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P A W S LOGIC SYMPOSION Proceedings of the Logic Symposion held at Patras, Greece, August 18-22,1980 Edited by GEORGE METAKIDES Department of Mathematics University of Patras Greece 1982 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK OXFORD ' NORTH-HOLLAND PUBLISHING COMPANY - 1982 All rights reserved. No part of rhis publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without rheprior permission of the copyright owner. ISBN: 0 444 86476 8 Published by: North-Holland Publishing Company- Amsterdam New York Oxford Sole distributors for the U.S.A.a nd Canada: Elsevier Science Publishing Company, Inc. S2 Vanderbilt Avenue New York, N.Y. 100 17 Library ot c ongrras Cataloging in Publication Data Logic Symposion (1980 : Patrai, Greece) Eatras Logic Symposion. (Studies in logic and the foundatians of mathema:ics : V. 109) Includes bibliographical references. 1. Logic, Symbolic and mathema.t ical--Congresses. I. Metakides, George, 1945- 11. Title. 111. Series. QA9.AlL65 1980 511.3 82-14107 ISBN 0-444-86476-8 (u.s.) PRINTED IN THE NETHERLANDS ix PROLOGOS ‘After over two milennia Logicians are returning to meet in Greece’. Thus the headlines ran in the Patras news media during the Logic Symposium from August 18 to August 22, 1980. From all over the world they met indeed, as 23 countries were represented from five continents. In keeping with such cos- mopoly we decided to invite a representative selection of topics from most areas of Logic, rather than to focus on a particular aspect. So, h&e you fmd loosely arranged, a broad spectrum of papers ranging from algorithmic troubles to the morasses of set theory. This was the first International Logic Meeting to be held in Greece. It was sponsored by the International Council to Scientific Unions and the Asso- ciation for Symbolic Logic and held under the aegis of the Greek Ministry of Culture and the University of Patras. Having the 02-group mept in Patras just before the conference helped to ensure the participation of many eminent Logicians. The Organizing Committee consisted of J.E. Fenstad, A. Kechris, A. Levy, G. Metakides, S. Negrepontis, and G. Sacks. The University officials, the Town officials and its various industries and news media, they all offered their support and hospitality with a generosity that would have pleased Xenios Zeus. Special thanks are owed to Alexandra Pliakoura and Ioanna Riga of Patras who handled the local organizing with efficiency and charm. To Roberto Colon and Marion Lind of Rochester N.Y. belong the credits for the careful typing of this typing together with a personal debt of gratitude. One notable result of the meeting was a proof of the compatibility between having a good time and getting work done. The lectures were very well attended in spite of the lure of the beaches. Theorems were proved and conjectures refuted in classrooms as well as in tavernas and Mediaeval castles. As Professor Kleene was widely quoted saying to the Press: ‘When I was helping to found the A.S.L. 46 years ago, little did I dream that some day we would have meetings as fine as this one’. May 1982 George Metakides PATRAS LOGIC SYMPOSION G. Metakides led.) 0 North-Hollandhblishing Company, 1982 1 P.ECURSIVE FVNCTIONALS AND QUANTIFIERS OF FINITE TYPES REVISITBI) I11 S. C. Kleene University of Wisconsin Madison, Wisconsin 53706 9. INTRODIICTION ANP REVIEW. 9.1. RFQFTR I 1978 and RFQFTR I1 are the published versions of my lectures of June 13, 1977 at the Second Symposium on Generalized Recursion 1 Theory at Oslo and June 19, 1978 at the Kleene SvmDosium in Madison. In RFQFT I 1959 and RFQFT I1 1963, there were limitations on the substitution of A-functionals and on the use of the first recursion theorem. In 1978 I proposed to overcome those limitations by altering the list of schemata and introducing new computation rules. The new schemata include ones that provide directly for substitution of A-functionals and the first recursion theorem. The new computation