
Since be
is a special case, no - s is added. Instead, the lexicalization rule for verbal categories refers us back to the lexical item be__, where a paradigm search finds i s as a match, whereupon be is replaced with is. The next rules are examples of constituent structure rules:
179
(b)
{ { b e Adjective} { 1 } }
-----> {{be {Adjective}} < 1 } }
Rule (21b) must be restricted to declarative and imperative sentences. Note that it orders itself intrinsically after (21a), and that it is formulated in a sufficiently general way to accommodate positives, adverbially expanded adjectives, and other kinds of adjective constructions as well, because they are all categorically relabeled as Adjectives. In other words, in a more comprehensive grammar of this kind, we would have to systematically lable constituent braces with category labels.
For example, rule (21b) presupposes that the
total expression {{'-er Adjective} {than j[2}}}
is labeled as an Adjective.
Now we must sketch the serialization process leading from the still unordered though lexicalized and constituent-braced string to the surface representation (2). Some of the processes outlined here are of greater generality than comparison sentences require. We omit the information which is drawn from the modality to determine whether or not a rule applies. We use brackets [ ] to indicate ordered strings. Note, however, that an string X can be referred to as either [xJ or
ordered
The notation { X } simply abstracts
away from the order. Thus, the serial order imposed by the following rules is entirely determined by their structural descriptions and the inputs they operate on, but independent of their order of application. (22)
{ - e r Adjective~
~, EAdjective-er]
(23)
{than
(24)
{ { - e r Adjective} {than ~[ 2 } } }
{2}~ ~ [than 2}~ > ~{-er Adjective} {than
(25)
(a)
{be {Adjective}}
(b)
<~1 t
2 }~
:.-~e {Adjective}~
{be {Adjective}}} ----> ~{ 1} { be{Adjective}}~
This group of rules formulates special applications of the general English word order principles that in declarative clauses the verb precedes its complement, and the subject its verb. After all rules have been applied, we need to apply a convention which erases all non-phonetic information from the string. We indicate this by replacing constituent symbols
180
by t h e i r u n d e r l i n e d f o r m s ,
(Recall the r e l a t i o n between d and d defined above, )1
We s u m m a r i z e the d e r i v a t i o n of s e n t e n c e (2) in T a b l e 1. It m u s t be kept in m i n d that the p a r t i c u l a r o r d e r showing in t h i s d e r i v a t i o n is in p a r t a r b i t r a r y . not t h e m s e l v e s o r d e r e d .
R u l e s of g r a m m a r a r e
They apply w h e n e v e r t h e i r s t r u c t u r a l d e s c r i p t i o n is met.
This
i m p l i e s that a r u l e cannot apply a s long a s its s t r u c t u r a l d e s c r i p t i o n is not m e t , which m a y i m p o s e a c e r t a i n a m o u n t of i n t r i n s i c o r d e r on the a p p l i c a t i o n of the r u l e s , a p p l i c a t i o n n e e d not, h o w e v e r , be s t a t e d in the g r a m m a r ,
T h i s o r d e r of
b e c a u s e it i s c o m p l e t e l y a c o n s e -
quence of the s t r u c t u r e of the r u l e s and t h e i r input. F i n a l l y , we s k e t c h the r u l e s needed for the o t h e r c o m p a r i s o n c o n s t i t u e n t s r e s u l t i n g f r o m the a p p l i c a t i o n of (9).
(9a) r e q u i r e s no f u r t h e r r u l e s .
Since P o s is not l e x i c a l i z e d by
any r u l e of E n g l i s h , it will finally d i s a p p e a r f r o m the s t r i n g t h r o u g h the a p p l i c a t i o n of the n o n - p h o n e t i c f e a t u r e - d e l e t i o n convention. The f u r t h e r d e r i v a t i o n of s u p e r l a t i v e t e r m s , cf. (9c), i s difficult to d e s c r i b e in a g e n e r a l way, b e c a u s e so m u c h depends on the way in which Y is defined and l e x i c a l i z e d : J o h n is a) the t a l l e s t of P e t e r ' s s o n s defined a s { y : Son(y, P e t e r ) } .
b) the t a l l e s t son of P e t e r ' s .
A s s u m e t h a t Y is
Now ( x ) ( x (Sup d) { y : Son(y, P e t e r ) } )
a s ( t h e (Sup d) { y : S o n (y, P e t e r ) } } .
The s e t can then be l e x i c a l i z e d e i t h e r a s a g e n i t i v e of
(Sup d), o r a s a head noun with g e n i t i v e for (Sup d).
The f o r m e r yields a) the t a l l e s t of
P e t e r ' s sons (equivalently: the t a l l e s t of the s o n s of P e t e r ) , the l a t t e r , Peter.
(Sup d), o r m o r e p r o p e r l y ,
a r e a n a l o g o u s to (16) and (22). of P e t e r ' s s o n s
Ithe t a n e s t
c a n be l e x i c a l i z e d
{Sup d } ,
b) the t a l l e s t son of
is l e x i c a l i z e d and s e r i a l i z e d by r u l e s w h i c h a r e
In o r d e r to get f r o m h e r e to s e n t e n c e (4), J o h n i s the t a l l e s t
we have to i n t r o d u c e the s e m a n t i c r e p r e s e n t a t i o n of the s u p e r l a t i v e t e r m
of P e t e r ' s s o n s l , cf. (9c), into an equality r e l a t i o n John = I
John I
I ), which is lexiealized and serialized as John is I
The a s . . . as c o n s t r u c t i o n s of (gd) p r e s e n t no new p r o b l e m s .
I (or
] One way of l e x i c a l i z i n g
the r i g h t - h a n d side of (9d) is by" r u l e (26):
(26)
(~Equald}
1
2
The serialization rule for as is straightforward. our study of the comparative
1
1
2 All other rules needed are available from
sentence (2).
Note that the e r a s u r e of all n o n - p l ~ n e t i c i n f o r m a t i o n a t the end of a d e r i v a t i o n m a y well be an o v e r s i m p l i f i c a t i o n . F o r e x a m p l e , i t m a y well b e the c a s e that the a r g u m e n t m a r k e r s 1 and 2 a r e s u p e r f l u o u s a f t e r the a p p l i c a t i o n of the s e r i a l i z a t i o n r u l e s (24) and (25) in E n g l i s h , while in a language r e l y i n g of m o r p h o l o g y r a t h e r than position (e. g. Sanskrit) the a r g u m e n t m a r k e r s b e c o m e s u p e r f l u o u s a f t e r the a p p l i c a t i o n of the c a s e m a r k i n g r u l e s (which a r e l e x i c a l i z a t i o n r u l e s in o u r system}.
181
TABLEI i
2 1
(13) 2 c~.
(14)
(9b'),cf. (15)
{{Comptall}join Ma2ry} -er tal_~John Mary 1 2 {{-er tall}John{than(Mary}}}
(16a)
{{-ertall}be{jolhn}{than{M2ary}}}
(19)
{(-er
(18)
tail}[}irdul:r:on3jolhn{than{Ma2ry}}}
1 2 ~V~:dPersonlJohn{{-er tall%~than{Mary}}}} LSingular d be 2 1l [k L,Singular -J
(20) (21a)
(21b)
~3:dPerso: I[tall -er](than{Ma2ry}}}}jolhn} [' LSingular _
(22)
{Ib3:dPerson1 {[tall -er] [than{Ma2ry}]}}Joln} l_Singular A ~V~:dPersonl [[tall -er~ ~than(Ma2ry}~}jolhn} LSingular J {Ip3:dPerson][[tall-er][thanfM2ary}~l jolhn} kSingular J [j1oho i-be3rdPersoo lit a l l - e r~ Vth~n ~Mary 2 }]?]] Singular
(23) (24) (25a) (25b)
_
%e ohn
Singular
Present Indicative
iI-eo
han
[ jolhn[is [[tall -er] ~than{M2arY}l~l? John is taller than Mary
ar
(unnumberedrule) (unnumberedrule)
182
Although we have not i l l u s t r a t e d t h i s , it i s obvious t h a t o u r r u l e s work in both d i r e c t i o n s , f r o m s e m a n t i c s t r u c t u r e to s u r f l c i a l s t r u c t u r e and c o n v e r s e l y .
Our f r a g m e n t
of a g r a m m a r i s thus n e u t r a l a s to p r o d u c t i o n o r i n t e r p r e t a t i o n . This d i s c u s s i o n of l e x i e a l i z a t i o n and s e r i a l i z a t i o n h a s b e e n s o m e w h a t t e c h n i c a l a n d tedious.
We feel that i t s e x p l i c i t n e s s m a y be useful, b e c a u s e it r e p r e s e n t s the f i r s t f a i r l y
c o m p l e t e d e s c r i p t i o n of a d e r i v a t i o n of c e r t a i n l a n g u a g e - s p e c i f i c s u r f a c e r e p r e s e n t a t i o n s f r o m l a n g u a g e - i n d e p e n d e n t ( u n i v e r s a l ) s e m a n t i c s t r u c t u r e s in a g r a m m a r without o r d e r e d deep s t r u c t u r e s , without m o v e m e n t t r a n s f o r m a t i o n s , and without e x t r i n s i c r u l e o r d e r (natural generative grammar). 5.
F u r t h e r E v i d e n c e and Special A p p l i c a t i o n s . W i t h i n o u r f r a m e w o r k , we can a c c o u n t for a n u m b e r of p h e n o m e n a r e l a t i n g to
r e l a t i v e a d j e c t i v e s a n d c o m p a r i s o n which have puzzled e a r l i e r a n a l y s t s .
F o r r e a s o n s of
t i m e l i m i t a t i o n s , we c a n h e r e onIy s k e t c h o u r s o l u t i o n s . 5.1
S e n t e n c e s c o n t a i n i n g p o s i t i v e s t y p i c a l l y c o m p a r e a m e a s u r e m e n t value of an individual
to an a v e r a g e o r n o r m p r e s u p p o s e d a s p a r t of the c u t t u r a l b a c k g r o u n d of s p e a k e r s of the language.
Sentences c o n t a i n i n g c o m p a r a t i v e s t y p i c a l l y c o m p a r e a m e a s u r e m e n t value of one
individual to t h a t of a n o t h e r w h i c h i s not p a r t of the l i s t e n e r ' s c u l t u r a l b a c k g r o u n d , but h a s to be g e n e r a t e d in h i s m i n d f o r the specific p u r p o s e of the c o m p a r i s o n .
We t h e r e f o r e c o n -
s i d e r the c o m p a r a t i v e a s s e m a n t i c a l l y " m a r k e d " , the positive a s s e m a n t i c a l l y " u n m a r k e d " . It i s t h i s s e m a n t i c a s y m m e t r y which i s r e f l e c t e d in the m o r p h o l o g i c a l " m a r k e d " : " u n m a r k e d " c o n t r a s t of the c o m p a r a t i v e v s . the positive, such a s in E n g l i s h , tall + e r : tall + ~. 5.2
J u s t as two i n d i v i d u a l s can be c o m p a r e d in one d i m e n s i o n , two d i m e n s i o n s c a n be
c o m p a r e d for one individual.
5.3
(27)
T h i s house i s w i d e r ( o r : m o r e wide) than high.
(27a)
xfM(D1) > x~(D2)
A m e a s u r e m e n t v a l u e fM(x, D) c a n be quantified in two d i f f e r e n t s c a l e s , with units
1M 1 and 1M 2. We e x p r e s s t h i s by r e p r e s e n t i n g the m e a s u r e m e n t v a l u e a s a function o v e r s c a l a r units. (28)
M a r y i s h e a v i e r in A m e r i c a n pounds than in C o n t i n e n t a l pounds.
D)) (1M t) > (Y(x.D))(1M 2)
( 28a) 5.4
One and the s a m e o b j e c t m a y have d i f f e r e n t m e a s u r e m e n t v a l u e s in the s a m e d i m e n -
sion at d i f f e r e n t t i m e s . (29)
t2 a)
The U . S . was l a r g e r in 1900 than in 1800.
1)
183
5.5
5.6
J u s t a s in the c a s e of t i m e , a m e a s u r e m e n t value can also be a function o v e r p l a c e s . (30)
E x p l o r e r 5 is l i g h t e r on the moon than on e a r t h .
(30a)
(~(x,D))(p 1) <(~(x,D))(p~)
A m e a s u r e m e n t value m a y f u r t h e r v a r y a c c o r d i n g to d i f f e r e n t p r o p o s i t i o n a l attitudes
of d i f f e r e n t individuals. We r e p r e s e n t this with the c o n c e p t of " p o s s i b l e w o r l d s " developed in model t h e o r y (ef. Hintikka, 1969). (31)
John b e l i e v e s he is t a l l e r than he i s .
(31a) CP(x.D))(ix, believe) > (P(x,D))(i0) 5.7
Again, the m e a s u r e m e n t value of an individual in a d i m e n s i o n can be u n d e r s t o o d a s a 1 function o v e r c i r c u m s t a n c e s u n d e r which the M e a s u r e m e n t is taken. (32)
M a r y i s l i v e l i e r with h e r l o v e r s than with h e r p a r e n t s .
(33)
M a r y is p r e t t i e r in a nightgown than in a r a i n c o a t .
(3~.33a) (~(x,D))(~ 1) > (~(x,D))(~) 6.
Conclusion. " C o m p a r a t i v e c o n s t r u c t i o n s in any language have proven t h e m s e l v e s r e s i s t a n t to
s a t i s f a c t o r y a n a l y s i s " ( J a e o b s and R o s e n b a u m , 1970, viii). We a g r e e with this a s s e s s m e n t of the r e s e a r c h situation and conclude that linguists in the p a s t have not fully a p p r e c i a t e d the r o l e of s e m a n t i c s in d e t e r m i n i n g syntactic s t r u c t u r e s .
We c o n s i d e r the s e m a n t i c p r o p e r t i e s
(in production) and the p h y s i c a l p r o p e r t i e s (in reception) of a s e n t e n c e a s "given"; the r o l e of syntax is d e t e r m i n e d by the e x i g e n c i e s of s e m a n t i c s t r u c t u r e on one side and by the c o n s t r a i n t s a r i s i n g f r o m the physical n a t u r e and r e q u i s i t e efficiency of c o m m u n i c a t i o n on the
1
We would like to point out that for s e n t e n c e s like t h o s e in Sections 5 . 4 - 5 . 7 , a l t e r n a t i v e i n t e r p r e t a t i o n s a r e p o s s i b l e in which the indices t, p, i, j a r e indexical to the individuals, o r in the c a s e of p o s i t i v e s , a l s o to the a v e r a g e o r n o r m . F o r e x a m p l e , s e n t e n c e (29) could also be i n t e r p r e t e d a s in (29a').
(29a)
Car g e~ x)(x=~S)(tl)) > Carge((x)(x=US)(t2))
In this c a s e , the t e r m s left and r i g h t of the c o m p a r a t i v e sigh have the s a m e denotations in the two i n t e r p r e t a t i o n s . The s a m e i s not true of the following e x a m p l e : The p r e s i d e n t of the U . S . was o l d e r in 1965 than in 1971. In the a b s e n c e of factual knowledge this s e n t e n c e could be i n t e r p r e t e d a s a single i n d i v i d u a l ' s having grown younger in t i m e :
(¢(x)J(tl) ~ (¢(x))(t2) However, the intended i n t e r p r e t a t i o n is, of c o u r s e , "Of the two d i f f e r e n t individuals who held the office in the y e a r s 1965 and 1971, r e s p e c t i v e l y , the f i r s t was o l d e r in 1965 than the second w a s in 1971":
(f~((x)(P(x)~tl))) (tl) > (¢~ x)(P(x))(t2))~t2) D e t e r m i n i n g the conditions u n d e r which such i n t e n s i o n a l l y d i f f e r e n t i n t e r p r e t a t i o n s a r e e x t e n s t o n a l l y equivalent goes beyond the scope of the p r e s e n t p a p e r .
184
other.
Our analysis of c o m p a r i s o n sentences s t a r t s with an analysis of t h e i r (language-
independent) semantic p r o p e r t i e s .
We explain t h e i r seemingly paradoxical o v e r t p r o p e r t i e s
by making explicit some of the applicable constraints imposed by communication. Specifically, we propose that semantic r e p r e s e n t a t i o n s a r e logical f o r m s which a r e d i r e c t l y mappable f r o m the level of logical syntax onto models of states--of-affairs.
