LOGIC COLLOQUIUM '88
STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 127
Editors:
J. BARWISE, Stanford H.J. KEISLER, Madison P. SUPPES, Stanford A.S. TROELSTRA, Amsterdam
AMSTERDAM
NORTHHOLLAND NEW YORK 0 OXFORD 0 TOKYO
0
LOGIC COLLOQUIUM '88 Proceedings of the Colloquium held in Padova, Italy August 2231, 1988
Edited by
R. E R R 0 C. BONOTTO S. VALENTINI A. ZANARDO University of Padova Padova, Italy
1989
NORTHHOLLAND AMSTERDAM 0 NEW YORK 0 OXFORD 0 TOKYO
0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying. recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. Physical Sciences and Engineering Division, P.O. Box 103.1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U S A .  This publication has been registered with the Copyright Clearance Center Inc. (CCC),Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science Publishers B.V., unless otherwise spec$ed. No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise. or from any use or operation of any methods, products. instructions or ideas contained in the material herein.
pp. 37 1374: copyright not transferred ISBN: 0 444 87455 0 ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, lo00 AE Amsterdam, The Netherlands Distributors for the USA. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.
PRINTED IN THE NETHERLANDS
V
EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC PADOVA, 1988 The 1988 European Summer Meeting of the Association was Logic Colloquium ' 88, held at the University of Padova, Italy, from August 22 to August 31. It was organized by the Department of Pure and Applied Mathematics of the University of Padova. Financial support was received from IUHPS, the Consiglio Nazionale delle Ricerche, the Ministero della Pubblica Istruzione fondi 40%, the Camera di Commercio of Padova, the Cassa di Risparmio di Padova e Rovigo, the Giunta Regionale del Veneto. Further help was received from the Provincia di Padova, the Comune di Padova, the Comune di Piazzola sul Brenta, the Banca Antoniana di Padova e Trieste, Italsiel. There were 237 registered participants and 46 accompanying persons, from 30 different countries. 48 participants were supported by the organizing committee. The organizing committee consisted of: C. Bonotto, R. Fern, A. Zanardo (University of Padova); D. Mundici, S. Valentini (University of Milan); M. Dalla Chiara (University of Florence); A. Marcja (University of Trento); F. Montagna (University of Siena). The program committee consisted of: F. Drake (University of Leeds, UK), J. E. Fenstad (University of Oslo, Norway), R. Ferro (University of Padova, Italy), J. Gurevich (University of Michigan, USA), Y. N. Moschovakis (UCLA, USA), D. Mundici (University of Milan, Italy), A. Scedrov (University of Pennsylvania, USA). Professor John Steel, of UCLA, gave a series of three onehour invited lectures on "Determinacy, large cardinals and inner models". Other invited onehour lectures were as follows: W. Craig (Stanford) "Logical partial functions and extensions of equational logic" E. Casari (Florence) "Comparative logics and abelian L groups" J. C. Mitchell (Stanford) "Typed lambda calculus and logical relations" I. Moerdijk (Amsterdam) "Models fro the geometry of S. Lie and E. Cartan" M. Dickmann (Paris VII) "The model theory of chain fields" C. Bohm (Rome) "Solving equations in lambda calculus" D. N o m n (Oslo) "Kleene  spaces" D. Mundici (Milan) "The language of projective modules over AF C* Algebras" S. Friedman (M. I. T.) "The lllzsingleton conjecture" W. Maass (Illinois) "Some problems and results in the theory of actually computable functions" E. Bouscaren (Paris VII) "Model theory of elementary pairs of models" JY. Girard (Paris VII) "Geometry of interaction" A. Baudisch (ADW der DDR) "On stable solvable groups of bounded exponent" R. Chuaqui (Stanford) "Probabilistic models" M. Magidor (Hebrew) "Aronszajn eees and successors of singular cardinals" S. Feferman (Stanford) "Finitely presented logics" T. A. Slaman (Illinois) "Bounded time reducibility Turing degrees" A. Macintyre (Oxford) "A Lefschetz principle for algebraic integers" G. Rota (M. I. T.) "Logic and invariants theory" A. Kechris (CalTech) "Definability problems in analysis" G. Sambin (Padova) "Intuitionistic formal spaces and their neighbourhood"
fieface
vi
Also two panel discussions took place, one on "The teaching of logic" with panelists M. Davis (Courant) and D. Watt (Leeds), the other on "Trends in logic" with panelists M. Davis (Courant), S. Feferman (Stanford), M. Magidor (Hebrew), A. Macintyre (Oxford).
In addition there were 25 sessions for contributed papers at which 89 papers were presented, and a further 37 papers were presented by title. The editors, members of the local organizing committee, take this opportunity to thank very warmly all those who contributed to the success of the meeting, and in particular Stefan0 Baratella, Enrico Gregorio and Ugo Solitro. The editors Ruggero Ferro Cinzia Bonotto Silvio Valentini Albemzanardo Padova, December 30,1988
Vii
TABLE OF C 0 " T S Preface
V
On the model theory of free metabelian groups of bounded exponent A. Baudisch
1
D.0.P and ntuples of models E. Bouscaren
11
Algebra and model theory of chain fields: an overview M. A. Diclanann
29
A Lefschetz principle for integral closures A. Macintyre
53
The C*algebras of threevalued logic D. Mundici
61
Some problems and results in the theory of actually computable functions W. Maass, T.A. Slaman
79
Kleenespaces D. Normann
91
On bounded time Turing reducibility on the recursive sets T.A. Slaman
111
The nisingleton conjecture: an introduction S. Fnedman
113
The descriptive set theory of 0ideals of compact sets A.S. Kechris
117
Solving equation in lambdacalculus C. Bohm, A. Piperno, E. Tronci
139
Comparative logics and abelian 1 groups E. Casari
161
Finitary inductively presented logics S. Feferman
191
Geometry of interaction 1: interpretation of system F J. Y. Girard
22 1
viii
Table of Contents
Intuitionistic formal spaces and their neighbourhood G . Sambin
261
Probabilistic models R. Chuaqui
287
Logical partial functions and extension of equational logic W. Craig
3 19
Panel discussion on: TRENDS IN LOGIC Relation with computer science M. Davis
357
Some remarks S. Feferman
361
Trends in logic A. Macintyre
365
Remarks concerning the comments of A. Macintyre V. Ham&
369
Remarks on logic in mathematics and in computer science G . Longo
37 1
Comments in the panel discussion on trends in logic A. Ranta
375
Concluding remarks in the panel discussion Y.N. Moschovakis
377
Panel discussion on: THE TEACHING OF LOGIC The teaching of logic C. Bernardi
38 1
Teaching the incompleteness theorem M. Davis
385
LTF  a logic teaching framework D. J. Watt
393
Logic Colloquium '88 Ferro, Bonotto, Valentini and Zanardo (Editors) 0 Elsevier Science PublishersB.V. (NorthHolland), 1989
I
On the model theory of free metabelian groups of bounded exponent by And reas Baud i sch KarlWeierstraBInstitut
fur
Mathematik der AdW der DDR
1) Introduction Let F (q,p) (or F only) be the free group v v with p free generators in the variety A A of all groups G 9.p with a normal subgroup H such that H is abelian and of exponent q and G/H is abelian of exponent p. p and q are primes. In the elementary theory of FP with xo 4 P we interprete the monadic second order theory of an infinite vectorspace over the field K with p elements, where the P second order variables range over finite subspaces. For this interpretation G/H gives the desired vectorspace and the elements of H code the finite subspaces. One subspace is given by several points in H. The main tool from algebra is the Magnusembedding [ 8 ] . That it can be used also for Fp(q,p) is proved by N. Blackburn [ G I . Using the methods of W. Baur [ 5 ] Rnd the undecidability of the universal theory of finite groups, due to A.M. Slobodskoi [ 121, we obtain the undecidability of the elementary theory Th(FP) of Fp (x, 4 p ) . Furthermore it follows that Th(Fv) is unstable. It has the strict order property and the independence property. This example has led us to prove that every stable solvable group of bounded exponent is nilpotent by finite. But this theorem follows already from results of R.M. Bryant and B. Hartley C71. Let us look at the elementary theory of relatively free groups of other varieties of groups. For absolutely free groups it is only known that they are not superstable (Gibone, see C113). The stabilityproblem and the famous problems of Tarski concerning the decidability and the elementary equivalence are open. For free nilpotent of length c ( > 1 ) groups [9] and free solvable of length
A . Baudisch
2
c ( > 1 ) groups [ l o ] Malzew gave interpretations of the elementary theory of the natural numbers. Undecidability and unstability follow. From Szmielews elimination of quantifiers [13] decidability and stability for free abelian groups are obtained. I n the case of free abelian groups of bounded exponent n we have even ostability and nonmultidimensionality. The same properties are proved for infinite free nilpotent of length c groups of exponent pm with c < p [ 2 ] , [ 3 ] . I am very grateful to Elena and Ralph Sti;;hr for helpful discussions.
2 ) The MagnusEmbedding
extended by N. Blackburn [ 6 ] , The work of W. Magnus [ a ] , gives us a presentation of the groups F free in the P variety aqAP of all groups G with a normal subgroup H where H is abelian of exponent q and G/H is abelian of exponent p ( p , q primes): Let B be an abelian group of exponent p generated by {bi : i < P I . Let Kq be the field of q elements and K (B) be the group ring of B with 9 coefficients in I< . Assume that A is a free I< (B)module 4 9 freely generated by {ai : i < p } . Then the sernidirect product BA is the group of all pairs (b,v) where b f B and 'p f A with the following group operation (b,qp?(c,v)= (bc,;pc + w). (1,O) is the tinit of this group and ( k .  ' , y b  ' ) is the inverse element, of (b,v). Sending y of A to ( 1 , q ) we can consider A as a normal subgroup of BA and BA/A B. It is eas i ly computed :
(1)
(b,q)'
= ( l , ' p ( l + b + b2 +
. . .
+ bp')),
The group BA is an elernent of the variety A A . Let 9 P {ci : i < p} be 3 set of free generators of F . Then it is CL possible to extend f(cij = (bi,aij to a homomorphism f of FP into BA. By Blackburn ( 6 3 f is an embedding of F in P
Model Theory of Free Metabelian Groups of Bounded Exponent
3
BA. T h e r e f o r e w e i a e n t i f y FP w i t h t h e c o r r e s p o n d i n g LetH be t h e subgrnup F A o f FP. By s u b g r o u p o f BA. P ( 3 ) H is a n o r m a l s u b g r o u p o f F . Then FP/H B. If we P work in H w e o f t e n w r i t e o n l y 4 i n s t e a d o f ( 1 , ~ ) .We c a n c o n s i d e r H as a Kq(B)module. Con j u g a t i o n d e f i n e s an a c t i o n o f B on H . I n o t h e r words ( l , q j ( b J v ) = ( 1 , q b ) is i n d e p e n d e n t from w . E v e r y e l e m e n t i n A has a u n i q u e p r e s e n t a t i o n 'p = I: r i a i w h e r e r i € K ( B ) . B y t h e r e s u l t s
9 i ( l , 4 ) f H i f f Z ri(lbi) = 0 i n K ( B ) . 9 i By (1) ( l , S i ) = ( b i , a i ) ' C H and by ( 2 ) ( 1 , ~ .. ) = 1J [ ( b i , a i ) , ( b ., a . ; I € H . W e use t o d e n o t e t h e J J s u b g r o u p g e n e r a t e d by g l , . . . , g n .
of B l a c k b u r n [ S ]
Lemma 2 . 1 . I f w e c o n s i d e r 13 as a I< (B)module, t h e n i t is cl g e n e r a t e d by t h e s e t {Si : i < p } IJ . : E . . : j < i < p } . If 1J h f H f l < ( b l , al), . . . , ( bn, srlj > t h e n h = Z E  .r. + 1 6 . s  where r  . € K ( < b l , . _ . b , n>) l<j. Then P V ( h ) !z + H / H .
C o n s i d e r i n g t h e MagnusEmbedding l e t ci be ( b i , a i ) as above i n s e c t i o n 2. Then h = I: a i r i ( i n module n o t a 1$ i < n are e l e m e n t s o f t h e g r o u p r i n g t i o n ) where t h e ri
7
Model Theory of Free Metabelian Groups of Bounded Exponent
K q ( < b l , . . . ,b n > ) (Lemma 2 . 1 . ) . K q ( < b l , . . . , b,>) i s t h e g r o u p r i n g o v e r t h e s u b g r o u p < b l , . . . ,bn> of B w i t h c o e f f i c i z n t s i n I< . W e show t h a t i f d f t h e n d f V ( h ) . By 4 t h a~t c l , . . . , c ~ , c ~is+ ~ Lemma 3 . 1. t h e r e i s some c ~ s u~c h +
a s u b s e t of a s e t of f r e e g e n e r a t o r s of Fp and c , + ~ is i n L e t c n + l = ( b n + l , a n + l ) . Then t h e c o s e t of d modulo H. bn+l = d . W e o b t a i n t h a t h ( b r l + l  l ) = 0 i s i m p o s s i b l e s i n c e Thir; is a t h i s would i m p l y r i ( b n + l  l ) = 0 f o r 1 G i < n . c o n t r a d i c t i o n b e c a u s e r i € I . Then by C l a i m 1 D r V ( h ) L b 1, . . . , bn) = D, as d e s i r e d . 2, there is a sum of 2"th powers which is not a sum of squares (Becker[2; Thm. 16, p. 351). It follows that fields having exactly one order, and algebraic extensions of them, do not carry orderings properly of higher level; for example, no algebraic extension of Q and no real closed field carries such an ordering. In [2] Becker worked out the extension theory of higherlevel ordered fields; the reader may also consult the survey 13). The correct notion is that of a faithful extension:
DEFINITION 1.3. Let K L be fields and P, Q orderings of higher level on K , L, respectively. is called a faithful extension of < K , P > iff Q n K = P, and P, Q have the same level. 0 Note. The relation Q n K = P only guarantees that the canonical map K'/p. > L'/ is welldefined and injective; hence the level of P is less than or
Q'
M.A. Dickmann
32
equal to that of Q.
Now the main results of Becker's extension theory are easily stated: THEOREM 1.4. (1) A n y higherlevel ordered field has a maximal faithfil algebraic extension. (Any such extension is called a higherlevel real closure or, simply, a real closure, of . 1. and
K2) and any n 2 2, Pn = k,zn U a 2nII{2n ,n 2nl n1 =Ii U a is the unique ordering of level Zn, and I? (b) For any element a E Ii