rules do not require the values of parts of expressions being computed to be computed unless and until they are needed. The starkly formal character which this gives to the computations gave reason to seek a semantics for the language used in them. In 1980, I drew the lines I proposed to follow in this semantics. Here I shall con- tinue along those lines. I shall endeavor to repeat enough material here from the two earlier papers 1978 and 1980 of this RFQFTR series to enable persons unfamiliar with them to follow in the main the present one, though not to appreciate fully how it fits into the continuing investigation. (An RFQFTR IV is projected.) 2 Readers familiar with may skip to 510. 9.2. In 1978, the primary objects were of the following finite t,ypes: 2 S.C. KLEENE type 0 = {O, 1, 2,...1 (the natural numbers); t.ype it1 = the (total) one- place functions from type into lo, 1, 2, ...) . &, b, c, ... range over the natural numbers. SL, yL,. . . range over type i (i= fl, 1, 2, 3). For ;i = 1, the superscript 'lL" may be omitted. Then I considered functions (into (0, 1, 2 ,...> ) +@), $(w) ,... ... where 91,8, are each a list of variables each ranging over one of the finite types. These functions are "partial". A partial function $(a) is one which, for each choice of values of its variables from their respective ostensible ranges, either takes as value a natural number or is undefined. 3 A partial function which is defined for each such choice is total. I defined when a function +@I) is partial recursive - i8~, w here e = (0l ,...,el) with -7, -> 0 is a list of "assumed" partial functions - (variable or constant), each of a given list of variables of our finite types, or equivalently (with e variable) when a functional $(ea) is partial recursive. Toward developing a semantics to clothe the bare bones of formal computations a la 1978, my .198.0 p.rop.osa l called for representing the types 0, 1, 2, 3 within types 0, 1, 2, 3. (Another semantics was developed by Kierstead in 1980, 1982.) b ,..., @ Here type = EO, 1, 2 @I where = undefined. For 1 = 0, 1, 2, type (Ltl). = the "unimonotone" partial one-place functions ai t1 from type 1. into the natural numbers. I shall proceed to explain "unimonotone" for successive values of J+l. o o = 0 , Ishallwrite @ ' = A & O @ , @2=li1@, o3 0 = A A ~ , yo = type i - I@> = EO, I, 2,...1 = type 0, l1= type i - {@'I, N~ = type i - {Q21>3= , type 3 - ~ ( 3 ~ 1 . d is, Take > 0 for the moment, Remember that and being partial functions, need not be defined for all members of their ostensible Recursive Functionals and Quantifiers of Finite Types Revisited I11 3 &i=$ domain type (i-1)'. Let extends have its usual set- ( I d @I1) theoretic meaning, considering the functions as sets of ordered pairs with second members E 10 . Whenever &i+ 1 (8* J) is. defined for a given ii, I '+1 .i shall wish that the fact and the value of aJ (B ) to depend only on defined values of $-, never on the absence of definition of values. So if kJ' +(BI. i) is defined and k i 3 $-, then &J'+ 1 (.ai ) shall be defined with the same value. This being so, I is monotone. For i = 0, I define when monotone likewise, after putting 8" :& O = 80 v b0 = is monotone means that, 0 &' if kl( ) = E E 1". then a- 0 E type 0, i.e. a* 1 is the constant function h&'n. &' &' is unimonotone means simply that is monotone. &' is unimonotone means that &' is monotone and, for each for -2 -1 which a (a ) is defined, there is a Eique lntrinsically determined basis &' &.' 8' for w. r. to &2 (with 8' c and i2(8') = &'(&')), as I shall &' i' explain. The basis represents the information about that is used in determining the value a- 2( 'a1 ), and 8' being intrinsically determined means that it is determined by working from within 8' without looking at &' outside of its subfunction 8'. 