The
reason for positing this kind of semantic r e p r e s e n a t i o n is twofold: 1. the conviction that s e m a n t i c s is u n i v e r s a l , in the sense that only the basic words, the morphology and the word o r d e r of sentences such as (0-8) a r e specifically English, but t h e i r meanings,
which r e p r e -
sent e x c l u s i v e l y the s t r u c t u r e and the r e s u l t s of conceptual operations, a r e independent of any p a r t i c t ~ a r language; 2. the inferability principle, i . e . , the postulate that all valid i n f e r e n c e s , and only these, must be possible on the semantic level by no other means than logical inference r u l e s (and the l a n g u a g e - s p e c i f i c meaning postulates for iexical i t e m s ) : e, g. while the surface s t r u c t u r e s of (2), (21), and (1) below de not r e v e a l that (2) does not entail (1) while (2') does, the semantic s t r u c t u r e s m u s t r e v e a l this according to this principle, and do so in our notation: (34)
(2) John is t a l l e r than Mary.
(35)
(2') John is even t a l l e r than Mary.
(34a)
f~D(X) > ~DD(Y) / > a>b
(35a)
/>
> (1) John is tall.
i~Dl(X) > ND, y
a>c
f~D(X)> f~D(y) & a>b
/ > (1) John is tall.
ND, y>
g~ b>c---->a>e
~ IDa(X)> ND, y
.
We consider comparing (measuring) as a universal capacity of the human mind, and r e p r e s e n t it s e m a n t i c a l l y by the g e n e r a l m e a s u r e function which is a two-place operation, with individuals as f i r s t a r g u m e n t s and dimensions as second a r g u m e n t s .
While positives
and c o m p a r a t i v e s a r e , in our s y s t e m , both based on c o m p a r i s o n , they a r e defined independently of each other.
It follows that they a r e not based conceptually one upon the other.
thus eliminate the basic defect of the syntactically oriented analyses.
We
We c h a r a c t e r i z e and
e x e m p l i f y the syntactic r u l e s r e l a t i n g these s e m a n t i c r e p r e s e n t a t i o n s in t e r m s of t h e i r o v e r t manifestations.
The rules l e x i c a l i z i n g c o m p a r i s o n s by m e a n s of r e l a t i v e adjectives a r e of a
type that has not been d i s c u s s e d in the l i t e r a t u r e before.
We argue that the semantic r e p r e -
sentations we provide a r e not inherently l i n e a r l y o r d e r e d (a consequence of both the u n i v e r sality and the inferability principles), and we have found no r e a s o n to postulate e x t r i n s i c o r d e r for our syntactic rules.
185
It seems significant to us that e a r l i e r approaches, in addition to leaving doubts as to their semantic adequacy, have invariably left - or created - residual problems of even g r e a t e r magnitude than they solved. By contrast, all of these problems a r e resolved in our theory in a uniform way without any ad hoc apparatus.
It seems to us that our approach
reveals these problems to be pseudo-problems created by models of language which treat syntax as the generative component of a grammar.
We therefore feel encouraged to propose
that our mode of analysis, which is based on the universality and inferability principles for semantic representations, be extended to other syntactic problems and to the foundation of language theory in general. References Bartsch, R. and T. Vennemann (1972), "Relative Adjectives and Comparison", in P. Sehacht e r andG. Bedell, e d s . , UCLA Papers in Syntax, 2. Univ. of Calif., Los Angeles. Bierwiseh, M. (1971), "On Classifying Semantic F e a t u r e s " , in D.D. Steinberg and L.A. Jakobovits, e d s , , Semantics: An Interdisciplinary Reader in Philosophy, Linguistics and Psychology, Cambridge: University P r e s s , 410-435. Dik, S.C. (1971), " E l s j e is twee jaar ouder dan twee jaar geleden", MS, Univ. of Amsterdam. Fillmore, C . J . (1965), "Entailment Rules in a Semantic Theory", Pola Reports, 1 0 6 0 - 8 2 (Columbus: The Ohio State University). Hintikka, J. (1969), "Semantics for Propositional Attitudes", in J.W. Davis and D . J . Hockney, e d s . , Studies in Philosophical Logic, Dordreeht: D. Reidel, 21-45. Jacobs, R.A. and P. S. Rosenbaum (Eds.) (1970), Readings in English Transformational G r a m m a r , Waltham, Mass. : Ginn. Reichenbach, H. (1947), Elements of Symbolic Logic, paperback edition, 1966, New York: F r e e Press. Ross, J . R . and D.M. P e r l m u t t e r (1970), "A Non-source for Comparatives", Linguistic Inquiry, 1, 127-128. Russell, B. (1945), A H i s t o r y of Western Philosophy (Clarion paperback), New York: Simon and Schuster. Sapir, E. (1944), "Grading: A Study in Semantics", Philosophy of Science, 11, 93-116. Seuren, P. (1970), "The Comparative", MS, Magdalen College, Oxford. Wierzbicka, A. (1972), "The Deep or Semantic Structure of the Comparative", Lin~uistische Berichte, 1_66, 39-45. Wunderlich, D. (1970), "Vergleischssatze "" " , MS, F r e i e Universit~t, Berlin.
A SIMPLE
HIERARCHICAL
MODEL
OF NATURAL
SELECTION
David J. Winter The University of Michigan
i. Introduction The purpose of this paper is to describe a hierarchical model of an evolutionary process resembling natural selection. In the model, one begins with an initial "population" of "genetic structures".
Each genetic structure of a population reproduces "progeny" and
the progeny of these genetic structures m a k e up the next population. The degree of success of reproduction of progeny by a particular genetic structure Ks determined by its "fitness" in the "environment".
One expects fitness of a g~ven genetic structure to correlate highly
with s o m e of i t s c h a r a c t e r i s t i c s in the e n v i r o n m e n t , the c h a r a c t e r i s t i c s of that genetic s t r u c t u r e b e i n g s e e n t h r o u g h the e y e s of " d e s c r i p t o r s " . highly fit types of genetic s t r u c t u r e s to
With the p a s s a g e of t i m e , one e x p e c t s
e m e r g e and e x p e c t s s o m e of t h e s e highly f i t types
to be i n c r e a s i n g l y " s t a b l e " , s t a b i l i t y b e i n g defined with r e s p e c t to d e s c r i p t o r s and t h e r e f o r e r e p r e s e n t i n g the continuance of c h a r a c t e r i s t i c s . In t h i s paper, we p r e s e n t only a v e r y g e n e r a l model.
The genetic s t r u c t u r e s t h e m -
s e l v e s m i g h t r e p r e s e n t o r g a n i s m s within a p h y s i c a l e n v i r o n m e n t o r a r t i f i c i a l s t r u c t u r e s within an a r t i f i c i a l e n v i r o n m e n t .
It is hoped that r e a d e r s f r o m a wide v a r i e t y of b a c k g r o u n d s
m a y extend and i m p r o v e t h i s model and develop v e r s i o n s and v a r i a t i o n s of it which e n a b l e t h e m to effectively p r e d i c t the r a t e and d i r e c t i o n of c o n v e r g e n c e of genetic s t r u c t u r e s to highly fit s t a b l e types of genetic s t r u c t u r e s of i n t e r e s t to t h e m .
(See K i m i r a and Ohta, 1971;
Holland~ 1973; and Holland, to a p p e a r . ) 2.
G e n e t i c S t r u c t u r e s , Genetic Operators~ P o p u l a t i o n s and F i t n e s s We s t a r t with a s e t X ( s u c h a s the s e t of all s t r i n g s of l e t t e r s ) .
c a l l e d a genetic s t r u c t u r e . of two v a r i a b l e s on X.
An e l e m e n t x of X is
A genetic o p e r a t o r is a n o n n e g a t i v e r e a l valued function w(x, y)
F o r any two genetic s t r u c t u r e s x and y, w(x, y) i s the p r o b a b l e
f r e q u e n c y of o c c u r r a n e e of y a m o n g the p r o g e n y of x.
Given a s e t w 1 .
. . . .
w m of functions
f r o m X to X (which m a y be thought of a s c r o s s o v e r , i n v e r s i o n , mutation, e t c . ) and a s e t of n o n n e g a t i v e r e a l valued functions ~rl(x) . . . . ,7rm(X) (which m a y be thought of a s the p r o b a b l e f r e q u e n c y of o c c u r r a n c e a t x of the w 1. . . . . w m) we obtain a genetic o p e r a t o r m
w = (Wl, 7rI . . . . . Wm, v m) defined by letting w(x, y) = 7 . i:1"=
~
7ri(x).
A population i s n o n -
y =w (.x ) 1
n e g a t i v e r e a l valued function p on X which a s s i g n s to e a c h genetic s t r u c t u r e x i t s f r e q u e n c y of o c c u r r a n e e p
in the population p. An e n v i r o n m e n t i s a n o n n e g a t i v e r e a l v a l u e d function x u on X which a s s i g n s to e a c h genetic s t r u c t u r e x i t s f i t n e s s u(x) in the e n v i r o n m e n t u.
187
A genetic o p e r a t o r w(x,y) d e t e r m i n e s for e a c h population p in an e n v i r o n m e n t u the p r o b a b l e c a n s t i t u e n c y of the next population p', the p r o b a b l e f r e q u e n c y of o c c u r r a n c e of y in p' b e i n g py' = E
u(X)PxW(X,Y}"
X
3.
D e s c r i p t o r s and Stability A d e s c r i p t o r on a s e t X of genetic s t r u c t u r e s i s a n o n n e g a t i v e r e a l valued function
d{x, y) of two v a r i a b l e s on X such t h a t 1. d(x,x) = 0
for x in X ;
2.
d(x,y) = d ( y , ~ f o r x , y in X ;
3.
d ( x , y ) + d ( y , z ) > _ d(x,z) for x , y , z in X.
If d(x, y) i s s m a l l , x r e s e m b l e s y with r e s p e c t to d. t i n g u i s h a b l e with r e s p e c t to d.
If d(x, y) = 0, x and y a r e i n d i s -
Note t h a t if d(x, y) = 0, then d(x, z) = d(y, z) for all z.
L e t t i n g d(x, y) be a d e s c r i p t o r on X, we obtain a n e q u i v a l e n c e r e l a t i o n d(x, y) = 0 on X. The e q u i v a l e n c e c l a s s { y e X I d ( x , y ) = 0 } of x i s denoted x and c o n s i s t s of all genetic s t r u c t u r e s i n d i s t i n g u i s h a b l e f r o m x with r e s p e c t to d. The s e t { x l x e X } of x is denoted by X / d o r by .K. One s e e s e a s i l y t h a t the d e s c r i p t o r s d on X a r e the functions on X of the f o r m d(x, y) = m (f(x), f(y)) w h e r e f is a function f r o m X into a m e t r i c s p a c e with m e t r i c m. /
If d t . . . . .
d m a r e r e a i v a l u e d functions on X, then dIx,
(di(x -di(Y 2 defines 1 such a d e s c r i p t o r d which is denoted d = (d 1 . . . . . rim). If S i s a finite s e t of r e a l valued 0 1 functions d 1 . . . . . d m on X, we let d~ denote the d e s c r i p t o r (d . . . . . . d ). If d and d a r e . l m I d e s c r i p t o r s on X such t h a t d0(x, y) >_ d (x, y) for all x, y, which we i n d i c a t e by w r i t i n g d O >_ d 1, then d 1 m a y be r e g a r d e d a s a d e s c r i p t o r on the s e t X 1 = X / d 0 = :~ of e q u i v a l e n c e c l a s s e s of X with r e s p e c t to d O, letting d l ( x , y ) = d l ( x , y ) for x , y in X.
tf d O, d 1 . . . . , d m
a r e d e s c r i p t o r s w i t h d O >_ d 1 >_ . . . >_ d m, we m a y t h e r e f o r e r e g a r d d i a s a d e s c r i p t o r on X i for 0 < _ i < m
where X0=X,
X 1 =X0/d 0,...,X m=Xm-1/d
m-1.
If SO,S 1 . . . .
Sm a r e
finite s e t s of r e a l valued functions on X and SO > S 1 _> . . . _> Sm, t h e n ds0 >_ ds1 > _ . . . >_ d s
, m
so t h a t we m a y r e g a r d ds.~ a s a d e s c r i p t o r on X i = x i - 1 / d S i _ l
for i = 0, 1 . . . . .
m.
Finally, we n e e d the concept of stability, which i n d i c a t e s r e c u r r a n c e of a c h a r a c t e r i s t i c in s u c c e s s i v e g e n e r a t i o n s .
F i x i n g a d e s c r i p t o r d on X and s m a l l p o s i t i v e n u m b e r s 5 a n d e,
we s a y that a genetic s t r u c t u r e x is s t a b l e with r e s p e c t to d if
E w(x, y)/ ~_~ w(x, y) d(x, y) >_6 y < E, that is, if the r a t i o of the n u m b e r of p r o b a b l e p r o g e n y of x not r e s e m b l i n g x with r e s p e c t to d to the n u m b e r of all p r o b a b l e p r o g e n y of x is s m a l l .
188
4.
The E v o l u t i o n a r y P r o c e s s We now d e s c r i b e the e v o l u t i o n a r y p r o c e s s of the p r e s e n t model, which p r o c e e d s at
one level, then p a s s e s to a n e x t and so on. w h e r e X i s a s e t of genetic s t r u c t u r e s ,
We begin with an 8-tple ( X , w , u, S, 5, c, n, p(0))
w i s a genetic o p e r a t o r on X, u i s a n e n v i r o n m e n t
for X, S is a finite s e t of r e a l v a l u e d functions on X, 5 and c a r e s m a l l p o s i t i v e r e a l n u m b e r s , n is a positive n u m b e r and p(0) is an initial population.
Within the p r e s e n t level,
the i n i t i a l population g i v e s r i s e to s u c c e s s i v e populations p(t) (t = 1, 2 . . . . . by the p r o b a b a l i s t i c a l t y applied genetic o p e r a t o r .
i) a s specified
The genetic s t r u c t u r e s in the populations
p(t) a r e e x a m i n e d for f i t n e s s and for s t a b i l i t y with r e s p e c t to the d e s c r i p t o r s d T w h e r e T r a n g e s o v e r all s u b s e t s of S which a r e obtained f r o m S by r e m o v a l of a single e l e m e n t of S. A s soon a s n s t a b l e genetic s t r u c t u r e s a r e found, t h a t T for which f i t n e s s of the s t a b l e genetic s t r u c t u r e s is m a x i m a l is c h o s e n .
At t h i s point, a t r a n s f e r is made to the next level
of the p r o c e s s by p a s s a g e to the next d-tpte (x, w, u, S, 5, ~, n, p(0)). be X / d T. ia T.
The s e t X is taken to
The s e t S of functions on X is the s e t of f u n c t i o n s on X induced by the functions
The v a l u e s 5, c , n c a n be taken a s b e f o r e o r modified in a c c o r d a n c e with the r a t e of
convergence.
F o r each x in X, w(x,y) is t a k e n to be w(x,y) = 1=. 1T ~ w(x,y) and u(x) x y XCX
ycy i s t a k e n to be u(x) = l'~ ~
u(x), w h e r e I x l , lyl a r e the n u m b e r of e l e m e n t s of x , y
XEX
respectively.
Finally, the initial population p(0) of the new l e v e l is defined in t e r m s of the
population p(i) at the end of the p r e c e d i n g level, the definition being p(0)y~, =
~ _ p(i) x. XEX
A d r a w b a c k to the p r e s e n t model is the s y s t e m a t i c s e a r c h t h r o u g h all of the T.
An
i m p r o v e d model m i g h t be found by r e g a r d i n g the d e s c r i p t o r s d T with T any s u b s e t of S a s genetic s t r u c t u r e s and producing a s u i t a b l e genetic o p e r a t o r and e n v i r o n m e n t so t h a t t h e s e d e s c r i p t o r s could quickly evolve into d e s c r i p t o r s which a r e effective in l o c a t i n g highly fit and s t a b l e genetic s t r u c t u r e s in the o r i g i n a l s y s t e m . References Holland, J. (1973), " G e n e t i c A l g o r i t h m s and the Optimal Allocation of T r i a l s " , Comp. 2._, 88-105.
SIAM J.
Holland, J. (to appear) Adaptation in N a t u r a l and A r t i f i c i a l S y s t e m s , U n i v e r s i t y of Michigan P r e s s , Ann A r b o r (1974). K i m i r a , M. and T. Ohta (1971), T h e o r e t i c a l A s p e c t s of Population G e n e t i c s , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , Ch. 5.