(I?
U
Algebra and Model Theory of Chain Fields: A n Overview
33
In particular, these identities hold in higherlevel real closed fields (and hence also in chainclosed fields; see Proposition 2.7 below). U No account of higherlevel orderings, however sketchy, could omit mentioning the following brilliant application to number theory. THEOREM 1.6 (Becker [2; Thm. 25; pp. 6 2 4 3 1 ) . Let K 6e an injnite field and n, m be integers. Given arbitrary xl, ...,xr E li, there exist yl, ...,ys E li such that
r 2n 2m s 2n+m (EXi ) = c yj . i=l j= 1 In particular, for li = Q ( X l , ...,X I $ this proves the existence oJi.ationa1functions F I ,...,Fs E Q(Xl ,...,X J such that
'f
The key step in Hilbert's celebrated solution to Waring's problem, [ 171, consist,ed in proving a special, though slightly different, form of these "generic" identities, namely for n = 1, r = 5 , but arbitrary integral exponent on the lefthand side, instead of 2m; Ililbert's proof was based in geometrical considerations. For further information, consult Ellison's survey [13].
52. CHAIN FIELDS.
The nonuniqueness phenomenon accounted for in Theorem 1.4(4) leads to the question of classifying the Kisomorphism types of the real closures of a higherlevel ordered field < K , P > . A solution to this question is contained i n Becker [2; Thm. 13, p. 1631, under a slightly more restricted form than that which we will consider here. Indeed, Becker's conditions for &isomorphism of two real closures of < K , P> led Harman to the notion of a chain of higherlevel orderings: DEFINITION 2.1 (Chain). A chain on a field I< is a sequence
of
M.A. Dickmunn
34
(i) Po, P I are different (usual) orders of I 0, r a E P } . (c) The residual image o f P, p = { a / M I a t P} (where M = M(IC, P)), is an archimedean (usual) order on the residue field B(K, P ) = B(K, P)/Mcli,pi. (d) P is compatible with B(IC, P), that is, 1 + M ( K , P ) P. (e) B(K, P) is the unique valuation ring A of I\' compatible with P (in the sense of (d)) such that the image pis an archimedean order on the residue field 2. (fl [2; Thm. 18, p. 48 and Thm. 24, pp. 5859] If is a higherlevel real
Algebra and Model Theory of Chain Fields: A n Overview
31
closed field, then B(K, P) is Henselian and the residue field B(K, P) is real closed under the (usual) order P. 0 Remarkably, it turns out that this setting transfers without modifications to the context of chain fields: PROPOSITION 3.2 ([lG; p. 144, Thni. 1 3 , and Prop. 4.41). Let iEw is chainclosed, then B(li, Pi) is Heitselinn and B(K, Pi) is real closed under the order p r b r i E w.
0
Notation. We shall denote by B(K) any of the valuations R(K, Pi), for a chain field < K , Pi>.
0
c
The valuations B(I 2 (cf: 3 3 3 RobinsonZakon [23; Dej 3.31). (d) For v E Y(K), IrJ I = 2. 0 2rU This result is proved in Dickmann [la; Prop. 1.10 and Prop. 1.7(e)],except for the assertion about B(I 2) are (implicitly) definable from Po, PI. Using Proposition 1.5 one may dispense with Po and P 1 as well, and get an axiomatisation for chainclosed fields in the language LF. However, a neater axiomatization was obtained by Gondard in [14], [15]:
PROPOSITION 4.2. Cliainclosed fields are exactly the modek of the following set of axioms in LF (which we call CCF): 
Axioms for commutative gelds.
 Pythagorean axioms: "a sum of two squares is a square" and "a sum of two
fourthpowers as a fourthpower".  3y[Vw(y# w2 h y # w 2) h Vdz(z = z2 v x = z 2 v z = yz2
vx
= yz2)].
 "Evey polynomial of odd degree has a root".
A n alternative axiomatization for CCF is obtained by adding the axioms for real
42
M.A. Dickmann
fields and omitting the first Pythagorean axiom.
0
2
2
Remark. The third axiom says that, for any a E K  ( K UK ), the sets 2 2 2 2 Po = K U aK and P1 = K U aK are positive cones for total orders on K; these sets do not depend on the choice of a , cf. Proposition 1.5. For several alternatives to the Pythagorean axioms, see [15] and [16; Cor. 2.41. The class of chainable fields (i.e., fields admitting a t least one chain), an obvious analog in our context to the class of real fields, was proved by Gondard to be axiomatizable i n the expansion LF(c) of tlie language for fields by a new individual constant c. PROPOSITION 4.3 (Gondard [15; Thm. 111, [14; Tlrm. 11). Chainable fields are exactly the models of the following set of axioms in LF(c):  Axioms for commutative real Jields. . not a sum of fourthpowers''  "c2 as
0
It is an open problem whether the class of chainable fields is axiomatizable in LF.
A third class of chain fields having an obvious analog in tlie context of ordered fields is the class of fields carrying a unique chain. An axiomatisation for this class in the language LF is only known for the case of fields which, in addition, are Pythagorean and have exactly two orders; Gondard [15; Tlim. 11121. Let us now turn to the modeltheoretic mentioned a t the beginning of this section.
results for chainclosed fields
THEOREM 4.4 (Dickmann (12; Thm. 1.111). The theory CCF is complete and decidable. Hence so is CCCF. Proof. The Jacob valuation of a chainclosed field is Henselian with real closed residue field. The firstrder theory of its value group is the theory of regularly dense, odddivisible ordered abelian groups r such that I l?/2r I = 2; cf. Theorem
Algebra and Model Theory of Chain Fields: A n Overview
43
3.8. This theory was proved complete by RobinsonZakon [23; Thm. 4.71. The result follows, then, by the AxKochenErshov transfer principle. Completeness of CCCF follows immediately, since this is an expansion of CCF by definable predicates. Decidability is a consequence of completeness and the fact that both 0 theories are recursively axiomatizable. In order to discuss modelcompleteness we need to enrich the language LCF; we will use the language LCF(A) introduced in 4.1. Let us call CCVF (for chainclosed valued field) the theory CCCF augmented by the following axiom, which interprets the predicate A as the Jacob ring:
A(x)
(X
j! *P2 A 1
+ x E P2) V [X E *P2 A Vy(y j! *P2 A 1 + y E P2 > 1
+ xy E P2)].
The choice of P2 is immaterial; P2 can equivalently be replaced by any Pn with n > 2, by Po n P1, or by I< 2 ; see Proposition 3.5. THEOREM 4.5 (Dickmann [12; Thm. 2.31). The theory CCVF is modelcomplete in LcF(A). 0.
The proof proceeds along a line similar to that of Theorem 4.4, using variants of the RobinsonZakon and AxKochenErshov results adapted to modelcompleteness. One needs showing that the (lifting of) divisibility predicates Dm(a) ++ 3p(ry = mp), m > 2, i n the language for the value groups, are quantifierfree definable in LCF. In an unpublished paper [lo; $21 Delon deals with several questions related to modelcompleteness of chainclosed fields. Among other things, she gives a useful characterization of elementary inclusion: PROPOSITION 4.6 (Delon [lo; Prop. 2.11). Let Zi7c L be chainclosed jelds. The following are equivalent: (a) I( is existentially closed in L (for the language LF).
44
M.A. Dickmann
(b) K 5 L (in LF). (c) K is relatively algebraically closed in L and J(L) n li = J(1i).
0
This characterization has two important consequences: PROPOSITION 4.7 (Delon [lo; Prop. 2.41). (a) CCF has no ezistentially closed model. (b) The chainclosed fields
>
the counterexample mentioned after Theorem 5.4 below. Now we turn our attention to certLin results on polynoiriials over chainclosed fields suggested by Hilhert's 17th problem. Hilbert's classical problem was that of characterizing the class of rational functions f E W(X1, ...,Xn) which are nonnegative definite on W, i.e. which take nonnegative values whenever the denominator does not vanish. ArtinSchreier's celebrated theorem proved: 
f is nonnegative definite on I< iff f E C K(X)
2
,
41
Algebra and Model Theory of Chain Fields: A n Overview 
for rational functions f E K ( X ) = K(X1, ...,Xn) with coefficients in a real closed field K. A similar problem for higherlevel real closed fields was solved by Becker and Jacob in [ 5 ] ; they characterized the class of rational functions f E K(X1,,..,Xn) whose values lie in P for lionvanishing denominators. The equivalence (*) can also be read as a characterization of rational functions in C I ts cp(n,1nf:39
and
pm=U , x ) E
= (a(O),p(O), ...). For a, a' E
3 m Vn
2
A ww
m ( a ( n ) = a'(.)).
If u E ww w e can also view u as a s t r a t e g y for player I p u t t i n g
put
The Descriptive Set Theory of aIdeals of Compact Sets a(ao...anl) where
< ... >
= (I(< .a ,..., a n  l > ) ; a i E
is a recursive coding of finite sequences.
r E ww as a s t r a t e g y for player 11.
We d e n o t e by u
131
w
Sinmilary we c a n view
*
r t h e m e m b e r of w w
resulting in t h e r u n of a g a m e in which I uses u a g a i n s t I1 using r , i.e. u*r(O) = u(0), u * r ( l ) = r ( ( a * s ( O ) ) ) , u * r ( 2 ) = u ( ( u * r ( l ) )u, * r ( 3 ) ) = r ( ( u * r ( O ) ,u*s(2))), ....
We h a v e t h a t x E' I iff I h a s a winning s t r a t e g y in t h e g a m e I1
I
"
0
I wins iff ( < a , o > , x ) E
A.
Using a trick from [I
(7)issolvable w ( V l S i C j S t ) y i s y j
a Mir,Mj.
The relation 9 helps to express the main theorem of [CDR 781 in a more condensed way but the content is essentially the same. To be more precise it is the separability which is characterized in [CDR 781, and not the weak separability. Notice fiat ( 4
not every set
T is separable ,
as exemplified by [ W, , Ax. x n 1. Since here i t j may coexist with yi= yj, we may say that (7*) [(5*)]holds iff the partitions induced by the identity on the multisety = ( y l ,...,yI] are unions of the zF [ = ] classes Ptl
partitioning the indexed multiset F = (MI,. ..,M,}.
We will say that
is weakly Separable [Tro 871 iff
(7 ) is solvable, where: (V 1 5 i <j 5 t) yi = y,
@
Mi qP M, .
We obtain the result:
(e)
every multiset F is weakly separable ,
which, together with
(0
every set l'J is weakly separable ,
may be thought of as an alternative way to express the separability problem. The meaning of the definition 'weakly separable' is that we take into consideration only the subsets of systems (7) with maximal refinement, i.e. where the panitions induced by the identity on the multiset
y
= ( yl, ....yl} are exactly the rFclasses of the indexed
multiset F = [MI,. ..,MI]. Let us stress that the characterizations (5*) and (7*)are both constructive, in the sense that if the rhs condition is fulfilled the proof supplies a solution to the corresponding system. Generalizing (5'1and (7) in the Dcalculus Let us examine the following set of equations solved by Rosser in the sixties. The aim was to find a single combinator acting as a basis of CL:
Solving Equations in LarnbdaCalculus
147
xx=s P
XS=K.
P
Its solution was X = 1 t . t (K*K)K(KS). Selfapplication of the unknown term inside system (5) is not admitted
. We
may
generalize both Rosser and (5) systems asking if there exists a discriminator possibly distinguishing the elements of a multiset including itself!
Let yo=yi (i = 1, ...,t) and the same assumptions on the Mi's in (5) be still valid. Then the answer [BT 871 is that the system (8)
xx=yo
P
X M i = y i ( i = l , ...,t) P is solvable iff (5) is solvable. Notice that defining FOE W,
Fi=hx.xMi
( i = 1, ...,t)
system (8) acquires the shape (8')
Fix=yi ( i = O , ...,t).
P
Notice that system (8'),under the assumption that Fi's are arbitrary closed Pnfs, may be viewed as an extension both for equation (4')and system (5) in the Pcalculus. A further generalization of (7) and (8') is achieved in the pflcalculus if we consider
systems with n unknown terms, i.e., starting with a multiset F = ( F l , ...,FJ of combinators in Panf all having the same number n of initial abstractions. We may then construct a system of shape = yi ( i = 1,...,t) . Pn Such family of systems have been studied in [BT 871, [BP 881, [BP 8?] (the case without
(9)
F i x , ...X,
f2)with Y ,f as sets and in [BT 8?] as multisets. Refemng to the set of the initially abstracted variables
C. Bbhm, A . Piperno and E. Tronci
148
x=( X I ,...,XJ the solvability of system (9) was called the Xseparabiliry problem [BT 871, [BP 8?] and it was constructively characterized in [BP 881. Relaxing every restriction, i.e.allowing multisets both in lhs as in rhs of (9). here we will talk of a new problem, the Xweak
separabiliry problem. Notice that (8)
not every set Fis Xseparable
,
as exemplified by any inseparable set. We will say that f is Xweakly separable iff
(9*)
(9) is solvable, where: (V 1I i c j I t) yi = yj a Mi p7. Mj .
Warning: The problem of deciding whether F is Xweakly separable is a particular Xweak separability problem and must not be confused with the whole problem. Let us note also the following fact (h)
not every multiset Fis Xweakly separable,
as exemplified by ( I , W,) which is even discriminable. Relationshim between XseDarabilitv and discriminability
In order to understand the inherent difficulties of the Xweakseparability problem let us return to the Xseparability problem where no fl occurs in F (see equation (9)). Looking at the structure (6) of the solution of ( 5 ) (which always exists) we recognize that if F is Xseparable then it is discriminable with s = n (n being the number of initial abstlactions of each element of F). The last constraint may be removed, and in [BP 8?J it is proved that if F is any set of
0
qnfs and n is the maximum number of initial abstractions in F, then lF is discriminable withs=n+l.
In other words, studying Xseparability allowed improvements on discriminability. A sufficient condition on F to be discriminable with s = n is [BP 8?] that
Solving Equations in LambdaCalculus (V 1 < i < j l t) Fiz
does not occur in Fjz]...z,
149
.
Other aDplications of the Xseuarabilitv: oneside invertibilities of nt maDoings Applying to (9) the same reasoning as for equation (47, (9) (without f2)is solvable iff (10)
R
R
Fi (F ,yl.. .yJ.. .(F,,yl., .yJ R
is solvable where P = (Fy, ...,F:)
= yi
P
(i = 1,...,t)
is an unknown set of combinators. Viewing the set P
as the specification of a combinatory mapping from n to t dimensions (a nt mapping) we may interpret the X separability as the problem of finding a tn mapping PR which is the right inverse of a nt mapping F. Moreover, since in relation (10) we may well equally thinkof
FRas known and asking for F, we may also view (10) as a left invertibility
problem. In [BP 881 such a problem has been proved to be decidable, generalizing the methods described in [BD 741 and [MZ 831 for single combinators.
The main result for the Xweak separability A characterization of the Xweak separability problem (solvability of (9)) is still an open problem: it may even be undecidable. Nevertheless, the main outcome of this paper is a decidability result which recapitulates and improves all the previous results concerning discriminability, separability and Xseparability:
Theorem: The predicate F is Xweakly separable , i.e. (9*), is decidable. The meaning of the theorem is that given any multiset F,it is decidable whether, in the case where the partition of T induced by the ZFclasses is reflected by the identity classes of Lj, system (9) is solvable. The main results reviewed or contained in this paper may be summarized by the following statements:
Weak discriminability Separability Weak separability Xseparabiliry Xweak separability
:decidable
F is weakly discriminable : true
:decidable
F is separable
:decidable
: decidable
7 is weakly separable 7 is Xseparable F is Xweakly separable
: hue
:decidable :
?
:decidable :decidable
C. Bdhm, A . Piperno and E. Tronci
150
Part 2: The main theorem We assume the reader to be familiar with the basic definitions and properties of kcalculus; for a complete treatment of them see e.g. [Bar 841 and [HS 861. However, some notions (Bohm trees, useful paths, context and others), which have capital importance throughout this part of the paper, will be recalled in Appendix 2. Given a finite set F of terms, we first introduce an equivalence relation between elements of F, which generalizes the notion of indistinctness for finite sets ([CDR 781).
2.0. Definition: (Findistinctness) L e t F c A and M,NE F. We define the relation % C F x F (calledFindistinctness),as follows: M p N
3PGfs.t. ((M,N)GP
@
A
1(3o!usefulforP)).
Q
It is easy to verify that 3 is an equivalence relation. The problem of solving the class of systems of equations where rhs's are arbitrary variables, whose distribution reflects ZFequivalence classes of lhs's, is called Xweak separability problem (Xw separability in short): 2.1. Systems of eauations and T hee
We denote by A,,* the set of AfTee terms in pflnormal form, i.e. free
F E Apn
@
F = SMl. ..M,
A
F is in P0normal fonii.
free
LetF' ( F 1,...,F,) C h o n , X = ( x l ,_.., x,) c U and let Y = (yl ,..., yt) be a multiset of arbitrary variables; let T be a semisensible theory; we say that F is I"Xwseparable iff there exists a substitutive context (2.1.0)
(2.1.1)
where (2.1.2)
D[ J = (Axl.. .x,.[
I
I) X,. ..X,
such that:
D[F,]=(Ax l...x,.Fl)X1...X,; y1 . . . . . . . . . . . . . . . . . . . D[F,l=(Ax l...x,. Ft)X1... X , ; y t
V i , i E ( 1,...,t), F ; z F F j
y i = y J. .
Solving Equations in LambdaCalculus
151
2.2. Remarks; (i) From this point onwards we will assume that there do not exist F, ,FzE F such that
F, occurs in F2 In fact, as in [BP88, $31, it can be proved that this assumption
does not cause any restriction in characterizing 'ITXw separability. (ii) The relation 9 cannot be refined by any context C[ 1 E A[ 1, as stated by the
Theorem; ([BT 8?, 3.51) Let P c A. Then: V C[ I E A[
(2.2.1)
I
1
V P ,Q E F , P ZF Q
C[PI 2cwlC[QJ.
0
As a consequence of (2.2.1), (2.1.2) identifies the finest partition (wrt the 'd relation) which can be reached over right hand sides of (2.1.1).
Let F c A:;
0
when using substitutive contexts, we are forced to consider a path y only if
y = < j > * p and j does not exceed the degree of elements of F. Furthermore, it will be important to take into account the subtrees of elements of P whose head is a free variable. 2.3. Definition: (effective path) A path y is said efective for T (E HNF) iffy= cj> * p
A
deg(T) 2j .
0
2.4. Definition: Let F c A:;
the set N, is defined in the following way:
N, = ( N E hpn I N is a proper subtree of some F E
A
h e a d 0 E FV$) ) . 0
We are now able to define the notions of regularity and quasiregularity, which are the central issues in characterizing Xwseparability. 2.5. Definition: (regularity)
Let F'cA.:
0.
In order to introduce the notion of regularity, we first define a new equivalence relation s F ~ F ~ N , ~ F ~ N , :
M zF N
Q
head(M) = head(N)
(M,N) L P, UP,
A
A
,
3P (# 0 )c F,P
N, s.t.
(301 useful for P u P, and effective for P
C. BBhm, A . Piperno and E. Tronci
152
We say that F is regular iff the following conditions are both satisfied:
1.
(2.5.1)
i ( 3 F € F , N € N p S.t. F=,N);
(2.5.2)
V M,N
E F,
M =, N a deg(M) = deg(N)
0
.
2.6. Definition: (quasiregularity) 0. Let A(F) be the set of the approximants of F C A;
which preserve the =+equivalence
classes: (2.6.1)
A(F)= [g$)
I g:F+A A
A
VMEF,g(M)CM
A
V M , N € F , g(M),(p)g(N)
3
M,N1
.
1. We say that F is quasiregular iff 3F#E A(F) s.t. F' is regular .
2.7. Theorem: (main theorem) Let F c A"= and let T be a semisensible theory; then: Pn
F is TFV(F)weakly separable
CJ
F is quasiregular
.
A sketch of the proof will be given in Appendix 3. The
(e) is proved in a
constructive way. We exhibit here the shape of the solution of (2.1), if it exists: (2.7.0)
D [ ] = ( h x l...~,.[])X1...Xn , where, f o r i = l , ...,n :
(2.7.1)
Xi = (
Ufyh
(a€M, lShSt)
' ~z~...z,.z,X~,~...X~,~Z~.Z~ , where for j = 1,...,r , X i j has shape (2.7.1) again.
2.8. ExamDle: LetF= ( F1,...,F5), where
F ~ E x ( ~ K ~ ) ( ~ v . x v ( vFKZ )= )~,I ( y f l x ) F, ~ = X O ( Y K XF) ,~ ~ Y xFsyCIy. Y , #
#
We observe that F is quasiregular, since F' = { F l,. ..,F5], where #
#
F = x fl (hv.x v (v K)) , F = Fi (i = 2,. . .,5), is regular. Hence F is PFV(F)weaklysepable. In this case, (2.1) is solved by: 9
D[l=(hxy.[ l)(hab.b(U:ly,)fl(U :4y2)ab)(hab.bCI(huvz.zuvz)(U 1 ~ 3 ) a b ) .
Solving Equations in LambdaCalculus
153
It is easy to verify that: DIFll = y1 , D[F21 = y2, D[F31 = y2, D[F41 = y3, DIFSl = y3. Q P P P P P 2.9. Corollarv:
Let F C A:;
and X C U ;let T be a semisensible theory;
let us denote by F, the set obtained from F substituting n for elements of FV(F)  X and then reducing to anormal form; then:
F is TXweakly separable a Fx is quasiregular
.
Sketch o f the proof; As in [BP 88, $31, we observe that:
F is TXweakly separable a Fx is TXweakly separable a
Fx is quasiregular. 0
2.10. Examule: LetF= (Fl,,..,F5), where F 1 ~ x z ( L . v . x v ( v K ) ) ,F ~ = X I ( Y Z XF) ~, = x ~ ( ~ KFx~ )= ,Y X Y F ~, = Y z Y . since F(x,yl = F' where F' is the one We observe that F is (3(x,y)weaklyseparable, appearing in Example 2.8. Then D[ 1 of example 2.8 satisfies our requirement.
0
It is easy to verify that a weakseparability problem can be stated as a particular ( x ) weakseparability one; the following corollary shows that a weakseparability problem can be always solved by means of an ntuple, for a suitable n.
2.1 1. Corollarv: 0
0 .
Let F= ( F1,. . .,Ft) c Apn , where Apn is the set of closed (3flnormal forms and let Y= (yl,.. .,yJ be a multiset of arbitrary variables. Then:
(2.1 1.1)
3n E
M,3 Xi ,...,X,
EA
such that:
V i e (1,...,t ) , < X 1,...,X,>Fi = F i x l...X, = y i ,
(2.11.2)
P V i , j E ( 1,...,t). FizFFj a y i = y1. '
Sketch of the aroof: It is sufficient to choose n large enough, so that the set
F*= { F1 XI.. .x,,,.
free.
..,F,xl.. .x,) c Apn IS quasiregular.
P
where
C. Bdhm, A. Piperno and E. Tronci
154
0
It is easy to verify that such n exists. 2.12. Corollarv:
(About discriminability) 0
L t F = (F,, ...,Ft) c A P , , , w h e r e n o is thesetofclosedpqnormalfomsandlet Ptl
Y = ( y,,. ..,yJ be a multiset of arbitrary variables. Let n = max(ord(F) I FE F 1. Then: (2.12.1)
3 X, ,...,Xn+, E A such that: V i E ( 1,...,t),
(2.12.2)
< X , ,...,Xn+, > F i = FiX1...Xn+, = yi, P P V i , j E ( 1,...,t), Fi = F. e y i = y j . Prl
where
'
Sketch of the nroot It is easy to verify that: F*= (F,x,.. .x,,+,.. ..,F,X~. .. x , + ~1 c
free .
is quasiregular,
since f2 does not occur in F*.
0
Note that, if V i j E (1,. ..,t) (i#j), Fi does not occur in Fj, then 2.12.1 holds with n in place of n+l.
CONCLUDING REMARKS Summarizing, it has been shown that, for any finite set F of hfree terms in PRnf and within any semisensible theory, it is possible to decide whether F is Xweakly separable. Such a characterization has led to some improvements in solving weak separability and discriminability problems. The case where the elements of F are not hfree still remains open.
ACKNOWLEDGMENTS The authors are grateful to M.DezaniCiancaglini and RStatman for the permission of
relating their results about undecidability of some family of equations.
Solving Equations in LambdaCalculus
I55
REFERENCES
rsar 841
Barendregt. H.P., The lambda calculus, Nonh Holland, 1984
[BD 741
BCihm, C. and DezaniCiancaglini, M., Combinatorial problems, combinator equations and
normal forms, in: k k x (4.)Aut.,Langu. and Progr.2th Colloquium, LNCS 14,1974, pp. 185199 [BDPR 791
BOhm,C.,DezaniCiancaglini.Peretti9.and Ronchi della Rocca,S., A discrimination
[Bdii 681
algorithm inside hPCalculus, Theor. ComputSci. 8, (1979). pp. 271291 Berarducci,A., Programmazione funzionale e rapprescntabilitain alcuni sistemi di Logica Combinatoria. Tesi di Laurea in Matemauca, 1983 (supervisor C.BBhm) Bdhm, C., Alcune propriea delle forme Pqnormali nel IK calcolo, IAC Publ. n. 696,
[Bbh 881
Roma ,I9 pp.. 1968 B6hm.C.. Functional programming and combinator1 algebras, MFCS 88, to appear in
lBer 831
LNCS
[BP881
B6hm.C. and Pipern0.A.. Characterizing Xseparability and oneside invertibility
P P 8?]
pp. 91101 B0hm.C. and Piperno.A.,Surjectivity for finite sets of combinators by weak reduction, in "Logik in dcr Informatik", Karlsruhe 1987, to appear in LNCS
in h~flcalculus, LICS 88, Edinburgh, June 58.1988, Computer Soc.of the IEEE,
[BT 871
B6hm.C. and TronciE., XSeparability and LeftInvertibility in LambdaCalculus,
[BT 8?]
LICS 87, Ithaca ,N.YJune 2225.1987. Computer Soc.of the IEEE. pp. 329328 B6hm.C. and Tronci,E., About systems of equations, Xseparability and leftinvertibility in
[CDR 781
the hcalculus, submitted to publication Coppo, M., DezaniCiancaglini, M. and Roncbi della Rocca, S., (Semi) separability of finite sets of terms in Scott's D, models of the Icalculu:, in:
[Dez761
Ausiello and Edhm (eds.),Aut.,Lan. and Prog, 5Lh Colloquium,LNCS 62,1978, pp.142164 Dezani,M., Characterizationof normal forms possessing inverse in the 10qcalculus, Theoretical Computer Science 2,1976, pp. 323337
[Dez 871 [HS861 [Mz 831
[Sta 87al [Sta 87bl F r o 871
Dezani,M., Private communication Hind1ey.R. and SeldinJ., Introduction to Combinaiors and Xcalculus, Cambridge University Press, 1986 Margaria, I., Zacchi, M., Right and Lect Invertibility in Apcalculus, RAIRO Th.Inf. 17, 1983, pp. 7188 Statman,R., On sets of solutions to combinator equations, to appear in Theor. Comp. Sci. Statman,R., Private communication Tr0nci.E.. RisolubilitA di sistemi di equazioni nel hcalcolo, Tesi di Lauea in Ingegneria Eleuronica, 1987 (supervisor C.BOhm)
C. Bdhm, A . Piperno and E. Tronci
156
APPENDIX 1 AI.1: Undecidability of M X = S [Sta 87b]:
P If M is a hterm, let M#x be the result of replacing each redex (hy.Y)X in M by x(hyY)X. In particular, M#x is normal and [I/x] (M#x) =M. P For each Giidel number e construct a term P, uniformly in e, such that:
'
P=(
I
if ( e ) (e) converges
unsolvable
if { e )(e)diverges .
This can be done in the usual way (see [Bar84,$8.2]) . Now let
M = Axhabc. a (Pc#x) (b (x c)).
It is easy to see that there is a normal form X st. M X =S if and only if ( e )(e) converges. P In particular, if (e)(e)converges then M I = S. P On the other hand if M X =S then X c = c, hence X = I. Thus P c = c, so P is solvable and P
( e )( e ) converges. A1.2: Undecidability of M X =I [Dez 871:
P In the case of M X =I, consider: P M= hxy .x (x (P#x) (xy)) and proceed as in A 1.1.
I57
Solving Equations in LambdaCalculus
Basic definitions  We denote by FVO the set of free variables of a h e m T.  WedefineHNF=(hzl ...z& TI ...Tm IT1 ....,T m ~ A , n , m ~ O J , S O L = ( M ~ A 1 3 N ~ H N F s . t . M ~ N ) . APPENDIX 2:
 If T ; L1..z,,.tT1.. . .TmE SOL, we will define the order, degree and head of T to be respectively:
BTO’~). . otherwise, if T B SOL: BT(T)=
. . B?(T,,J
n .(’)
 We will sometimes identify a term T with its Bdhm tree.  A p a f h y @>O) starting from the root of BT(T) is a (possibly empty) finite sequence of positive integers uniquely identifying a @ss. non proper) subtree of B T O ; we will then write YE B T O and we will denote by Ty the subterm of T identified by y. Moreover, we will write YE,, B T O iff there exisls an qexpansion T’ of T such that Y E BT(T’). In this case we will say that y is a virtual path for T. W e will use * as a concatenation symbol for sequences. Lct
[email protected]); we will write y€(,,)BT(F) iff y€(,,)BT(T) , for every FEF. Let T1, T ~ E HNF and y be a path (y€(,,)BT(Ti) , i=1,2) ; we define the following equivalence relations: euuivalencc between terms [Wh 681: T1 T2 iff head(T1)=head(TZ) A deg(T1) ord(T1)=deg(T+ y eauivalence betwccn terms [CDR 781:
ord(Tz).
T1yT2 iff (T1)y Vdy. LetPcAand,foreveryFEJ’.yE,,BTO; wesay that [CDR 781 y is & for F iff ( V F E P , F ~ # ~A~ () ~ F F , , F ~s .Et .F FlqF2)
A
(V~ * p be a path effective for Fxand useful for (F urn,),.
Then there exists C E (e,r) E N2such that all the following clauses are valid: (A3.3.0)
If F is regular then DJFI
(A3.3.1)
If P V
(A3.3.2)
is regular .
(FuN,), is (x,e)minimal. then:
M.N E P, head(D,JMI)
head(D,,JNl)
M,,,
N .
If length(@ > 1, then: 3 a path y s.t.: y
is useful for Dx,[(P u N,),] and effective for D,,[FI
;
 length(y) < length(ol) . (A3.3.3)
If M E (F uN,), (D,.,IMl)d,
,%ct&?f
E
and N E NF,with M, (WDJ
I)  WF) )
= N, then: 3 h E
P
A
deg(D,,,"I)
M' s.I.:
: wehave: M = h u . x M ,...M , ~ ( F u N ~ ) , a n d M , ~ , = h a l...a,.yG1...Gw ( y s x ) . P P Then D,,,[Ml ;hu.y ~ x , ~ ~ G t l . . . ~%x+.I ~. . .~%~D,,,~MII...D~.,[M,I wl .