9.3 To formulate this, I shall proceed at once to the notion of an "oracle" for a type-; function &2.4 This notion served in 1980 for the final definition of type (bottom p. 15). The reader should have no difficulty in extrapolating from the following account of oracles for type-2 objects to oracles for type-i objects, and indeed a description of them is substantially included in it. The major objective of this paper is to characterize oracles for type-3 objects (in 11.2). An oracle for a type-i object i2or ,br iefly an a0 2- oracle, is an agent (shall we say an agent of Apollo, and use the feminine gender after 4 S.C. KLEENE the oracles of Delphi?) who responds to questions, as follows. W e are en- ;' i. titled to ask her "What is &'(&')?", for any E type To ask such a question, we put an oracle for i1 in a closed envelope (or chest) and 5 present the envelope to her. =-CASE i2: The i2-oracle pays no attention to our envelope (stands mute). Then &' is the totally undefined function ;.? @ = @. -2 CASE 'L2: Without opening our envelope, the a -oracle pronounces that A2(&') = m. Since she answers "c" without knowing what type-; object a* 1 is, &* is the total constant function ?.&'m. - CASE 3': The &'-oracle opens our envelope, revealing that she will re- &', quiresome information about indeed some values of a-1 , if she is to answer our question ll&'(&l)?ll. (Were she willing to answer %'(k')?'' without learning some values of i1, she would do so under Case T'.) AS her first step toward obtaining such information, she asks the &'-oracle -1 *O who emerges from our envelope a preliminary question 'la (a )?" using an empty envelope (6' = @ ). Subcase m2:Th e a* 1- oracle does not re- 0. spond; she stands mute (Case T'). Then so does the a-1 -oracle; a*' (a-1 ) = Subcase 3.2': Without opening the envelope, the &'-oracle declares that P a* 1( a-0 ) = n (Case 7'). Thus the i2-oracle learns everything about &', namely that 6' = X.o!i Depending in general on the n, she may then 0) -2 -1 3.3': stand mute (&2(&') = or declare that a (a ) = m, Subcase *1 *2 The a -oracle opens the envelope (Case 3'). The a -oracle, observing this, 0. -2 -1 may stand mute, making a (a ) = (She could have been hoping to get an m'.) answer from the &'-oracle under Subcase Or she may pose a first non- r, preliminary question 'I&'(-0r )?" with an E (passing over the fact that the ;'-oracle, finding the envelope for the preliminary question empty, -1 did not answer that). Suppose the latter. As we know, the a -oracle opens Recursive Functionals and Quantifiers of Finite Types Revisited I11 5 all envelopes. Opening this one, and finding r+ , inside, she may stand 0. -2 01 mute. Then so does the &'-oracle, making a (a ) = Or the &'-oracle -1 +I ., -2 may declare that a (r ) = In the latter event, the a -oracle may de- 4 -1 *1 r, cide that the information that (the a -oracle opens envelopes) & a (%) = -2 -1 is sufficient to rule out a (a ) being defined, and accordingly stand mute. Or she may decide that it justifies her declaring that a- 2( *a1 ) m. Or she *I( q)?". may decide to seek more information by asking another question "a 3.3', Altogether, in this Subcase the &'-oracle questions the ;'-oracle with a series of distinct integers (possibly extending into the transfinite), q), El, ..*, %a -.- -1( q)), -2 (i.e. r+ = a until either (a) the a -oracle first asks a question -1 -2 *1 which the a -oracle does not answer, making a (a ) = @,or,with all questions % for 5 < some E < o1 answered, (b) the &'-oracle then stands mute (-a2 ( -a1 ) -- @ ) or (c), with 5 > 0, the i2-oracle then declares that a* 2( -a1 ) = m. Throughout the process described, the &.-oracle operates determin- istically, always doing the same thing under the same conditions as she knows them. Thus, in Subcase 3.3' with her not standing mute at the be- ginning, the questions %, t--, ..., r , ... are determined by her, the -5 r, first one outright (the same of all envelope-opening ;'-oracles), and each one r on the basis of the earlier questions and answers (the same -5 of all envelope-opening ;'-oracles with h1 {<q,q><l-n 5 )). 3 -2 -1 We question the a -oracle with an a -oracle in an envelope. But &.' each type-; object has an oracle who is unique (ifn ot in her person, &' a'. 