ON THE NOTION O F A R U L E Thomas Olshewsky U n i v e r s i t y of Kentucky In his "On the Notion 'Rule of G r a m m a r ' " ,
Noam C h o m s k y g i v e s an exposition of the
n e e d f o r a p r e c i s e f o r m u l a t i o n of a r u l e of g r a m m a r .
In so doing, he c a u t i o n s us not to take
too r e a d i l y a g e n e r a t i v e g r a m m a r a s a m o d e l for the s p e a k e r o r the h e a r e r of a language (Chomsky, 1961, Note 16). himself.
T h a t caution h a s not b e e n widely heeded, not even by C h o m s k y
His tone in Language and Mind, a l r e a d y p r e s a g e d in A s p e c t s , is quite d i f f e r e n t .
While noting t h e r e that o t h e r f a c t o r s a r e involved in l a n g u a g e p e r f o r m a n c e , he m a i n t a i n s t h a t l a n g u a g e a c q u i s i t i o n i s a n i n t e r n a l i z a t i o n of a s y s t e m of r u l e s and t h a t l a n g u a g e p e r f o r m a n c e is t h e i r e m p l o y m e n t (Chomsky, 1968, p. 23).
One m i g h t e x p e c t t h i s c l a i m to invite
an e x p l o r a t i o n of the c o n c e p t u a l r e l a t i o n s h i p bet~veen r u l e s of language and p a t t e r n s of v e r b a l behavior.
R a t h e r , m a n y p s y c h o l o g i s t s have t a k e n C h o m s k y ' s l a t e r t h e s i s a s a given f o u n d a -
tion for i n q u i r y , not a s a p r o b l e m to be d e a l t w i t h (ef. McNeill, 1970; M i l l e r and I s a r d , 1963; Saporta, 1967).
I hope t h r o u g h e x p l o r a t i o n s of the notion of a r u l e to r e i n s t i t u t e C h o m -
s k y ' s e a r l i e r caution. B a s e d on C h o m s k y ' s t h e s i s , l i n g u i s t s , p s y c h o l o g i s t s and p h i l o s o p h e r s a l i k e h a v e found a r e n e w a l f o r the contention t h a t v e r b a l b e h a v i o r in p a r t i c u l a r and h u m a n b e h a v i o r in g e n e r a l i s r u l e - g o v e r n e d b e h a v i o r . John S e a r l e (1969) t a k e s a s the t h e s i s f o r h i s book, "Speaking a language is e n g a g i n g iu a r u l e - g o v e r n e d f o r m of b e h a v i o r " .
G . A . M i l l e r (1963)
n o t e s , " T h e question r e m a i n s open w h e t h e r a d e s c r i p t i v e s c i e n c e , s u c h as psychology a s p i r e s to be, c a n i n c o r p o r a t e s y s t e m s of r u l e s into a f r a m e w o r k p r o v i d e d b y the m o r e t r a d itional f o r m of s c i e n t i f i c l a w s " .
I find the o r i e n t a t i o n inviting, but for it to b e c o m e a w o r k -
able one t h e r e i s the n e e d to e x p l o r e not only the r e l a t i o n of language to speaking, but a l s o to u n d e r s t a n d the notion of r u l e - g o v e r n e d
b e h a v i o r in t h e s e c o n t e x t s .
The p r o b l e m s have not
gone u n r e c o g n i z e d (el. Quine, 1970), but p o s i t i v e ways of dealing with t h e m r e m a i n to be devised.
In e x p l o r i n g the notion of a r u l e , I will e x p o s e s o m e q u e s t i o n s I find r e l e v a n t to the
notion of a r u l e in the p s y e h o l i n g u i s t i e e n t e r p r i s e , and s o m e t h e s e s t h a t m a y give s o m e d i r e c t i o n for d e a l i n g with the p r o b l e m s . I
One way of getting a t the notion of a r u l e is to c h a r a c t e r i z e how r u l e s o p e r a t e in d i f f e r e n t c o n t e x t s of a c t i o n and i n q u i r y .
A first general characterization is whatrules are
. Henley (1969) s u g g e s t e d the s t a r t i n g points t h a t I have t a k e n , but they have u n d e r g o n e c o n s i d e r a b l e r e w o r k i n g in m y hands. He is in no way r e s p o n s i b l e f o r - - a n d p r o b a b l y would not a g r e e w i t h - - m u c h of the o u t c o m e . The w o r k of Gumb (1973) is a l s o p e r t i n e n t to t h i s study, but I have not yet had the o p p o r t u n i t y to study it.
190
c o n t e x t - d e p e n d e n t . A r u l e is always a r u l e of s o m e t h i n g , a r u l e of c h e s s , a r u l e of etiquette, e t c . , and that s o m e t h i n g d e t e r m i n e s the context in w h i c h the r u l e is o p e r a t i v e .
A rule need
not be l i m i t e d to a single context, but for it to be a r u l e , it m u s t o p e r a t e in s o m e context. This gives a s our f i r s t t h e s i s : T-l:
R u l e s a r e n e c e s s a r i l y c o n t e x t - d e p e n d e n t , and it i s always p o s s i b l e to specify the r u l e - c o n t e x t w h e n e v e r Y'rule" is p r o p e r l y u s e d .
A p a r t of u n d e r s t a n d i n g a r u l e , then, i s to u n d e r s t a n d the context in which it o p e r a t e s . Not only a r e r u l e s c o n t e x t - d e p e n d e n t , but they also s e r v e d i f f e r e n t r o l e s in r e l a t i o n to t h e i r c o n t e x t s .
This g i v e s us our s e c o n d t h e s i s :
T - 2 : R u l e s o p e r a t e a c c o r d i n g to d i s t i n g u i s h a b l e and d e t e r m i n a b l e r o l e s within any given context. The d i s t i n c t i o n of r o l e s r e q u i r e s a typology and any one o f f e r e d will inevitably be s u b j e c t to debate.
The c a t e g o r i e s that follow give at l e a s t a w o r k a b l e typology, and will facilitate our
e x p l o r a t i o n of the notion of a r u l e in p s y c h o l i n g u i s t i c s : A ~ e n e r a l i z a t i o a r o l e i s one in which the r u l e s e r v e s to f o r m u l a t e r e g u l a r i t i e s of the context; e . g . , r u l e s of safe d r i v i n g o r r u l e s of p r o p e r diet.
A r e g u l a t i v e r o l e is one in which the rule s e r v e s to r e g u l a t e o p e r a t i o n s w i t h -
in a context, o r b r i n g o r d e r to a context, e . g . , r u l e s of the r o a d o r r u l e s of table etiquette. A constitutive r o l e is one in which the rule s e r v e s to e s t a b l i s h a context o r define it; e . g . , r u l e s of c h e s s o r of s o m e o t h e r g a m e .
T h i s logically l e a v e s the way open f o r a single r u l e
to play d i f f e r e n t r o l e s in d i f f e r e n t c o n t e x t s .
Indeed, it may be p o s s i b l e for a single r u l e to
play d i f f e r e n t r o l e s in the s a m e eontexL but t h i s r e q u i r e s f u r t h e r e x p l o r a t i o n . Regulative r o l e s for r u l e s p r e s u p p o s e the context as given, and thus the r u l e s a r e d e p e n d e n t upon the context.
F o r g e n e r a l i z a t i o n r o l e s and constitutive r o l e s the e x i s t e n c e of
c o n t e x t s and r u l e s a r e c o - d e p e n d e n t . Regulative r o l e s a r e o r d i n a r i l y developed upon c o n ventioaal foundations, while g e n e r a l i z a t i o n r o l e s d e r i v e f r o m knowledge of the n a t u r e of the context, o r d i n a r i l y b a s e d upon e x p e r i e n c e .
H e r e we have an echo of the Humean c a t e g o r i e s
for knowledge which lie at the b a s e of s t a n d a r d i n t e r p r e t a t i o n s of the a n a l y t i c / s y n t h e t i c d i s tinction. W e r e constitutive r o l e s a l s o c o n s t r u e d a s conventional, we would have a n e a t s e t of d i f f e r e n t i a e by which no r u l e could unambigously s e r v e m o r e than one of the t h r e e r o l e s in any given context. move.
So long as we l i m i t o u r s e l v e s to g a m e s for p a r a d i g m s , this s e e m s a safe
The question is w h e t h e r o r not the Kantian m a n n e r of t r e a t i n g c o n s t i t u t i v e r u l e s i s an
a p p r o p r i a t e one. al s c i e n c e ,
To ask waht c o n s t i t u t e s human e x p e r i e n c e , o r even m a t h e m a t i c s and n a t u r -
is not like asking what c o n s t i t u t e s c h e s s o r b r i d g e o r baseb&ll.
The a n s w e r s
cannot be d e r i v e d f r o m conventions, nor g e n e r a l i z e d f r o m e x p e r i e n c e s , but m u s t be found in an exposition of the p r e s u p p o s e d foundations for any p o s s i b l e o p e r a t i o n within the context. is a g a i n s t the b a c k d r o p of t h i s expanded conception of a c o n s t i t u t i v e r o l e that c o n t r o v e r s i e s
It
191
o v e r n a t i v i s m and "innate i d e a s " , a s well a s those o v e r l i n g u i s t i c u n i v e r s a l s , begin to make s e n s e . With such a conception of constitutive r o l e s , we can a t l e a s t u n d e r s t a n d the c l a i m that a linguistic m o d e l m a y s e r v e a s a p s y c h o l o g i c a l one.
The c l a i m i s that the r u l e s w h i c h
have a generalization r o l e in a t h e o r y of language will a l s o have a constitutive r o l e in i t s u s e . All t h r e e kinds of r o l e f o r r u l e s have both d e s c r i p t i v e and n o r m a t i v e ( p r e s c r i p t i v e ?) force.
How t h e s e f o r c e s r e l a t e to one a n o t h e r and to p a r t i c u l a r r o l e s f o r r u l e s in c o n t e x t
s e e m s a worthwhile
topic for study unto i t s e l f .
P e r h a p s d e s c r i p t i v e f o r c e d o m i n a t e s in the
g e n e r a l i z a t i o n r o l e , but n o r m a t i v e f o r c e d o m i n a t e s in the constitutive one.
Certainly there
i s a n o r m a t i v e f o r c e to the f o r m u l a t i o n of r e g u l a t i v e r u l e s a s well a s to t h e i r e m p l o y m e n t . W h a t e v e r the d e t a i l s , such a study b a s e d on t h e s e s t a r t i n g points would facilitate M i l l e r ' s c o n c e r n with the status of r u l e - g o v e r n e d b e h a v i o r in s c i e n c e , without getting hung up on the notions of d e s c r i p t i v e vs. p r e s c r i p t i v e m e a n i n g s for r u l e s . L e s t we too r e a d i l y a s s i m i l a t e r u l e s to l a w s , it i s i m p o r t a n t to note that r u l e s can always be broken.
So long a s we deal with the notion of a law in a l e g i s l a t i v e context, it can
be a s s i m i l a t e d to the notion of a r u l e .
Statutory laws a r e r e g u l a t i v e r u l e s and constitutional
laws (presumably) a r e constitutive r u l e s .
To b r e a k a r e g u l a t i v e r u l e i s i n a p p r o p r i a t e within
the r u l e - c o n t e x t , but to b r e a k a constitutive r u l e does v i o l e n c e to the definition of the context itself.
Action b r e a k i n g a constitutive r u l e will be ineffectual o r i m p o s s i b l e within the r u l e -
context, o r it will be judged not in that context at all.
Moving the king m o r e than one s p a c e
in a single move is ineffectual, t r u m p i n g with a king is i m p o s s i b l e , and stacking a king on top of a rook may be a p a r t of s o m e g a m e , but c l e a r l y not a g a m e of c h e s s .
Not all c o n s t i t u -
tive violations a r e so n e a t a s t h e s e e x a m p l e s , and not all c o n t e x t s a r e so neat a s a w e l l c o n s t i t u t e d g a m e , a s constitutional law will so r e a d i l y t e s t i f y .
M o r e difficult e x a m p l e s may
well s t r a i n T - 1 . The r e l a t i o n of g e n e r a l i z a t i o n r u l e s to n a t u r a l laws i s m o r e difficult. R u l e s of p r o p e r diet, for i n s t a n c e , a r e b a s e d upon b i o - c h e m i c a t , physiological and o t h e r n a t u r a l l a w s . In t h i s r o l e , the notion of a r u l e c o m e s c l o s e s t to the laws of the " d e s c r i p t i v e " s c i e n c e s . in t h i s r o l e , r u l e s a r e d e r i v a t i v e f r o m l a w s and not equatible with t h e m .
But
T h e s e r u l e s can be
violated without violating the r e l e v a n t laws; indeed, in the v e r y violation of the r u l e s , the r e l e v a n t laws a r e given negative v e r i f i c a t i o n .
They a r e not r e d u c i b l e to the l a w s , h o w e v e r ,
s i n c e they involve value c o n s i d e r a t i o n s not e n t a i l e d by the laws t h e m s e l v e s .
F o r r u l e s of
p r o p e r diet you n e e d the value c o n s i d e r a t i o n of good health; for the r u l e s of safe driving you need the value c o n s i d e r a t i o n of p e r s o n a l safety.
But it s e e m plausible that g e n e r a l i z a t i o n
r u l e s can be analyzed into the r e l e v a n t n a t u r a l l a w s , t o g e t h e r with the r e l e v a n t value c o n s i derations,
tf so, given the c l a i m s of r e c e n t l i n g u i s t i c s and p s y c h o l i n g u i s t i c s , t h i s cannot be
the notion of a r u l e we a r e seeking.
192
W i t t g e n s t e i n and his f o l l o w e r s have made much of t r e a t i n g s u b - c l a s s e s of languages on an analogy to r u l e s of g a m e s .
While this may prove s a t i s f a c t o r y for c e r t a i n a s p e c t s of
s e m a n t i c c o n c e r n , it s e r v e s a s a s i n g u l a r l y i n a p p r o p r i a t e model for a language a s a whole. G a m e s involve a c o n s c i o u s e n t e r i n g into, w h e r e a s a language p r o v i d e s a b a s i s for e n t e r i n g in.
F a r f r o m being the r e s u l t of convention, language is p r e s u p p o s e d by v i r t u a l l y all c o n -
ventional e n g a g e m e n t s .
F o r r u l e s of games~ constitutive r u l e s a s well as r e g u l a t i v e r u l e s
a r e made definite by conventions for s p e c i f i e d p u r p o s e s .
The r u l e s of Hoyte a s well a s the
r u l e s of G o r e n for b r i d g e c a m e into being by human decision~ w h e t h e r e x p l i c i t o r i m p l i c i t , singular or collective.
L i n g u i s t i c s t r u c t u r e s a r e c o n s t i t u t e d by physiological and p s y c h o -
logical a s well a s s o c i a l conditions.
The F r e n c h A c a d e m y may s e t f o r t h r u l e s , but t h e s e a r e
r e g u l a t i v e r a t h e r than constitutive. W h e t h e r positive m e r i t n a t i v i s m m a y have (and I think it has little or none), its negative t h r u s t is that the p o s i t i v i s t i c options of e m p i r i c a l g e n e r a l ization or conventional definition a r e inadequate.
R u l e s of l a n g u a g e s a r e not like r u l e s of
games. While the analogy is thus d e f e c t i v e , it points to a n o t h e r p r o b l e m .
However limited~
the analogue between r u l e s of a g a m e {say, bridge) and r u l e s of a language (say, English) would lead one to e x p e c t a c o m p a r a b l e analogue on the m o r e g e n e r a l level.
Yet, we a r e
n e v e r inclined to speak of r u l e s of Game in the way so many r e a d i l y speak of r u l e s of L a n guage.
If, indeed~ r u l e s a r e context dependent, we fail to find a context c a l l e d " G a m e " that
i s defined by any s e t of r u l e s .
What i s t h e r e to l e a d us to s u p p o s e that t h e r e i s a context
called "Language" that s e r v e s a s a context for r u l e s of language ? Such a c o n t e x t i s n e i t h e r p r e s e n t f o r o b s e r v i n g r u l e - g o v e r n e d b e h a v i o r n o r for engaging in it.
T-1 s e e m s to make
the notion of a rule of Language an anomalous one. Thus f a r , we have spoken only about r u l e s of language, net about r u l e s for speaking. The shift f r o m of to for s e e m s not an i n s i g n i f i c a n t one.
W h e r e the f o r m e r r e l a t e s r u l e s to a
context, the l a t t e r r e l a t e s r u l e s to g o a l - d i r e c t e d activity. tools.
This invites an analogy of r u l e s to
The h a m m e r i s a tool o f c a r p e n t r y , but is a tool fo_.~doing c a r p e n t r y work.