Lct y= ; we have: y is useful for D,,,[(f u NF),1 and effective for D,.,[FI and length($ < length(a)
.
(A3.3.3):LetM=Au.xM I...M, with M,,,= N = h a ,...a,.xG I...Gw ( w < e ) . P hu.&+v.wat...q D,,,[GtI...D,,,~GwI a,,+t...ar D,,,lMtl ... D,.,[M,I . We have: D,,,[MI P
P
=
B
D,.,"l
= hat...%L+l...
P
4..t at... % D,,, [GI].. .D,,,[G,I
L+~.. .t, ;
choosing h = r + e + 1, the thesis follows. By an easy induction we treat the case where deg(N) 2 e. (A3.3.0): The result can be proved taking into account that the existence of both useful and effective paths is preserved by D,,[
I (A3.3.2.3). An example will clarify the matter.
Let F = ( ~ ( x x,)x(xK)R , x(xxx)R R }. We have N, = ( x x ,x K ,X X X ) , hence F is regular. Let D,,"
I = ( A x . [ I)
(ht.tal
... a$).
D,~,[fl= ( a l al...a,al...qD,,,[x] ala
W e have:
al...a,D,~,[x Kl , a l a,... a,K al...a,D,,,[x
a,al...a,D,,,[x]D,~,[x]al...a,D,,,[xxx]
n4.aP1=(a l,....a,, a l a
K] R
,
0 0 ) and
q a l . . . a , D , , , [ x l , a l a 3 . . . a , K . a l a l . . . a , a l ... a,D,.,[xl D,.,[xl
It is easy to verify that D,,,[f] is regular.
1. 0
We are now able to prove the: A3.4, Theorem: (main theorem)
Let f C A:;
and let Tbe a semisensible Iheory; then:
F is TFV(P)weakly separable
t)
P is quasiregular
Sketch of the oroof: (*):By absurd. Suppose that P is not quasiregular. We distinguish two possible cases.
Case 0: 3F E f , N E N, s.t. F
N. Let D,,,[F]
=
3f*
y (denoting by
x * the largest semisensible
theory). Then it is possible to verify that head(D,,,[N]) = y or D,,[N] =, 0 . It follows that we can
w
C. Bohm, A . Piperno and E. Tronci
160
substitute fl for any wcurrence of N in P. The iteration of case 0 yields a set P#E A(P) which is regular, and this is absurd.
Case 1: 3 M . N e P s.t. M p N anddeg(M)#deg(N).Let M = x M , ...M , a n d N = x N P B with m < n. Then it is possible to verify that from D,,[M] =, y it follows D,,[Nl
= D,,,[xN,
x*
...N,l
,...N,
x
D,.,~N,+~l...D,,c~N,l
=* y Dx,c[Nm+ll...D,JN,I
x
, which is absurd.
(G): Wlog, we can assume that P is regular, otherwise we consider F#E A(P)
The thesis follows by induction, considering that at least one of the following events holds:  the maximum of the cardinality of the (x. e)minimal sets of DJF] the cardinality of the (x, c)minimal
SCIS o
is greater than the maximum of
ff;
 Lhc sum of the length of die uscful paths for P is greater &an the analogous amount in D,,[F]. 11 follows that a composition of context filling of shape A3.0.1 gives the result.
cl
Logic Colloquium '88 Ferro, Ebnotto, Valentmi and Zanardo (Editors) 0 Elsevier Science Publishers B.V. (NorthHolland), 1989
161
COMPARATIVE LOGICS AND ABELIAN IGROUPS Ettore Casari Dipartimento di Filosofia Universita degli Studi di Firenze Firenze, Italy
INTRODUCTION The ground insights of comparative logic go back to Aristotle. His Topics contains, for instance, (see Casari [1984a]) a highly systematic treatment of the nine kinds of propositions which arise by crossing comparisons of majority, minority and equality with situations in which 'one is said of two', 'two are said of one', 'two are said of two', namely: x is more (less, as much as) A than y; x is more (less, as much as) A than B; x is more (less, as much as) A t'nan y is B. It is possible to account for some significant features of the logical mechanism of comparison by simply enriching an elementary language with operators which, for instance, send the predicate A into the predicate A< (A(1). Suppose again that G is separated but not totally ordered. Let u be neutral; then clearly also u is neutral. Take x=unO, y=unO, z=O. We have: x+z=(u~O)+o=(u+O)~(O+O)=(u+O)~~O.But i t must be x+zEO, otherwise it would be oC_ u+O i.e. 0511 and so, by (S) K u against the assumption. Thus, by (S), x+GO. Similarly we get y+gO and so: (i) (x+z)u(y+z)&0. On the other side, xUy=(unO)U(unO) and so (xUy)+z=((unO)U(unO))+O= (by (A25)) ((uUu)~O)+o=((uUu)+O)~(O+O)=((uUu)+o)~O. By unperforatedness, 0Cuuu. Thus (ii)(xuy)+z=O. From (i) and (ii): G is not an mpregroup. (1)(4).Suppose G is separated and totally ordered. Let be C&x and g  x u y . If it were Ogr it would be yC0 and so s  0 , and also, being =, otherwise; (3) o=. We call the resulting structure S the 2sDlittinP of 6 with respect to w. Theor1 If 6.is an abelian g r o w . G ' some of its suberous a nd 2 s o m e zDseudoe roup then the 2snlittine of G with resDect to G'is a normal Dseudogroup havine G6l01 as the set of ordinarv elements and G'@fOZ 3s the set of principal elements.
m f . Just two examples: ((32.3) and (N2) A d (G2.3). If X E G', then  X E G' and so <x,a>=. Therefore: <x,a>+<x,O>=<x,a>+=. If x e G',then XEG'and a=O and so <x,a>= =. Therefore: <x,a>+<x,a>=<x,O>+=. (N2). Suppose <x,a>#<x,a>+ i.e. <x,a>#<x,O>. Then
XE
G' and also XEG'.
It is: (<x,a>O)=( <x,a>+( <x,a>+))= (<x ,a>+ <x,O>)= (<x,a>+)= =()=. On the other side: (<x,a>)O = <x , a > +  (  <x,a> + )= =+( +)=+=. We now show that any pseudogroup can be represented as the splitting of an abelian group. More precisely: Theorem 2 &pose : (1) G is a normal preerouD: (21 l*t is the meltin? grOUD of 6; i3I N is the suber o w of % of all penerared melting classes 0f G: (4) S is the Z(G)sDlittine of % with resDect to N. Then the mapping 6.defined on
G bv dfx)=<M(x).xO>. is an isomorphism between G and S.
m.(i) 0:G+S.
Let XE G. (a) M(x)={x}. Then M(x)E N and x=x+O and so xo=O. Therefore, <M(x),O>=<M(x),xo>~S. (b) M(x)#{x}. Then, by (Nl), M(x)E N and so {M(x)}@ZzS. As X O E Z, <M(x),x">E S. (ii) Injectivity: Suppose <M(x),x">=
I86
E. Casari
=<M(y),y">. Then M(x)=M(y) and xo=yo. If M(x)e N, then x=y. If M(x)E N, then, by (Nl) and Theorem 5.1.1, x=y. (iii) Surjectivity: Suppose <M(x),u>E S. (a) M(x)eN. Then M(x)={x} and u=O. It is $(x)=<M(x),x">=<M(x),u>. (b) M(x)E N. It holds X*+UEM(x); furthermore: @(x*+u)=<M(x*+u),(x*+u)">=<M(x), (x*+u)+(x*+u+O)>= =<M(x), (x*)"+u>=(by(Nl)) <M(x),u>. (iv) It is $(x+y)=<M(x+y),(x+y)"> and, on the other side, @( x)
[email protected](y )= <M( x) ,xo> + <M (y ) + yo> = <M( x)+M (y ) ,xo + y " > = =<M(x+y),xo+yo>.(a) F4(x+y)0 N. Therefore (x+y)"=O. As N is a subgroup, either y~. M(x)o N or M(y)e N and so either x"=O or y"=O. Thus, in any case ( ~ + y ) ~ = x " +(b) M ( x + y ) N. ~ It must be both M(x)E N and M ( ~ ) N. E Indeed, if N(y)e N, then, for all z ~ M ( x )z+y=z+y+O , and so M(x+y)PN. By (Nl) and Lemma 5.1.3.(2) we thus have: x+y=(x+y)*+(x+y)o,x=x*+xo
and y=y*+yo. Therefore ( x + y ) * + ( ~ + y ) ~ = x * + y * + x ~ + y. ~
By Lemma 5.1.2(2), x*+y* is principal and thus, by Lemma 5.1.3(1), x*+y*=(x+y)*. Therefore: (x+y)*+(x+y)"=(x+y)*+xo+y". Adjoining ((x+y)*+O) to both sides and using principality: (x+y)"=x0+yo. (v) It is @(x)=<M(x),(x)">=.(a) M(x)ct N. Then (x)"=O and also M(x)cr N because N is a group. Therefore x"=O and <M(x),x">=~M(x),O>=. From M(x)EN it follows that either x#x+O or x#x+O. Indeed, from x=x+O it follows xo=O, whereas frcm x=x+O it follows x=(x+O)=x* and so x0=(x*)"; however, by assumption, (x*)"=0. From x#x+O it follows, by (N2), (x)"=(x)"; from x#x+O it follows, by (N2), (x)"=(x)" and so (x)"=(x)" and then (x)"=(x)". Thus, in any case, (x)"=(x)" and hence <  M ( ~ ) , (  x ) ~ > = <  M ( x ) ,  ( x ) " > (vi) . $(O)=<M(O),O"s = =<M(O),O>=O. 5.3 MELTING AND SPLITTING WITH ORDER We now consider melting in PO and Ipregroups. L e m m a 1 In anv noprxroup G.: (1) M(x)=[y I x*gg*+O]; (2) if YE M(x), then yp*+yo; (3) " is order preserving from M(x) into M(@). Proof. (1) From y+O=x+O=x*+O and y g + O if follows yp*+O. From sy+O it follows (y+O)g aild thus, by Lemma 5.1.1(2), x * g . On the other side, if (x+O)g&(x+O)+O=x+O, then  (  x + O ) + O C j + ~ + O and so x + O C j + ~ + O . therefore %(x+O)+(x+O) and so (2) 0=(x+O)+(x+O)=(x+O)+(x+G)+O;
Comparative Logics and Abelion 1Groups
187
yg(x+O)+y+(x+O); but x+O=y+O and thus %*+yo. (3) Suppose y,zs M(x) and then y+(y+O)Ez+(y+O). But y+O=z+O; therefore y z z " . Theorem 1 In anv Dopregroup G. if M(xl is a generated melting class. then O is an prder isomorphism between M(x) and MCQZ Proof, By Theorem 5.1.1 and Lemma 1(3), O is, under the assumptions, an order preserving bijection from M(x) onto M(0). Let u,wcM(O) and G w . Let y,z be the unique elements of M(x) such that yo=u,zo=w. Therefore y Q o and so x*+yQ*+zo. Therefore, by Lemma 5.1.3(2), yEz. Remark that from Lemma l(1) it follows that all melting classes M(x) of an Ipregroup G.are sublattices of the lattice of G and from Theorem 1 it follows that if M(x) is generated, then O is a lattice isomorphism between M(x) and M(0). The melting structure of a PoDreclrouD G is the melting structure of G. as a pregroup, enriched by the relation gdefined by M ( x W ( y ) iff x+Otj+O. Theorem 2 The meltinp structure of apopregroup G is an abelian noeroup. If G is 1ordered. thus its meltine structure is an abelian 1grow with: M(x)nM(y)=M(xny); M(x)UM(y)=M(xUy). proof. We only prove the last equation. First of all remark that x+O=z+O and y+O=u+O imply (x+O)~(y+O)=(z+O)~(u+O) and thus also ((x+O)~(y+O))+O=((z+O)~(u+O))+O. By (A25) we then have: ((xUy)+O)+O=((zUu)+O)+O and so (x~y)+O=(zuu)+O. We conclude that the equation M(x)UM(y)=M(xUy) makes sense. Now: from x , y g d y it follows x+O,y+oC_(x~y)+Oand so M(x),M(
[email protected](xUy). Suppose M(x),M(y&M(z). Then x+O,y+c)Ez+O and so (x+O)U(y+O)E+O and also ((x+O)u(y+O))+!&z+O. Therefore, by (A25), ((xUy)+O)+Kz+O and so ( x l  J y ) + e + Oi.e. M(xUy)S(z). Lemma 2 If G. is an abelian DoclrouD and Z is a znowegroup. then the 2sDlitting . sub2 roup 6' is partially ordered bv t he gf G. with respect to its Iexicog_raphical order defined by <x,a>t iff x c y or (x=y and c b ) . The lexicographical order is clearly a partial order which satisfies (G6). & l (G7).Suppose Ecx,a>+. Case (i) XE G'. Then <x,a>= and so we have C; if 0=x+y, then x=y and Ea+b; therefore . Case (ii) xp G'. From x=y it follows y e G' and so =cy,O>. Therefore <x,a>=. Lemma 3 If G. is an abelian lerouD and Z is a zlDregrouD. then the 2splitting of G. with resDect to its subgrouD 6'is lattice ordered bv the IexicograDhical prder and it is:
E. Casari
That the so defined fl and u are the inf and the sup of the lexicographical ordering can be established by computations. As an example, we prove that if
[email protected] and f i x then: (1) If x u y G', ~ then both (1.1) <x,a>,E<xUy,O> and (1.2) <x,a>,E implies <xuy,O>&z,c>; (2) If x u y G', ~ then both (2.1) <x,a>,E<xUy,O> and (2.2) <x,a>,C_implies <xuy,O>E. Suppose
[email protected], ygx. Then: (i)
x,ycxUy, and (ii) <x,a>,E implies x c z and y c z . From (i) it follows that (1.1) and (2.1) are certainly true. From (ii) it follows that if the premiss of (1.2) and (2.2) holds, then x U y k . Now if xUycz, then, clearly also the conclusions of (1.2) and (2.2) hold. If xuy=z, then: if x u y c G', then <xuy,O> exists and being &c for all c, the conclusion of (1.2) holds; if x u y e G', then, being xuy=z, also ZE G' and so c=O. Therefore, the conclusion of (2.2) holds. (G10) It is (<x,a>+)n(+)=<x+z,a+c>nn)+=<x+z,a+c>n. (i) x=y. Then the first member is
<x,anb>+=<x+z,(anb)+c>=<x+z,(a+c)n(b+c)>.
From x=y it follows x+z=y+z
and so the second member is also <x+z,(a+c)n(b+c)>. (ii) x c y . Then the first member is <x,a>+=<x+z,a+c>. As G is a group, from x c y it follows x+zcy+z and so the second member is just <x+z,a+c>. (iii) y c x . Simmetrically. (iv) x Q and fix. Then the first member is <xny,O>++ C+; then <x+O,a+O> C. As ci is a group and Z a zpregroup: <x,O>C and thus necessarily, xcy. But then, for any c,d, <x,cL and so, in particular, <x,a>E.
Theorem 4 Suppose : (1) ci is a nonormal noDreerouD (IDreerouDk (2) N is the meltine noerour, (IProuD) of 6:(3) 1*( is the suberouD of ?% of all generated melting classes of 6:(4) S is the ZG)splittine of 7% with res~ect 10 N.Then the mamine 6(x)=<Mfx).xo> is an isomorphism between ci and S as nopreeroups (as Ipreeroups). G and CJ. m f . We just prove that 41 is an order isomorphism. Suppose x , y ~ Therefore x+OCj+O and so M(x&M(y). If M(x)CM(y), then, in any case, <M(x),xa>g<M(y),ya>and so $(x)Q(y). If M(x)=M(y), then, by Lemma 1(3), C J implies xQ0. Thus: $(x&$(y). On the other side, suppose <M(x),z>E<M(y),u>. Let v and w be the elements of G such that @(v)=<M(x),z>,$(w)=<M(y),u> i.e. such that M(v)=M(x), vo=z, M(w)=M(y), wa=u. From <M(v),vo>g<M(w),wo>it follows: either M(v)CM(w) or M(v)=M(w) and v a ~ w aIf. M(v)CM(w) then v+Ocu+O and so, by (N3), E w . If M(v)=M(w), then either M(v) is a singleton and thus v=w or M(v) is proper and then, by (Nl), it is generated and therefore, by Theorem 1 v x w a implies Cw .
Remark. The comDarative systems or Janus semiprouDS of Casari [ 1984],[1985],[1987] are precisely the 8splittings of abelian togroups with respect to their trivial subgroup {0}, where B is the two elements boolean algebra.
ACKNOWLEDGMENTS The author is indebted to Pierluigi Minari and to Daniele Mundici for several stimulating discussions on topics covered by this paper. REFERENCES Anderson, A.R. and Belnap, N.D. Entailment, vol.1, Princeton University Press, Princeton. [ 19751 Avron, A. The semantics and Droof theorv of linear 1oFiL Preprint. [1987] Bigard, A,, Keimel, K. and Wolfenstein, S.
E. Casari
190 [ 19771
Groups et Anneaux RCticulCs,, Lect. Notes in Math. 608, Springer, Berlin.
Birkhoff, G. u t i c e Theorv. 3rd ed.,Am. Math. SOC.Colloq. Publ. 25, Providence, R.I. [ 19671 Casari, E. &marks on comparison and superlation, Atti Convegno Naz. Logica (1979), [1981] pp.261271, Bibliopolis, Napoli. [ 19841
KomDarationstheorie und Mehrwertiekeit, Tagungsberichte des Math.
[ 1984al
Forschungsinst. Oberwolfach, 1, p.4. Note suIla loeica aristotelica della comparazione, Sileno 10, pp.131146. Loeica e ComDarativi, Scienza e Filosofia (ed.C.Mangione), pp.392418,
[ 19851
Garzanti, Milano. ComDarative Logics, Synthese 73, pp.421449. [1987] Chang, C.C. Aleebraic Analvsis of manv valued l o e i a T.A.M.S.88, pp.467490. [1958] Dunn, J.M. [1986]
Relevance Logic and Entailment, Handbook of Phil. Logic (ed. D.Gabbay and F. Guenthner), vol .III, pp. 1 17224, Reidel, Dordrecht.
Mangani, P. [ 19731
Su certe algebre connesse con logiche a pih valori, Boll.U.M.I.8(4), pp.6878.
Minari, P.L. [ 19881 Qn the semantics of comparative loeic, Z.M.L.G.M., to appear. Mundici, D. [ 19861
Interpretation of AF C*Algebras in Lukasiewicz Sentential Calculus,
J. Functional Analysis 65, pp.1563. Saeli, D. [1975]
Problemi di decisione Der algebre connesse a loeiche a pih valori, Ac. Naz. Licei, Rend. Sc. fis. mat. e nat. 59, pp.218223. Schmidt, K.D. A common abstraction of boolean rines and lattice ordered groups, Comp. [ 19851 Math. 54, pp. 5162.
Logic Colloquium ’88 Ferro, Bonotto, Valentini and Zanardo (Editors) 0 Elsevier Science Publishers B.V. (NorthHolland), 1989
191
Finitary inductively presented logics by Solomon Feferman’ Departments of Mathematics and Philosophy Stanford University
Abstract
A notion of f i n i t a r y i n d u c t i v e l y p r e s e n t e d (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems) met in practice. A f.i.p. theory FSO is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FSo is a conservative extension of P R A . The aims of this work are (i) conceptual, (ii) pedagogical and (iii) practical. The system FSo serves under (i) and (ii) as a theoretical framework for the formalization of metamathematics. The general approach may be used under (iii) for the computer implementation of logics. In all cases, the work aims t o make the details manageable in a natural and direct way. W h a t is a logic? The question here is not “What is logic?”, which (tendentiously) seeks to canonize some one distinguished system of reasoning as being the only true one. But also, here, we are not after a n y logiconly those that are syntactically described, or formal, as distinguished from those that are semantically described. For the latter, a reasonable general basic notion has evolved, that of modeltheoretic logic; cf. e.g. BarwiseFeferman (19851. Curiously (for a subject so devoted to foundational matters), there is no corresponding generally accepted basic notion for the formal logics. Such should cover as special cases propositional and predicate calculi of various kinds (classical, intuitionistic, manyvalued, modal, temporal, deontic, relevance, etc.) and styles (Hilbert, Gentzennatural deduction, Gentzensequential, linear, etc.), as well as equational calculi, lambda calculi, combinatory calculi (typed and untyped), and various applied logics (theories of arithmetic, algebraic systems, analysis, types, sets, etc.) and logics of programs.’ The first answer usually given is that by a syntactically described logic we mean a f o r m a l system, i.e. a triple consisting of a language, axioms and rules of inference, all of these specified by their syntactic form. But what does this last mean? And what
For Logic Colloquium ‘88, Padova, 2330 August 1988. (The talk itself was presented under the title, “Finitely presented logics”.) Research supported by a grant from the National Science Foundation. Thus the approach here is neutral as t o the reasons for choosing any particular logic for study, or for choosing one logic i c preference to another.
192
S. Fefemzan
is a language anyhow? In practice, languages are themselves systems of interrelated syntactic categories including such notions as sorts, variables, terms, propositional operators, quantifiers, abstraction operators, atomic formulas, formulas, etc. So what do we mean by a f o r m a l language system? Some answers to these questions have been proposed, but none so far has gained wide acceptance. Either they are too general, so that lots of things we would not ordinarily call logics are lumped in together and nothing interesting about them can be proved, or they are too specific, so that many logics met in practice are excluded, or they are too abstract, so that many significant particularities of logics in practice are simply ignored, or they are too coercive, so that everything is forced into one uncomfortable mould. The aim here is to propose a comprehensive definition of a finitary inductive s y s t e m which includes both languages and logics and which covers all examples met in practice, in a reasonably natural and direct way. As is to be expected for work of this nature, many of the ideas will have already been employed in the literature, in one way or another, for this and related purposes. The novelty here lies mainly in the particular combination of ideas, although there are some definitely new concepts which axe introduced, the main ones being that of a presentation of a finitary inductive system and thence of a finitary inductively presented logic. The most closely related work by others is, on the one hand, that of Smullyan I19611 (following Post [1943]cf. also Fitting (19871) and, on another hand, that of Moschovakis [1984], where the question dealt with“What is an algorithm?”is answered in terms of a more general class of inductive systems. The direct predecessor of this paper is Feferman [1982], “Inductively presented systems and the formalization of metamathematics”, which I refer to in the following as [IPS]. What is done below corrects, extends and improves [IPS] in certain respects, but is based on the same leading ideas; I shall refer to it frequently in the following. There are three main reasons for pursuing our leading question: (i) Conceptual. The topologists have their topological spaces, the algebraists have their algebraic structures (the v e r y abstract algebraists have their categories), the analysts have their normed spaces, the probabilists have their measure spaces, the modeltheorists have their modeltheoretic logics, so the formal logicans ought to have their formal logics. But the matter on the conceptual level goes deeper. Many subjects have been transformed by the search for the “right way” to provide a manageable systematic development of their part of mathematics, even though some of the basic ideas and results are very intuitive. Examples are the exterior differential calculus as the framework for the ordinary and higher dimensional versions of Stokes’ and Gauss’ theorems, or the categorical theory of homology as the framework for combinatorial topology. Where an exact notion of formal system is particularly needed in our subject is in the formalization of metamathematics, beginning with Godel’s Second Incompleteness
Finitary Inductively Presented Logics
193
Theorem. As we know, this is very sensitive to how the formal systems dealt with are presented (cf. Feferman [1960], Kreisel [1965], 153154). While the twoline informal argument for Godel’s theorem is very convincing3, there is no really satisfactory presentation which is both detailed and sufficiently general. But, as anyone knows who has gone into proof theory at all, this is just the tip of the iceberg. For a number of other examples, see the concluding section of this paper. (ii) Pedagogical This is simply an extension of (i). A good conceptual framework is necessary to explain the formalization of metamathematics in a reasonable and convincing way without excessive handwaving. Students gain confidence when they see that various steps can be worked out systematically and in detail. This does not mean that the subject must be presented entirely at such a level, just that the mechanics of the details are at hand when confidence falters. As intuition and experience take over, the need for such recedes, but the existence of a manageable underlying framework is always reassuring. (iii) Practical. There has been much talk in recent years of implementing various kinds of logics on computers, especially for the purposes of proofchecking. Examples for relatively specific logical systems are provided by projects of de Bruijn, Boyer and Moore, Constable, Ketonen, and others. The ELF (Edinburgh Logical Framework) project led by Plotkin (see Harper, et al [1987] and Avron, et a2 [1987]) aims to handle an enormous variety of formal logics, by a kind of reduction in each case to the typed lambdacalculus. However, in the ELF approach much preparatory work must be done by hand for each logic so that it can then be implemented on a computer. I believe the step from a logic, as it is presented in usual (humanly understandable) terms to its computerready presentation should be as natural and direct as possible. The use of some notion such as that of (presented) finitary inductive system developed here, seems to me to be essential for this goal. Because there is much ground to be covered, the work in the following concentrates on statements of notions and results; by necessity, proofs are omitted or only sketched.
See p.137 of Kleene’s introductory note to G6deZ 1931 in Godel [1986].
S. Fefemzan
194
1. The universe of tree expressions. The collection V of individuals dealt with, is taken to be built up from a class U of urelements by closure under the operation P of pairing. We write Pl,P2for the left and right inverses of P , resp., so that Pi(P(z1,z2)) = zi,and z is a pair just in case z = P(Plz, Pzz). Thus U is the class of z such that z # P(Plz,P,z). U has a distinguished element 0, and VOdenotes the subclass of V generated from 0 by closure under pairing. Elements of V are called (binary) tree ezpressions over U , and elements of VOare called the pure tree ezpressions.
The standard alternative approach to syntax takes the universe of expressions to be the class U 8 of finite strings, or words, uo . . .unl with u , E U , thinking of U as a set of basic symbols or alphabet (cf. e.g. Smullyan [1961),Fitting [1987]). These are represented here instead by finite sequences, defined below in terms of iterated pairing. The present approach builds in the tree structure of expressions met in practice, with z1,22 being considered the immediate subezpressions of P ( z l , z 2 ) . The main advantage is that it provides for a more natural and direct introduction of syntactic functions, typically introduced by recursion (from subexpressions to expressions). The same approach is followed in the familiar listprocessing programming language LISP.415 For simplicity in the following we shall work entirely with pure tree expressions. There is no loss of generality for syntactic applications over any finite alphabet U , by representation of such U within Vo. The more general situation ( V over any U ) is useful if we want to develop recursion theory over an arbitrary structure (U, . . .), as done e.g. in Moschovakis [1969], as well as treat general model theory (as initiated in [IPS]). All of our work extends directly to the more general situation. Abbreviations.
(2,
y ) := P ( z , y )
(%I) : = X I
9
(z~,..>zn,zn+~) :=((z~,...,zn),zn+l).
Note that this representation makes each ntuple ( 2 1 , . . . , z,) an mtuple for each m 5 R , e.g. ( z 1 , ~ 2 , z 3 , ~ 4 , = ~ 5( () ~ 1 , ~ 2 , z 3 ) , ~ 4 , zFinite g ) . sequences will be represented below in a modified form in such a way that each finite sequence has a definite length. As further abbreviations, we take z‘ := (z,O)
Note z’
# 0 (by
,
1 := 0’
( z , y ) # 0), and z’ = y’
,
2 := I’, etc.
+z = y.
A third alternative (ti) the theory of trees as here, or to concatenation theory) is to work in some form of hereditarily finite set theory. However, sets must still be represented by trees or lists when it comes to effective implementation. Another reason for working with trees rather than sequences is that the former have a natural generalization to syntactically described infinitary logics; cf. also the concluding remarks to this paper.
195
Finitary Inductively Presented Logics 2. Functions, classes, relations.
Functions. We use f , g , h (with or without subscripts) to range over arbitrary unaxy functions from Vo to V,. Each unaxy f determines an nary f ( " ) by f("'(11,. . . , I " ) = f x for I = ( 1 1 ) ..., I " ) , i.e. f(") is the restriction of f to the class of ntuples. We do not distinguish f(") fr3m f ; thus each f is simultaneously an naxy function for all n. (This is an immediate advantage of the pairing structure of the universe.) Capital letters F, G, H are also used for specific functions.
:;:
Classes. We use A, B , C, X , Y ,2 to range over arbitrary subclasses of VO. The characteristic function C A : VO + VOof A VO is given by C A I = .Here,and
(!
in the following, '0' represents 'True' and '1') 'False'. Relations. As with functions, each class A determines an nary relation A(") given by:
. . ,x,) E A(") iff x E A
(Ii,.
,
for x = ( 2 1 , .
. . ,I " )
Again we shall not distinguish A(") from A ; thus each A acts simultaneously as an nary relation far each n. Following usual relational notation, we shall also write A ( z 1 , .. . , x n ) for (xi,.. . , x n ) E A . Functionals and collections. At the next level, script letters like Q, 3t will be used for certain specific functionals on and to functions or classes, as e.g. G ' ( f , g ) = h, 'H(A,B)= C, etc., and script letters like F(K)will be used for certain collections of functions (classes).
3. The explicit functions. The basic functions (on Vo) are I , K O D, , P I ,Pz, where
Ix=x,
and
PI,
I(Ox=O
allx,
Pz are the inverses to pairing from $1.
The ezplicit compounding functionals are P (for pairing) and C (for composition), given by P ( f , g b = (fz79.) 9
C(f,S). = f ( P ) . Wealsowrite(f,g)forP(f,g) andfogforC(f,g).
A class 7 of functions is said to be closed under explicit definition, or Eclosed, if it contains I )K O , D, PI, Pz and is closed under P and C. We denote by & the least &closedclass. Throughout the following ?.= is any Eclosed class.
S. Feferman
196
(i) Pn,i E E for each i, 1 5 i
5 n, with
Pn,i(zl,.. . , x n ) = xi.
(ii) K , E E for each a E VO,where K,2 = a. (iii) (iv)
F F
is closed under ntupling of functions: (91,.. . ,gn)z = (9x2,.. . ,gnz). is closed under general composition:
(v) E E E , where E ( 2 1 , z z ) =
0
21 = 2 2
1 otherwise
(and EO = 0)
(vi) E, E E for each a E Vo, where
E,x
=
( 01
x=a otherwise
(vii) Q , E E where Q,z = (I,a) for each a E Vo. (viii) The propositional functions Neg, C n j , Dsj E E , where, for x , y E (0,1} we have
Neg 0 = 1, Neg 1 = 0 , C n j (2,y) = 0 Hsa: = O A y = 0 , Dsj (2,y) = 0 2 =0 v y =0 .
*
All these (i)(viii) are easily established. R e m a r k . Speaking logically, every propositional combination of equations between terms built up from variables and 0 by the €closure conditions reduces to an equation of the form t = 0, by (v) and (viii).
4. Explicitly d e t e r m i n e d classes a n d relations. A class A is said to be explicitly d e t e r m i n e d , and we write A E E , if cA E E ; we do the same for a class B considered as a relation. The following are easily checked:
6 ) (0) E E (ii) I f f E E and A E E then f  ' A ( = (zlfsa: E A } ) is in E . (iii) E is closed under n, U and  (complementation).
Finitary Inductively Presented Logics
197
Exercises. 1. For each f E E , f  ' { O } ( = {zlfz = 0)) is in E. Hence & = Kr'{O}, 0 = K,'{O} and {u} = E;'{O} axe in E.
2. A , B € & * A x B € E , by A x B = { x
IZ
= (Pi., Pzz) A P ~ E z A A P21 E B }
= { z p ( z , ( P l x , P z Z ) ) = o } nP ;' AnP ;' B
Abbreviations. A' := A , A"+' . An x A .
A collection K: of classes is said to be Fclosed if it contains {0} and is closed under f' for each f E F,U,n and . X: is said to be an F+closed collection if it satisfies the same closure conditions except possibly for the complement operation. Note that & is the least &+closed collection since if A E & then A = cil{O}. Most closure conditions stated in the following apply to any F+closed collection. We shall represent m
+ 1tuples of classes (Ao,... , A m ) by disjoint sums,
U A, x {i}.
(Ao,..., Am) :=
ism
Thus A; = Q;'(Ao,. . . , A m ) . We also write ( A i ) i s m(or simply ( A , ) )for (Ao, .. . , A m ) . 5. Primitive recursive functions and classes. The functional R for definition by primitive recursion is given by
R(f , g) = h where
{
hO = 0 h ( z ,0 ) = f x h ( z ,(Y,2)) = dr,Y, 2 9 h ( z ,Y), h ( z ,2)) .
We consider this as primitive recursion with one parameter x. By the representation of ntuples in $1,the very same functional yields primitive recursion with nparameters for any n 2 1: h ( z 1 , . . . ,zn,O) == f ( Z 1 , .
.. , z n )
h ( z l , .. . ,rn,(~,z))=g(21,...,xn,~,~,h(ll,...1rn,~)1h(x1,...1zn,2)). To obtain recursion with no parameters, i.e.
hO = u
h(Y, one applies R to suitable
(f0,go).
2)
= 9(Y,2, hY, h z )
9
S.Feferman
198
A collection F of functions is said to be PR(primitive recursively) closed if it is &closed and closed under R: The least such collection is denoted PR. A class A is said to be primitive recursive, and we write A E PR, if CA E PR. The PR classes are also €closed. For any 3,P R ( 3 ) denotes the least PRclosed collection of functions (and thence of classes) which contains 3; members of P R ( F )are said to be primitive recursive in F. 6. Presentation of primitive recursive functions. Informally, by a presentation of any specific F E PR we mean a description of how F is defined in some particular way from the basic functions by successive application of the compounding functionals. Later, this will be specified by a function term in the formal system FSO . However, there are other more ad hoc means of presentation, such as provided by the following coding system C.
Vo such that:
C is defined as the least class X ( 0 , i ) E X for i = 0,...,4, and c1,cz E X
With each
c E
+ (j, (cl, cz )) E X
f o r j = 1,2,3
C is associated a function [c] E PR by:
[(O,O)] := I, [(0, l)] := K O , [(0,2)] := D , [(0,3)] := PI, [(0,4)] := p z , [(L(Cl,CZ))l := ~ ( [ ~ l I , [ ~ Z[(2,(Cl,CZ))l I), := ~(~Cl1,~CZl)l [(3r(Cl,CZ))l := R ([cll, [cZ l)~
Every P R function F is [c] for some c E C (in fact for infinitely many c E C). Presentations of explicit functions are obtained from the subclass Co of C generated from the (O,i)(i = 0, ..., 4) by closing under (l, (cl,c z )) and (2,(clrc2)) only. Thus F E € just in case F = [c] for some c E Co. It is interesting to note the following results concerning C, though they are not needed below. (3) Substitution ("s11") theorem. There is an operation S E P R such that c E C implies S(c, a ) E C, and [S(c,a)]z = [c](a, z) for all a , 2. (4) Recursion theorem. For each P R function [f [f](e, z)for all z.
we can find e E C with [e]z =
Proofs. The function S for (3) is simply given by S(c,a) = (2, (c, (1, ( K ' a , (0,O))))) where IPO = (0, l ) , K'(a, b) = (1,( K ' u , K'b)), so that [h"a] = I
that is, N = Z l ( { O } , { ( ~ , y ) l= ~ y'}). Hence we have: Closure. 0 E N AVx(x E N
+ I' E N ) . + I' E A ) + N C A .
Induction. 0 E A A VZ(I E A
Next note that N is primitive recursive. Its characteristic function c~ is defined by the primitive recursion (with no parameters):
This is because ( y , z ) E N H y E N A z = 0. Primitive recursion on N . We can define an operator R N from R so that
h = R N ( f r 9 ) =+ h ( z ,0) = f. and h ( 2 , Y') = f(Z7 Y, h ( I , Y)) From this one obtains recursion on N with n parameters for any n Note. h is total on Vo; the equations only show how h ( z , y ) acts for
.
2 0. I
E VO,y E N .
205
Finitary Inductively Presented Logics
It is immediate that all numbertheoretic primitive recursive functions and relations are in PR. But also many closure conditions on PRN extend to arbitrary functions. For example, we have: Bounded quantification. With each f is associated g such that for each y E N and x E Vo: f(2,Z) =O) . g(z,y) = 0 * V Z ( Z < y
*
g is defined by recursion on N , with
0 ifg(x,y)=OAf(x,y)=O 1 otherwise Similarly with each f is associated g such that for each y E N and g(z,y) = O
3z(z
I
E VO,
< yAf(x,z) = 0).
Finally, we can introduce the bounded minimum
foryEN, x ~ V o .
11. Sequences. Here we follow [IPS] 53.3: each z E Vo represents a sequence, with 0 representing the empty sequence and if y represents (yo,. . . ,y,,1) then (y, z ) represents (yo,. . . ,ynl, 2).
Sequences from a class. For each class Z, the class Seq(Z), or Z < w , of finite sequences, all of whose terms belong to Z, is defined as the least class X satisfying:
OEX,
YEX z=(y,z)AzEZ Z E X
That is, Seq(Z) = 21({0}, {(x,y)lr = (y,Pzz)APzx E Z}). Again we have the closure and induction principles for Z. The following properties of the adjoint are easily checked : I* = 1 u** u (u v)* = u* v*
+
+
(X.U)* = xu* (uv)* = V*.U* N(u) = N(u*) N(uu*) = N(u)2 The last equality is the crucial one, and has a lot of consequences. For instance, if w* = 0, then N ( w * ) = 0 = N(v)2, so N(v) = 0, hence v = 0. A C*algebra is a complex Banach algebra together with an involution * enjoying the properties just listed. Standard examples are : the algebra C(X,d) of continuous maps from a compact space X into Q:; the involution is just pointwise complex conjugation. This examp!e is known as the general form of a commutative C*algebra.
226
J.Y. Girard the
algebra &(C)
of
il x
n
square
matrices;
the
involution
is just
transconjugation. norm and *closed subalgebras of B(M) where M is a Hilbert space. T h s is known to be the general form of an abstract C*algebra. If we keep in mind that a Hilbert space is determined up to isomorphism by the cardinality of its Hilbertian basis, then fl is so to speak "the" Hilbert space (denumerable basis), the other ones being either too small or too big for current needs : so our starting example is very general indeed, and we can always keep in mind that our operations in a C*algebra should be easily understood in terms of fl (or any other isomorphic space : the proviso is not trivial, since the isomorphism between two spaces may be complicated). 1.2. zoology of operators It is useful to distinguish among operators of a C*algebra K, certain classes : i) u is said to be unitary when uu* = u*u = 1. In B(H) this means that u is an isometry of H onto itself. If u is unitary, then the map v H u*vu of K into itself is an automorphism of K. The unitaries of the trivial C*algebra 4 are just the complex numbers of modulus 1, i.e. the unit circle. ii) u is said to be h e m i t i a n when u = u* ; expressions of the form vv*, v+v* are always hermitian. In Q, the hermitians are just the real numbers. iii) u is said to be a projector when u is hermitian and u2 = u. In M, given a closed subspace E of DI and its orthogonal complement F , there is a unique decomposition of any vector x as
+
x = XI XI!, xt E E, x" E F, and the map u(x) = xt is clearly a projector of B(H). Conversely, every projector of B(M) is of this form, whch means that, in C*algebras, we have an internal way of speaking of closed subspaces. When two projectors commute, then their product is easily seen to be a projector ; similarly, 1u is a projector when u is, and (1u)(x) = X I'. This shows, that commuting projectors behave like a boolean algebra... but it is not in t h s way that we want to connect C*algebras with logic. iv) when u is both hermitian and unitary, u is said to be a symmetry : u2 = 1, u = u*. Symmetries are deeply related to projectors : in the case considered in iii), u(x) = X I  x" is easily seen to be a symmetry, and every symmetry of B(M) is obtained in t h s way. In fact, when p is a projector, then 2p  1 is a symmetry ( (2p  1)2 = 4p2  4p 1 = 1, etc.), and conversely, given a symmetry s, 1/2.(1 s) is a projector, which means that symmetries are just another way of internalising subspaces in a C*algebra. v) thngs become more serious with socalled partial isometrics : u is a partial isometry when both uu* and u*u are projectors ; in fact it is enough to require that one of these products is a projector, because of the
+
+
Geometry of Interaction I : Interpretation o f System F
227
LEMMA 1 : If uu* is a projector, so is U*U. PROOF : let v = UU*U  u ; then VV* = ( u u * ) ~ 2(uu*)2 + uu* = 0 ; it has been remarked that this forces v = 0, hence uu*u = u,so (u*u)2 = u*u. QED In particular, if u*u = 1, then UU* is a projector, that nothing forces to be equal to 1. The following example is very basic : for this it is convenient to introduce the canonical base (bn) of F, by bn(rn) = 0 if n # m, = 1 if n = m. The sequence (2,) of I can therefore be written as L' zn.bn ; define operators p and q by : p(C s,.bn) = C zn.b2n q(c z,.bn) = C zn.b2n+l it is easily checked that p*(C zn.bn) = C zZn.bn q*(C 2n.b') = C ~2,+i.bn Obviously, p and q do not erase information, they display it on a smaller space, whereas p*