6 at least in her performance), except for = Using this fact and 6 S.C. KLEENE the determinateness of the behavior of an A2-oracle, her behavior reveals the values of a partial one-place function i2 of a variable ranging over -1 type-i objects $’ (not just over oracles for type-: objects a ). Is this function necessarily monotone? It is easily seen that the -2 ‘1 only way it could fail to be is when the i2-oracle declares that a (a ) =~IJ under Subcase 3.3’ with n = n for all 5 < 5, if she does not then also -5 -2 7 3.2’ z1 declare that a (a ) = ~IJ under Subcase when = xion-. I (and -2 *o m. Apollo) now mandate that in this circumstance she declare that a (l )aI I = This mandate completes our specifications for an oracle for a type-2 u- object. A partial one-place function $‘ from type-i into No is monotone (and thus of type 2) iff i t is the function whose values are re- 7 vealed by a (monotonicity-assuri8) oracle for a type-2 object. For any $’-oracle for whom an a* 2- oracle declares that a.2 (-a1 ) = m, the basis 6’ for i1y . 1. & i2 is the subfunction of $’ which has exact- ly the values of $’ which the k2-oracle discovered before declaring that k2(i1) = ~IJ (so 8’ c i1 and i2(8’) = n~). Thus in Case F2¶ the k2-aracle 0’ discovered nothing about 6’; so the basis 6’ is = Xio@ (the 3.2’ empty set of ordered pairs), the same for all ;’Is. In Subcase with &*(&’) = 111, the A2-orac1e in one fell swoop learned the whole of A’ = Xa* O2 , so the basis 8’ is $’ itself. In Subcase 3.3‘ with a‘2 (a01 ) = ~IJ, the basis is E’ = I<qaryIs< 51 (with 5 > 0). Now I prove that, although I used oracles just now, the basis for a-1 w. r. to k2 (whena- 2( =a1 ) is defined) is, as the name suggests, Eique -; for i1 and i2 as type-; and objects. That the basis is jntrinsically determined (cf. end 9.2) follows from the manner in which an ;?-oracle operates. 6 The independence of the choice of the ;’-oracle is immediate. To prove the independence of the choice of the k2-orac1eY8 take given Recursive Functionals and Quantifiers of Finite Types Revisited 111 I type-i and -;o bjects i1and a* 2, suppose that a* 2( -a1 ) is defined, and consider a given choice of the a* 2- oracle. Then i2# @, so Case i2 applies to no choice of the a- 2- oracle. If Case F2 applies to the given choice, then a‘(@’) is defined and the oracle is unique. If Case 7‘ applies to the given choice, and thus to all choices, and Case 1’ to 4’ (i.e. Subcase 3.2’ applies), then the basis is i1irres pective of the choice. It remains to treat Subcase 3.3’ for two choices of the &oracle. I 92 01 first establish the following remark: f,w ith a given i2-oracle, a (a ) -is- de-fi-ne d under Subcase 43.3 w ith the basis 8.1 , then, for each 6 c l’ El with a* 2( 7B1 ) defined, 3 8’. Assume the hypotheses. By Case 3’ for 0; 0; il,;’( a) = so by 8 il,k( @) = so ;2(g)1~’being defined is under Subcase 3.3‘. Now if k $ 8’, the given i2-oracle, ques- tioned with the oracle for F1, would (since F1 c il)ask of her the same 01 questions and receive the same answers as of and from the a -oracle, until, i’, as must happen for some question from the domain of she fails to answer, .2 “1 making a (B ) undefined! 2 Applying this remark with 8’ and the bases for i1w.r. to two k2-oracles under Subcase m‘, we get 2 2 b’ and 8’ 3 78 1, i.e. 2 = 8’. One more observation will complete the picture. Suppose that 3 = . i1 and a-2 (-a1 ) = 5. Then by the monotonicity of i2,a* 2( ral ) = m. W i l l ra l have the same basis as G1? Clearly so, if the ;‘-oracle operates under 0’). Case F1 (all &’Is have the same basis Likewise, if Subcase 3.2’ applies to both a0 2 (*a1 ) and a* 2( ’1a ); for, 2 3 C? & A’ = AAoc+ g1 = A’. 3.3‘ Likewise if Subcase applies to both; for, then the A2-o+acle, ques- z1 tioning (,11) will ask the same questions as she asks of &’, receiving m, the same answers, up to the point at which she declares the value to be But in the circumstance in which I mandated monotonicity above, the basis