H e r e , we
m u s t not be h a s t y to a s s i m i l a t e r u l e - g o v e r n e d b e h a v i o r to g o a l - d i r e c t e d activity, or to equate r u l e s with tools.
T h e r e a r e h a z a r d s enough with such m o v e s a s we shall s e e p r e s e n t l y .
Rather~ the point that I would make h e r e is that the r o l e s of r u l e s in language a r e d i f f e r e n t in kind f r o m the r o l e s of r u l e s in s p e a k i n g - - i n d e e d , even the s e n s e s of "in" h e r e s e e m d i f ferent.
Adequate explication of the n a t u r e and r e l a t i o n of t h e s e r o l e s is a condition for m e e t -
ing the c o n c e r n s with w h i c h we began. We m i g h t hope for an explication of r u l e s for speaking in s p e e c h - a c t a n a l y s e s such a s t h o s e of S e a r l e (1969), but r u l e s for s p e e c h - a c t s a r e not equatable with r u l e s for speaking. In s e m i o t i c t e r m s , s p e e c h - a c t a n a l y s e s a r e p r a g m a t i c .
They a r e c o n c e r n e d with what the
193
s p e a k e r can do in and t h r o u g h the a c t of speaking.
P h o n e t i c , s y n t a c t i c and s e m a n t i c r u l e s
m a y provide the s p e a k e r with the e q u i p m e n t fo..~rthat doing i n a n d through. as tools treatment seems plausible. well.
So f a r the r u l e s
Yet, t h e r e is the a c t o f s p e a k i n g to be c o n s i d e r e d a s
The doing of the s p e a k i n g i t s e l f c a n n o t so p l a u s i b l y be t r e a t e d a s a g o a l - d i r e c t e d a c t i -
vity a s doing s o m e t h i n g in a n d t h r o u g h t h a t act. l e a d i n g on t h i s point.
Some d i s c u s s i o n s h a v e b e e n h o p e l e s s l y m i s -
F o d o r , for e x a m p l e , m a i n t a i n s t h a t a c h i l d " a p p l i e s " o r " e m p l o y s "
l i n g u i s t i c r u l e s to p e r c e p t u a l input ( F o d o r , 1966, p. 117).
Such m e a n s - e n d s c h a r a c t e r i z a -
t i o n s w h i c h a r e a p p r o p r i a t e to g o a l - d i r e c t e d a c t i v i t y s e e m s i n g u l a r l y i n a p p r o p r i a t e for l o c u t i o n a r y a c t s , while quite s a t i s f a c t o r y for i l l o c u t i o n a r y and p e r l o c u t i o n a r y ones. F o r l i n g u i s t i c s , g e n e r a l i z a t i o n r o l e s s e e m a p p r o p r i a t e for c o n s i d e r a t i o n of r u l e s of a language.
F o r psychology, r e g u l a t i v e and c o n s t i t u t i v e r o l e s s e e m a p p r o p r i a t e f o r c o n s i d e r -
ation of r u l e s for speaking.
A t t e m p t s to conflate g e n e r a l i z a t i o n r o l e s with c o n s t i t u t i v e r o l e s
a r e a t t e m p t s to provide a b r i d g e b e t w e e n l i n g u i s t i c s and psychology. b a s i s for conceptual confusion o v e r the n a t u r e of a t h e o r y of language.
They a r e a l s o the I n s o f a r a s one r e -
g a r d s the r u l e s in g e n e r a l i z a t i o n r o l e s , the conception of a h y p o t h e t i c o - d e d u c t i v e model is a p p r o p r i a t e , s i m i l a r to a n e m p i r i c a l s c i e n c e .
I n s o f a r a s one r e g a r d s the r u l e s in c o n s t i t u -
tive r o l e s , the conception of a n a x i o m a t i c s y s t e m i s a p p r o p r i a t e , s i m i l a r to a f o r m a l s c i e n c e . C o n t r o v e r s i e s o v e r w h i c h conception i s a p p r o p r i a t e to a t h e o r y of l a n g u a g e have b e e n g e n e r a t e d out of the a t t e m p t to have the t h e o r y b o t h w a y s .
T h i s i s not to s a y that we c a n n o t have
it both ways, but how t h i s could be s o r e m a i n s a c o n c e p t u a l p r o b l e m . T h e s e p r e l i m i n a r y e x p l o r a t i o n s of p r o b l e m s with contexts and r o l e s for r u l e s l e a v e us with a t l e a s t two q u e s t i o n s r e l e v a n t to our c o n c e r n s : Q - l : How c l o s e l y knit a f a m i l y is the s e t of c o n c e p t s e x p r e s s e d by " r u l e " ? (e. g . , r u l e of a g a m e , r u l e of t h u m b , r u l e of i n f e r e n c e ; o r r u l e of g r a m m a r , r u l e of l i n g u i s t i c a c q u i s i t i o n , r u l e of action) Q - 2 : How, if at all, do r u l e s of language function as r u l e s f o r s p e a k i n g ? To deal with s u c h q u e s t i o n s we n e e d to m a k e f u r t h e r e x p l o r a t i o n s into the c o n c e p t of r u l e g o v e r n e d b e h a v i o r , and into i s s u e s r e l a t e d to the l a n g u a g e - s p e a k i n g d i s t i n c t i o n r e l e v a n t to a t h e o r y of language. II In h i s d i s c u s s i o n of r u l e s , W i t t g e n s t e i n d i s t i n g u i s h e s b e t w e e n a p r o c e s s b e i n g in a c c o r d a n c e with a r u l e and a p r o c e s s involving a r u l e ( W i t t g e n s t e i n , 1958, p. 14). e x a m p l e is the n u m b e r s e r i e s "1, 4, 9, 1 6 , . . . ".
His
A r r i v i n g a t the next n u m b e r in the s e r i e s
m a y be in a c c o r d a n c e with the r u l e of s q u a r i n g , but it n e e d not involve t h a t r u l e .
I might,
for i n s t a n c e , see that by adding 3 to 1, I g e t 4; 5 to 4, I g e t 9; 7 to 9, I g e t
16; so t h a t
t94
by adding
9 to 16, I will get 25.
This rule works
as well for generating
the series as does
the squaring rule.
That they have proved to be logically equivalent only assures
work
Clearly,
equally well.
my
calculations
me that they
using the one rule will be a different kind of
operation from what it would be with the other, even though my activity in employing or another,
or none (on the implausible
ly be characterized
as being in accordance
to speak in accordance speaking.
basis of persistent lucky guesses) with both.
In like manner,
either,
wou~d appropriate-
it is plausible for me
with the rules of a language without those rules being involved in that
If our sense of rule-governed
behavior
is to be behavior
tain rules, then we would have a clear, but innocuous
in accordance
sense of how to answer
with cer-
Q-2.
It would
j u s t a s c l e a r l y be inadequate to the c o n c e r n s of the r e l a t i o n of linguistic t h e o r y to p s y c h o logical inquiry.
By the s a m e token it would r e c o n c i l e r u l e - g o v e r n e d b e h a v i o r to n a t u r a l l a w s
s i n c e any r e g u l a r i t y could be counted as in a c c o r d a n c e with a r u l e . A p r o c e s s involves a r u l e , a c c o r d i n g to W i t t g e n s t e i n , if "the symbol of the rule f o r m s p a r t of the c a l c u l a t i o n " .
But this s u g g e s t s that the p r o c e s s u n d e r c o n s i d e r a t i o n i s a g o a l -
d i r e c t e d a c t i v i t y in which r u l e s a r e tools, in which the r u l e s of language a r e m e a n s to the end of a p r o c e s s of speaking.
We can h a r d l y say that a s p e a k e r u s e s his language to s p e a k
( r a t h e r , we say he s p e a k s inn his native language).
The alien may indeed use his knowledge
of the r u l e s of a f o r e i g n language as a m e a n s to speaking it, but the m a r k if the native is that, i n s o f a r as he i s c o m p e t e n t in speaking, this m e a n s / e n d s d i s t i n c t i o n is a b s e n t f r o m his p r o c e s s of s p e a k i n g - - t h a t it is not a calculation.
W h e r e " a c c o r d i n g to a r u l e " p r o v e d too weak
f o r r u l e - g o v e r n e d b e h a v i o r , this s e n s e of "involving a r u l e " p r o v e s too s t r o n g . The notion of following a r u l e s e e m s t r a p p e d ambiguously between t h e s e two. Suppose the s p e a k e r is following a r u l e without knowing it.
This s t r a i n s our m e t a p h o r of
following, and l e a v e s us with nothing m o r e than acting in a c c o r d a n c e with a r u l e .
On the
o t h e r hand, to explicitly know he is following a r u l e puts us back with g o a l - d i r e c t e d activity, this t i m e with the r u l e a s analogous to a guide t o w a r d a goal. s a m e s o r t s of counts as the tool analogy.
The guide analogy f a i l s on the
If we appeal to s o m e s o r t of t a c i t knowledge about
following a r u l e , we have made no gain until the c h a r a c t e r of both the knowing and the f o l l o w ing a r e s o r t e d out.
Thus, in s e e k i n g an a n s w e r to our initial q u e s t i o n s , we have picked up
another: Q-3" What s e n s e of "involving a r u l e " o r "following a r u l e " is i m p l i e d by " r u l e governed behavior" ? T h e r e is a s e n s e in which this whole m a n n e r of t r e a t i n g the p r o b l e m m i s s e s the m a r k . R u l e s that s e r v e a s tools o r a s guides in r u l e - g o v e r n e d b e h a v i o r a r e o r d i n a r i l y r e g u l a t i v e r u l e s , not constitutive o n e s .
They s e r v e , t o make the activity m o r e e f f e c t i v e o r m o r e efficient
o r m o r e a p p r o p r i a t e , but they do not s e r v e to make the activity what it is as an activity.
The
195 c o n s t i t u t i v e r o l e in r e l a t i o n to a c t i v i t y within i t s c o n t e x t i s u s u a l l y one of s e t t i n g l i m i t s on a c t i v i t y within the context, r a t h e r than p r o v i d i n g guidance for action.
This provides a basis
for a new t a c t with a new t h e s i s : T - l : R u l e s a c c o u n t for opposition by s e t t i n g l i m i t s for action. T - 3 p r e s u p p o s e s t h a t r u l e s a r e f o r m u l a t i o n s in action r a t h e r than f e a t u r e s of r e a l i t y . f e a t u r e s of r e a l i t y a r e the o p p o s i t i o n s c o n f r o n t e d in action.
The
It f u r t h e r p r e s u p p o s e s t h a t t h e
f o r m u l a t i o n s will, i n i t i a l l y a t l e a s t , be n e g a t i v e , s e t t i n g l i m i t s on a c t i o n within the context of t h a t r u l e - g o v e r n e d b e h a v i o r .
Thus the r u l e s g o v e r n the b e h a v i o r by r e s t r i c t i n g its scope
and c h a r a c t e r , A c o m p a r i s o n to m o r a l c o n t e x t s m a y be helpful. into t e l e o l o g i c a l and deontological t h e o r i e s .
M o r a l t h e o r i e s a r e u s u a l l y divided
The f o r m e r a r e c o n s t r u e d in t e r m s of e n d s o r
goals, the l a t t e r in t e r m s of r u l e s o r m a x i m s .
T e l e o l o g i c a l t h e o r i e s invite f o r m u l a t i o n of
o b l i g a t i o n s in t e r m s of what one ought to do; deontological t h e o r i e s i n v i t e f o r m u l a t i o n of o b l i e a t i o n s in t e r m s of what one ought not to do.
It is in t h i s way t h a t r u l e - g o v e r n e d b e h a v i o r
h a s a q u a s i - m o r a l or deontologieal c h a r a c t e r .
R u l e s tell u s what we m a y not do.
T h i s u n d e r s t a n d i n g of r u l e - g o v e r n e d b e h a v i o r r u n s c o u n t e r to the r o l e s r u l e s play in the f o r m a l s c i e n c e s .
T h e r e , the r u l e s a r e both c o n s t i t u t i v e and t e l e o l o g i c a l .
They both
c o n s t i t u t e the s t r u c t u r e of the s y s t e m and s e r v e a s e x p l i c i t m e a n s for new f o r m u l a t i o n s , by formation or transformation.
However, i n s o f a r a s t h e s e s y s t e m s a r e c o n s t r u e d l o g i c i s t i c -
ally, the r u l e c o n t e x t s a r e d e t e r m i n e d c o n v e n t i o n a l l y a g a i n s t the b a c k d r o p of p u r p o s e s beyond the s y s t e m (cp. Carnap~ 1950).
On an i n t u i t i o n a l i s t u n d e r s t a n d i n g , the r u l e c o n t e x t s a r e
d e t e r m i n e d by the l i m i t s of r e a l i t y , giving again a deontological b a s e , e x p r e s s e d n e g a t i v e l y as limits.
However one c o n s t r u e s the foundations of m a t h e m a t i c s , t h e s e a l t e r n a t i v e s r e v e a l
d i f f e r e n t u n d e r s t a n d i n g s of the c o n s t i t u t i v e r o l e for r u l e s .
E i t h e r the c o n s t i t u t i v e c o n -
s t r a i n t s a r e f o r m u l a t e d t e l e o l o g i e a l l y by convention, o r they a r e c o n f r o n t e d deontologically a s l i m i t s to action.
W h a t e v e r o n e ' s m e t a p h y s i c a l c o m m i t m e n t s , the f o r m e r i n t e r p r e t a t i o n
h a s m a n y h e u r i s t i c v a l u e s for m a t h e m a t i c s .
So long a s we c l a i m to be w o r k i n g w i t h n a t u r a l
l a n g u a g e s , the f o r m e r is c l e a r l y i m p l a u s i b l e both t h e o r e t i c a l l y and p r a c t i c a l l y for s t u d i e s of language and of speaking. With T - 3 as a s t a r t i n g point, we have a b a s i s for u n d e r s t a n d i n g r u l e - g o v e r n e d b e h a v i o r in the w e a k e s t p o s s i b l e s e n s e : Rule-governed behaviorl=Df"
Habituated a c t i v i t y within a d e l i m i t e d context,
w h e r e the h a b i t u a t i o n i s d e t e r m i n e d n e g a t i v e l y by t h e s e l i m i t s , but c a n be d e s c r i b e d a s b e i n g in a c c o r d a n c e with a s e t of r u l e s t h a t m a k e the l i m i t s definite. On t h i s i n t e r p r e t a t i o n , it is p o s s i b l e for d i f f e r e n t s e t s of r u l e s , l o g i c a l l y e q u i v a l e n t , to define the context for action.
It i s a l s o p o s s i b l e f o r m a n y p a t t e r n s of action to be g o v e r n e d
196
b y t h e s e r u l e s , without t h e s e a c t i o n s e v e r confronting the l i m i t s of the c o n t e x t for which the r u l e s give an accounting. involving a r u l e .
Thus, it is p o s s i b l e for a c t i o n s to be c o n s t r a i n e d by l i m i t s without
This m a k e s the n o n - s m o k e r " g o v e r n e d " in his actions by the r e g u l a t i o n of
a n o - s m o k i n g r u l e , s i n c e the act of smoking would be p o s s i b l e for him independent of the context, and the r u l e s of the context would p r o h i b i t that action.
Still, while we might be
willing to concede that he w a s g o v e r n e d by the r u l e in t h i s weak s e n s e , we would find it odd to say that he was obeying it~ and l u d i c r o u s to say that he was following it. In the light of such c o n s i d e r a t i o n s as t h e s e , Saporta, 1967, p. 22) r e j e c t s such a weak s e n s e of r u l e - g o v e r n e d b e h a v i o r .
He m a i n t a i n s that "in any c a s e w h e r e the s i m p l e s t s i m u -
lation of the o r g a n i s m r e q u i r e s c o m p u t a t i o n s that appeal to the r u l e , it follows that the o r g a n i s m w h o s e b e h a v i o r i s s i m u l a t e d m u s t know the r u l e . "
This would at l e a s t be a plausible
c l a i m , w e r e a given s e t of r u l e s uniquely a p p r o p r i a t e to a given context. the example of the s q u a r i n g and additive r u l e s , this is not the c a s e ,
But as we saw in
To s i m u l a t e the p r o c -
e s s e s r e q u i r e d to develop that n u m b e r s e r i e s does r e q u i r e c o m p u t a t i o n s that appeal to a r u l e , but not to a uniquely d e t e r m i n e d one.