and q* erme information. It is also easily checked that p*p = q*q = 1, and that pp*(C 2n.b') so that
= C Z2n.b"
qq*(C Zn.bn) = C ~2~+l,bZn+'
+
pp* qq* = 1. Such operators u are called partial isometries because, when they act on concrete Hilbert spaces, u sends isometrically the subspace defined by the projector u*u (initial projector) onto the subspace defined by the projector uu* (Jnal projector). Typically, p sends isometrically (z onto the space of sequences "without" odd coefficients. It might be of interest to give an alternative description of p and q : consider the Cantor space X = {O,l}' of denumerable sequences of 0 and 1. This space is equipped with a measure p, and one can form the Hilbert space L2(,u) of square integrable complex functions on X. The
= [gdp ; define
scalar product is the familiar P(f)(O # 8) = f(s).J2 = q(r)(l
# 8)
P(f)(l # s) = 0 = q(r)(O # where O#s denotes the result of prefixing 0 t o the sequence s, etc. ; it is immediate that q * ( W = f(l # S ) . W P*(f)(S) = f(0 # s).s1/2 the coefficients $2, $212 are needed to keep the norm constant : for instance, f is send by p on g which is "the same", but on a space twice smaller, which means that the integrals have to be renormalised, and the square root comes from the fact that the norm is defined via a quadratic expression. This example can be used to demonstrate how close C*algebras can be w.r.t. computation : imagine an infinite stack of 0 and 1 : p has the meaning of "push O", q means
J.Y. Girard
228
"push 1" ; the meaning of p' is "pop O", q* means "pop 1". However one would first argue that "pop 0" means nothing, since we don't know what is the stack. But "pop 0" and "pop 1" together mean "pop", i.e. by combining p* and q* it will be possible to describe an action depending on the last value on the stack. A much more serious objection is that a concrete stack may be empty, and that this modelisation has the same defects as the algebraisation of school arithmetic, which may yield a negative answer for the capacity of a bottle. Here we stumble on the most fascinating question raised by this land of approach : when p* is to be applied to an empty stack, the operation cannot be performed, until the stack is filled. One has only to make sure that there is no deadlock, namely that there is somewhere another operation that can be performed. The study of this clearly belongs to part C of our program, and should contribute to answer the question "what is time in computation". Now, in order to make our idea work it will be necessary to replace the condition pp*+qq* = 1 by the weaker p*q = 0. T o end with partial isornetries, they have an unpleasant feature, namely that the product uv of two partial isometries is not always a partial isometry (strong difference with unitaries !) ; but when the initial projector u*u of u commutes with the final projector w* of v, then uv is a partial isometry.
1.3. direct sums The direct sum of Hilbert spaces (e.g. M and MI) is defined by talung their direct sum as
vector spaces, i.e. formal expressions x ex',with the scalar product <x Bx',y By'> = <x,x'> ; the Hilbert space F 0 P has a natural basis formed of the vectors bn 0 0 and 0 @ bm ; since this basis is denumerable, there are isomorphisms between P and fi @ f i ,typically those induced by a bijection between N and PI PI. If Q is such an isomorphsm, it is immediate that Q can be written Q(X)= p*(x) Bq*(x), for some operators p and q on @. Now it is immemate that
+
+
PP* + qq* = 1 p*p = q*q= 1. So to speak, p and q internalise a,i.e. enable us to use 0,while staying withn endomorphisms of H = fi. For obvious reasons, combinations of p and q will be enough to speak of isometries between B1 and any finite direct sum of copies of M, M", n # 0. We basically need to define (1) (2)
PI,...,pn,such that (1') r, pipi* = 1 (2') pi*pi = 1 for i = 1,...,n There are a lot of solutions, e.g. for n = 5, randomly : p1= P2, PZ = PW, P3 = pq2, P4 = 9 2 , P5 = gP, or P I = q, pz = pq, p3 = p2q, p4 = p3q, ps = p4 ; the equations (1) and (2) are enough to
Geometry of Interaction I : Interpretation of System F
229
establish their generalisations (1') and (2') to 5 partners in both cases. Once we have access to the pi we can speak of operators on Hn, but stay within operators of H. If we denote by Hi the ith factor of the direct s u m Hn, then any operator u on Hn is faithfully described by a matrix (uij) made of operators of H, uij being obtained from by restricting the domain of u to bl, and its codomain to Hi. In fact it is easy to see that if = U V, (resp. A.U, UV, U*), then W i j = U i j V i j (resp. .\.Uij, C UikVkj, Uji*), i.e. operators on Hn are the same thing as n x n matrices with as coefficients operators on H. It follows from W
+
+
the isomorphism between H and Hn induced by the pi that the algebra II,(B(H)) of operators on isomorphic to B(M), namely to a matrix (uij) associate the operator
Dln is
$ ( ( ~ i j ) )= C piuijpj*. The inverse of 9 is defined by : U(U) = (Pi*uPj). The fact that 9, Q are isomorphisms is obvious from the way we found these formulas. It may be of interest to try to verify some properties of 9 and '4 directly by algebraic manipulations ; for instance, one will need the property pj*pi = 0 for i # j (proof : first observe that, if u+v, u, v are projectors, then uv = 0, since (u+v)2 = u+v+2uv. From this, it is easy to get a nary analogue : if ul,,..,un are projectors and 1 ui is a projector, then uiuj = 0 for i # j. Applying this to ui = pipi*, we get for i # j pjpj*pipi* = 0, hence pj*pi = pj*(pjpj*pipi*)pj = 0. QED) In fact, the situation is quite general : as soon as a C*algebra X contains operators p and q enjoying (1) and (2), then one easily defines isomorphisms between the algebras &(x) and X ; this can also be used to define isomorphisms between .Nn(X) and U,(X).In the sequel we shall meet the following construction : contract the last two rows/columns of a matrix, which amounts to define an isomorphism between Hn+l(X)and &(X) : in fact to a n + l x n+l matrix (uij) we associate a n x n matrix (vij) as follows : vi, = uij for i,j # n vin = uinp* uinilq* for i # n vnj = pun, qu,, for j # n
+ +
+
+
vnn = PunnP* pUnn+lq* qUn+lnP*+ quntln+lq* Conversely, the process of splitting an row/co[umn, i.e. passing from (vi,) to (uij) is defined by : ui, = vi, for i,j # n, n + l uin+1= vinq for i f n,n+l U i n = V inP unil, = q*vnj for j # n,n+l = p*vnj unn = P*vnnP U n n + l = P*vnnq U n t l n = q*vnnP U n + l n i l = q*vnnq. In fact, in our algebra A*, we shall prefer to work with the weaker condition p*q = 0. When we no longer have pp* + qq* = 1, it is still true that there is an isomorphism which contract Unj
230
J;Y. Girard
rows, but this isomorphsm is not onto ; moreover, it does not send the unit of unit of kn(K),but only on a projector of
.Mn+l(Q on the
&(a.But this is enough for our purposes. There is no
longer any kind of isomorphism which splits rows. II.4.tensor product The tensor product of Hilbert spaces H and H' is defined in a more complex way : we first form the algebraic tensor product of H and HI, i.e. the space of all linear combinations of expressions x e x ' with coefficients in 6,quotiented by the relations : (x y) @ X I = x @ X I y @x'
[email protected](x'+y')
[email protected]'
[email protected]' (A.x) @ X I =
[email protected](A.x') = X.(x a x ' ) that we equip with a scalar product defined by < x @x',y @y'> = <x,x'>. 2. We thus obtain M2+i12+l= D''i+iz+j
M~+ir2+n = n"l+i,n MZtn,ztj = n " n , ~ all other coefficients null. If we consider M as a n x n matrix, by
M2+n12+n = II''11 removing the indices 1,2, and renumbering 3, ...,2+n into 1,...,n, then we get exactly the matrix corresponding t o the proof that would come from II' by ad hoc exchanges. But, this proof is what is currently taken as the normalised version of lI. Hence, as long as this case (which is by the way the essential one) is considered, then our formula corresponds exactly to cutelimination. Good start !
J7Y. Girard
236
From now on, we shall concentate on simple cases of cuts : namely that of
II' +A,
II"
r I
I
[A],
r,
Al, A
A
a cut between two sequents, each of them being proved in a cutfree way ; we shall assume
that the last rules (R') and (R") applied to II' and II" are (up to exchange) logical rules for A or A l . In that case, we have a way to replace the cut by other ones, and this process is the heart of Gentzen's proof, and of all its variants, such as our proofs in [2] or [3]. If we denote by E the proof obtained by this process, our goal is to relate EX(IIa,u) with EX(Sa,7) where 7 is the partial symmetry expressing the new cuts of E. We shall meet 5 cases ii) A = B 0 C, so that A1 = BI'B C l ; hence (up to exchanges, that we once for all ignore), II' comes from proofs II1 and IIz of sequents I B,I'l and I C, (with = rl, rz ) by means of a *rule, whereas II" comes from a proof II3 of I Bl, Cl,A , by a %rule. 8 is defined by making a cut between III and II3, which yields IIo, proof of I [B], CL, I'l, A, and a second cut between I I 2 and IIo yields a proof E of I [ B , q , A. To see what happens, we shall assume that rl, rz and A all consist of one formula, so that we can write a matrix, which is much more visual than indices : the matrices IIp and II3. (2 x 2, 2 x 2, 3 x 3) are given : a b e f i j k c d g h 1 m n
r
rz,
r,
X
Y
Z
Now II.is 5 x 5 :
0
pap*+qeq*
0
0 CP* gq*
0 whereas fo is 7 x 7 : a O O O i O O O c O O Moreover,
0
0 xp*+yq*
O j
b O
h 0
0 O O
O e O O f l O m O O O O O d O O g O O h x O y O O whereas u exchanges 1 with
0 2
O k
O n O O e 2,
7
exchanges 1 with 2, and 3 m t h 4 Define an
Geometry o f Interaction I : Interpretation of System F
237
isomorphism 9 from UT(B(B1))into Us(B(D1)) by contracting indices 1,3 into 1 and inmces 2, 4 into 2 by means of p, q in both cases, the indices 5,6,7 being renamed 3,4,5; then, "(Z') = Do, 9(T ) = u.(pp* qq*). So uII0 = u(pp* qq*)IIo is nilpotent exactly if TZ. is
+
+
and in that case E X ( P , u ) = ~( EX( ? ,T) ) . Now if we restrict our attention to the last 3 x 3 squares of both matrices: since 4 is identical on this square, it makes sense t o say that
EX(?,T) = EX(IIo,u). iii) assume that A = VaB, so that A1 = 3aBA ; this means that II' is obtained from a proof II1 oft B, by means of a Vrule (so a is not free in r), whereas 11" is obtained from a proof IIZ of + Bl[C/a], A by means of a %rule ; in that case, II" = D 1., II1'. = .',I Z is defined as follows : we first form 113, proof of * B[C/a], then a cut with 11, yields a proof Z of +[B[C/a]],,'I A. Now there is no change in the size of matrices involved and u = 7 ; moreover, an immediate induction (which is uniformative, since all steps are trivial) shows that II3. = El', hence EX(Eo,u) exists iff EX(Eo,7) does and in that case they are equal. iv) assume that A = !B, so that A1 = ?BI ; then II' comes from a proof 111 of I A, (with r of the form ?PI), by means of a !rule. Assume moreover that (R") is the contraction rule, so that II" comes from a proof II, oft ?Bl, ?BI, A. 2 is obtained by first makmg a cut between
r
r,
r
III and IIz, so t o get rid of (the first occurence of) ?BL, yielding thus a proof II3 of [!B],?BI, r, A , then another cut between II1 and 113 yields a proof IIo of + [!B,!B], r, r, A ; finally a sequence of contractions yields a proof Z of c [!B,!B], r, A . In fact our formula holds only when r is empty (this is not as bad as it looks !). To see what happens, let us assume that r is empty and that A consists of exactly one formula, so that we can use a matricial representation : by hypothesis IIl0 is a 1 x 1 matrix, and I,. is 3 X 3 :
I
bed
a
e h
f i
g 2
and IIo is therefore 1 @a
0
0
0 p ' bq' *+p 'cq' * +q'ep' * + q ' f q ' *
0 hp'*+iq'* j with p' = p @ 1, q' = q @ 1 ; on the other hand, Em 1s the matrix 0 0 0 0
[email protected] a b 0 C d 0 0
[email protected] a 0 0 0 0 e 0 f g
0
h
0
i
J
:
J7Y. Girard
238
Consider the isomorphsm \k from h(B('8)) to &.(B(H)) described informally as follows : the index 5 is renamed 3, and the indices 1,3 and 2,4 are respectively contracted into 1 and 2, by means of p' and q'. It is immediate that \ k ( ~ )= u.(p'p'* q'q'*). Now ly(E.) is almost IIo ; the only difference lies in its first diagonal coefficient, which is now (pp* qq*) @ a . But
+
+
+
+
+
remark that ono is nilpotent iff (1 @a)(p'bp'* p'cq'* q'ep'* q'fq'*) is, and this last expression is nilpotent iff ((PIP'* q'q'*) @a)(p'bp'* p'cq'* q'ep'* q'fq'*) is in turn q'q'*)\k(Eo) is nilpotent. It is also easy nilpotent, which is another way to say that u(p'p'* to see that, in case of nilpotency, EX(IIo,u) = EX(\k(Zo),u.(p'p'* + q'q'*)), so EX(IIo,u) = \k(EX(Z0,7)), but if we restrict to the last 2 x 2 squares on which V is identical, we get EX(IIo,cr) = EX('.,T). v) as in iv), but assume that II' comes by dereliction from a proof IIz of I BI, A ; in that case, II is defined as the result of cutting III with IIz, so that to get a proof of I [B], r, A. Here again we shall work with the extra hypothesis that r is empty, and illustrate the proof in the particular case where A consists of one formula. Assume that IIio and IIzo are respectively a b C d e
+
+
+
+
+
Then IIo is obviously
0
0
[email protected] a
0 0 whereas Zo is
PbP* dP*
PC e
a 0 0 0 b C 0 d e and u = T . The nilpotency of 76. means that ba is nilpotent. on the other hand, the nilpotency of IIo is the same as the nilpotency of (pbp*)(l @ a )= pbap*, using the fact that p*(l @a) = ap*. But p(ba)p* is nilpotent iff ba is nilpotent. Then 7E0 and uIIo are silmutaneously nilpotent. If one of them is nilpotent, then the unique coefficient of EX(E0,7) is e dac* dabac* dababac* +..., whereas the unique coefficient of EX(IIo,u) is e dp*(l @a)pc* dp*(l @a)pbp*(
[email protected])pc* ... which is equal, using p*(l @a)p= a, to e + dac* dabac* ..., i.e. once more EX(?,r) = EX(II*,u). vi) as in iv) but II" is obtained from a proof II2 of I A by means of a weakening. E is defined as follows : since all formulas of 'I begin with ?, simply apply weakenings to IIz, so that to get a cutfree proof of I A. Here again we shall assume that is empty and that A consists of one formula. Hence IIl0 and Zo have both dimension 1, and 7 = 0. Let a and b be their respective coefficients. Then E* is a 1 x 1 matrix consisting of b, whereas IIo is 3 x 3 :
+
+
+
+
+
+
+
r,
+
r
Geometry oflnteractwn 1 : Interpretation of System F
a 0
0 0
0 0
O
O
b
239
It is immediate (school computation) that IIooIIo= 0, hence if we ignore the first two rows/columns, EX(II.,u) is equal t o b, hence to Zo ; but since 7 = 0, EX(Z0,7) = So, and the property holds in that case too. vii) there is yet another interesting test case : consider a proof
II of I [!B], A, !C ending with a !B (proved by II', which comes from a proof II1 of I B by a !rule) and I ?BI, ?A, !C (proved by II", which comes from a proof IIz of I ?BI, ?A, C by a !rule). Here Z is classically defined as the result of first cutting II' with II2 so that to get a proof II3 of c [!B], ?A, C to which a !rule is then applied so that to get a proof Z of c [!B], A, !C. As usual we shall assume that A consists of one formula, so that we start with the following matrices €or III. and nz0: cut between
I
a
so
that IIo is :
[email protected] a
0 0 0
0 t ( l @b)t* t(1 @e)t* (1 @h)t*
and Zo is : t(1 @ (
[email protected]))t*
0 0
b
c
d
e h
f i
g j
0
0
t ( l @C)t*
t ( l @d)
t ( l @f)t* (1 @ i)t*
t(1 "€9
[email protected] j
0 t( 1 e b)t* t( 1 @ e)t*
0 t ( l @C)t*
0 t(l ad)
t(1 "f)t* t(1 "d (1 @ h ) t * (1 @ i)t*
[email protected] j 0 moreover, u = 7.But t ( l @ (
[email protected]))t*= (1 @ 1) @ a= 1 @ a ,and so IIo = Zo. 11.3. the main theorem : statement and discussion THEOREM 1 : i) if (II*,u) is the interpretation of a proof II of a sequent + [A], r, then oIIois nilpotent. ii) if r does not use the symbols *?'I or "3",and E is any cutfree proof of I 'I obtained from II by using standard Gentzen reduction steps in any order, then EX(IIo,o) = Z. (As usual this makes sense with the abuse consisting in removing from EX(IIo,o) the rows/columns corresponding to A , which are filled with null coefficients). The theorem must be dscussed, since there is an important restriction as t o the form of. ' l This restriction is due to the fact that in many of the cases yet considered, we had to
J.Y . Girard
240
require that the context of a !rule is void. However let us remark the following points : 1 The nilpotency of U Pis established without restriction, hence EX(II.,u) always makes sense, although it may be very far from 2'. 2Typical formulas involving "?" are the usual data types, e.g. tally integers, whose type, in linear l o g c is rnt = Va(?(a8 01) 'B (a1 18 a ) ) . Therefore, any program ending with an output of type int is not covered by the theorem ! First observe that this limitation does not apply when the result is an intermemate one, since we can only meet problems with final results. But we have here to remember that the result has to be displayed somewhere, by means of a side effecf (screen, printer, noise, etc.). These side effects are not part of logic, which does not mean that they cannot be modelised via functional analysis (but the operators involved may lose some property, e.g. maybe not longer hermitian). Here we shall content ourselves with a very primitive method, which is enough for the crude purpose of showing that integer computations are covered by our theorem. Define a new boolean type truth = Va((a 8 a) O ( a 8 a ) ) ; the truth values true and false are respectively defined as the two basic cutfree proofs of truth whose associated operators (1 x 1 matrices) are respectively : p"*q* qp(p*)2 pq(q*)z qzq*p*, and p2(q*)2 q2(p*)? pqp*q* qpq*p*. Moreover, there is a function of definition by cases of type V a ( ! a4 ( ! aO (truth 4 a ) ) ) , coming form the proof of the sequent ~  ? a l?, a l , truthl, a whose matrix is :
+
+
+
+
0
0
ip*P
0
0
0
q"p*
0
+
+
P2 Pq 0 qP2 0 0 (P*)%* 0 and which is written in traditional syntax as :
I
?al, ! a
I
?al, ! a
I 7 0 1 , 7a1, 3 a ( ( ' a 8 ' a ) 8 (7al 'g d ) ) ,a This functional proof corresponds to an IF THEN ELSE instruction the IF part is the 3rd row/column of the matrix, the parts THEN and ELSE occupy indices 1 and 2, whereas the result is given by index 4 All this shows that truth is a perfectly legtlmate boolean type Now, everythng that comes as a result can be eventually seen as a sequents of bits of fixed length, hence can be represented by a program of type some (maybe quite big) tensor power of truth, say truth", and since such
Geometry of Interaction I : Interpretation of System F
24 1
a type is free from 'I?", we are done. Theoretically speaking, it is possible to construct an object of type int 9 int transforming the (badly shaped) result of an execution of integer type into a wellshaved integer : see chapter IV. 3 To understand why our approach strongly differs from the standard syntactical "symbol pushng", remember that symbol pushing mainly rests on socalled pconversion, (Xxt)u I+ t[u/x], in which the global entity u is duplicated or erased, and moved. If we see this as a physical process, and if we imagine that such a term occupies a very big space, this operation can only be performed by some omnipotent God (or in more concrete terms, we postulate a global time for the operation). This is why traditional syntax has been of very little help for parallelism, since as soon as we need a global time, we need some synchronizing device, and such devices may be more costly than the improvement due to parallelisation. Our approach refuses any land of global time, i.e. we can only make local moves, with no need for synchronisation. Sometimes these moves are not yet possible, but some other move will make them possible. Now, OUT conception of time is roughly the same as the one coming from relativity theory, namely causality : a move p is before a move Y when Y has contributed to v. So most of moves will be temporally unrelated, and a snapshot of our kind of execution will therefore show something quite far from what syntax usually yields. The difference is so big that we have not been able to state clearly what happens in the general case (restriction on l'?l'). In fact from a purely syntactical viewpoint, the execution makes 'histakes", but it is precisely because of these "mistakes" that we can free ourselves from the need of a universal time ! 4 A last word to clarify what we mean by causality : a move p* (or q*) is a "pop" move, and is by definition impossible, unless the stack is not empty, i.e. p* makes sense only in the contexts p*p ( = 1) or p*q ( = 0). When, in the development of EX(II',a), we find a monomial q * P , the part /3 has first to be brought to the form py or qy (one has to take thls as a possible definition of first). The reader will argue that booleans are then not doable at all, since they involve a lot of p* and q* ; but we must use a side effect : to know the value of a boolean p, compute pq2, and depending on the value (qp or p2) found, deduce the truth value of p (true or false). By the way, the meaning of removing pp* qq* = 1 is to prevent 1 to occur as a sum of monomials. Concretely, this equation says that if we pop and push again what we poped, then we get the same thing : what a physical nonsense ! (think of an empty stack). In fact we are simply refusing very strongly sccalled qconversion : when we remove
+
pp* + qq* = 1, then the two proofs of I (A @ B)I, A a B gven by the identity or given by the identies on the components and logical rules are distinct operators (both are antidiagonal 2 x 2 matrices) ; but the former has coefficients equal to 1, whereas the latter has coefficients equal to pp*
+ qq*
242
J.Y. G irard
II.4. some rudiments of theory We first start with some lemmas concerning EX : LEMMA 2 : Let p, p' be two monomials of B(M) (using p,q,t,s,l,@,*),without scalar coefficients ; then they are partial isometries, and their initial projectors commute. PROOF : here we come back to the concrete interpretation of our operations in B(R) ; all the partial isometries which are our primitives are induced by partial functions from N t o #, namely, when s is one of these partial isometries, there is a partial function s from W to W such that s(C (,,.bn) = C (n.bs(n) ; moreover, we have that (ss') = sss", and when s, s' both come from partial functions, so does s a s ' , namely (s @s')() = <s(n),s(m)>. In fact the initial (and final) projectors of these isometries are all obtained from some idempotent partial function and t h s is why they commute. QED LEMMA 3 : Let u be any partial isometry of B(M) of the form I. or u (here U,,(B(BI)) has been for a moment replaced by B(H), using monomials in p and q to contract the indces), and let K be a projector belonging to the Boolean algebra B generated by initial projectors of monomials (see lemma 2). Then there is in B a smallest projector u[ir] such that : u[x].u = ux. PROOF : looking carefully at the construction of, say, ,.I we discover that it is a sum of monomial partial isometries 0; with pairwise disjoint domains, pairwise disjoint codomains. Ths means that I. also comes from a partial function from N to N. In particular, if 111x1 is defined as urn*, then u r n * is a projector in B, and um*u = uu*ux = uir. QED LEMMA 4 : If EX(II.,u) exists, then it is a partial symmetry, induced by a partial function from N to N PROOF : let x = luz, and consider the monomials pn = nIIo(uII0)nn. Then pn = pn*, moreover, pn2 = irDo(uDo)nir(Iiou)nIIoir = r ~ ~ I i ~ ( u I I ~ ) n ( I I ~ u=) nx I7 ,~xi r= "7, from some projector x' of B, hence is a projector. So pn is a partial symmetry. The initial projector KT, of pn are such that TT" TT, = 0 for n # m : if m = n+p+l, then 7" = IIo(uII.)~[(uIio)~+l[~~], whereas T, = LIo(uIi')"[r], but since (uII.)p+'[r]begms with u, then its product with ir is null, hence T,T, = 0. Finally the pi's are partial symmetries with pairwise disjoint supports, hence their sum is of the same nature. It is also induced by a part.ial function from N to N. QED LEMMA 5 (associativity of cut) : we asume that II is a proof of a sequent I [r,A], A , and that o and 7 are the partial symmetries corresponding to r and A ; then i) II.(U+T) is nilpotent iff uII. and TEX(II.,U) are nilpotent
Geometry of Interaction 1: Interpretation of System F
243
ii) in that case EX(II0,u+7) = EX(EX(II0,u),~). PROOF : let 8 = u+7, r = 1 4 , and introduce, for n E tN and q C {l,...,n} the monomial
= Uplupzu...UpnU,with u = 'I and pi = 7 for i E q, pi = u otherwise. When n is fixed, the 2n monomials pn,q are partial isometries with pairwise disjoint domains, pairwise disjoint codomains : in particular, since u( eU)n is the s u m of these 2n monomials, Bu is nilpotent iff for pnrq
some n all pn,q are null. Since ~ ( Q u=) pn,O ~ (with 0 the void set), QU is nilpotent if€some pnro is null. NOW if QU is nilpotent EX(IIo,o)(~EX(IIa,u))n splits into a finite s u m of monomials (lU2).pn,i.7(1U2).p.,j7(1U2) .... .(1U2)pp,k.(Iu2) = r.pn,i.T./im,j.T ....T.)lp,k.X (with n times 7 ) and these monomials are partial isometries of the form Ir/ra,bn. They have pairwise disjoint domains, pairwise disjoint codomains, hence if 7EX(II',u) is nilpotent too, there is an integer n such that all monomials r . j ~ , , ~are n null : but then any monomial U P ~ ~ , , ~ Ois~ + ~ U null using pir = p i , which means that for some m all monomials pm,pare null. Conversely the existence of such a m entails in a trivial way nilpotency of QU and TEX(U,O).Putting things together, we just proved claim i). Now EX(u,B) is easily shown t o be the sum of all monomials ~ p , , , ~whereas r, EX(EX(u,o),?) is easily shown to be the sum of all monomials
( 1r2).x.pn,i.T+,,,,~.T. ...7.,up,k. r.(l  ~ ~ ; but ) these monomials are exactly the monomials T ~ , , ~ using x, (14)r = r , and we established ii). QED We shall now develop in a C*algebraic framework the exact xialogue of OUT proof of normalisation of [2] (more precisely of its adaptation to the case of linear logic), as worked out in [3]. DEFINITION 1 :
The message space B is the Boolean algebra generated by initial (or final) projectors of monomials as in lemma 2. A n observable operator is a partial isometry a such that : for all n in the Boolean algebra 8 used in the previous lemmas, a*m and a m * belong to B. X will denote the set of observable operators. Composition of two observables is an abservable, sum of two observables with disjoint domains and dlsjoint codomains, is an observable. Let a and b be two observable operators ; then a is said t o be orthogonal to b exactly when ab is nilpotent (notation a I b) Orthogonality is obviously symmetric, moreover 0 is orthogonal to everything. DEFINITION 2 : Given a subset X of K, define X I = {a ; Vb ( b E X a L b)}. A type is any subset X of X equal t o its biorthogonal, i.e. X = X l l . Since Y l = Y u l for any Y, X is a type iff X is equal to Y l for some X C K ; clearly 0 belongs to any type. DEFINITION 3 : Let A be a formula of the language of linear logic under study ; let Q be a sequence of
J.Y. Girard
2 44
free variables includlng all free variables of A, and let X be sequence of types of the same length. We define the type OA[X/a] by induction on A as follows : i) if A is ai (one of the variables of a),then OA[X/a] is Xi (the ith type in X) ii) if A is ail, then OA[X/a] is Xil, (the ith type in X) iii) if A is B eC, then consider the set Y made with all operators pap* qbq*, when a and b vary through OB[X/a] and BC[X/a] respectively. Define BA[X/a] = Yll. iv) if A is B 'B C, then consider the set Y made with all operators pap* qbq*, when a and b vary through (OB[X/a])l and (eC[X/u])L respectively. Define OA[X/a] = Yl. v) if A is !B, then consider the set Y made with all operators 1 @ a when a varies through BB[X/a]. Define BA(X/a] = Yll. vi) if A is ?B, then consider the set Y made with all operators 1 @ awhen a varies through (OB[X/u])l. Define OA[X/u] = Yl. vii) if A is V B , then consider the set Y which is the intersection of all types eB[X,T/a,pl, when T varies through all types. Define BA[X/o] = Y. (Here Y l i = Y, since (.)I commutes
+
+
with intersection.) viii) if A is 3PB, then consider the set Y which is the intersection of all types (OB[X,T/a,fl)l, when T varies through all types. Define OA[X/a] = Yl. LEMMA 6 : (substatution lemma, see e.g. [2])