If the p s y c h o l i n g u i s t finds his own r u l e s uniquely
d e t e r m i n e d , it i s b e c a u s e they a r e so d e t e r m i n e d by the t h e o r e t i c a l s y s t e m in which he o p e r a t e s , not by the activity c o n t e x t in which his s u b j e c t o p e r a t e s .
This t h r o w s us back to the
p r o b l e m s of i n f e r r i n g f r o m the g e n e r a l i z a t i o n r o l e of linguistic r u l e s to the constitutive and r e g u l a t i v e r o l e s of r u l e - g o v e r n e d b e h a v i o r in speaking.
Even if we concluded on s o m e
ground o r a n o t h e r that the s q u a r i n g r u l e i s t h e o r e t i c a l l y p r e f e r a b l e , the s u b j e c t would not have to know it in any s e n s e of "know" to c a r r y out the activity r e q u i r e d .
That he would have
to know s o m e rule in s o m e s e n s e s e e m s to be r e q u i r e d for t r e a t i n g p a t t e r n e d b e h a v i o r a s r u l e - g o v e r n e d , and this does r e q u i r e a s t r o n g e r s e n s e of r u l e - g o v e r n e d b e h a v i o r than our f i r s t , weak s e n s e . A s t r o n g e r s e n s e of r u l e - g o v e r n e d b e h a v i o r is one in which the o r g a n i s m knows w h e r e
the limits are, though he may not know what the limits are. Rule-governed
behavior2=Df"
Habituated activity determined
negatively by the
limits of the context and deriving its patterns from confrontation with those limits. The limits are learned by breaking the rules in the sense of activity exceeding the limits of the context, and thus proving in a variety of possible ways unsuccessful text.
action for that con-
This need not involve the formulation or "internalization" of rules for the context, but
only an accommodation
of action to its limits.
Such a trial and basis for habituating action
to the limits of a context is for a variety of reasons both inefficient and inadequate. individual's society cooperates with him in the development positive reinforcement the organism
may
But an
of his action patterns, both by
and by derivative negative reinforcement.
By such social constraints,
be habituated to certain patterns of action in accommodation
to derived
197
l i m i t s i m p o s e d by his s o c i e t y .
Such c o o p e r a t i o n would not r e q u i r e the o r g a n i s m to know the
u l t i m a t e l i m i t s of the context, m u c h l e s s r u l e s that g o v e r n action in r e l a t i o n to t h e m ; but i n s o f a r a s i t w e r e i n t e n t i o n a l , such c o o p e r a t i o n would r e q u i r e his s o c i e t y to know what the l i m i t s w e r e and s o m e s e t of r u l e s t h a t would a c c o u n t for t h e m . N e i t h e r of the s e n s e s offered so f a r gives us a b a s i s for talk about i n t e r n a l i z i n g , knowing, e m p l o y i n g o r applying r u l e s . R u l e - g o v e r n e d behavior3=Df"
F o r t h i s , we n e e d a yet s t r o n g e r s e n s e . Habituated a c t i v i t y u n d e r s t o o d by the a c t o r a s in
a context d e t e r m i n e d by r u l e s a c c o u n t i n g for the l i m i t s of t h a t context.
Such
r u l e s m a y be e m p l o y e d by the a c t o r as guides for f u r t h e r activity. T h i s is the f i r s t s e n s e which s e e m s adequate to the i n q u i r i e s of l i n g u i s t i c s , m a t h e m a t i c s and c o m p u t e r s c i e n c e ; it i s s t r o n g enough to s u p p o r t c o n c e r n s with e x p l a n a t i o n , p r e d i c t i o n and i n f e r e n c e ; it is too s t r o n g , a s we have s e e n , to i m p u t e to the s p e a k e r / h e a r e r a s a c o n dition for his c o m p e t e n c e , and i t is s t r o n g e r than n e c e s s a r y for a s o c i e t y to a c c u l t u r a t e i t s m e m b e r s to p a t t e r n s of a c t i o n s within r u l e - o r d e r e d c o n t e x t s .
B e c a u s e m o d e l s for l a n g u a g e s
in l i n g u i s t i c s , m a t h e m a t i c s and c o m p u t e r s c i e n c e p r e s u p p o s e s o m e t h i n g like t h i s u n d e r s t a n d ing of r u l e - g o v e r n e d b e h a v i o r , they a r e c l e a r l y i n a p p r o p r i a t e for m o d e l s for p s y c h o l o g i c a l d e v e l o p m e n t and p e r f o r m a n c e .
None of the s e n s e s o f f e r e d give an adequate u n d e r s t a n d i n g of
r u l e - g o v e r n e d a c t i v i t y for the s p e a k e r / h e a r e r ;
and if we had such, we would s t i l l n e e d an
a c c o u n t of how the o r g a n i s m m o v e s in h i s d e v e l o p m e n t f r o m r e a c t i o n to action to r u l e - f o r m a tion to r u l e - e m p l o y m e n t .
In t e r m s of T - 3 , t h i s l a t t e r q u e s t i o n b e c o m e s :
Q - 4 : How does an o r g a n i s m develop in i t s o p e r a t i o n s f r o m c o n f r o n t a t i o n with l i m i t s to f o r m u l a t i n g l i m i t s a s r u l e s for use in s u b s e q u e n t a c t i v i t y ? III While the n a t i v i s t would a g r e e t h a t t h e r e is d e v e l o p m e n t , he would m a i n t a i n t h a t the r u l e s t h e m s e l v e s a r e not the c r e a t i o n s of h u m a n a c t i o n , but a r e in s o m e s e n s e built into the o r g a n i s m f r o m the o u t s e t .
T h i s d i s c u s s i o n b a s e d on T - 3 i s thus c l e a r l y a n t i - n a t i v i s t i c .
does not i m p l y t h a t it is therefore~ a s so m a n y would i n f e r , b e h a v i o r i s t i c . has t h r o u g h o u t p r e s u p p o s e d that a c t i o n is i n t e n t i o n a l action.
This
This exploration
The e m p l o y m e n t of " b e h a v i o r "
in psychology is u s u a l l y a s l i p p e r y one, s i n c e it c l a i m s h o n o r i f i c a l l y to be e m p i r i c a l , and yet s e r v e s to r e f e r to a c t i o n i n t e r p r e t e d a s i n t e n t i o n a l .
For most purposes this is harmless
enough, but will not do when the notion of b e h a v i o r i t s e l f is in question.
Philosophers, work-
ing on the concept of action, usually d i s t i n g u i s h action f r o m b e h a v i o r , only to be left w o n d e r ing how to u n d e r s t a n d intention, and how to r e l a t e the notion of a c t i o n to t h a t of b e h a v i o r . One p a t e n t m o v e i s to r e d u c e action c a t e g o r i e s to b e h a v i o r c a t e g o r i e s .
Such a t t e m p t s i n e v i -
t a b l y r e s o r t to s o m e d i s p o s i t i o n a l c o n c e p t s ( " b e l i e f " , " a t t i t u d e " , o r m o r e g e n e r a l l y " h a b i t s " ,
198
"propensities")
as reduction devices.
tions require ana analysis in terms
Work
with reduction sentences has shown that disposi-
of contrary-to-fact
conditionals,
tionals cannot be completely reduced to perational considerations T-4- Reduction of dispositional statements to observation
and that these condi-
(cp. Hempel,
1965).
statements is systematic-
ally impossible. With such a conclusion,
it should become
clear that the "behavior"
erned behavior" is not strictly speaking observational,
spoken of in "rule-gov-
but requires interpretation as
intentional action or dispositions to action. I am inclined to maintain a stronger thesis than T-4; namely, only be understood as potentials for actualizations of some with in a teleological framework clude the development
Even this stronger version would not pre-
of the conceptual complexities
involved.
to call for,
To deny that such
takes place would require that we posit either dispositions or intentions as
starting-points. some
ends, and must therefore be dealt
of intentions from dispositions from events, as Q-4 seems
but it does expose something development
of explanation.
that dispositions can
Pe~:haps this is what is embedded
explanation.
in the nativist claim, but that too requires
In the end, I think we need an adequate understanding
to get at the notion of a r u l e in r u l e - g o v e r n e d activity. d i r e c t e d action is inadequate. n e c e s s a r y one.
of intentional action
To say that intentional action is g o a l -
G o a l - d i r e c t i o n i s a sufficient condition f o r intention, but not a
It is quite a p p r o p r i a t e to call an action intentional even when it is not done
"on p u r p o s e . " Indeed, it was j u s t a t this point that we found our a n a l y s i s of " g o v e r n e d " i n adequate.
F o r a r u l e to be "involved" does not r e q u i r e it to be a tool o r a guide to activity
any m o r e than intention r e q u i r e s p u r p o s e .
The s e n s e in which " r u l e - g o v e r n e d " activity i n -
volves a r u l e , but not as a tool for calculation, is just the s e n s e in which a c t i v i t y i s i n t e n t i o n al, but not g o a l - d i r e c t e d .
It i s in t h i s a s yet inadequately e x p l o r e d r a n g e of m e a n i n g that the
s e n s e of a notion of a rule r e l e v a n t to c o n c e r n s in psychology of language l i e s .
IV So f a r , q u e s t i o n s and t h e s e s r e g a r d i n g r u l e - g o v e r n e d b e h a v i o r have been exposed by
treating "rule", "governed"
and "behavior" in turn.
Another tact of exposition might be to
e x a m i n e o t h e r s e t s of c a t e g o r i e s r e l e v a n t to p s y c h o l i n g u i s t i e s with which r u l e s and action s e e m to have s o m e a f f i n i t i e s .
T h e r e is s o m e t e m p t a t i o n to draw analogies along the l i n e s of
rule/action = structure/function = competence/performance = explanation/observation.
The
extent to which such equations of the r e l a t i o n s of c a t e g o r i e s can be run would make an i n t e r e s t i n g c o m c e p t u a l e x p l o r a t i o n in itself.
We can e x p l o r e the m a t t e r h e r e only so far as the
v a r i o u s s e t s of c a t e g o r i e s s h e d s o m e light on the notion of a r u l e .
199
S t r u c t u r e s s e r v e a s a b a s i s f o r function a s r u l e s s e r v e as a b a s i s f o r a c t i o n s . s t a t e m e n t is patently f a l s e , but invites f u r t h e r exploration. be d i s t i n g u i s h e d .
The
F i r s t , kinds of s t r u c t u r e s m u s t
Social, linguistic and p h y s i c a l s t r u c t u r e s d i f f e r in kind f r o m one a n o t h e r
p r e c i s e l y with r e f e r e n c e to the r o l e played by r u l e s .
The s t r u c t u r e s of language and s o c i e t y
a r e d e s c r i b a b l e in t e r m s of r u l e s f o r the formulation of s t r u c t u r e s and of r u l e s for the o p e r ation of functions within t h o s e s t r u c t u r e s .
P h y s i c a l s t r u c t u r e s by c o n t r a s t can be d e s c r i b e d
without r e c o u r s e to r u l e s , and it m a k e s no s e n s e to s p e a k of the r u l e s of f o r m a t i o n f o r physical structures.
P h y s i c a l functions a r e d e s c r i b e d in t e r m s of l a w s , not r u l e s .
How like
laws r u l e s may be and to what extent the l a t t e r can be r e d u c e d to the f o r m e r a r e s a l i e n t p r o b l e m s in philosophy of s c i e n c e , and on t h e m t u r n many of the q u e s t i o n s about how s o c i a l and b e h a v i o r a l s c i e n c e s a r e r e l a t e d to p h y s i c a l and biological s c i e n c e s .
But taws a r e not r u l e s .
An a n a l y s i s of how laws d i f f e r f r o m r u l e s might be v e r y r e v e a l i n g for how c o n c e p t s of s t r u c t u r e and function and the r e l a t i o n between s t r u c t u r e and function d i f f e r between a p h y s i c a l and a social context. In a quite d i f f e r e n t m a n n e r , linguistic s t r u c t u r e s differ f r o m social s t r u c t u r e s .
In
the s e n s e that f o r m a t i o n and function r u l e s for s o c i a l c o n s t i t u t i o n s can be changed by human d e c i s i o n and action, they a r e conventional. same sort.
On a TG a n a l y s i s , l i n g u i s t i c r u l e s a r e not of the
The p r e v a l e n t t h e s i s is that they a r e built in; built into the o r g a n i s m for b a s e
s t r u c t u r e and into the language for t r a n s f o r m a t i o n s to s u r f a c e s t r u c t u r e .
The s e n s e s of
s t r u c t u r e h e r e s e e m a m e n a b l e n e i t h e r to the p h y s i c a l m o d e l n o r to the s o c i a l one.
Language
s t r u c t u r e s a r e analogous to p h y s i c a l s t r u c t u r e s e x c e p t that they r e q u i r e r u l e s for t h e i r description.
SpeaMng functions a r e analogous to s o c i a l functions e x c e p t that t h e i r c o n s t i t u -
tive foundations a r e not c l e a r l y conventional. between p h y s i c a l laws and social n o r m s .
L i n g u i s t i c r u l e s hang in conceptual d i s c o m f o r t
J u s t a s t h e i r s t a t u s r e l a t i v e to the f o r m a l / e m p i r i c a l
divide is p r o b l e m a t i c , so is it, r e l a t i v e to the p h y s i c a l / s o c i a l divide.
However the analogue
between r u l e s / a c t i o n and s t r u c t u r e / f u n c t i o n may w o r k out, the notion of a r u l e a s a p s y c h o linguis tic phenomenon will r e m a i n p r o b l e m a t i c . T h e r e is one c l e a r s e n s e in which s t r u c t u r e s a r e like, r u l e s .
Both s e t l i m i t s to
function.
But, a s we have s e e n , t h e r e a r e r u l e s in and f o r action a s well a s action a c c o r d i n g
to r u l e s .
R u l e s , a s human f o r m u l a t i o n s , s e r v e a s tools a s well a s l i m i t s .
differ f r o m physical s t r u c t u r e s and f r o m physical l a w s .
As such, they
As such, they a l s o differ f r o m l i n -
guistic r u l e s , a t l e a s t f r o m that notion of r u l e that we have been s e e k i n g . A l s o like r u l e s , c o m p e t e n c e s e t s l i m i t s to action, and in its r e l a t i o n to p e r f o r m a n c e s e e m s to have a s i m i l a r r o l e to that of s t r u c t u r e in r e l a t i o n to function.
The c o m p e t e n c e /
p e r f o r m a n c e c a t e g o r i e s a l s o s e r v e a s a conceptual bridge between l i n g u i s t i c s and psychology. The p r e v a i l i n g analogy between c o m p e t e n c e / p e r f o r m a n c e and l a n g u e / p a r o l e i s a m i s l e a d i n g
200
one, h o w e v e r .
C o m p e t e n c e , unlike s t r u c t u r e , i s aa action c a t e g o r y .
Competence is a con-
dition f o r p e r f o r m a n c e in d i f f e r e n t w a y s f r o m s t r u c t u r e a s a condition for function, and f r o m language a s a condition for speaking. (presumably) s t r u c t u r e is not.
C o m p e t e n c e i s t e l e o l o g i c a l l y defined in ways
C o m p e t e n c e can only be u n d e r s t o o d in t e r m s of the p e r f o r -
m a n c e it is the c o m p e t e n c e for, while s t r u c t u r e can be u n d e r s t o o d independent of function. So too with language and speaking, but in a d i f f e r e n t way. does not p e r f o r m a c o m p e t e n c e .
One s p e a k s a language, but one
One m u s t s u r e l y have c o m p e t e n c e with a language in o r d e r
to have the p e r f o r m a n c e of speaking it, but t h i s d o e s not w a r r a n t equating the r e l a t i o n of language to speaking with that of c o m p e t e n c e to p e r f o r m a n c e .
W h e r e we can speak of r u l e s
of language and r u l e s for speaking and p e r f o r m a n c e , we cannot speak of r u l e s for c o m p e t e n c e . A n o t h e r p r o b l e m with the r o l e of " c o m p e t e n c e " on p s y c h o l i n g u i s t i c d i s c o u r s e i s a s y s t e m a t i c ambiguity between i t s c a p a c i t y - s e n s e and its a b i l i t y - s e n s e (the d i s t i n c t i o n h e r e i s
a technical one~ since "capacity" and "ability" are used interchangeably in most discourse, which facilitates the ambiguity). "Capacity" has a passive connotation and "ability" an active one. Capacities are "built in" to the organism, but abilities are developed.
One may have
the capacity to speak Language, but he acquires the ability to speak a language, and it is in the acquisition of this ability that he is said to acquire a language.