&'AIx,&'"/QI/a,W = ~("/fl)[X/QI PROOF : the lemma states that, if we compute &'Awith OB[X/a] as the type associated with p, or if we compute directly BC, where C is A[B/PJ, we find the same result. The lemma is proved by a straightforward induction on A . It uses strongly the fact that S(B[X/a]l) = (OB[X/u])l, which is an obvious consequence of the definitions. QED DEFINITION 4 : Let I r = + Al, ...,An be a sequent, let Q be a sequence of free variables including all free variables of A, and let X be sequence of types of the same length ; a datum of type &'r[X/a]is a n x n matrix hl = (ail) such that : i) it is a partial isometry ii) all aij are in L (so to speak, M is an observable matrisc) iii) for any E OAj[X/a]l, ..., fin E OA,[X/a]l, the matrix (Piaij) is nilpotent. An algorithm of type &'r[X/a]is a matrix hl = (ai,) of dimension 2m+n x 2m+n, for some integer m, enjoying conditions i) and ii) above, and such that, if u denotes the partial symmetry exchangng indices 1 and 2, 3 and 4, ..., 2m1 and 2m, then iv) crM is nilpotent v) the n x n matrix obtained from EX(hl,u) = (lu2)M(luM)l(lu2) by removing the first 2m rows/columns, is a datum of type Or[X/u].
Geometry of Interaction I : Interpretation o f System F
245
THEOREM 2 : If II is a proof of ,'I then IIo is an algorithm of type Br[xla]. PROOF : by induction on the proof II : to simplify notations, we shall make the proof in the case Q is empty i) if II is an axiom, so that = A, A1 ; then IIo is the antidiagonal matrix ; take any a, b respectively in BA and BAl ; we want to show that the matrix M :
r
O a ab 0 b O is nilpotent. But its square is 0 ba and ab and ba ase nilpotent, so M 2 n = 0 for some n. Before going on, let us prove a very useful lemma : LEMMA 7 ; Given a n x n matrix M = (aij), with n # 0, and a E 8(H), define CUT(a,M) as the matrix (&) : PI, = a q j , pi, = oij when i # 1 and let rl be the projector of the first row/column. Then an observable matrix M belongs to B(A,r) iff for any a in BAl, aall is nilpotent, and the nx n matrix (extracted ex(CUT(a,M)) = ( l  q ) . M . ( lr+XT(a,M))l.( 1rl), is in Or. PROOF : assume for simplicity that n = 2, so that is a formula B and M is
from)
r
o P 7 6 M E O(A,B) iff for any a and b in BAl and OBI, the matrix P[a,b] aa
a5
by b6 is nilpotent. But introduce N[a,b] as a 0 0 0
o
o
o
p
O
O
b
O
o
y
o
s
and the partial symmetries u (which exchanges 1 and 2) and T (which exchanges 3 and 4). It is immediate that P[a,b] is nilpotent iff (u+r)N[a,b] is nilpotent. Now, assume that M E B(A,B). Then (u+~)N[a,b]is nilpotent, and by associativity of cut, uN[a,b] and T.EX(N(a,b),o) a r e nilpotent. Now, if we forget the first 2 indices, EX(N[a,b],u) can be written as b
O
0
6
for a certain 6 independant of b. The fact that T.EX(N(a,b),u) is nilpotent is the same as the nilpotency of bb, hence 6E BB. (The fact that OBI is non void has been very heavily used). But this can be restated as the nilpotency of the first Aagonal coeficient of CUT(a,M), and b
246
J.Y. Girard
is easily shown to be ex(CUT(a,M)). The other direction is proved in the same way. QED
r"
ii) assume that II is proof of c [A!, A'{, A], r', coming from two proofs II' and II" of respectively + [A'], I", A and I [A1'], AL, I'l, using a cutrule. By hypothesis I I'. and Ill'. respectively belong to B(A,r') and B(A1,r"). We assume for simplicity of notations that both r' and consist of single formulas, B' and B". We first investigate the case where both A' and All are void. By lemma7, given any b' in OBI, then CUT(b',II'.) is nilpotent and a = ex(CUT(b',IIt0)) E BA. Then CUT(a,II''.) is nilpotent and ex(CUT(a,II".) E BB". Now an easy use of associativity of cut yield that uII. is nilpotent and the first diagonal coefficient of CUT(b,EX(II.,u)) nilpotent, and ex(CUT(b,EX(II.,u)) = ex(CUT(a,II".)) E BB". Using lemma 7 once more, we get that E X ( P , u ) E BB". It remains to consider the case where A, and/or At! is non void. But by an easy use of associativity of cut, it can be reduced to the case just treated. By the way, from now on, we shall ignore the cuts in the proofs, since it is always possible to first eliminate them by EX, then prove the statement and then apply associativity of cut iii) if II ends with an exchange rule, then there is very little to do. iv) if II ends with a @rule, applied t o two subproofs II' of I r', A and II" of I I'", B. Then, we can easily, by uses of lemmas 5 and 7, reduce the problem to the problem of showing that,
+
given a in BA, b in BB, pap* qaq' E B(A 8 B), which is immelate. v) if II ends with a % eu,rl' applied to a subproof II' of c r, A, B, then the problem is easily reduced to showing that, whenever a
P
Y 6 belongs to B(A,B), then c = pap* pPq* q?p* q&q*E B(A 'BB). But by hypothesis we have that, for any a in BAl, any b in BBl, a a a/? b r b6 is nilpotent, hence for any such a and b (pap* + qbq*).c is nilpotent, whch shows that c 6 B(A 7 B). v1) if II ends with a Vrule, applied to a subproof II' of I r, A, then the problem is easi!y reduced to showing that, whenever a belongs t o BA[X/a] for all X, then it belongs also to NaA. Thls is immedate. vii) if II ends with a >rule, applied to a subproof II' of + r, A[B/a], then the problem is easily reduced to showing that, whenever a belongs to BA[B/a], then it belongs to WoA. But (lemma 6) BA[B/aJ is of the form BA[X/a],hence the property reduces to the trivial fact that the union of all BA[X/a] is included in B3aA. viii) if II ends with a !rule, applied to a subproof II' of I I', A, with r of the form ?P,then
+
+
+
Geometry o f interaction I : Interpretation o f System F we must be cautious; for simplicity we consider the c u e where formula B. Assume that
I'.
r
247
consists of the
and IIo are respectively
t( l @ a)t* t ( l @ a) ( 1 @ Y)t*
[email protected] take any b in BBI, hence 1 @ b E B!(Bl). The hypothesis says that M U
P
7
8
(1 @b) P 6 is nilpotent and that ex(M) E @A.Consider N (1 @ b)t(l @ a)t* ( 1 @b) t(
[email protected]) (1 @ 7)t* 1 el5 since 1 @ b= (1 @ l @b, ) we get 1 @ b = t ( l @ (
[email protected]))t*,so N is equal to t(1 @ ( (
[email protected])a))t* t(1 @ ((1 @ b)P)) (1 @ 4 t *
[email protected] Consider the isomorphism 4 of 12(8(M)) into iself which transforms .'I into IIo ; then it is clear that Q(M) = N , so N is nilpotent, and ex(N) = rU(ex(M)) = ex(M) E BA : the hypothesis of lemma 7 holds for any element 1 @ b of .B!(Bl), and those are dense in this type w.r.t. biorthogonality ... so the property holds (easy analogue of lemma 7 ) for all objects of d!(BI). ix) if II ends with a weakening rule, applied to a subproof II' of * r , then the problem is easily reduced to showing that 0 E B?X,which is obvious. x) if II ends with a dereliction rule, applied to a subproof II' of I r, A, then the problem is easily reduced to showing that, whenever a belongs to BA, then pap* belongs to d?A. For this take a in BAl, so that a a is nilpotent. Now ( 1 @a)pcup* behaves like p*(l @.)pa w.r.t. nilpotency, but p*( 1 @ a)p = a, hence pap* I (1 @a ) ,and we are done. xi) if II ends with a contraction rule, applied to a subproof II' of I r, ?A, ?A then the problem is easily reduced to showing that, whenever a matrix (1 @ b ) a
7
a P 7 6 belongs to B(?A,?A),then ~ = ( p @ l ) ~ ( p * @ l ) + ( p @ 1 ) 1 3 ( q " @ l ) + ( q @ 1 ) ~+p( *q
[email protected] 1) ) 6 ( q * @ 1 ) B(?A). ~
The
hypothesis yields that (1 @ a ) a
(1 @a ) P
( 1 @a)y (1 @ a ) 6 is nilpotent for any a in BAl, and this can be rewritten, using contraction of indices as
+
0 = e.((p @ 1)(1 @ a)(p* 1) (q @ 1)( 1 b a)(q* @ 1)) = t.(pp*+qq*) @ a = c.((pp* qq*) @1).(1 @a),but c.((pp* qq*) @ l )= t, so < . (l @ a ) = 0, and we are done.
+
This ends the proof of theorem 2.
+
248
J.Y. Girard
II.5.the main theorem : proof First observe that theorem 2 contains part i) of theorem 1, namely the nilpotency of
[email protected] have to prove part ii), and we shall indeed content ourselves with a sketch. First we shall use the proofnets of [3], and the basic result is that, if two proofs yield the same multiplicative proofnet, then their interpretations are the same. We shall not prove it, since, SO to speak, the functional analysis model is a generalisation of proofnets, and the proof can only be boring and uninformative. Then, proofnets have been generalised in [5]so that boxes for universal quantifiers are removed. In that case, the fact that our interpretation depends only on the underlying proofnet is even more obvious, since the interpretation of quantifiers is particularly trivial. So we are left with a proof containing only !boxes. Now, due to the form of the result (without 3 or ?), we know that all possible normalisation strategies lead to the same thing, since the result does not contain any rule for I'?''.Now, it is an easy exercise to show that the following normalisation strategy can be followed : use only the contractions (in the terminology of [3], chapter 4) (AC), (@/%SC), (!/W?SC), (!/D?SC), (!/C?SC), (V/ESC), (!CC) ; (by (V/ESC), we mean its obvious adaptation to the case where there is no longer Vboxes). Moreover in all the !contractions, the !box is without context. The justification is as follows : due to strong normalisation, it is possible to use all contractions which are not !zontractions, up to the moment where all cuts are on formulas !A, ?Al, the part !A being the main door of a !box. If the result is not cutfree, then among all these boxes, there is one which is without context (this comes from the hypothesis on the shape of the conclusion), let say the one leading t o !A0 ; then we look at the last rule yielding ?A01 ; this rule is not an axiom (because of (AC)), hence must be either a weakening, a dereliction, a contraction, or a !box : in this case, ?A0 is a side door of this other box. All these cases are handled respectively by (!/W?SC), (!/D?SC), (!/C?SC) and (!CC). Hence we can make a new normalisation step, and we got one step closer to the final result (strong normalisation). Now, the rules we have been considering have been examined in 11.2. as respectively cases i), ii), vi), v), iv), iii) and vii), and in each case, we were able t o show that EX is invariant. However, the sketch is not rigourous, since in the cases we were considering, the cut under elimination was the last rule But observe that, so to speak, EX commutes with logcal rules. The typical example is when our proofnet II is a!box proving +[A],r , coming from a proofnet II' proving I [A], r', A, with r' beginning with *?'I,and r is r', !A. Now, if we assume for simplicity that both A and r consist of one formula, II" has a matrix : a b c d e f g h i j k l w x y z
Geometry of Interaction I : Interpretation of System F whereas IIo has the matrix t(1 @a)t* t ( l @b)t* t(1 @e)t* t ( 1 @f)t*
t ( l @C)t* t ( l @g)t* t ( l @k)t*
249
t ( l @d) t(l a h ) t(l e l )
t( 1 @ i)t* t(1 @j)t* (1 @W)t* (1 @X)t* (1 @Yh* 1 0 2 Consider the isomorphism O of k4 into itself whch precisely changes matrices in this way ; it is immediate that O(u ) = u, hence aIIo is nilpotent iff uDt0 is, and in that case
EX(II',u) = 't(EX(Dt0,u)). But 't(EX(II'.,u)) is precisely the result of the interpretation of the box I'!l' applied to EX(IIt0,o) and we are done. There is still another difficulty, namely that in II.2., we have only considered situations where the premises of the cut are cutfree. But this is an easy application of associativity of cut, and we are done. This ends our proof, or rather ow sketch of proof.
111. the C*algebra A* Our first intention was to introduce A* by means of an axiomatic description ; but this has two essential drawbacks : first we shall have to worry about completeness, namely making sure that some equation is not missing, and if this kind of work is difficult, it is of very little interest ; second, we would like to find a very concrete physical meaning to our operations, since we have not in mind that the execution could run through something as ugly as implementing the syntax of A*. The difficulty is that in traditional settheoretic terms, partial isometries are monsters. But this is maybe because the primitives of settheory are wrong... In fact, quite surprisingly, we can modelise our operations in a nice finitary way. The idea is that is to use (continuous) step functions on the Cantor continuum, the values being G, D, or I. To do that, we have just to push the dichotomy to a certain point. Once the values are gwen, dichotomising further will change nothing. Our primitives will be the basic moves on such step functions. In settheory, any space in which such moves make sense is infinite ; but can we dream of something more finite than that ? The Cantor continuum can of course be alternatively described by means of finite binary trees, but then we have to spent a lot of energy on changing these trees and this is why we stick to the very basic intuition of continuity. DEFINITION 5 : i) the diods are defined as the two formal objects G, D and I, equipped with a partid composition relation : G2 = G, D2 = D, I2 = I, GI = IG = G, DI = ID = D, GD and DG undefined. So to speak G is the left diod, D is the right one, and I is neutral.
250
J.Y. Girard
ii) the Cantor space is the space R of infinite sequences of 0 and 1, i.e. (0,l)" ; concatenation of sequences is denoted by #, and we shall write 0 # s instead of (0) # s etc. ; equipped with the product topology, the Cantor space is a compact space. Its basic open sets are of the form 0, = {q # s ; s 6 R}, where q is any finite sequence ; moreover, due to compactness, every clopen set is a finite union of sets 0,. iii) a pure message is a continuous map from R to the (Ascrete) diod space. A pure message can therefore be seen as a finite tree of 0 and 1 whose leaves are labelled with diods. Such a representation is not unique, but among all possible representation of the same pure message, there is one with the smallest possible tree. iv) if M and M' a r e pure messages, one may (try to) define their product as their pointwise product, which, due to the fact that GD and DG are not defined, is not always defined. v) if M and M' are pure messages, one may define their tensorisation MI' = M @ M I by M"(0 # s) = M(s), M"(1 # s) = M'(s). vi) an atomic message is a message M such that M(s) # I for all s E R. To say that M is atomic means that for any MI, either MM' = M or MM' is undefined. DEFINITION 6 : i) a pure observable (M','p,M) consists in the following data pure messages M (source message) and M' (target message) a continuous function 'p which is a bijection between iM and iM', where iM is defined as {s ; M(s) = i}, etc., enjoying the property : (P) there exist finite sequences a],..., a, (resp. bl, ...,b,,) of 0 and 1, such that the domain (resp. the codomain) of 'pis the dsjoint union of the 0 (resp. Ob.) and for i = 1,...,n and s E R, we ai I have 'p(ai # s) = bi # s. ii) another way to represent pure observables is to consider 3tuples (U,'p,T), where T and U are finite binary trees made of 0 and 1, with leaves labelled with G, D or I, and 'p is a bijection between the set of Ileaves of T and the s e t of Ileaves of U. Of course such a representation is never unique. iii) let (M','p,hf) be a pure observable, and let N be a pure message ; we (try to) define the product (M',(p,M)N aa follows : if M N is undefined, then (M','p,M)N is undefined otherwise, define 'p' as the restriction of 'p to i(MN), and the message MI' by M"(s)= M'(s) except for s 6 iM'  rg('p'), in which case, M"(s) = M('pl(s)). iv) in particular, when N is atomic, then 'p' is always void, and MI' is atomic ; to each pure observable (M','p',M) is therefore associated a partial function f from atomic messages to atomic messages, which satifisfies : f(N) defined iff (M',(p,M)N defined and in this case
Geometry of Interaction I : Interpretation of System F
25 1
(M','p,M)N = (f(N),@,MN). Two pure observables with the same induced function on atomic messages must obviously be equal, v) the product of two pure observables (M','p,hl)(N',$,N) is defined exactly when MN' is ; in that case, we form the products (M',p,M)N' = (M",p',MN') and M(N',$,N) = (MN',$',NIl), and we define (M',p,M)(N',$,N) = (M",'p'$',N"). The definition used M(N',$,N), whose definition can be imagined easily. The product is defined in such a way that it corresponds to to the composition of the associated partial functions. The product is undefined exactly when the composition of the associated partial functions would we nowhere defined. We can think of pure messages as a particular case of pure observables, mamely by representing M by (M,'p,M), where 'p is the identity on the Ileaves of M. This identification is compatible with the two definitions of product involving messages. As an observable, a message has an underlying partial function, which is idempotent, i.e. can be seen as a set of atomic messages. vi) the tensor product (M','p,M) @(N',$,N) of pure observables is defined as (M' @N','
[email protected]$,
[email protected] N ) , with ( p a $)(O # s) = 0 # ds), ('pa $)(1# s) = 1 # $(s). This definition extends the one already gven for messages. As to the induced partial functions on atomic messages, it is immediate that if f , g are associated with two pure observables, then h defined by h(M @MI) = f(M) @g(M')is associated with their tensor product. vii) given a pure observable (M','p,M), one can define its adjoint as (bl,p',M'), so that pure messages are selfadjoint. In terms of induced functions, adjunction is just inversion. Here we have to remark that the partial function associated with a pure observable is a bijection between its domain and its codomain. It would now be the room to check endless properties ; the best is to remark that we came as close as possible to our intuitions of chapter 1. Consider a bijection n, m H between N x P( and N ; it does not cost much to assume that = 0, = 1 and that all integers are generated from 0 and 1 by means of .Then to each atomic message M, we can associate a unique integer $M,namely $G = 0, $D = 1, $(M @ N ) = (by G we mean the message constantly equal to G etc.). So atomic messages are in bijection with N, and we can say that to any pure observable is associated with a partid bijection from PI to N, hence with a partial isometry of C When we translate our definitions of product, tensor, adjoint into these partial isometries, we just find the notions we introduced in chapter 1 ; when the product is not defined, then the associated partial isometry is 0. This remark is enough t o understand what has been done so far. Among pure observables, let us mention i) t = (1,pJ) : the initial and the final message are both I (constant function with value I) and rP(0 # s) = (0,O) # s, 'p((1,O) # s) = (OJ) # s, d ( L 1 ) # s) = 1 # s. ii) s = (1,4,1) : fl0 # s) = 1 # s, dl # s) = 0 # s (sorry for the mismatch of letters ; more
252
J.Y. Girard
seriously, remember that we never used s, and that its eventual removal from A* is not excluded). iii) p = (G @I,O,I), q = (D @I,O,I) with O(s) = 1 # s. For instance it is immediate that , if M is an atomic message, then pM = G @ M , qM = D @M, hence p and q induce the functions n H and n H < l , n > , as ewpected. DEFINITION 7 : i) the space PA* (p for "pre") is defined as the set of all formal finite linear combinations C Xi.Pi, where the Pi are pure observables and the Xi are complex coefficents. If by convention, we decide that PP' = 0 (the null linear combination), when their product was not defined, there is no difficulty in extending by (bi) linearity the operations of product, tensor and adjoint, so that, together with the obvious notion of sum we get a C*algebra, without yet a norm. The object I plays the role of the unit ii) since there is an obvious interpretation of.our linear combinations as operators of P , there is at least one possible C*norm on PA*, namely the norm of the associated concrete operators. We didn't look serously at the marginal question of determining whether or no this norm is the only posible one, but there are two simple facts, namely first that there is, as usual a greatest C*norm, and second, that there is a smallest one, namely the norm N coming from the P representation. (see below) Then we norm pA* by means of N, and complete so to get A*. LEMMA 8 : N is the smallest C*seminorm on PA*. PROOF : let (m,) be the enumeration of atomic messages which has just been introduced ; if we define the pure observables bn as bn = (m,,@,mo), and if N' is any C*seminorm on PA*, then N'(bn) = fl'(bO), hence with n = 0, N'(b0) = 0 or 1 ; but mg = 0 would induce mo @ I= 0, and then we would get pp* = 0, hence p*p = 0, but p*p = I = 1. So N'(bn) = N'(b0) = 1. In particular, N'(C Xi.bi)2 = N'(C Xi.bl.CXi.bi*) = 1 X i . 1 , . Now, if u is a n y operator of PA*, then u(C Xi.bi) = C pi.bl, and we pass from the sequence ( X i ) to the sequence ( p i ) by means of the operator of P that we associated with u. Hence N(u) = sup{N(C pi.b') ; N(C Xi.bi) = 1) = sup{N'(Cpi.bl) ; N'(C 1i.b') = 1)
5 sup(N'(uv) ; N'(v) = 1) = N'(u). QED The space A* is incredibly concrete. To finish with
OUT definitions, DEFINITION 8 : i) a message is any fimte sum of pure messages whose pairwise products are null. ii) an observable is any sum of pure observables whose domains are pairwise incompatible, and
Geometry of Interaction I : Interpretation o f System F
253
whose codomains are pairwise incompatible. Observe that in the boolean algebra genrated by pure messages, we only consider those projectors which have positive coefficients ; in the same way, ii) is stronger than saying that an observable operates on messages. It is easy to see that we were in fact using this refinement in chapter 2. By the way, let us remind the reader that there is no way to say that 1 is the s u m of all atomic messages (such a denumerable sum cannot converge in norm, and only makes sense w.r.t. some kind of weak topology, typical of socalled Von Neumann algebras, which are another world). Finally, the only question at this moment is to decide whether or not OUT kind of computation is feasible. But the result is a sum of monomials ; we already discussed this issue in 11.3. and reduced it morally (more refined studies should be made later) to the case of a boolean result, and we remarked that it was enough to compute the expression EX(IIo,a)q2, and depending on the result pq or p2, the answer was found. Now, our expression is a finite sum of observables, and therefore all summands but one are null. This is one of the main reasons for deleting the equation pp*+qq* = 1, which would have induced possible synthesis of observables. So one of the monomials is 1, whereas the others are zero. In particular, if we do the execution from right to left, starting with 92, and then making the development : qz, II.(l$)q2, (lo2)IP(la2)q2
[email protected]*(1u2)q2, (luz)II*( 1o2)q2 + ( 1u2)IIouIIo( 1oz)qZ IIooIIooIIo(l02)q~, etc., then each step consists of exactly one monomial. This is because everything is isometric, and in order to get two monomials, we should therefore have a choice between multiplying on the left by, say, ap* and pq* at some moment. But then the only way t o recover a monomial pq or p2 at the end is that the right part is of the form py or qy etc. So there is at least a way to execute. This way is by no means the best, and part C of the program should be concerned with the study of efficiency, as long as this remains a mathematical problem. From the moment on, we obtained what we were longing for, namely
+
+
a finitary dynamics free from syntax.
N . example : tally integers Integers in tally representation are interesting from an abstract vlewpolnt because they behave to some extent like sequences, but are simpler. We shall introduce their representations in A*, and demonstrate the dynamic power of (EX) on simple functions like iterators.
J.Y. Girard
254
DEFINITION 9
:
i) the type intis defined as Va.(!(a4 a) 4 ( a9 a)). ii) the integer No is defined as qp(q*)2 q2p*q*, for k # 0, the integer
+
Nk
is defined as
c (PP'itlPq(P*)'P'i*P* + PP'iP'q*P*P'itl*P* + + dP*)'P'k*P* + PP'kP2P*9* + q2q*p*p10*p* + pp'opq(q*)2, where p'o, ...,p'k are pure observables (to be defined below) of the form ri
@ 1 such
that p'i*p', = 0 for i
# j, and the sums are taken oven 0, ...,k1. The ri's
are not yet defined, since they come from contractions inside the kth canonical proof of the formula int, and therefore, there are several possible choices, depending on the order of the contraction. Their actual values will be obtained by means of the successor function.
DEFINITION 10 : The successor function is the 2
x
2 matrix coming from the proof of
I
infl, id,which
precisely defines the successor. We therefore get :
0 P(P Q l)P* + P(q
Q
P(P* Q l)P* l)pYq*)2
+ WP%*
+ q'(p*)'(q*
P(9 Q I)pq(q*)'
Q
l)P*
+ qPP*q*
+ q%*p*(q*
Q
I)P*
LEMMA 9 : It is possible t o chose the p'i in such a way that the successor of Nk is N k + l . PROOF : let's write the successor as O B B* C then the successor of Nk is EX(M,a), where u exchanges the first two indices of the matrix M : Nk 0 0 O O B 0 B* C and EX(M,o) is easily shown to be equal to C + B*NkB, i.e. to :
c
P(q Q l)Pq(q*)' + q2q*P*(q* Q I)P* + P(P Q l)p'i+lpq(p*)2p'i*(p*Q I)P* + C P(P Q l)P'iP2q*P*P'itl*(P* Q I)P* + P(P Q l)p'oPq(P*)2(q* Q l)p* p(q Q l)p2q*p*p10*(p*Q l)p* p(p @ l)p'kp2p*q* qp(p*)2p'k*(p* 8 l)p*. ro = q, r;+1= pri. Then our expression rewrites as :
+
pP'OPq(q*)'
+
+
Now
define
+ q2q*P*p'oxp* + C P P ' ~ + ~ P ~ ( P * ) ~ P ' +~ +CI PP'i+lP2q*P*p'i+2*p* *P*
+ PP'lPq(P*)2P'O*P* + PP'OP2q*P*P'l*P* + PP'k+lP'p*q* + W(P*)2P'k+1*P*, 1.e. is equal to
Nk+l.
QED
DEFINITION 11 : Take any type A , m an object of type A, M (a square matrix) of type
I
A l , A ; then the
iterator It(m,M) is a square matrix deduced from the logical proof corresponding to iteration (and whch is of type I t n t l , A) whose coefficients are, provided M is
Geometry of Interaction I : Interpretation of System F a c ~ ( 1 (pap* @
255
b d
+ pbq* + qcp* + qdq*))p* + qpmp*q* (q*Y
q2
0
Typical examples are :
i) with A = int, m = No, M = successor : It(m,M) is called the shaving functional ii) with A = !id, m = 1 8 No, M being the matrix 0 t ( l @B) (1 @ B*)t*
[email protected] C deduced from the successor matrix by means of the !rule : It(m,M) is called the linearising functional . THEOREM 3 : Let A , M and m be as in definition 11 ; then the result of applying It(m,M) to the integer Nk is exactly Mk(m), if we use the notation M(m) to denote the result of applying M to m (i.e. making a cut on A , then executing), etc. PROOF : add a new symbol A to the formulas of linear logic, together with axioms (i.e. syntactical boxes) t A ( a denumerable family, the kth being interpreted by Mk(m)) and + A l , A (to be interpreted by M). Then we can adapt the machinery of theorem 1 t o this case : syntactically speahng, a cut between the kth axiom c A and our new axiom c A l , A being reduced to the k+lth axiom + A. Then everythng works, since the result is of atomic type. The result of cutting the kth axiom with the new identity axiom is Mk(m)
0
0
0 0
a b c d and after execution, the result is Mk+'(m), i.e. the only new feature of the syntax is interpreted in the right way. QED
REMARKS : i) the result still holds (associativity of cut) when we don't plug in Nk, but the operator obtained by execution of a program of type int, whose syntactical result is the kth integer. This simply comes from the fact that semantically speaking, the final result of type A, must be Mk(m). ii) in particular, if we feed the shaving functional with a (semantic) integer coming from the execution of (the semantic translation of) a proof of type ant, then we find a Nk. iii) it is easily checked that the linearisation functional does two thngs : first shave the input into some Nk, then replace this Nk by 1 @Nk. The name of this functional comes from
J . Y . Girard
256
the fact that it can be used to replace a general function defined on integers (type t ?inti, B for some B) by a linear one (type I int, B). We shall here suggest an exercise, namely compute the shaving of Nk ; there is nothing new here, but toying a little with this objects, which, on the whole are very concrete ones, may be illuminating : the question is to compute EX(P,u), where u exchanges the first two lines of P : Nk 0 0
+
+
0 ~ ( 1 b0B q * + qB*p* qCq*))P* d J o P * q * 42 0 (q*P 0 The concrete computation must rather be done on a blackboard than inside a mathematical paper ; good luck !
V. two ideological themes V. 1. communication by nilpotency