Insofar as competence
sets limits in the capacity-sense, it is related to performance just as structure is to function. But this is so just so far as we can say that physiological structure determines the capacity for speaking, and the structures involved here are not linguistic; nor does a notion of rule in any way come into play. While there may be some sense in which the acquisition and employment of abilities for speaking is rule-governed, the sense is not clear, nor is it clear that such psychological rules are of the nature of linguistic ones. An understanding of how the
competence~performancecategories serve as
then, requires some sorting out of these issues.
a bridge between linguistics and psychology, Far from illuminating the role of linguis-
tic rules in the act of speaking, the presuppose an understanding of that role. I noted at the outset G.A. Miller's concern with the role of rules in descriptive science. The contrast of descriptive/normative motivated his concern, since it is in the context of the normative that discussion of rules most readily appears. We might equally well ask about the role of rules in an explanatory gcienee. Is "rule-governed behavior" an explanatory term in psyeholinguistics to give an account of observed and predicted acts of speaking? This seems a specification of the broader question of analogues between explanation/observation and rules/action. To pursue the matter, we need an additional thesis: T-5: Explanation influences description and is not reduceable to it. The thesis is necessary because unenlightened neopositivists and neobehaviorists have continued to maintain a descriptivist view of theories (i. e . , that theories are shorthand deserip-
201
t i o n s of o b s e r v a t i o n s ) , and this h a s r e m a i n e d m o s t p r e v a l e n t in psychology, of all the s c i ences.
The move away f r o m such a view, i r o n i c a l l y , has been led by s u c h logical p o s i t i v i s t s
a s C a r n a p (1936) a n d Hempel (1965) t h e m s e l v e s .
The l a t t e r p o r t i o n of T - 5 i s but a c o r o l l a r y
to T - 4 , s i n c e a n a l y s i s of t h e o r e t i c a l t e r m s i n e v i t a b l y r e s u l t s in d i s p o s i t i o n a l f o r m u l a t i o n s . The f o r m e r portion is but a c l a i m t h a t o b s e r v a t i o n s t a t e m e n t s t h e m s e l v e s a r e t h e o r y - l a d e n , and r e q u i r e for t h e i r u n d e r s t a n d i n g s o m e t h e o r e t i c a l o r i e n t a t i o n on the p a r t of the o b s e r v e r . Once t h i s t h e s i s is g r a n t e d , d i s c o m f o r t with h y p o t h e t i c a l c o n s t r u c t s , i n t e r v e n i n g v a r i a b l e s , m e c h a n i s m s and the like n e e d no l o n g e r d o m i n a t e c o n c e r n with the r o l e of l i n g u i s t i c r u l e s in psychology of language.
The a i m is no l o n g e r to m a k e " r u l e " o b s e r v a t i o n a l o r e v e n o p e r a -
tional, but r a t h e r to u n d e r s t a n d i t s r o l e in the u n d e r s t a n d i n g of psychology of language. L e a v i n g d e s c r i p t i v i s m behind, we a r e s t i l l not out of t h e woods.
R e a l i s m and
i n s t r u m e n t a l i s m r e m a i n v i a b l e a l t e r n a t i v e s for ways of u n d e r s t a n d i n g the r o l e of t h e o r i e s in s c i e n t i f i c explanation.
On a r e a l i s t i n t e r p r e t a t i o n , t h e o r e t i c a l e n t i t i e s m a y not be o b s e r v a -
ble, but they a r e n o n t h e l e s s r e a l (e, g . , s u b - a t o m i c p a r t i c l e s a r e not d i r e c t l y o b s e r v a b l e , but t h e y a r e t h e r e ) .
It i s t h i s s o r t of i n t e r p r e t a t i o n t h a t has l e d to c o n t e n t i o n s by p s y c h o -
l i n g u i s t s t h a t l i n g u i s t i c r u l e s a r e i n v o l v e d in v e r b a l d e v e l o p m e n t and b e h a v i o r . guistic r u l e s a r e r e q u i r e d to give an adequate a c c o u n t of v e r b a l b e h a v i o r .
Certain lin-
Since the notion
of a r u l e i s r e q u i r e d for such explanation, t h e r e m u s t r e a l l y be s u c h a r u l e o p e r a t i n g in the d e v e l o p m e n t and p e r f o r m a n c e of the s p e a k e r / h e a r e r .
On m y i n t e r p r e t a t i o n of T - 3 , " i n v o l v e -
m e a t of a r u l e " is a n o m a l o u s for the r e a l i s t , since I r e q u i r e d denying the " r e a l i t y " of a r u l e , j u s t w h e r e he r e q u i r e s a f f i r m i n g it.
Even if m y i n t e r p r e t a t i o n can be gotten a r o u n d , the
r e a l i s t s t i l l has the p r o b l e m of t e l l i n g us j u s t what s o r t of t h e o r e t i c a l r e a l i t y a r u l e i s . linguistic, psychological, or physiological? other?
Is it
Can one o r a n o t h e r of t h e s e be r e d u c e d to a n -
With t h e s e q u e s t i o n s we a r e c l e a r l y thrown back upon e a r l i e r q u a n d i r e s . My i n t e r p r e t a t i o n of T - 3 i s c l e a r l y i n s t r u m e n t a l , t r e a t i n g r u l e s a s h e u r i s t i c d e v i c e s
of i n q u i r y f o r u n d e r s t a n d i n g , p r e d i c t i n g and planning.
The r o l e of r u l e s in e x p l a n a t i o n a r e
thus s e e n a s the tools of the i n q u i r e r , not the r e f e r e n t s of the i n q u i r y .
T h i s l e a v e s the
question of the r e l a t i o n s of v a r i o u s c o n c e r n s of l i n g u i s t i c s , psychology and physiology r e g a r d i n g r u l e s still in question.
It does, h o w e v e r , show t h a t the q u e s t i o n of w h e t h e r they a r e
r e l a t e d i s a n open one, and it a l s o shows t h a t the q u e s t i o n of w h e t h e r t h e y a r e r e d u e e a b l e to 1 one a n o t h e r Ks a n u n n e c e s s a r y one. But if it so a l l e v i a t e s s o m e of the p r o b l e m s of a r e a l i s t 1
Something of t h i s l a s t point has a l r e a d y been s e t f o r t h in J o h n L a m e n d e l l a ' s c o n t r i b u t i o n to this c o n f e r e n c e , although we m a y d i f f e r in the d e t a i l s of f o r m u l a t i o n and i m p l i c a t i o n . An i n s t r u m e n t a l i s t i n t e r p r e t a t i o n of t h e o r i e s r e q u i r e s no s i n g l e u n d e r s t a n d i n g of the s a m e o b j e c t s and e v e n t s , but allows f o r a v a r i e t y of e x p l a n a t i o n s a c c o r d i n g to a v a r i e t y of c o n c e r n s . The r e a l i s t r e q u i r e s r e d u c t i o n , for e c o n o m y of i n q u i r y and s i m p l i c i t y of ontology.
202
i n t e r p r e t a t i o n , it b r i n g s with it p r o b l e m s of its own.
If " i n v o l v e m e n t of a r u l e " a p p e a r s
a n o m a l o u s f r o m a r e a l i s t p e r s p e c t i v e , " i n v o l v e m e n t of a r u l e " a p p e a r s m y s t e r i o u s f r o m an instrumentalist perspective.
If the notion of a r u l e i s r e q u i r e d for e x p l a n a t i o n of the p s y -
chology of l i n g u i s t i c d e v e l o p m e n t and p e r f o r m a n c e , to say t h a t it is only r e q u i r e d for the u n d e r s t a n d i n g of the i n q u i r e r s e e m s to deny t h a t he is r e a l l y getting at what is going on in the s p e a k e r / h e a r e r situation.
A r e r u l e s involved f o r the a c t o r s a s well a s the s p e c t a t o r ?
If so, the question of how they a r e i n v o l v e d r e m a i n s .
That they a r e involved d i f f e r e n t l y for
the a c t o r s f r o m t h e i r e x p l a n a t o r y r o l e for the s p e c t a t o r only a c c e n t s o u r e a r l i e r q u a n d r i e s about how r u l e s a r e i n v o l v e d in speaking. We u n d e r t o o k t h i s e x p l o r a t i o n of s t r u c t u r e / f u n c t i o n , c o m p e t e n c e / p e r f o r m a n c e , and e x p l a n a t i o n / o b s e r v a t i o n , w i t h the hope t h a t exposition of s i m i l a r i t i e s and r e l a t i o n s to r u l e s / a c t i o n would a l s o r e v e a l a m o r e adequate u n d e r s t a n d i n g of the notion of a r u l e in p s y c h o l i n quistic c o n c e r n s , and with that a n s w e r s to o u r e a r l i e r q u e s t i o n s .
W h a t we have found a r e
new q u e s t i o n s , m a n y of which s e e m to be r e f o r m u l a t i o n s of the q u e s t i o n s e x p o s e d in o u r e a r lier explorations.
Indeed, an a d e q u a t e e x p o s i t i o n of the notion of a r u l e s e e m s to be r e q u i r e d
for an adequate u n d e r s t a n d i n g of t h e s e s i m i l a r i t i e s and r e l a t i o n s . V In this p a p e r , I have p r o d u c e d no m o r e t h a n I p r o m i s e d .
In e x p l o r i n g the notion of a
r u l e , I have offered a few q u e s t i o n s t h a t give s o m e o r i e n t a t i o n to c o n c e p t u a l p r o b l e m s , and a few t h e s e s that give s o m e d i r e c t i o n for f u r t h e r e x p l o r a t i o n .
Negatively, we have found t h a t
it m a k e s no s e n s e to talk of r u l e s of L..anguage; that t h e r e is a shift in r o l e s f r o m r u l e s of language to r u l e s fo__rs p e a k i n g and t h a t t h e i r r e l a t i o n s h i p r e q u i r e s e x a m i n a t i o n ; t h a t the notion of a r u l e is without an e s t a b l i s h e d conceptual home within l i n g u i s t i c and p s y c h o l i n guistic discourse.
P o s i t i v e l y , I have p r e s e n t e d s o m e t h e s e s a n d s o m e e x p l o r a t i o n s of " r u l e -
g o v e r n e d b e h a v i o r " t h a t s u g g e s t t h a t an a d e q u a t e notion of a r u l e in t h e s e c o n t e x t s m u s t wait upon a m o r e adequate u n d e r s t a n d i n g of i n t e n t i o n a l action.
The notion of a r u l e p r e s e n t s s o m e
e x c i t i n g p r o s p e c t s in t h e s e c o n t e x t s , a m o n g t h e m a b r i d g e between the f o r m a l and e m p i r i c a l s c i e n c e s (but unlike e n g i n e e r i n g ) and between the n a t u r a l and social s c i e n c e s (but unlike physiological psychology).
F o r such b r i d g e s to stand, h o w e v e r , we m u s t u n d e r s t a n d the
notion of a r u l e on which they a r e b a s e d . References C a r n a p , Rudolf,(1950), " E m p i r i c i s m , S e m a n t i c s , and Ontology", Revue I n t e r n a t i o n a l e de P h i l o s 0 p h i e , 4 , 20-41. R e p r i n t e d with r e v i s i o n s in h i s Meaning and N e c e s s i t y ; A study in s e m a n t i c s and modal logic, 2nd E d . , Chicago: U n i v e r s i t y P r e s s (1956) and in Olshewsky, T ( e d . ) , P r o b l e m s in the P h i l o s o p h y of Language, New York: Holt, R i n e h a r t and Winston (1969).
203
Carnap, Rudolf (1936), "Testability and Meaning", Philosophy of Science, 3 , 419-471. Carnap, Rudolf (1937), "Testability and Meaning", Philosophy of Science, 4, 2-40. Chomsky, Noam (1961), "On the Notion "Rule of G r a m m a r ' ", Proceedings of the Twelfth Symposium on Applied Mathematics. Reprinted in Fodor, J . A . and J . J . Katz (eds), The Structure of Language; Readings in the philosophy of language, Englewood Cliffs: Prentice-Hall (1964). Chomsky, Noam (1968), Language and Mind, New York: Harcourt, Brace and World. Fodor,
J. (1966), "How
to Learn to Talk: Some
Simple Ways",
in Smith, F.
and G.A.
Miller (eds.), The Genesis of Language, a P sych01inguistic Approach, Proceedings of a Confrence on Language Development in Children, Cambridge: MIT P r e s s (1966). Gumb, Raymond (1972), Rule-governed Linguistic Behavior, The Hague: Mouton. Hempel, Carl (1965), "Empiricist C r i t e r i a of Cognitive Significance", Aspects of Scientific Explanation and Other Essays in the Philosophy of Science, New York: F r e e P r e s s . Reprinted in Olshewsky, T. (ed.), Problems in the Philosophy of Language, New York: Holt, Rinehart and Winston (1969). Henley, Kenneth (1969), Rules and Language, Unpublished M a s t e r ' s Thesis, University of Virginia. McNeill, David (1970), The Acquisition of Language: the Study of Developmental Psycholinguistics, New York: Harper and Row. Miller, G.A. and S. Isard (1963), "Some Perceptual Consequences of Linguistic Rules", J~ Verbal Learning and Verbal Behavior, 2 . Reprinted in Jakobovits, L.A. and M.S. Miron (eds.), Readings in the Psychology of Language~ Englewood Cliffs: Prentice-Hall (1967). Quine, Willard (1970), "Methodological Reflections on Current Linguistic Theory", Synthese, 2...1.1. Saporta, Sol (1967), Psycholinguistic Theories and Generative G r a m m e r , Eugene, Oregon: State System of Higher Education. Searle, John R. (t969), Speech Acts: An E s s a y in the Philosophy of Language, London: Cambridge University P r e s s . Wittgenstein, Ludwig (1958), P r e l i m i n a r y studies for the "Philosophical Investigations", generally known as the Blue and Brown Books, New York: Harper.
~f
EMPIRICAL
RESTRICTIONS
ON
THE
POWER
OF
TRANSFORMATIONAL
GRAMMARS
R o y a l Skousen U n i v e r s i t y of I l l i n o i s Abstract In t h e i r a r t i c l e "On the G e n e r a t i v e P o w e r of T r a n s f o r m a t i o n a l G r a m m a r s " ,
Peters
and R i t e h t e (1971) a r g u e t h a t given any r e c u r s i v e l y e n u m e r a b l e s e t t h e r e e x i s t s a t r a n s f o r m a t i o n a l g r a m m a r t h a t will p r o d u c e t h a t set.
T h e y c l a i m t h a t the d e l e t i o n r u l e u s e d in
t h e i r p r o o f (b--->~) i s a r e c o v e r a b l e d e l e t i o n r u l e .
This deletion rule obligatorially deletes
all o c c u r r e n c e s of a t e r m i n a l s y m b o l b f r o m e v e r y s e n t e n c e of a c o n t e x t - s e n s i t i v e language r e l a t e d to the o r i g i n a l r e c u r s i v e l y e n u m e r a b l e s e t . t h e r e e x i s t s no s u r f a c e s e n t e n c e with b.
A s a r e s u l t of such a t r a n s f o r m a t i o n ,
The t e r m i n a l s y m b o l b is only r e c o v e r a b l e b e -
c a u s e in t h e i r f o r m u l a t i o n the g r a m m a r t h a t p r o d u c e s the b ' s i s a l r e a d y d e t e r m i n e d .
There
i s , h o w e v e r , no e v i d e n c e in the s u r f a c e s t r i n g s of the language f o r any b. Given t h e a s s u m p t i o n t h a t s p e a k e r s do not have r u l e s f o r w h i c h t h e r e i s no e m p i r i c a l e v i d e n c e , the type of d e l e t i o n r u l e found in P e t e r s and R i t c h i e ' s p r o o f would n e v e r be found in any h u m a n language. able.
In t e r m s of language a c q u i s i t i o n , the deletion r u l e i s not r e c o v e r -
Given s u c h an e m p i r i c a l condition on g r a m m a r s ,
the t y p e s of r u l e s found in the g r a m -
m a r s of n a t u r a l l a n g u a g e s would be r e s t r i c t e d on the b a s i s of e m p i r i c a l e v i d e n c e .
Peters
and R i t c h t e ' s p r o o f f a i l s to m e e t t h i s e m p i r i c a l condition and i s t h e r e f o r e invalid. In A s p e c t s of the T h e o r y of Syntax t C h o m s k y (1965) s u g g e s t s t h a t h u m a n l a n g u a g e s are recursive sets.