Our basic claim is that nilpotency expresses the absence of loops inside the information flow. To illustrate this, we have the mathematical development already made, but also a toy example, namely the paradigm o f the dictionary : consider a (technical) dictionary, where a set of words, WI, ...,wk is explained, by use of current language. Concretely, some terms are defined using other ones, e.g. k = 4, and w1 is defined in terms of w2 and w3, w2 is defined just by means of current language, the definition of w3 involves w2, whereas w4 is defined by means of w1 and w3. Now, our dictionary works like a miniprogram, namely, to understand w4, we are reduced to w1 and w3, which are in turn reduced to w2 and "3, which are in turn reduced to w2, and w2 is reduced t o nothing so that wq is eventually understood. Now we write a 4 x 4 matrix expressing the dependency (a,,= 1 when w , is defined in terms of w,, = 0 otherwise), then we obtain : 0 0
1 0
1 0
0 0
0
1
0
0
1
0
1
0
Now, to say that our dctionary is sound means that we never loop when trying to find the
meaning of a word. In other terms, it is possible to relabel OUI words as w'i, so that each w'i IS defined using only the previous ones. Typically, in our example, relabel 1,2,3,4 as 3,1,2,4, and the matrix becomes :
Geometry of Interaction 1 : Interpretation o f System F
257
0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 and we have obtain a new matrix which is obviously strictly triangular. Such a matrix is nilpotent, and this nilpotency is independant of the numbering of rows/columns. Observe that very often the relabelling is far from being unique, hence the formulation of nilpotency which does not involve any sequentialization is much more manageable. To some extent, it is what we have been doing, except that instead of coefficients 1, we were using partial isometries. T o illustrate how these kind of considerations could be of possible logical interest, consider the old question of Henkin quantifiers. These quantifiers express, like all quantifiers, dependencies of existential variables over universal variables. But here Q~x'yy'.A means that one can simultaneously find the value of y in terms of x and the value of y' in terms of X I , Traditional (sequential) quantifiers cannot express this lund of dependency, e.g. if we write Vx3yVx13y'A, then y' is supposed to depend on both x and x', and therefore, sequential logic (the only kind of extant logic) will therefore fail to express this subtle situation. We know by abstract modeltheoretic considerations (essentially Lindstrom's theorem) that there is no way to fix this defect without destroying the main properties of classical logic. With linear logic, the situation becomes slightly better, since we are building the l o s c on a symmetry proojs/counterproojs and the arguments of model theory do not apply. The only problem is now to find out what could be the abstract dual form of dependency (coHenkin quantifier) Qlyy'xxl of y and y' over x and X I such that, when doing normalisation, the actual value of the terms is eventually found. Two quantifiers R and S (one expressing the dependency of the yj over the xi, the other expressing t,he dependency of the xi over the yj) will be said to be orthogonal when, on the sole basis of the dependencies, the values of the xi and the yj will eventually be found. A quantifier is a way to speak of functions, without carrying them, only remembering the dependencies : this is why the logical approach should consider the question independently of the concrete functions involved. Now, if we form a dictionary with the xi and the yj as entries, we are reduced to the previous problem, namely t o look for a nilpotent dependency matrix. Coming back t o Henlun, Q I will be a set of possible dependencies of the y over the x, w h c h can be expressed by matrices a b c d (columns : y,y', rows : x,x') ; the Henlun dependency being expressed by
J.Y. Girard
258
1
0
0 1 (columns : x,x', rows : y,y'), and we are trying to find out which possible values 0 and 1 for a,b,c,d will make the square matrix O O a b
O
O
c
d
1 0 0 0 0 1 0 0 nilpotent. I t is immediate that the only possibilities are : a=b=d=O c = l a=c=d=O b = l a=b=c=d=O in other terms, QLyy'xx'A says t h a t y depends on x', y' depends on x , and one of these dependencies is fake. One may ask what is the biorthogonal of Q,and it is easy t o verify that Q is its own biorthogonal. Then this inhcates the way of handling parallel dependencies in logic. This should eventually be of interest for those questions related t o parallel execution, where usual logics force us t o stick t o sequential dependencies V.2. the absence of understanding, or genericity This has been one of the main underlying themes of our modelisation. The basic idea is that typing is not only a way of making sure that a program will eventually terminate, will produce the right answer : it is a measure of the degree ofgenencaty of the unit we have typed. Second order typing possesses two ways of abstraction, the one coming from exponential connectives, the one coming from quantifiers. When we pass from a proof of + A [ B / a ] t o a proof of I 3aA, B is irreversibly lost ; we can argue that, anyway, our formulation does nothing in that case, but observe that replacing B by an isomorphc type B' would change n o t h n g to nilpotency (and nothing to the result of an execution with a cut on 3aA). In the same way, the rules of dereliction and contraction involve partial isornetries, whose choice is random : typically, if we introduce po = p , p, = q, pk = (p; @ l ) p j when k = < i , j > , then we can use any pk (not always the same) in any dereliction, and any two pairs pi 0 1, p, d 1 (with i # j) in any contraction without any need to use always the same pair. This is because the dual operations (Va, !) do not at all know in advance what land of say duplication isometry will be used, and is prepared therefore to treat them generically, without understanding them. T h s genericity does not extend to the other operations : typically, we have decided once for all that p and q should be used in this order for all @ a n d all 7, and if we were changng our mind, then we would have t o make a new uniform choice. So t o speak, the typing, which basically inhcates to which depth we are analysing things, tells us that anything beyond this
Geometry of Interaction I : Interpretation of System F
259
degree of analysis, is "up to isomorphism". Thus genericity is ultimately the greatest warranty for modularity : in a cut (communication) between A and A1 there is a finite coon language, namely the operations whch correspond to the logical decomposition of the type A, and all other instructions are internal to the two participants. (In real life we communicate
using words, but this limited interface does not fully expresses our thougths). Beyond the logical level, every protagonist is unable to do anything but generic operations on the instructions send by its opponent, and this is what we call communication without understanding
This philosophical (or ideological) point has been too important during the genesis of our program not to have been developed here. In later works to come, this thesis will be formulated in precise mathematical terms. Just for the moment, let us remark the following, namely, that if we accept understanding, then there is a lstep predecessor function on the integer type, namely the 2 x 2 matrix 0
B*
+ q2(q*)2
B + s2(q*? 0 where B has been defined in lemma 9. Let us call it P ; if we compute in ths way the predecessor of Nk, then the result is (B + q2(q*)2)Nk(B* + q2(q*)2) ; then, when k = 0, we obtain No ; when k = i + 1, so that N k is C + B*NiB, we find BB*NiBB*, whch is equal to N,. The computation is lstep since the development of the formula (EX) is cut after the second monomial. But this computation is possible only because we know the combinations (e.g. p @ 1, q @ 1) used in the contraction rules ; if we were making other choices, then the result would be inpredictible, for instance it might lead to 0, i.e. erase everythng. However, one may argue that would be posssible to work with int as a new atomic type with two primitives, zero and successor, the successor function being no longer defined up to some isomorphism (i.e. p @ 1 and q @ 1 rigdified), and w.r.t. this new primitive type, there would be no possibility of mistake.
J;Y. Girard
260
BIBLIOGRAPHY [I]
Cuntz, J .
Simple Ctalgebras generated by isometries, Comm. Math. Phys. 57, pp. 173185, 1977.
121
Girard, J.Y.
Une extension de l’interpr6tation de Godel b l’analyse et son application ir l’klimination des coupures dans l’analyse et la thkorie des types, Proc. 2nd Scand. Log. Symp., ed. Fenstad, pp. 6392, N OrthHolland 1971.
[3]
Girard, J.Y.
Linear Logic, Theor. Comp. Sc. 50, pp. 1102, 1987
[4]
Girard, J . Y .
Muitiplicatiues, to appear in Rendic. Semin. Univ. Polit. Torino, 1983.
[5]
Girard, J.Y.
Quantifiers in lanear logic, to appear in the proceedings of the Congress SILFS, held in Cesena, January 1987.
[S]
Girard, J.Y.
Towards a geometry of interaction, to appear in AMS volume dedicated to the congress ”category theory and computer science”, held in Boulder. June 1987.
[7]
Kelly, G.M.
On Mac Lane’s condatrons f o ~coherence of natural assoctatrurties, Jour. Algebra 1, pp. 397402, 1964
Logic Colloquium ’88 Ferro, Bonotto, Valentini and Zanardo (Editors) 0 Elsevier Science Publishers B.V. (NorthHolland), 1989
26 1
Intuitionistic formal spaces and their neighbourhood
GIOVANNI SAMBIN Dipartimento di Matematica Pura ed Applicata Via Belzoni 7  35131 Padova
Formal topologies and formal spaces are the basic notions of an intuitionistic “pointless” approach to topology, which grew out of a suggestion by P. MartinLof and collaboration since then (definitions and fist results have been given in [IFS]); however, apart from the foundational interest of building up topology within MartinLof’s theory, the technical tools developed in that work revealed to be flexible enough to be used in other areas. In particular, as shown in the present paper, after minor adjustments they provide with simple and complete propositional semantics for Girard’s linear logic, as well as for usual intuitionistic logic. The central notion here is that of pretopology, which roughly speaking is obtained from that of formal topology by isolating and disregarding those conditions which correspond to the structural rules of weakening and contraction (section 1). In a certain precise sense, pretopologies are the weakest possible structures in which an operation corresponding to implication is definable (section 2); they give a sound and complete semantics for a ground logic, here called minimal linear. Completeness of intuitionistic logics, with or without either structural rule, is then an easy corollary (section 3). By restricting to those pretopologies in which every open, or saturated, subset is regular, one obtains exactly Girard’s phase semantics, and hence completeness for the classical system, but also an extension to the linear case of the usual double negation interpretation (section 4). It is also possible to treat exponentials, an essential feature of Girard’s linear logic, in the pretopology framework, but a firm grasp is still to come (thus only a sketch of technical results is given in section 5, together with some final comments). The whole paper, like [IFS], is written in the framework of MartinLof’s constructive set theory [ITT], with subsets defined as in [IFS], but it can be read independently of both of them (I have occasionally put some remarks on the connection with [IFS]in the footnotes). The definition itself of pretopology grew out after attending some lectures on linear logic given by Michele Abrusci in Padova, June 1987. I thank him for that, and also for writing the paper [A] in which some of the problems I met were already solved. Of course, I a m also indebted to Girard’s [LL], which turned some isolated results (see [A] for some bibliographical references) into an active field of research. Finally, I thank JeanYves Girard and Per MartinLijf for helpful conversations in the days of the Logic Colloquium, but most of all for creating (or exploring) fascinating
G. Sambin
262
regions in the country of logic, still not so far from each other that I could not connect with an easy and, I hope, natural road. Silvio Valentini has collaborated to build such road directly by spending together with me several pleasing days of work on the topics here treated. 1. The definition of pretopologies.
The notion of pretopology is the combination of two notions, which we define separately. 1. A (formal) base is a structure S = (S,., 1,I)where: DEFINITION 1. S is a set with a binary operation and a distinguished element 1 such that (S,,1) is a commutative monoid, that is:
associativity commutativity one hold for any a , b, c E S ; 2. Iis any unary predicate, alias a subset of S. We will always use a , b , c . . . for elements of S, and U, V,W,2 for subsets of S. An intuitive picture has helped my imagination, and might help the reader as well: the set S is considered to be the universe of (formal, mental, concrete or whatsoever) objects a , b, c . . . of an individual mind, 1 is the object carrying no information, a . b is the result of putting a together with b. Then I is the subset of impossible objects (a magic wand, a sixlegged dog, ...), and a . b E Imeans that a and b are incompatible (which of course may hold also when a , b are not impossible).
2. A relation 5 between objects and subsets of S is called a precovering DEFINITION relation, or a precover, on the base S if it satisfies aEU
reflexivity
transitivity
a),
K,]
fF1(B).
B, then fG1(A)KA, fG1(B).
R . Chuaqui
312
We have the following theorem, which is a generalization of the process described above for UB1 and UBZ. We assume, as earlier, that all our structures are finite. This theorem, however, can be generalized to infinite structures, but the definitions and proof become more involved.
Theorem 18 FOT each finite equiprobability structure H there is a super classical equiprobability structure K and a homomorphism f of K onto H. Proof. Let T be the causal ordering of H. We define K,, Kx,, Kt], K,, ft,f ~f i ~ l, and fm by induction on T . We start with To. Notice that for t E To, A, = t . Let m be the minimum common multiple of #Ht for t E To. Put as Kt a set of cardinality m, for every t E To. Suppose that #Ht = n. Then n divides m. Partition Kt into n parts of m / n elements each. Define ft (which is also f ~from ~ Kt ) onto H, such that the elements of each class in the partition are sent to a distinct element of H,. Suppose that f, and Kt are defined for t E T,. Similarly as above, let m be the minimum common multiple of #H,, for t E T, and p E H, and for X E K,, let F be a set of cardinality m ,and let Kx = F . We assume that for each X E K , , ft(X) E Ht. We define fx, for X E K,, a function from Kx (= F ) to Hft(x)as ft was defined for the case n = 0. For each t E T,, define
Kt] = { X
n
w : X E Kt and w E Kx }
and
ft](X
W)
= f4X) A
fA(W)
For s E T,+l, define K, by
K, and
={
U{h ( t ) : t i.p. s } : h E n ( K t l : t i.
fs,by f s ( a )=
p. s) }
U{ftl(at1) : t i.p. s 1.
Finally, define
and
fn(X) = U{.ft](At~) : t 1.e.
U{T, : i 5 n } }.
We can prove by induction on T that f is a homomorphism of equiprobability structures.
,
313
Probabilistic Models
8
Induced measures
In this section, we shall study measure transported by a homomorphism from one equiprobability structure to another. We now define a provisional measure prK, which for super classical structures coincides with the definitive measure PrK. prK is the invariant measure defined on the disjunctive field of sets generated by the measuredetermined subsets of K .
Definition 19 Let K be an equiprobability structure. Then FKis the disjunctive field generated by the memuredetermined subsets of K and PrK is the invariant measure defined on FK.The provisional probability space of K is (K,FK, PrK). For a classical structure K we know that FK= P(K) and PrK = PrK.
Definition 20 Let H and K be equiprobability structures with causal universe T , f a homomorphism from K onto H, and (K,FK, prK) the provisional probability space of K. We define P ~ K H ( A= ) pr~(f'(A)), for any A E H with f'(A) E FK,and, for K super classical, Prj(xt)(A)= PrK~,(f;:t)(A)), for A
G H~(A with , ) ff;:,)(A) E FK~,.
prf(x,)is well defined because Kx, = K,, for any A, p E K , and f ( A t ) E Kt. Notice that PrKH depends also on f. If K is super classical, then PrKH is defined on d subsets of H. w e now prove that the main properties proved for the measure PrK, for K super classical, are transferred to PrKH. so for the next lemmas, we assume that K is super classical and that f is a homomorphism from K onto H.
Lemma 19
If X E H
and
t E T,,with 0 < n, then
PrKH({At}') = n(P'KH({As]}o) :
i. P. t ) .
Thus, PrKH is the product measure when restricted to Ht. Proof. We have, by Theorem 17, f'({At}o)
=
U{f'({Xs]}O) : s i. p.
t }.
Hence, by Theorem 16,
By the definition of PrKH we obtain the conclusion of the theorem.
R. Chuaqui
314
9
The external invariance principle
We are now ready to introduce the definitive definition of the disjunctive field of events and the measure. The basic idea is that two homomorphic equiprobability structures should determine the same probability measure. This is what I call the External
Invariance Principle. Definition 21 Let H be an equiprobability structure with causal universe T . 1. A subset A of H is Hweakly measureddetermined if and only if for every equiprobability structures K and K' and every homomorphism f from K onto H and f' from K' onto H such thai prKH(A) and prK,H(A) are defined, we have that prKH(A) = prK,H(A). 2. The family of events of H, FH,is the disjunctive field of subsets of H generated by the family of weakly measureddetermined sets. 3. PrH is the measure defined for A E homomorphic to H.
&
by PrH(A) = prKH(A), for any K
4. The probability space of H is the triple ( H , F H , P ~ H ) . First we state that it is enough to consider in the definiiion of weakly measureddetermined sets, super classical equiprobability structures K and K'. The proof depends on the fhct that we can always find a super classical structure homomorphic to any structure.
Pmbab ilist ic Models
315
Theorem 21 Let H be a n equiprobability structure and A 5 H . T h e n A is weakly measureddetermined if and only if f o r every pair of super classical structures K and K' homomorphic to H, prKH(A) = PrK#H(A). It is not difficult to prove that the notion of weakly measured determined extends the notion of measuredetermined.
Theorem 22 Let H be a strict chance structure and A 5 J where J is a basic block of H. T h e n if A is Jmeasureddetermined, we have that A" M Hweakly measureddetermined. By Theorems 16, 18, 19, and 20, we obtain,
Theorem 23 Let K be a n equiprobability structure with causal universe T , t E T , and m less t h a n the height of T . T h e n the probability space of Kt i s the product space of the probability spaces of K.,], f o r s a n immediate predecessor o f t , the probability space of K, i s the product space of the probability spaces of K,I f o r s a last element of U{ T,: i 5 m } , and the probability space of K,] is the average probability space of the spaces of Kx,, f o r X E K , with respect t o the space of Kt. In particular, for classical structures the theorem is true with the power set as field of events and the counting measure as the probability measure. We next analyze the measures obtained for the examples. The structure IC of Example 1 is classical, so its measure is the counting measure for the hyperfinite approximation. The measure for the continuous IC results, as it should, the product of Lebesgue measure for each circle. Consider UB of Example 2. UB is, in general, not classical. For instance, if there are two urns, one with 2 ball, one white and the other black, and the other urn with three balls, 2 black and 1white, then UB is not classical. The super classical structure homomorphic to this UB can be obtained by taking the model of the choosing of two urns, each with six balls. It can be easily seen that with this structure the obvious measure is obtained.
10
Probability structures
We use function to the real numbers for describing properties of outcomes. We need functions similar to stochastic processes. So we call them measurement processes. In order to avoid notational complications when we write R, the real numbers, we mean R or 'R,the nonstandard hyperreal numbers.
Definition 22 Let K be a chance structure with causal universe T . Then 1. An mary measurement process over K is a function m is a natural number.
6 : "T
x K + R, where
R . Chuaqui
316
6 be a system (& : k E K ) and 6 a function from K into N. Then E is a measurement system over K of type (T,6),if & is a 6(k)ary measurement process over K, for each k E X.
2. Let
3. A probability structure of type (T,6) is a pair K = (K,E), where K is a chance structure with causal universe T , and E is a measurement system of type (T,6) over K, such that if m = 6(k), then [&(tl,.. . , t m ) = r ] E FK for every tl,. . . ,t, E T and r E R. Here,asit isusual, (&(tl,...,t , ) = r ] = { X € K : € k ( t l , , t , , X ) = r } . We can also introduce a natural “filtration” on the probability space of a chance structure.
Definition 23 Let K be a chance structure with causal universe T and (K, FK, PrK) the probability space of K . The filtration of K , FK,*is defined by
F K ,= ~(A
E FK : A = (A,)”}.
It is clear that if s 5 t , then FK,, F K , ~ . All the notions for stochastic processes can now be introduced. Consider Example 1. We could introduce the binary measurement process fined for t , s E In and X E IC by
de
[o(t, s, A) = the distance between the points of the circle selected at t and s in outcome
x Then, (IC,(0) is a probability structure. In [3, Section 61, a language that is interpreted in probability structures, simlar to one defined in [5], was introduced. The probability structures assign probability to the sentences of this language, instead of determining their truth or falsehood.
References [l] R. Chuaqui, A semantical definition of probability. In: Nonclassical Logics,
Model Theory, and Computability, Arruda, da Costa, and Chuaqui (eds.), NorthHolland Pub. Co. Amsterdam, 1977, pp. 135168. [2] R. Chuaqui, Foundations of statistical methods using a semantical definition of probability. In: Mathematical Logic in Latin America, Arruda, Chuaqui, and da Costa (eds.), NorthHolland Pub. Co., Amsterdam, 1980, pp. 103119. [3] R. Chuaqui, Sets of relational systems as models for stochastic processes. In: Contemporary Mathematics, vol. 69, American Mathematical Society, 1988, pp. 117148.
Probabilistic Models
317
[4] I. Hacking, Logic of Statistical Inference, Cambridge U. Press, Cambridge, 1965.
[5] H. J. Keisler Probability quantifiers. In: ModelTheoretic Logics, Barwise and Feferman (eds.), SpringerVerlag, Heidelberg, 1985.
This Page Intentionally Left Blank
Logic Colloquium '88 Ferro, Bonotto, Valentmi and Zanardo (Editors) 0 Elsevier Science Publihers B.V. (NorthHolland), 1989
319
Logical Partial Functions and Extensions of Equational Logic William Craig Department of Philosophy University of California, Berkeley
Abstract Equational treatment
of
logic for total
partial
functions is extended
functions and
some encoding of
connectives. The extension is equational.
to allow sentential
It uses Kleene equality, a
binary total projection function, and two restrictions of i t complementary domains.
to
We prove completeness, bring out an implicit
equational definability of logical functions of a certain kind, and discuss, but do not characterize, modeltheoretic strength. Introduction.
For the purpose of including partial functions in the
treatment
of
or
expressing
certain
sentential
combinations
of
equalities or of doing both one may extend the equational logic of total functions in a variety of ways.
One of these proceeds by
applying only changes of the following two kinds to the underlying language and results in extensions that are eauational:
First, one
interprets the nonlogical function symbols as ranging over partial functions, instead of ranging over total functions only, and one often *
The author should like to use this opportunity to record his gratitude to the University of California, Berkeley, for useful yearly assistance from its Committee on Research and for its generous and enlightened policies or practices that allowed him repeatedly to teach seminars with little or no official enrollment.
W. Craig
320
adjusts the notion of equality accordingly; second, one adds symbols for certain ones among those functions on a given universe A that are lonical (or invariant) in the sense that each permutation of A maps the function onto itself. In [C] we introduced an equational extension that involves only one logical function, namely the binary projection function Ae that maps every pair of elements of the given universe A into a
0'
A
second equational extension considered there is more comprehensive.
I t involves, in addition, the partial function d = el{ E 2A: A A .a = al} that results when Ae is restricted to the diagonal set {
E
2A:
. a = a 1} on A.
We will be concerned here with a
further equational extension, involving a third logical function in addition to Ae and Ad, namely the partial function I A3 = Ael{ E 2A: a0 # a1} that results when Ae is restricted to the nondiagonal set {E 2A: a
0
*
al} on A.
The following equalities may suggest uses of these extensions when one does or studies algebra. They may also give some idea of the relative
strength
of
suggestiveness, we use in (1)(5).
0 ,
the 1
three
, 1,
0
underlying
languages.
For
for the nonlogical function symbols
Logical Partial Functions and Extensions of Equational Logic
An
32 1
equality q = r will be regarded as holding in a partial
algebra iff whenever one of the terms q, r is defined so is the other and, when both are defined, then they have the same value. holds iff the 2ary function that is denoted by universe A of the algebra).
Also, (2) holds iff
0
1
Then ( 1 )
is total (on the
denotes a partial
1ary function that assigns to any object al that is in its domain an (Here, and elsewhere, we downplay or
object a2 such as aloa2 = 1.
ignore the usemention distinction. 1
Further, (3) holds iff the
domain of the 1ary function denoted by
1
such that a
#
0.
consists of the objects a
Thus (2) and (3) together characterize the inverse
function on the multiplicative semigroup of a division ring. Equality (4) expresses a right cancellation law, while ( 5 ) asserts that no
object other than zero is a zerodivisor.
The set of four equalities
in ( 6 ) implicitly defines that partial majority function on the universe A whose domain consists of the triples such that ‘.ao.a1.a2} contains at most two objects. Speaking in a more general vein, when one extends equational logic by means of e. d, and
3,
then, as we shall see, one can treat
much, but not all, of the Wlogic of partial functions equationally. More specifically, when all the nonlogical functions involved are total, then an Wsentence is always equivalent to a conjunction (and thus to a finite set) of equalities in the extended language.
In
contrast, for partial functions in general, not every Vsentence is equivalent to a conjunction of equalities.
W.Craig
322 Further
equational
extensions
can
be
constructed
by
the
introduction of symbols for further logical functions, such as the majority function implicitly defined by
In a sense, many
(6).
extensions of this kind are included in our extension by means of e.
3.
d, and
3
More precisely, when e. d, and
are available, then many
such functions can be implicitly defined by a set of equalities. completeness
result
below,
therefore,
applies
to
such
Our
further
extensions as well. Completeness.
Although we hope that reading the paper will by itself
suffice to give a general idea of its content, on matters of detail we shall somtimes depend on 9 1 and 97 of [Cl.
Our language here will be
like one of the equational languages of [Cl. except that it contains at most denumerably many function symbols and that e. d.
3
are now the
symbols fo, fl, f respectively, where p ( 0 ) = p ( 1 ) = p ( 2 ) = 2. 2
When
f
needed, we shall add a phrase to distinguish between use of d. for example, for denoting U
2ary function f2. shall mean any
U =
and its occasional use for denoting another
By a (partial) algebra = such that
p
we
A
is
any set, which may be empty, and such that each gk is a partial p(k)ary
function on A.