A r e c u r s i v e s e t is a s e t of s t r i n g s s u c h t h a t t h e r e e x i s t s an a l g o r i t h m
t h a t c a n e f f e c t i v e l y d e t e r m i n e if a given s t r i n g i s in the r e c u r s i v e s e t o r not.
Chomsky
a s s u m e s t h a t s p e a k e r s p o s s e s s j u s t such an a l g o r i t h m with r e s p e c t to t h e i r n a t i v e language. S p e a k e r s can c l a s s i f y l i n g u i s t i c s i g n a l s into two w e l l - d e f i n e d s e t s : s e n t e n c e s and n o n s e n t e n ces.
T h i s a s s u m p t i o n h a s b e e n b a s i c to C h o m s k y ' s l i n g u i s t i c a n a l y s e s . A w e a k e r a s s u m p t i o n would be t h a t h u m a n l a n g u a g e s a r e r e c u r s i v e l y e u u m e r a b l e s e t s .
A r e c u r s i v e l y e n u m e r a b l e s e t i s a s e t of s t r i n g s for w h i c h t h e r e e x i s t s a p r o c e d u r e (or T u r i n g m a c h i n e ) t h a t can e f f e c t i v e l y e n u m e r a t e the s t r i n g s in the s e t , but m a y be u n d e e i d a b l e with r e s p e c t to s t r i n g s not in the set.
Of c o u r s e , for s o m e r e c u r s i v e l y e n u m e r a b l e s e t s
t h e r e will e x i s t p r o c e d u r e s t h a t will effectively e n u m e r a t e the s t r i n g s not in the s e t a s well a s t h o s e in the set.
Such p r o c e d u r e s a r e , by definition, a l g o r i t h m s .
s i v e l y e n u m e r a b l e s e t s for which no a l g o r i t h m e x i s t s .
But t h e r e a r e r e c u r -
F o r such a r e c u r s i v e l y e n u m e r a b l e
s e t , the p r o c e d u r e m a y be able to d e t e r m i n e t h a t s o m e s t r i n g s a r e not in the set, but t h e r e will e x i s t a t l e a s t one s t r i n g for which the p r o c e d u r e will fail.
Such s t r i n g s a r e i n d e t e r m i n -
able with r e s p e c t to the p r o c e d u r e . ~Appeared
in: York Papers
in Linguistics
2 (1972),
61 - 7 0 a n d P a p e r s
in Linguistics
205
Chomsky admits that the t r a n s f o r m a t i o n a l theory that he proposes in A s p e c t s may be too powerful in that it might be able to d e s c r i b e any r e e u r s i v e l y is looking for a theory that will d e s c r i b e human languages.
n u m e r a b l e set.
Chomsky
And if human languages a r e r e -
c u r s i v e s e t s , then a c o r r e c t theory of t r a n s f o r m a t i o n a l g r a m m a r should only d e s c r i b e the set of r e c u r s i v e languages, not the l a r g e r set of r e c u r s i v e l y e n u m e r a b l e languages (Chomsky, 1965, p. 62).
In such a c a s e the p r o p e r solution will be to devise f o r m a l ways to r e s t r i c t the
theory of t r a n s f o r m a t i o n a l g r a m m a r s so that they d e s c r i b e only the set of r e c u r s i v e languages (p. 2081. P e t e r s and Ritchie (1971) argue that C h o m s k y ' s observation about the power of his t h e o r y of t r a n s f o r m a t i o n a l g r a m m a r is, in fact, c o r r e c t .
Having f o r m a l i z e d the A s p e c t s '
theory of t r a n s f o r m a t i o n a l g r a m m a r , they prove that e v e r y r e c u r s i v e l y e n u m e r a b l e language is generated by some c o n t e x t - s e n s i t i v e t r a n s f o r m a t i o n a l g r a m m a r .
T h e i r proof is based up-
on the following relation of r e c u r s i v e l y e n u m e r a b l e languages to c o n t e x t - s e n s i t i v e languages: If L 1 is any r e c u r s i v e l y e n u m e r a b l e language (that is, any language generated by an u n r e s t r i c t e d r e w r i t i n g s y s t e m (a type 0 g r a m m a r ) ) , then there e x i s t s a conte;~t-sensitive language L 2 and a r e g u l a r set R such that L 1 = L 2 / R .
The only difference between a string in L 1 and the corresponding string in L 2 is that the L 2 string may include a string in R. The following method I d e s c r i b e s how to c o n s t r u c t a g r a m m a r G 2 that wilt g e n e r a t e L2, given L 1 and G 1 (the g r a m m a r that g e n e r a t e s L 1): Let G 1 = (VN,VT, P l ' S1) be the u n r e s t r i c t e d r e w r i t i n g s y s t e m that g e n e r a t e s L1.
We will c o n s t r u c t a corresponding g r a m m a r G 2 =
(VN U { S 2 , B ] , V T U ~ b , ~ } , P2,$2), w h e r e S2 and B a r e not in VN, and b and ~ a r e not ia V T.
The production r u l e s in P2 a r e d e t e r m i n e d as follows: (1) If a---->/3 is a production rule
of G 1 and the length of a is not g r e a t e r than the length of /3, then a--->/3 is a production r u l e of G 2.
(2) If a--->/3 is a production r u l e of G 1 and the length of /3 is l e s s than the length of
a, where the difference in length is n, then a-->/3B n is a production rule of G 2.
(3) F o r
each symbol X in VN and x in VT, BX--->XB and Bx--->xB a r e production r u l e s of G 2. (4) B~--->b~ and Bb--->bb a r e in P2"
(5) $2--->$1 ~ is in P2"
The n o n - t e r m i n a l B's can be converted to the t e r m i n a l b ' s only after the B ' s have been moved to the end of the s t r i n g since the ~ symbol which t r i g g e r s the change of B to b o c c u r s only at the end of the string.
Thus we add production r u l e s to P2 to allow B to move
o v e r any other symbol in the vocabulary of G1. r e w r i t i n g rule. the same in G 2. 1
Each production rule of G 2 is a type 0.5
If a-->/3 is a l r e a d y a type 0.5 r e w r i t i n g r u l e in G1, then the rule is exactly If a --->/3 is not a type 0.5 r e w r i t i n g rule in G 1 (that is, the length of /3 is
Hopcroft and Ullman (1969, pp. 132-133). This method v a r i e s somewhat in notation f r o m P e t e r s and R i t e h i e ' s . Nonetheless, the f o r m of the proof is the s a m e .
206
l e s s than the length of ~), then a m i n i m a l n u m b e r of B ' s a r e added to fi to m a k e it a type 0.5 rewriting rule.
By a t h e o r e m of K u r o d a (1964, p. 211), e v e r y type 0 . 5 r e w r i t i n g r u l e is
c o n v e r t i b l e into a context s e n s i t i v e (type 1) r e w r i t i n g r u l e . type 0 . 5 r e w r i t i n g r u l e .
E v e r y r u l e of G 2 is a t l e a s t a
Hence, G 2 i s a c o n t e x t - s e n s i t i v e r e w r i t i n g s y s t e m .
In the d e r i v a -
tion of a s t r i n g in L2, B (and c o r r e s p o n d i n g l y b) can be i n t r o d u c e d only when a r u l e t h a t is not a type 0 . 5 r e w r i t i n g r u l e i s u s e d in d e r i v i n g the c o r r e s p o n d i n g s t r i n g in L 1.
The only
d i f f e r e n c e then between a s t r i n g of L 2 and the c o r r e s p o n d i n g s t r i n g in L 1 will be t h a t the L 2 s t r i n g will end in the r e g u l a r e x p r e s s i o n b* .
By d e l e t i n g the e x p r e s s i o n b*4~ f r o m e v e r y
s t r i n g in L 2, we will have a language equal to o u r o r i g i n a l language L 1.
Thus L 1 = L2/b*#.
P e t e r s and R i t e h i e do not, h o w e v e r , p o s t u l a t e a v a r i a b l e d e l e t i o n r u l e t h a t will delete the r e g u l a r s e t
b~I-~ in one step.
R a t h e r , they p r o p o s e a c y c l i c a l a p p l i c a t i o n of a c o n s t a n t
deletion r u l e t h a t will delete a single b when i t is the r i g h t - m o s t t e r m i n a l s y m b o l in a s e n tence.
T h i s deletion t r a n s f o r m a t i o n will apply a n u n l i m i t e d n u m b e r of t i m e s until all the
final b ' s in a s t r i n g a r e d e l e t e d 1. definition of r e c o v e r a b i l i t y .
P e t e r s a n d R i t e h i e m a k e t h i s d e c i s i o n b e c a u s e of t h e i r
A d e l e t i o n is r e c o v e r a b l e (1) if a copy o r r e p l i c a of the s t r i n g
that is d e l e t e d is left s o m e w h e r e in the t r e e , or (2) if the d e l e t e d s t r i n g is a specific, finite s t r i n g of w o r d s ( P e t e r s and R i t c h i e , 1971, p. 8).
Since all b ' s a r e deteted~ no copy o r
r e p l i c a of b is e v e r left in any s e n t e n c e of the l a n g u a g e . finite s t r i n g m a d e up of t e r m i n a l s y m b o l s in G 2.
But the single b is a specific,
Thus b c a n be d e l e t e d by a r e c o v e r a b l e
deletion r u l e : " . . . although [the t r a n s f o r m a t i o n ] does not n e c e s s a r i l y p r e s e r v e a copy of the s y m b o l b, it s a t i s f i e s the condition of r e c o v e r a b i l i t y of d e l e t i o n s by specifying t h a t the d e l e ted s t r i n g m u s t in e v e r y c a s e be a single b . " ( P e t e r s and R i t e h i e , 1971, p. 33) If the d e l e tion r u l e w e r e allowed to delete any s t r i n g of b ' s r a t h e r t h a n a single b, the t r a n s f o r m a t i o n would not s a t i s f y the condition on r e c o v e r a b i l i t y of d e l e t i o n s ; although the deleted s t r i n g would a h v a y s be s o m e s e q u e n c e of b ' s , i t would be i m p o s s i b l e to tell j u s t how m a n y b ' s had b e e n d e l e t e d by a given a p p l i c a t i o n of the t r a n s f o r m a t i o n .
In o r d e r to p r e s e r v e t h e i r notion
of r e c o v e r a b i l i t y of d e l e t i o n s , only one b at a t i m e c a n be deleted. of the t r a n s f o r m a t i o n , a single b is always r e c o v e r a b l e .
F o r a given a p p l i c a t i o n
The u n l i m i t e d u s e of a c y c l i c a l
d e l e t i o n r u l e g u a r a n t e e s t h a t any n u m b e r of b ' s c a n be deleted. T h e r e a r e o t h e r o b s e r v a t i o n s t h a t we can m a k e about P e t e r s and R i t e h i e ' s proof. e x a m p l e , B n e v e r i n t e r f e r e s in the g r a m m a r G 2.
For
In o r d e r to u n d e r s t a n d w h a t it would m e a n
for B to i n t e r f e r e in the g r a m m a r G2, c o n s i d e r a d e r i v a t i o n of a s t r i n g q in G I :
1
A c c o r d i n g to the n o t a t i o n u s e d in the proof in t h i s p a p e r , I n e e d a final a d - h o c t r a n s f o r m a -
tion t h a t will d e l e t e the ~ s y m b o l a f t e r all the b ' s have b e e n deleted.
207
Step in the derivati0u
Rule used
S1 x1
Ra
xi_ I
R.J
x.1
Rk
xi+ i
R~
x
R n
q
R
y
z
where R represents the rewriting rule used to derive each step of the derivation from the preceding step.
(The use of subscripts is only for notational purposes and does not imply
that Rj, R k, and R~, for example~ cannot in fact by an application of the same rewriting rule.)
The corresponding derivation in G2 will be of the following form:
S2
Xlb*
xi_ib* ~:}
x.b* l
xi+ib*~:
208 x b*~ n
qb*# The first string produced
must be SI~: because
is the only rule in G 2 that rewrites first rule that applies produces
S 2.
S 2 is the start symbol
Of course,
SI~
string xi_ 1 as ~!b, x.1 would be ~@.
where
is of the form
a string ending in the regular expression
in G 1 the rule R k used in the derivation of q rewrites
responding
in G 2 and $2--->SI~
~ as ~.
Let us assume
Slb-*~. b*#.
Therefore,
Hence,
representing
that in G2,
Then the corresponding
string for ~
rule R k' (the rewriting rule in
G I) is of the form
o~-->/3B n, where
Derivation
xi_ I
~
xi
~
n >_ 0.
the
that the step in the G 2 derivation cor-
to the derivation of x.1 in the G I derivation ends in the regular expression
m >_ 0.
the
Suppose that
would be ~b
G 2 corresponding The corresponding
in G 1
m~.
Now
bin#,
let us suppose
to the rewriting rule R k in string for x i (or ~)
would
Derivation in G 2
~bm~ (Rk:(~-->~
~I3Bn~Bm# (Rk':~--->,~Bn)
Under what conditions could the next step in the G 1 derivation be blocked in the G 2 derivation?
This could only occur if the rule Ri
tive of B remains
between
rewrites
j3~ as %
but that a B or some
~ and @, thus blocking the application of rule Rg '. In such a case,
we will say that B interferes in the derivation of the string in G 2. in G 2 for the following reasons.
First of all, all B's can be moved
application of the rules in G 2 of the form
BX--~XB
or Bx-->xB,
The only other possibility then is that B be rewritten as something over each of the symbols rules B#
-->b#
in @.
and Bb -->bb.
since b cannot be moved
The only rules that rewrite
But B will never interfere over the string @ by where
XEV
over @.
which could not be moved
B into another symbol
are the
The first rule (B4~ --->b~) applies only to a final B. Similarly,
B could not be rewritten as b unless
B has already been moved
we have in G 2 the following derivation:
~/~n@bm_~
In
b's can only be followed by b's or over ~.
neither B nor b can block the application of the next rule, R~ '. Corresponding x i -->xi+ 1 in GI,
N and xeV T.
So if B could be changed to b, then rule R~' could not apply
this case then B could not be followed by @. 4~. Hence,
deriva-
Thus,
to the steps
209 ~Bn-l~Bbm#
~ B n-l ~ b m + 1
~l~#bn+m~ T h i s l a s t s t r i n g i s of the f o r m X i + l b * # . sponding d e r i v a t i o n in G2°
So g i v e n a d e r i v a t i o n in G1, t h e r e e x i s t s a c o r r e -
The i n t r o d u c t i o n of B into G 2 n e v e r c a u s e s the d e r i v a t i o n of
any s t r i n g q in G 1 to be blocked in G 2.
B n e v e r p r e v e n t s a r u l e f r o m G 1 f r o m applying in
G 2 s i n c e any B c a n be m o v e d to the e n d of the s t r i n g ; a n d B n e v e r e a u s e s a r u l e f r o m G 1 to apply in G 2 s i n c e B e a n only be m o v e d to the e n d of the s t r i n g .
Thus B n e v e r i n t e r f e r e s
in the d e r i v a t i o n of s t r i n g s in G 2. The e l e m e n t B c a n n o t have b e e n in G1; o t h e r w i s e t h e r e would be a r u l e in G 1 u s i n g B and t h a t r u l e could be u s e d in G 2 to i n t e r f e r e in d e r i v a t i o n s .
If B w e r e r e w r i t t e n in G1,
then B ' s i n t r o d u c e d by r u l e s in G 2 could be r e w r i t t e n a s in" G 1 i n s t e a d of b e i n g m o v e d t o w a r d the end of the s t r i n g .
If B i s p r o d u c e d by r u l e s in G1, t h e n s u c h B ' s could be
changed to b a n d e v e n t u a l l y d e l e t e d in G 2.
S i m i l a r l y , b c a n n o t have b e e n in G 1.
Thus, B
and b do not i n t e r f e r e in G 2 in p a r t b e c a u s e they a r e n o t found in G 1. P e t e r s and R i t c h i e ' s deletion t r a n s f o r m a t i o n d e l e t e s only a s t r i n g - f i n a l b.
If b could
b e deleted any place in a s u r f a c e s t r i n g , then t h e r e w o u l d n ' t be any way of d e t e r m i n i n g in what s e q u e n c e the b ' s h a d b e e n d e l e t e d .
If a s t r i n g c o n t a i n e d n n u m b e r of b ' s , the deletion
t r a n s f o r m a t i o n could d e l e t e the b ' s in n! d i f f e r e n t ways.