When g
When g2 = A$ then
g1 = Ad, and g
2
= A$,
then
U
U
shall be &standard.
shall be {e,d,&standard
or simply standard.
The clause in the inductive definition of r
[Cl) that pertains to is in
0
A:
3
U
(as given on p. 8 of then reads as follows, where a =
E
aE
a
dom ru iff
E

323
U U dom so n dom s 1 U
also, when a E dom r , then
dom g2;
When U is $standard and hence in particular when U is standard, this can be stated more simply:
a
E
dom(s(sl.sZ))
)0 ; ( ' s
#
u
sl(a);
iff also,
E
when
U
and
dom so n dom s
aE
then
dom($(so.sl)
($(so,sl))(a)= so(a). uFor any algebra U, whether standard or not, and for any term r
such that vb{r} E {v,....
*
}
vn1
we define, in addition to the partial
wary function ru or A, the partial nary function rUan on A in the obvious way. Whether or not standard, an algebra U shall be a model of r = s iff ru = s
U
.
In contrast, the consequence relation we want to capture
concerns algebras that are standard. 21
that
5
1r
We let E
= s iff, for every
is standard, if U is a model of each equality in E then U is a
model of r

s.
Two narrower consequence relations also will play a role. these, which we shall denote by U
is standard
{e,d}standard.
.I,
is replaced by
One of
is obtained when the condition that the
weaker
condition that
The other, which will be denoted by
1,
U
is
is obtained
when one uses, instead, the following intermediate condition: 21
is
{e,d)standard
N' s {., . . , is in C, then fk (i) iff fk(r) is in S and, if both are in S,
s
then
is in C.
(c) For any q = r in E and any in S and. if both are in S,
5
in " S , SIqI is in S iff
5111
then <s[ql,s[rl> is in C.
is
329
Logical Partial Functions and Extensions of Equational Logic (a)
Proof.
By
l(c),
since the following two conditions are
equivalent:
(c)
Consider
any
q = r
in
E and
any
s
in
Let
"S.
k k vb{q,r} G {vo , . . . , v ~  ~ } . Then e (q.vo.. . . , vkl) = e (r.vo,.... vkl) k is in E and therefore so is e (s[ql.so,... "k1) k = e (s[r],~~....,s~~). Since { S ~ , . . . , S ~ C~ S} , therefore if s[ql is k k so are e (s[ql.so . . . . , s ~  ~ e) (s[rl.s,.....~~~), , and s[rl.
in S ,
Likewise,
if
{ s o , .. . 'Skl} c
s[rl
is
in
s
S,
so
is
s[q].
n 
since
n e (~[ql,~,,. . . .P,,~)
therefore
= e (s[r],p, ,..., Pn1 1 is in E for some n
Further,
f
0.
Hence if s[ql and s[rl
Starting with the algebra Z = of terms whose universe is T and using the set S of the last two lemmas, we now form 6 the relative subalgebra 6 = <S.> of 2 whose universe is S by letting each :f
be the restriction of :f
to
{ s E P ( k ) S : );(:f
Since, by Lemma 3(b). C is a congruence on 2 , we quotient algebra Sn = Lemma 4.
a <M.> = 6/C.
(a) 3l is standard.
(b) W is a model of E. (c) There is some
m
m
1 m
in n{dom pn: new) which is not in dom(p )
W. Craig
330
Proof.
(a) By 2(e). 2(f) and 3(a),
2(g)
it
is estandard, and by 3(a) and
To
dstandard.
is
prove
&standard, we first consider any q/C in M.
that
9l
is
By 3(a), d(q,q) is in S.
Then, by 2(i), %(q,q) is not in S. so that is not in the domain of
*m
a6.
It fol ows that is not in the domain of d .
We now
consider any q/C and r/C in M that are distinct. Then is not C and hence, by 3(a), d(q,r) is not in S.
By 2(j), a ( q , r ) is in S.
Then by 2(h), d(q,3(q,r)) is in S and hence, by 3(a), is in C. =
It follows that is in the domain of
zm and d+m (q/C,r/C)
q/c. (bl According to 3(b),
follows that when
,... >
< s /C,sl/C 0
s‘
iff
C is a closed congruence of 6.
s = < s o , s l . . . .>
and
sf
= <s;,s;,
. ..>
= <s‘/C,s’/C , . . . > then, for any term t,
0
is in dom t
6
.
1
Also, by 2(d),
s
It
are such that
s
6 is in dom t
is in dom t6 iff s[tl
is in S
and, in that case, t6(s) = s[tl. Thus for any term t and any W m = <mO.ml,... > in M, the following holds no matter which W m is s = < so , s l , . . . > in S we choose such that m = <so/C.s /C ,... > : 1
in dom tm iff s[tl is in S and, further, if tm(m)
=
m
is in dom tm then
G[tl/C.
Now consider any q = r in E.

any m = <mO,ml,... > in
W
that <so/C,sl/C,... > =
m.
To show that qm = rm, we consider

M and choose some s
=
<so,sl,... > in
W
S such
By 3(c), s[ql is in S iff s[rl is in S and,
if both are in S. then <s[ql,s[rl> is in C and hence s[ql/C = s[rl/C. Now, as we saw, qm(m) =
s[ql/C.
m
is in dom qm iff s[ql is in S; moreover in that case Likewise,
m
is in dom rm iff
Err]
is in S; moreover
Logical Partial Functions and Extensions of Equational Logic
 = s[rl/C. in that case rm (m) moreover
if
i
Hence iis in dom qm iff
in
is
dom qm
 = s[ql/C = strl/C = rm (m). qm(m) m therefore qm = r .
and
i is
in
33 1 in dom rm ;
don r
Since this holds for any
then
i
in @M,
(c) Let ri = vi if v. occurs in some term in T and hence is in vb{po ,..., pml,p'}, and let r i be any term in S otherwise. By 2(c), w r = is in S. Let 6 = . Since F[tl = t 0 1
m
for any term in T, it follows from the proof of 4(b) that dom tm iff t is in S .
ii
2(a).
By 2(b),
I m
is not in (p
)
m
is in dom :p
is in
for each n < w and, by
.
0
We can now prove the converse of Lemma l(a).
Theorem 5. If E
1 r=s
Proof. Consider any not in E.
then E
r=s.
E that is closed under
First let us assume that r
2
1 and
any r =
s is not in E.
that is
s
Then we apply I
our construction by choosing r for po, 1 for m, and s for p .
m
then gives us an m and and
m
such that m is standard, m is a model of E,
is in dom rm but not in dom s
not a model of e(r.s) = let us assume that
s
2
s
m
.
Since 111 is estandard, it is
and therefore not a model of r
r is not in E.
but with the role of r and Ithat
s
s
equivalent to E
1 rcs.
1 r=s.
E
s.
Next,
interchanged, we again obtain a standard
is not in E while r
three conditions E

Then by a similar construction,
is model of E but not a model of r = s.
where r =
Lemma 4
,,
2
s and s
1 s_cr
2
and E
There remains the case
r are both in E.
1 r,s I A I . where I A I is the cardinality of A.
then ,[MI
= 0 for some
relations M = eqn M, including the identity relation on n.) n Henceforth, it will be taken as understood that a function be partial on the set A concerned. reference to A or to n.
Also,
we
g
may
shall sometimes omit
The following two assertions are essentially
well known and easy to prove.
W.Craig
334
An nary relation S is invariant on A iff there is a
Theorem 6.
perhaps empty set {M
S
= u(
[M 1: J A d
E
J:
j E J} of equivalence relations on n such that
J).
Lemma 7. An nary function g is invarlant on A iff there is a perhaps empty set { g . : j E J} of functions g . that are invariant on A such
J
J
l M . 1 for some M = eqvnUj. such t h a t
that the domain of each g. is M.
J
*
g =
and hence
M.,
J
Utg
J..
j E
J A J j f M . 1 n [M.,] = 0 whenever g * gj,, and such that A J A J
J).
We now turn to certain kinds of behavior a function may have. When 0 < n < w then an nary function g shall be a proiection iff for

every a = in dom g g(a) = a..
there is some i < n such that
If there is some i < n such that
Ad,
Aa is a 0projection.
ai for every
For 0
n Aei be the nary iprojection whose domain is "A.
outwardthrusting, iff g(a) Q ran g
is
g(G)
*
outwardthrusting
and

nary
function
a for every a
a =
5
It follows that
g
in dom g. is
i < n we let
in
shall
be
Thus, when dom g
then
a. for every i < n.
Given any nary function g on A, we let g'+ following complementary restrictions of g :
and g  j be the
335
Logical Partial Functions and Extensions of Equational Logic We
call
the
gat
inwardthrustinn component
outwardthrustinfz comwnent of 8.
of
and
g
the
g**
It follows that a function
invariant on a set A iff its two components g
'C
g
is
and g  j are invariant
on A.
For any n such that 0
n < w and for any A we let A On be that
5
nary outwardthrusting function on A which satisfies conditions (i) and IAl are the cardinality of ran
and (ii) below, where Iran
A respectively.
dom AOn =
(i)
If
(ii)
a
E
a or
n It is easy to verify that AO is invariant on A.
{a E
"A: Iran al = IAlI}
n n dom AO , then AO (a) E A n ran
a.
The domains of the functions AOn vary in a fairly complex manner n with A. in contrast to the domains of the functions Aei. n
t
0 and A be given.
First suppose that n+l < IAl.
of A ~ n is empty, since Iran that n+l z I A l
2
3.
GI < 1 ~ 1  1for every
n
A with Iran
= 1 < IAl1 and, when n
Iran
a
n in A.
NOW
suppose
= IAI.
Also, when n
However, there also are 2
a
in
IAI, then there also are
a
n
A
in
n Therefore AO is not a total function on A.
Next, suppose that n+l empty.
Then the domain
Then the domain of AOn is not empty since there
n are a in A with Iran al = IAI1. with Iran
a
Let any
n IAl = 2. Then again the domain of AO is not
L
IAl = 2.
2
= IAl. so that
then again there are
a
in "A with
AOn is not a total function on A.
a
since Iran al = 1 for every n = 1 = IAl1 then AOn =
A0
1 .
is
in 'A,
therefore when
IAl
However, = 2 and
a total function on A; in fact it is
the Iary function that maps each of the two elements of A into the other.
Let us now turn to the case where n+l
then there is exactly one
a
1
IAl = 1.
in "A: since Iran al = 1
*
When n
t
1.
IAl1 therefore
W.Craig
336 n the domain of AO is empty. 0
in nA =
A,
therefore
a
When n = 0, then there is exactly one
namely the empty function
0;
since Iran
is in the domain of AO0 and the value of
any n there exists only one nary function on A.
Lemma
0; in
this case therefore
Consider
8.
AIM1 C dom AOn.
n
any
2
0 = IAl1
0 0 for
a
is the
Finally, when I A l = 0 then, for
unique element of the singleton A.
function
A
GI =
a
namely the empty
= 0 for every n.
and
0
any
M = eqv M
such
that
Then ,OnlAIM1, considered as an (n+l)ary relation on
A, coincides with A[M u {)l.
Proof. AOn(a) ker(?
)
i < n.
Consider
= M u {) and hence
b
any
6
Also
ker
6
ker = M and E therefore Hence b
E dorn AOn
E
A [ M u {)l.
and
G
u {)l.
in =
Then
It follows that
Now assume that AIM1
=
e ran.
a = M. an
ker
Also
Therefore AOn~AIMl5 A[M u {)l.
in dom(AOnClA[M1).
M u {}
and
dorn AOn. Then hence
Since AIMl G dorn AOn, Iranl = IAI1.
is the unique element of A n ran. so that
AOn() = bn’
Thus
b is in ,0”lA[M1.
This proves the
other inclusion.
0
Given any M = eqv M, we let IM: = l{{j: E M}: i and
if I A I
f
0,
' 0 then IAl = 1 and g = .
A
(d) If I A l = 0 then g =
Proof.
(d) is obvious.
*
dom g = First g(a) A ran
f
Now
assume
0
that
there
is