F o r e x a m p l e , if a s t r i n g c o n t a i n s
t h r e e b ' s , the d e l e t i o n t r a n s f o r m a t i o n could d e l e t e the b ' s in six d i f f e r e n t o r d e r s : b1
b2
b3
b1
b3
b2
b2
b1
b3
b2
b3
b1
b3
bI
b2
b3
b2
b1
(b. s t a n d s for the i t h b in a b * ~ ; for e x a m p l e , in a s t r i n g containing t h r e e b ' s : ~ b l b 2 b 3 ~ . ) 1
By specifying t h a t b is deleted only in s e n t e n c e final position, the b ' s m u s t be d e l e t e d in the o r d e r b 3 , b 2 , b 1.
By d e l e t i n g only the final b, the o r d e r of the d e l e t i o n of the b ' s i s c o m -
ptetely recoverable.
210
Only enough B ' s a r e added to e a c h r e w r i t i n g r u l e of G 1 so t h a t no s t r i n g in any d e r i v a t i o n in G 2 will d e c r e a s e in length.
W h a t would happen if we d i d n ' t r e s t r i c t the n u m b e r
of B ' s t h a t could be added to e a c h r e w r i t i n g r u l e of G I ? e v e r y r u l e in G 2 had to b e a type 0 . 5 r u l e .
Suppose t h a t we only s p e c i f i e d t h a t
In t h i s c a s e , we w o u l d n ' t be able to tell how
m a n y b ' s had b e e n d e l e t e d f r o m the s t r i n g p r o d u c e d in G 2 e v e n if we knew the o r i g i n a l g r a m m a r G 1.
F o r e x a m p l e , a s t r i n g ~ad~ r e w r i t t e n a s ~c@ in G 1 by the r u l e a b - - > e would be
r e w r i t t e n as ~adB@ in G 2 if the c o r r e s p o n d i n g r u l e in G 2 w e r e m i n i m a l l y of type 0 . 5 , i s , ad --> eB.
that
But if no s u c h c o n s t r a i n t w e r e p l a c e d on B - - t h a t i s , the c o r r e s p o n d i n g r u l e
n e e d only be of type 0, 5 and not m i n i m a l l y s o - - t h e n the r u l e in G 2 could be ad ---> e B o r a d --> cBB o r ad --->eBB, and so on.
Only by s e t t i n g s u c h a r e s t r i c t i o n on the r e w r i t i n g r u l e s
of G 2 can the n u m b e r of b ' s deleted by the t r a n s f o r m a t i o n be r e c o v e r e d , given only the o r i ginal g r a m m a r G 1 and not the c o n s t r u c t e d g r a m m a r G 2. Of c o u r s e , the c h o i c e of the s y m b o l s B and b i s totally a r b i t r a r y . c h o s e n any two s y m b o l s not in the o r i g i n a l v o c a b u l a r y of G 1. c a u s e they do not i n t e r f e r e in the g r a m m a r of G 2.
One could have
B and b a r e not unique b e -
They a r e , in e s s e n c e , d u m m y s y m b o l s
and could be r e p l a c e d b y any o t h e r s y m b o l s not in G1, s u c h a s l e t t e r s f r o m the G r e e k o r R u s s i a n a l p h a b e t s , C h i n e s e c h a r a c t e r s , h i e r o g l y p h i c s , o r any o t h e r s y m b o l s i m a g i n a b l e . F i n a l l y , the deletion r u l e i s always o b l i g a t o r y , n e v e r optional, so that e v e r y b is deleted; in no s u r f a c e s e n t e n c e does b e v e r show up. f r o m any of the s u r f a c e s t r i u g s .
T h e r e is a b s o l u t e l y no e v i d e n c e for b
The only r e a s o n we know how m a n y b ' s w e r e o r i g i n a l l y p r o -
duced in a given s u r f a c e s t r i n g is b e c a u s e we know the g r a m m a r G 1 a n d can d e r i v e a d e r i v a tion for t h a t s u r f a c e s t r i n g in G 1 and c a n d e t e r m i n e the c o r r e s p o n d i n g d e r i v a t i o n in G 2.
In
o t h e r w o r d s , the notion of r e c o v e r a b i l i t y depends c r u c i a l l y upon the fact t h a t the o r i g i n a l g r a m m a r G 1 i s given.
If G t w e r e not known, the b ' s in G 2 would not be r e c o v e r a b l e b e -
c a u s e the n u m b e r of B ' s (and h e n c e b's) p r o d u c e d in G 2 depends upon what r u l e s a r e in G 1. If t h e s e r u l e s a r e not specified, we d o n ' t know how m a n y B ' s w e r e o r i g i n a l l y p r o d u c e d for a given s u r f a c e s t r i n g .
C o n s i d e r , f o r e x a m p l e , a language w h i c h c o n t a i n s s t r i n g s having a
s q u a r e n u m b e r of a ' s : L =
{a, anna, aaaaaaaaa
.....
a
n
2 ....
}
C o n s i d e r two g r a m m a r s of the f o r m G = V n = ( { S 1 , A , C , D , E , F , G , H } , t h a t could g e n e r a t e L.
VT = { a } , P , St)
One would g e n e r a t e s t r i n g s of the f o r m AFnGHn2: S 1 ---> AFGH FG ---> DGHH FD --> D F AD ~->AC
211
CD ---> F C CG --~ F FGH To g e n e r a t e the s t r i n g f o r the s u c c e e d i n g i n t e g e r , two H ' s would be added f o r e v e r y F (by the r u l e FG - - DGHH) and finally an e x t r a H and F would be added (by the r u l e CG ---> F FGH) : 2 2 AFnGH n ._>AFnFGHH2nH n Of c o u r s e , the l a s t s t r i n g is e q u i v a l e n t to A F n+ 1GH (n+l} 2.
In o r d e r to get a n 2 , we would
n e e d r e w r i t i n g r u l e s to delete AFnG and c o n v e r t Hn2 to an2: AF -->E EF --~E EG --->E EH --->H H--->a 2 In o t h e r w o r d s , t h e s e r u l e s would always delete n + 2 s e g m e n t s f r o m the s t r i n g AFnGH n in 2 o r d e r to get a s t r i n g w i t h n s e g m e n t s . The c o n t e x t - s e n s i t i v e g r a m m a r c o r r e s p o n d i n g to t h i s g r a m m a r would include the following r e w r i t i n g r u l e s : A F --->EB E F --->EB EG - - > E B EH --->HB H--->a In t h i s g r a m m a r ,
n + 2 s e g m e n t s would be c o n v e r t e d to B ' s (and e v e n t u a l l y b ' s ) , while n
2
s e g m e n t s would end up a s a ' s . A n o t h e r p o s s i b l e g r a m m a r f o r p r o d u c i n g L would g e n e r a t e s t r i n g s of the f o r m A F 2n+ 1GHn2: S 1 --->AFFFGH FG ---> DGH F D ---> DF AD --->AC CD ---> FC CG ---> F F F G To g e n e r a t e the s t r i n g for the s u c c e e d i n g i n t e g e r , a n H would be added for e v e r y F (by the r u l e F G - - > D G H ) and finally two m o r e F ' s would be added (by the r u l e C G - - - > F F F G ) :
212
AF 2n+l GH n 2 ___>AF2n+I FFGH2n+ This last string is equivalent to the same
AF2(n+I)+I
IH n2
GH(n+I)2.
2
In order to get a n , we would need H n2 n2 to delete AF2n+IG and convert to a :
rules as in our first grammar
AF ---> E EF--->E EG --->E EH -->H H--->a In this ease, however,
2n•3 segments
must be deleted rather than n+2.
the context-sensitive
grammar
if the first grammar
were postulated as generating
This means
that in
corresponding to this second grammar, 2n+ 3 segments would 2 be converted to b's, while as before, n segments would end up as a's. Given a string of n2 the form a , the b-deletion rule would delete n+2 b's from the underlying representation
ated L, then 2n÷3 how" many grammar
L.
But if the second grammars 1 b's would be deleted from the underlying representation. We
cannot tell
b's have been deleted from any given surface string unless we already know the G 1.
Peters and Ritehie's proof depends upon the fact that a grammar Obviously,
speakers don't have the grammer
transformation
is already given.
when they start to acquire the language.
postulated by Peters and Ritehte is not recoverable
quisition.
Of course,
formation:
"... the deleted sequence of factors had a member
VT*
gener-
The
in term s of language ac-
Peters and Ritchie's definition of recoverability allows such a transof finite, preassigned
subset of
as its terminal string." (Peters and Ritchie, 1971, p. 8). As evidence for this part of
the defition, Peters and Ritchie could refer to a deletion rule like imperative deletion in English, which deletes the underlying subject "you" and the auxiliary "will" (Lees and Klima, 1963).
The rule always deletes the sequence of words
"you will" and is therefore recoverable,
according to Peters and Ritchie's definition of recoverability. given for higher performatives lish (Sadoek,
1969).
are always deleted.
is evidence for postulating such deletion rules.
ad-hoc way.
Similarly,
Yet in both these cases, there
Various co-occurrence
have an underlaying
eral conditions on co-occurrence some
have also been
in English that never appear in any surface sentence of Eng-
Such performatives
given to show that imperatives
Arguments
"you will".
Otherwise,
will be unexplainable for imperative
the arguments
arguments
for higher performatives
1 Both grammars will, of course, need the ad-hoc transformation all the h's have been deleted.
have been
certain rather gen-
sentences except in are based on various
deleting the final ~
after
213
co-occurrenc~ relations.
The point is t h a t for e a c h of t h e s e putative r u l e s , t h e r e is e v i d e n c e
for such a r u l e in the s u r f a c e s t r i n g s of E n g l i s h . evidence.
A r g u m e n t s c a n be given only on the b a s i s of
T h e r e m u s t be s o m e s e n t e n c e s of the language t h a t c o n t a i n c e r t a i n s u r f a c e f o r m s
t h a t will l e a d s p e a k e r s to p o s t u l a t e s u c h r u l e s . E v e n though c e r t a i n finite s t r i n g s a r e : r e c o v e r a b l e , t h i s d o e s not m e a n t h a t any finite string is recoverable.
F o r e x a m p l e , c o n s i d e r a putative t r a n s f o r m a t i o n t h a t would a l w a y s
d e l e t e the i m p o s s i b l e finite s t r i n g hps f r o m e v e r y s e n t e n c e of E n g l i s h (or any o t h e r language, for t h a t m a t t e r ) .
And suppose in addition t h a t one of the r e w r i t i n g r u l e s of E n g l i s h i s : S ---> hps S
Now the p r o b l e m with t h i s deletion r u l e i s t h a t it is not r e c o v e r a b l e .
No s t r i n g of the f o r m
hps e v e r a p p e a r s in the s u r f a c e of a s e n t e n c e of E n g l i s h , and hps n e v e r i n t e r a c t s with the o t h e r r u l e s of E n g l i s h . rule.
hps is n e v e r r e w r i t t e n by a n o t h e r r u l e n o r i s i t p r o d u c e d by a n o t h e r
T h e r e is a b s o l u t e l y no e v i d e n c e for p o s t u l a t i n g an u n d e r l y i n g hps n o r is t h e r e any
e v i d e n c e for d e l e t i n g all o c c u r r e n c e s of the finite s t r i n g hps in E n g l i s h .
Yet a c c o r d i n g to
P e t e r s and R i t c h i e ' s definition of r e c o v e r a b l e d e l e t i o n , s u c h a r u l e would be r e c o v e r a b l e and hence allowable f o r a g r a m m a r of E n g l i s h .
Such a n e x a m p l e could be c o n s t r u c t e d f o r a n y
i m p o s s i b l e s e q u e n c e of sounds in any l a n g u a g e .
T h e r e i s s o m e t h i n g w r o n g with s u c h a notion
of r e c o v e r a b i l i t y when i t allows t o t a l l y u n e m p i r i c a l r u l e s in n a t u r a l l a n g u a g e s . In o t h e r w o r d s , P e t e r s and R i t e h i e ' s definition of r e c o v e r a b i l i t y i s not s t r o n g enough. It allows for totally u n e m p i r i c a l t r a n s f o r m a t i o n s , r u l e s t h a t would n e v e r be p o s t u l a t e d by s p e a k e r s of n a t u r a l l a n g u a g e s .
The f a c t that s o m e t r a n s f o r m a t i o n in n a t u r a l l a n g u a g e s delete
specific s t r i n g s of f o r m a t i v e s does not m e a n t h a t the deletion of specific s t r i n g s i s always recoverable.
L i n g u i s t s postulate c e r t a i n deletion r u l e s for E n g l i s h b e c a u s e t h e r e i s e v i d e n c e
for t h o s e r u l e s in the s u r f a c e s e n t e n c e s of E n g l i s h .
And s p e a k e r s a c q u i r e such r u l e s b e c a u s e
t h e r e is e v i d e n c e for the r u l e s in the s e n t e n c e s of E n g l i s h .
R e c o v e r a b i l i t y c a n be defined
then in t e r m s of e m p i r i c a l evidence. L e t us postulate then an e m p i r i c a l condition on g r a m m a r s - - t h a t no r u l e c a n be p o s t u l a t e d u n l e s s t h e r e i s s o m e e v i d e n c e for the r u l e . unrealistic.
Such a convention would not be too
In fact, t h i s convention i s the b a s i s of the s c i e n t i f i c method.
c o n s t r u c t e d in a v a c u u m , but a r e grounded in e m p i r i c i s m .
T h e o r i e s a r e not
G r a m m a r s a r e t h e o r i e s of what
s p e a k e r s know about t h e i r language and a s such m u s t be b a s e d upon e m p i r i c a l e v i d e n c e , Such a condition on g r a m m a r s h a s been a s s u m e d i n u e a r l y all l i n g u i s t i c w o r k .
For example,
the e m p i r i c a l condition on g r a m m a r s i s the b a s i s of K i p a r s k y ' s (1966) a l t e r n a t i o n condition; s p e a k e r s will not postulate u n d e r l y i n g s e g m e n t s for which t h e r e i s no s u r f a c e e v i d e n c e .
All
the a r g u m e n t s a g a i n s t the s t r o n g f o r m of K i p a r s k y ' s a l t e r n a t i o n condition s i m p l y a r g u e t h a t
214
in s o m e l a n g u a g e s t h e r e is e v i d e n c e for u n d e r l y i n g s e g m e n t s t h a t n e v e r show u p o n the s u r face.
The point is, in any c a s e , t h a t e v i d e n c e is found f o r such a b s t r a c t s e g m e n t s .
In g e n -
e r a l , l i n g u i s t s should not a r g u e for r u l e s for which t h e r e is no e v i d e n c e at all. P e t e r s and R i t c h i e ' s p r o o f v i o l a t e s the e m p i r i c a l condition on g r a m m a r s . no s u r f a c e s e n t e n c e in w h i c h b a p p e a r s . s e r v e b.
There is
A s p e a k e r l e a r n i n g the language would n e v e r o b -
M o r e o v e r , since B plays a v a c u o u s r o l e in G 2 (that i s , B n e v e r p r e v e n t s a r u l e
f r o m G 1 f r o m applying in G2, n o r does it c a u s e a r u l e f r o m G 1 to apply in G2), t h e r e i s no i n t e r n a l e v i d e n c e for B e i t h e r . hint of B o r b.
A s p e a k e r l e a r n i n g the language would n e v e r find e v e n a
A s a c o n s e q u e n c e , P e t e r s and R i t c h i e have not shown t h a t for any given
recursively enumerable set, there exists a context-sensitive transformational grammar that will produce t h a t s e t .
The p r o b l e m i s t h a t the d e l e t i o n r u l e i s not r e a l l y r e c o v e r a b l e .
Of
c o u r s e , t h e r e m a y be a way to c i r c u m v e n t t h i s e m p i r i c a l condition on g r a m m a r s a n d s t i l l a r r i v e a t P e t e r s and R i t c h i e ' s c o n c l u s i o n .
However, t h a t r e m a i n s to be shown.
The e m p i r i -
cal condition on g r a m m a r s m a y be sufficient, in fact, to show t h a t t h e r e will always be s o m e r e c u r s i v e l y e n u m e r a b l e s e t t h a t c a n n o t be p r o d u c e d by a c o n t e x t - s e n s i t i v e t r a n s f o r m a t i o n a l grammar.
T h a t too r e m a i n s to be shown.
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3_9, 17-28. P e t e r s , P. Stanley, J r . and R . W . R i t c h i e (1971), "On the G e n e r a t i v e P o w e r of T r a n s f o r m a t i o n a l G r a m m a r s " , Mimeo. See I n f o r m a t i o n Sciences, 6, 49-83 (1973) Sadock, J e r r o l d M. (1969), " S u p e r - H y p e r s e n t e n c e s ' ,
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