some
and hence such that n
*
f
0 and
a
in
dom g
J
j
for each j < n.
is invariant on A. therefore g(b) = n(g(a)) = n(ai) = bi. holds for any b in dom g, it follows that
g =
a
# 0.
Since dom g
that g(a) is in A n ran that b
f
iMj < IAI1, so
g(a).
a. that
f
0,
a
a
It follows that ;MI .C IAI1.
a
Since g
Since this
in dorn g for which
there is then some
A n ran
that
Ae~~AIMl.
There remains the case where there is no
n ran
such
0 and g(a) = ai for at least
some permutation n of A such that n(a.1 = b
g(a)
hence
Since dorn g = AIMl. there is
Consider any b in dorn g.
one i < n.
IAl
that
0.
assume
a
0.
would
in dom g such Now suppose
contain an
element
Then there would exist a permutation n of A which maps g(a)
W.Craig
338
into b and also maps any ai in dom
a
into itseif and therefore. by the
invariance of g , maps g(a) into itself and therefore not into b.
We
can conclude therefore that iM[ = IAl1 and that g(a) is the unique element of A n ran
:Mi
=
a.
IAl1 therefore A
ran
n
There is then a permutation
T[
A[M].
of A which, for any j in dom
J
= x(g(a)) = c.
g =
Since
contains exactly one element, say c.
into b. and which maps g(a) into c.
g(b)
b in dom
Now consider any
a,
maps a. J
Since g is invariant, therefore
Since this holds for any
6
in dom g, i t follows
nC that g = AO IAIM1
0
Consider any n
2
0.
A class
@
of structures such that g is
a partial nary function on A and such that for every set exactly one in @ shall be an n  m . be the unique g such that and , of IAI)
h ( M ) = n and ! M I
S' = {M:
When there is no M in either S o r S'
*
IAI11
then h shall be proper
for
A.
The restriction p = hb(SVS') of h to the complement of SVS' shall be the ad.iustment
of
In either case p
h
U
Thus either A = p o r ran hA =
A.
(A) = h
(0)
v ran ~1
A'
U
(A).
Combining this last remark with Lemmas 7 and 9, one readily obtains the following (cf. [ M I or [Sl).
Theorem 10. An nary function g is invariant on a set A iff g is the
aggregated Arealization h
U
(A)
of some nallotment h that is proper
for A. The following theorem describes a simple relationship between a certain class of nmaps and the class of nallotments.
Theorem 11.
(a)
isomorphisms and ( A
If h is an nallotment then h
U
is closed under
1 ' + is closed under substructures.
(b) If 0 is an nmap that is closed under isomorphisms and if
@.+ i.s closed under substructures, then there is some nallotment A such that 0 = h
U
.
Logical Partial Functions and Extensions of Equational Logic
Proof.
(a)
Given any M = eqvnM and any i < n.
34 1
the nmap that
assigns to each A the function ,e:tA[M1 is closed under substructures. U Since ( h )'+(A) = u{Aei(M)fAIMl: A(M) < n) for every A, it follows U U is closed under substructures. Evidently, h is closed that ( A under isomorphisms. (b) Let
@
be an nmap such that 0 is closed under isomorphisms
is closed under substructures.
and
dom h C {M: M = eqv M} and ran h
S
We define a function h with
n+l by distinguishing three cases
that may hold for a given M = eqv M.
n
A[M]
Case 1:
S
f o r some A such that IAl = n.
dorn (@'c(A))
there is then at least one i < n such that (@'t(A))(a) a E AIM1. Choosing one i among these we let h(M) = i. Case 1 does not hold but AIMl
Case 2:
C
dom(@'j(A))
=
By 9(a),
a. f o r every
for some A such
that IAl = fMi+l. Then we let h(M) = n. Neither case 1 nor case 2 holds.
Case 3:
Then M shall not be in
dom h . Given any A.
consider any
There is some B such that Since
@*+
(@'+(B))(a)
is
closed
aE
a
E
nA such that
"B, IBI = n, and either A
under
substructures,
Let M = ker
= (@'+(A))(;).
a E dom(@'+(A)).
a.
aE
Since
@
S
B or B E A.
d~m(@'~(B))
and
and therefore
@'+
is closed under isomorphisms, therefore BIMl G dom(@'+(B)).
Thus,
case 1 holds for M and h(M) = i f o r some i such that (@'+(B))(a)
= a
Since
a
n E dom ,eifAIMI
therefore = @'t(A)(a).
aE
dom(h
U
and (A))
(@'f(A))(a)
since and
It follows that @'+(A)
(A C
U
(A))(a) u
(1 )'+(A).
i'
= (@'f(B))(a)
= ai,
= (Ae:\AIMl)(a)
= ai
W.Craig
342 Given
aE
U
dom((A
and ( ( A
U
A,
any ).+(A)).
consider
E
= ai.
)'+(A))(a)
therefore (@'C(C))(b)
suck
that
Then, for some i < n. A(M) = i
(@'f(B))(6)
IBI = n and Since @.+
b
E
BIM1.
under isomorphisms, therefore (@'t(C))(c) Hence, in particular, (@'*(C))(a)
aE
= bi for' every
is closed under substructures,
= bi for every
substructures therefore
n a E A
any
By the construction of A . there is some B
Let C = A u B.
BIMl.
a.
Let M = ker
such that BIM1 5 dom(@'C(B)).
b
now
= ci for every
ai.
=
Since 4 ' + is closed
Since
E
CIM1.
is closed under
and (@'C(A))(a)
dorn(@'c(A))
c
= ai.
Hence
(AU).+(A) G @'+(A). Given any A, consider now any
a.
Let M = ker
Since
@'+
a
E
"A such that
a
dom(@'+(A)).
E
is closed under isomorphisms and hence
is invariant on A, therefore AIM1 G dom(@'+(A)).
@'+(A)
Also, by
Lemma 9. IAI = iM/+l. Further, case 1 does not hold for M = ker since otherwise
a
would be in dom hA(M) = dom(Aei(M) lAIM1 1 and then,
a would be in dom(@'*(A)). n = AO a E dom(A u (A)),
as we saw earlier, A(M) = n, hA(M) Since
aE
dom(@'j(A))
invariant @'+(A)
on
G (A
IJ
Given a E
dorn(X
u
A
and since @'+(A) therefore
A.
Therefore case 2 holds, and ( A
u
(A))(a) = ,On(;).
is outwardthrusting and
(@'+(A) 1
(a) = AOn(a).
Hence
(A).
any )*+(A).
A,
consider
Let M = ker
finally
a.
any
a
E
n A
Then IAl = iMi+l.
such
@
is
closed
(@'*(A))(a)
Since (@*+(B))(b)
under n = AO (a).
= BOn(b) for every
isomorphisms, therefore
It follows that ( A
u
)'+(A)
6
aE
E
that
Also, by the
construction of A , there is some B such that BIM1 E dom(@'+(B)) IBI = iMi+l.
a
and
BIM1 and since
dom(@'+(A))
E @'+(A).
and
o
343
Logical Partial Functions and Extensions of Equational Logic
We now turn to problems of expressing, within our language,
For any n
functions or relations on A.
r
1, there is a simple 11
correspondence between the nary relations on A and the nmy partial functions that are 0proJections on A.
To each relation S there
For corresponds that partial 0projection g whose domain is S. 2 example, to the 2ary relation 2A, { E A: a0=a1) , {
E
2A: a0+a1) there corresponds respectively the function Ae*
For any n
L
1 and any M = eqv M we now define a term tc(M) in
which there occur no variables other than vO,...,vnl and no function symbols other then e, d ,
%.
When n = 1 = (01, so that { < O , O > ) is the
only M such that M = eqvlM. we let tc(M) be vo. such that M = eqvn+1M.
Now consider any M
For each i < n we let qi be d(vi,vn) or
~ ( v . , v) according to whether is or is not in M and then let i n 2 n tc(M) = e(tc( nnM),e (qo....qnl)). Lemma 12.
Let n
2
0 and M = eqvnM.
II
Assume that Y = is
standard.
(a) If n
f
0 then (tc(M))’*”
If 0
5
i < n then (e(v i ’ tc(M))””
(b)
corresponds to = Ae:tAIM1.
L; dom AOn, then [tc(M~J(}))’’~+~ corresponds to L AOnIAIMl, considered as an (n+l)ary relation.
(c) If
Proof.
(a) By induction on n.
(b) By part (a). (c) By part (a) and Lemma 8 .
W.Craig
344 For any n
n*
P
d,
3.
h
0
we now choose some n* such that p(n*) = n and
3, so that fn* is an nary function symbol that differs from e.
For i s n+2 and M = eqvnM we then let ec(i.M) be the equality
shown below. ec(i,M): e(fn,(vo,. . . * vn1 ),tc(M)) = e(vi,tc(M)), 0
i < n
5
. . * Vnl ),tc(M)) = 3(vo,vo) e(fn,(vo. . . . , v ~ 1, tc(Mu{) 1) = e(vn. tc(Mu{) 1) e(d(f n*(vo , . . . , vn1 ),vn),tc(M)) 2 tc(Md))
ec(n,M): e(f n* ( v o , . ec(nt1.M): ec(n+Z,M):
Let n
Lemma 13.
2
0 and M = eqv M.
Assume that ll = is
standard.
(a) If
0
5
f:,bAIMl
i < n
then
ll
is
a
of
model
ec(i,M)
iff
= ,e:lA[M1.
is a model of ec(n,M) iff f:,bAIM1
(b)
ll
(C)
I f ll is a model of ec(n+l,M), then IAl1
=
0. 5
;Mi.
ll
(d) 21 is a model of ec(n+2,M) iff fn,/;\[Ml is outwardthrusting. (el ll is a model of {ec(n+l,M).ec(n+2,M)) iff IAl1
5
jM: and
Proof. (a) By 12(b). (b) By 12(a). (c) Assume that
IAl1 > ;Mi.
Then there are
a
and b t
bn.
a
and
respectively.
In
AIMu{)l such that ai = b.i for every i < n and such that a Thus, by 12(b), the function (e(vn’tc(Mu(.))))U’ntl has for
for b as arguments the different values a and b
in
contrast, by 12(a), since ai = b. for i < n, therefore the function
Logical Partial Functions and Extensions of Equational Logic
as argument or else has for each the same value. to U.
345
Hence, with respect
the terms on the two sides of ec(n+l,M) denote different
functions. (d) Assume that U is a model of ec(n+2.M). U bA[M] 1. dom(fn,
dom(e(d(f
n*
a

Then
= fn,(a).
( v ~ , . . . , v ~),vn).  ~ tc(M)) 1u’n+l.
ec(n+2,M), a d ran n
Let
U
an
a.
also
is
in
;“
is
in
in
Since U is a model of U, n+l
dom(tc(Mw{)))
It follows that f:,bAIM1
a
Consider any
Hence
is outwardthrusting.
The
converse is obvious. (e) Assume that U is a model of ec(n+l,M). is in dom( 0
First suppose that
a
Then, for ?< A On(;)> as argument, the functions U,n+l n+l, (e(vn.tc(Mu{) 1) ) , (e(fng(vo... . * vn1 ) , tc (Mu{) 1 1 ) U and (fn,(v,,. . . ,vn 1 ))U,n+l have the value On(;), so that E dom fng U n and fn*(a) = AO (a). Now suppose that is in dom(f:,bAIM1). Assume A nlA
[MI).
a
a
that U is a model of ec(n+l,M) and of ec(n+2,M). the
functions
(e(fn,(v
,....
*
U,n+l Vnl 1, tc(Mut)1) 1
(e(vn,tc(Mu{))))u,n+l
are the same.
is in dom(tc(Mu{)))u’n+l
any b in A n ran
a.
u 
and hence b = fn,(a). that
Then ?
. . ,vn 1 ))u*n+l, and
It follows that a” t$
ran
a.
Now consider
is in dom(e(v ,tc(Mu{))))U’n+l
u 
u f (a) = n*
Assume now that IAl1
( tc (Mu{)
(fn* (v,..
Thus f (a) is the unique b in A n ran n*
a is in dom AOn and
ec (n+2,MI.
and hence f : * ( ; )
Then, by part (d).
In
the
1” n+l =
0,
S
so
AOn(a).
:Mi.
case
a.
By part (d), U is a model of where
IAl1 < ;Mi
one
has
so that U is trivially a model of ec(n+l,M).
W.Craig
346 Now consider the case where Assume that
u fn,bAIM1
IAl1 = :Mi
= ,OntA[M1.
that AIM1
so
Then, by 12(c).
G
dom AOn.
is a model of
U
ec ( n+1, MI.
0
For any n
0 and any nallotment A ,
2
we
let eq(h)
be
the
following set of equalities: {ec(i,M): A ( M ) = i < n) u {ec(n,M): M
u {{ec(n+l,M))
u {ec(n+2,M)):
dom A )
h(M) = n) Both
In stating the following theorem we invoke Theorem 11. parts then follow from Lemma 13. nmaps can be implicitly defined.
According to part (a), certain According to part (b), for certain
other nmaps one can implicitly define a certain restriction. nmaps 0 for which there are A and B such that IAl and @ ' + ( B )
C
Theorem H.
0 we
For
I B I , @'+(A)
f
f
0.
have no results.
Let
@
be
an
nmap
such
that
is
@
closed
under
isomorphisms and # * + is closed under substructures so that, b y Theorem 11, there exists an nallotment h such that h
U
= @.
u
(a) If @ = a*+, then an algebra U =
aE
iff
dom r
u
n dom su
and
u 
21
r (a) = s (a). A formula r + s , to be called a stronq diversity, will
be
regarded
ru(a)
f
u s (a).
as satisfied by
s
or r
x
iff
aE
dom ru n dom su
Note that there are cases where
neither r = s nor r
rX
tU,a>
* s.
and
satisfies
[In contrast, < U , a > always satisfies either
s, where a
satisfied by iff either
eaualitv r
aG
dom 'r
*s
n dom s'
will be regarded as
 = s 21 (a).] . or r u (a)
We now introduce a language LH that consists of Hornformulas of the following two disjoint kinds, where m terms contains e, d. or %:
2
0 . n 2 0 and none of the
W.Craig
348
With each formula of one of the two above forms LH= or LH+ we now associate the following equality of L respectively.
E
The following is then easy to verify. 21
Theorem 15.
Let 41 = be any algebra that is standard and
aE
Then tzl,a> satisfies an LH formula iff i t satisfies the
let
OA.
k
associated LE equality. ll
When 91 is an algebra such that each fk is a total function on A, then < U , a > satisfies r = s iff it does not satisfy r
+ s.
In other
words, with regard to algebras U that are total, r w s is equivalent to the negation r = s of r = s .
Now, as is well known, every L VT
sentence is equivalent to a finite set of sentences, each of which is the Vclosure of a disjunction, each of whose components is of one of the two forms r = s. lr = s .
When r 4 s is equivalent to w = s , then
each of these disjunctions in turn is equivalent to an L
H=
also to an LH+
formula.
formula, or
Since we are regarding U as a model of an L H
formula iff it is a model of the Wclosure of that formula, therefore %T
is
modeltheoretically
equivalent
to
LH
whenever
interpretations of LH are confined to algebras U that are total.
the
Logical Partial Functions and Extensions of Equational Logic
For algebras
U
that are standard, confinement to algebras that
are total can be expressed in LE by a set of equalities.
is a model of the Vclosure of this equality.] U =
II
k
e(vo,fn,(vo , . . . . vn,)) = v
0
is
that
[As in the
5 equality
case of LH, we regard an algebra U as a model of an
algebra
349
iff it
For example, a partial
is
standard
a
model
of
U
iff fn* is a total nary function on A.
Theorem 15 therefore yields the following as a corollary. Theorem 16.
For any set S of sentences in LvT one can find a set S’
LE such that, for any algebra U
of equalities in
91 is total and a model of S’
that is standard.
iff U is a model of S .
Moreover, for
any S that is finite one can choose an S’ that is finite.
The following notion bears on the question of what functions one can define explicitly in LE but also will play a role in translating
A term of LE shall be in normal form iff it is of
from LE into LH.
the form shown below where m
2
1, n
2
. . . ,qmlIro, . . . ,rm1,
and none of qo, e, d , o r
0. q and ro are the same term, 0
So,
...
.
Snl, to,
. . . ,tnl
contains
3.
em+n (d(qo.ro).
. . . ,d(qml,rmll.z(so.to).. . .
.z(snl.tnl 1)
The following theorem implies that the closure under composition of a set of partial functions on A that includes the functions Ae?,
Ad,
Az can
be generated in a certain normal way.
The proof is by
induction. Theorem 17.
For any term p1 in LE one can find a term p2 in normal
form such that every standard 91 is a model of p1 = p2.
W.Craig
350
Suppose now that one is given an equality p = p’ such that p and p‘ are in normal form.
To be specific, suppose that p is the term
displayed above (before Theorem 17) and that p‘ is similar except that one now has numbers m’,
n‘
and terms qb,.. . .q;,l.rb.
. . . ,rk,l,
.
o’ . . . ,s;, 1, t;. . . . t;,
S‘
Lemma 18.
If U is standard, then U is a model of p
^.
p‘ iff
is a
U
model of the five sets of equalities given below.
Proof. Let Then
U
91
be standard.
Assume first that
is a model of the first four sets.
is a model of p = p‘.
U
Moreover since qo and r0
are the same term and qb and r; are the same term, U is also a model of the equality that constitutes the fifth set. a
model
of
3
these
five
sets.
Since
Now assume that 21 is a
is
U
model
of
3
e (q ,p,p‘) = e (q’,p,p‘) but also a model of the two sets on the 0 0 first of the three lines above, it follows that 3
2
e (qo.p) = e (qb.p.p’).
Since
U
is also a model of the two sets on
the second line above, it then follows that 2
2
e (q0,p) = e (q6.p‘).
of
the
U
is also a model of
Since qo and ro are the same term and qb and r’ 0
are the same term, it then follows that Each
is also a model of
U
equalities
p
associated with an L formula. H=
2
U
is a model of p = p‘.
d(qi,r;)
and
p’
2
0
d(qi,ri) is
By Theorem 15, whenever
‘LI
is
35 I
Logical Partial Functions and Extensions of Equational Logic standard, then LH= formula. p'
2
a(s
j'
t
J
)
Similarly, for each of the equalities p
satisfies the It
formula. 3
remains
3
e (q,,p,p')
2
s(s'
t')
J' J
and
one can find an LH+ formula such that, whenever U is
standard, then
same
satisfies this equality iff it satisfies this
e (qb,p,p').
equality iff it satisfies the L
Iw
to
consider
For standard U,
as the following equality:
the
equality
it is satisfied by the
e3(3(q,,qk).p.p')
2
~(v,,v,).
For this equality, in turn, one can find an LHr formula such that again, whenever
21
is standard, then satisfies one set
in the pair iff i t satisfies the other.
(a) {lp
2
If
q}, {e(vO.p) = vo, q = . a(vo.vo)}
W.Craig
352
We
conclude
with
the
following
theorem
which
shows
that
hp.
It contrasts with Theorem
n and p(m) = p(n) = 1.
For brevity, let g, h, w
modeltheoretically LE is weaker than 16 and also with Theorem 20.
Let m
Theorem 21.
*
be fm, fn, vo respectively.
Then for none of the formulas (a),..., (el
below does there exist a set S of equalities in LE such that, for every U that is standard, U is a model of S iff U is a model of the formula.
(a) g(w) = h(w) [w < g(w) v w
(c)
[ g ( w ) = w v h(w) = w)
(d)
[ g ( w ) = w v h(w) = w)
(el
[g(w)X
Proof. Let fz =
5
(b)
Bz 8
gB = h
53 =
h(w))
w v h(w)Y
w)
53
5
53
be such that IBI = 1, fO = Be, fl = Bd,
= 0, and
3
fk =
0
for every k
= 0. Then 3 is standard.
2
3 and hence in particular
Also, 3 is not a model of any of
the formulas (a), ( b ) , (c). (d), (el.
Now suppose there were a set S
for which the theorem fails. Then S would have to contain an equality p
I.
q for which 3 fails to be a model.
Let T
{e.d)
be the set of those
terms r of L such that r contains no function symbols other then e or
E
d.
Then rg
*
0
iff r E Tte,di. Assuming that 8 is not a model of
p = q i t follows that one of the two terms p. q is in T
{e,d}
and the
Logical Partial Functions and Extensions of Equational Logic
353
We shall let p be the term that is in Tte,d) and q the
other is not.
term that is not. Now consider any U = that is standard such that for gU = :f
and
h
dom gU n dom hU =
0.
= f
one
n
dom g
U
has
a
E
f
0,
U
dom h
f
0,
v dom hU = A, and also such that fk = 0
Since p is in T {e,d)'
for every k in K n {0,1,2,m,n). whenever
dom gU
w
A and Iran al = 1 then
aE
U
dom p
.
therefore
Since q is not in
Tte,d), therefore q contains at least one occurrence of a term of one of the following four kinds, where k is in K n {0,1,2,m.n) and where r and s are in T{e.d):
fk(tO'.''tp(k)l 1 , a ( r , s ) , g(r). h(r).
contains an occurrence of fk(tO,.. . ,tp(k)l) contains an occurrence of ~ ( I  , s ) , then that 
a
E
0
a ct A
U dom q . together
aE
WA
0.
If q
and Iran al = 1 imply
"A
If q contains an occurrence of g ( r ) o r h(r), then with
ran
a n dom gu = 0
a
U
respectively, implies that least one
aE
then qU =
If q
ct dom q
such that Iran
al
.
or
ran
a n dom hu = 0,
Thus in every case there is at
= 1 and
a
ct dom q
U
.
It follows
that U is not a model of p = q and hence not a model of S . In addition, assume now first that gU and hU are the identity functions on their respective domains. Then U is a model of (a), (b), (c), (d).
U Assume now instead that g (a) f a and hU (a) * a for every a
in dom gU o r dom hU respectively.
Then U is a model of (el.
Thus,
for each of the formulas (a), (b), (c), (d), (el there is a standard U that is a model of the formula but not a model of S .
0
354
W.Craig Bibliography
[Cl William Craig, Nearequational and equational systems of logic for partial functions, Journal of Symbolic Logic, forthcoming. [ L T ] Adolf Lindenbaum and Alfred Tarski. On the limitations of the
means of expressions of deductive theories, Logic, Semantics, Metamathematics, Papers from 1923 to 1938 by Alfred Tarski, Ch. XIII, Oxford (19561. 178183.
[MI
Edward Marczewski, Homogeneous operations and homogeneous algebras, Fundamenta Mathematicae, vol. 56 (1964). pp. 81103.
[Sl Agnes Szendrei, Clones in universal algebra, U. de Montreal (1986). 166 pp.
[TG] Alfred Tarski and Steven Civant. A formalization of set theory without variables, AMS Colloquium Publications, vol. 41, 1987, xxi+318 pp.
Panel Discussion
TRENDS IN LOGIC
This Page Intentionally Left Blank
Logic Colloquium ’88 Ferro, Bonotto, Valentini and Zanardo (Editors) 0 Elsevier Science Publishers B.V. (NorthHolland), 1989
351
Trends in Logic: Relations with C o m p u t e r Science
MARTINDAVIS Courant Institute of Mathematical Sciences New York University Abstract. It is suggested that computer science has borrowed much from logic and has repaid the debt,providing many new and interesting problems. The “culture” of computer science is contrasted with that of mathematical logic. The P = NP problem is discussed in this connection.
The connections between logic and computation are numerous. Historically, some understanding of this connection goes back at least to Leibniz (Davis [2], [3]). There is good reason to believe that the work of Turing, originally developed to settle Hilbert’s Entscheidungsproblem, played a decisive role in the development of the modern electronic computer (Davis [3]). As I have written elsewhere (Davis [4]): Today the connections between logic and computers are a matter of engineering practice at every level of computer organization. Companies with names like Logical Devices or Logicsoft abound. One can walk into a shop and ask for a “logic probe.” This is by no means simply a matter of terminology. Issues and notions that first arose in technical investigations by logicians are deeply involved, today, in many aspects of computer science. The contributions of logic to computing practice include: the very concept of formal syntax, various aspects of programming languages, the algorithmic setting of computational logic, aspects of computational complexity’the Acalculus, logic programming, and many others. In return, computer science has provided interesting new logical systems (dynamic logic, logics of knowledge, nonmonotonic logics) for logicians to study, and many new interesting problems, some with a very direct connection with practice. Computer science is a young subject, and its practitioners tend to be young and to display the characteristics of youth. They are enormously energetic and prolific in their output. They are very clever and innovative. However, the critical and contemplative side of intellectual work tends to be relatively absent. Propositions which seem intuitively to be correct, but which no one has been able to prove, are often used as though they were axioms. At any given time, a few topics enjoy the prestige of being “in,” and the program committees of conferences tend to be flooded with contributions in these areas. Typically, yesterday’s hot topics have been abandoned. The complexity classes P and N P have been the subject of extensive investigations. The impetus for studying these classes came from the observation that there are problems for which every algorithm known seems to require (in the worst case) consideration of a number of particular cases which asymptotically is exponential in the
M.Davis
358
length of the input. In particular, the satisfiability problem in propositional calculus was such a problem. Because the exponential function is asymptotically greater than any fixed polynomial, it seemed appropriate that functions which could be computed in a number of steps bounded by a polynomial in the input length should be regarded as “feasibly” or ‘‘actually’’ computable. Thus, in effect a “feasible” version of Church‘s thesis was proposed, which we may call the CookKarp thesis: The feasibly computable functions are the polynomial time (in input length) computable functions (cf. [5]).P is then the class of sets whose characteristic functions are polynomial time computable. In favor of this thesis, one could note that it was surely not too narrow. Moreover, the class of polynomial time computable functions has a certain robustness: like the recursive functions the class is invariant under changes in the model of computation and satisfies various closure conditions. Following the analogy with elementary recursion theory, it is natural to define a number of corresponding concepts: A set S is of course r.e. if and only if we have:

uE
s
(%)R(U,Y),
where R is recursive. Similarly, S is in N P if and only if we have: uE
s
(3Y)lvl~p(lul)R(U,Y),
where R is polynomial time computable and p is a polynomial. Post’s notion of manyS, is defined by the existence of a recursive function f such one reducibility, R that I 6R f(z) E S.
Z,n .)
be disjoint r.e. predicates. Then there is a formula y = y(z1,
smh that for all ml,mz, ..., m, E N :
22,.
. ., Z n h


U(ml,mz,.. . , m n )implies t y(m1 ,m2 ,. . . ,rn,.);  V(rnl,mz,.. . ,m,) implies I y(ml ,m2 , . . . , m n ^ )
387
Teaching the Incompleteness Theorem
and
.
~ ( 2 1 , . . >Z n )
= ( ~ Y ) ~3 (. . z.I*zn, Y).
To keep foimulas from getting too long, we adopt the following notation: When there is a given ntuple ml,mz,. . . , m n E N , for any formula C containing the free variables z l r . . .,z,, we write C" for the formula obtained by replacing zi by mi, i = 1,2,. ..,n. Thus,
6m(Zn+1)= am(Zn+l)A (iz)[z
5 zn+1 A pm(z)],
and
y(ml,mz,...,mn) = ( % ) 6 " ( ~ ) . Now suppose that U ( m 1 ,m2,. . . , m n ) is true. Then,
A(ml,mz,.
. . ,mnrqO)is true for some qo E
N,
and (since U and V are disjoint)
B(ml,mz,. . . , m n , r ) is false for all r E> N SO,
am(qO"),
and for all r E N , t TP"'(r). By easily established properties of
t  ( ~ z ) [ zI qo It follows that

t  ~ > ( m i ,m2
& (or &')
A P"(Z)].

,...,mn
).
Next suppose that V(ml,mz,. . . ,mn) is true. Then,
B(m1, mz,. . . ,m,,qo) is true for some qo E N ,
M.Davis
388 and (since U and V are disjoint)
A(m1, m z , . . , ,m,, r ) is false for all r E N .
Thus, I /3"(qo),
and for all r E N , F &"'(re).
As above,
(*I
I (%)[z
5 qo^ A ~ " ' ( z ) ] .
We wish to show that  7 ( m l , mz, . . . ,m"), i.e. that F (3y)6"(y). Proceeding in quasi natural deduction style, we take (3y)6"(y) as hypothesis and derive a contradiction. Recalling how P ( c ) was defined, we have Instantiating, we may write: F
am(.).
(**I
F aye)
and
F  ( ~ z ) [ z5 c A pm(Z)]. From this last, we have
I [qoA 5 c A p m ( ~ o ) ] . Using axiom 9 of Q, F (c 5 q o j .
Using (**) we have:
I [(c 5 Po^) A (~"'(c)]. Hence,
I ( 3 z ) [ z I qo A ~ " ' ( z ) ] .
By (*), this is a contradiction. It is an immediate corollary that all recursive predicates are binumerable. 2. Diophantine Predicates.. In order to obtain suitable bases, we define certain classes of predicates. We begin with ordinary mathematical expressions of the form: c. ~
' ' 112"'
...
~ " k ,
where c is an integer (positive or negative). If x l , x2,.. ., z k are variables (whose range is N ) and m l , m z , . . . , m kE N , we cdl the expression a monomial. If each of x l r1 2 , . . . ,Zk, m , , m2, . . . ,mk can be either such a variable or an element of N , the expression is called an ezponential monomial, The sum Ul+UZ+".+U,
389
Teaching the Incompleteness Theorem
is called a polynomial if each of u l , u2,. .. ,u. is a monomial; if we can only say that each u , is an exponential monomial, then the s u m is called an ezponential polynomial. We consider equations p(al,a2..
..ramrz1,22r... ,zn) = 0,
in the variables a1,a2,. . . ,arn,~ 1 ~ 2 2 . .,,znr . speaking of a l , a 2 , . . . , a , as parameters and of z l , 1 2 , . .., z n as unknowm. Such an equation defines a Correspondingpredicate: P ( a l , a z , . . . ,a,,,) which is true if and only if the equation
. . r a n z , ~ l r1 2 , . . .,2 , )
p(a1, a2,.
=0
has a solution in zI, 1 2 , . . .,I, E N . We use such equations to define two classes of predicates:
1. if p is a polynomial, then P is called Diophantine; 2. if p is an exponential polynomial, then P is called ezponential Diophantine. Evidently, Diophantine predicates are also exponential Diophantine, and exponential Diophantine predicates are r.e. The key results are: DavisPutnamRobinson Theorem (DPR). Every r.e. predicate is exponential Diophantine. Matijasevich’ Theorem (M). Every exponential Diophantine predicate is Diophantine. Corollary. Every r.e. predicate is Diophantine. The unsolvability of Hilbert’s 10th problem is an immediate consequence (see for example Davis [3]or Davis, Matijasevich and Robinson [4]). (It is the corollary that is usually called Matijasevich’s theorem in the literature.) For a Diophantine or exponential Diophantine predicate P we can write: P(al1.. .,a m )
* (311,.. 3zn){ .., a m , * ( ~ Y ) ( ~ s I., 3 z n ) < y { p(al . I
7..
9 . .
..
11,.
t
In)
. , a m , 21,.
= 0}
..’I n ) = 0 1’
where p is a polynomial or an exponential polynomial, respectively. This is true because y can simply be chosen to be the largest number among 2 1 , . . . ,I,,.We call this relation the bounding properly. Let A be the class of predicates of the form
(311,.. ., 32n)