On End-Extensions of Models of ¬exp

Math. Log. Quart. 42 (1996) 1 - 18

Mathematical Logic Quarterly

@ Johann Ambrosius Barth 1996

On End-Extensions of Models of Texp

Fernando Ferreira

Departamento de Matembtica, Universidade de Lisboa, Rua Ernest0 de Vasconcelos, Bloco C1,3, 1700 Lisboa, Portugal’)

Abstract. Every model of IAo is the tally part of a model of the stringlanguage theory Th-FO (a main feature of which consists in having induction on notation restricted to certain AC! sets). We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IAo + l e x p is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models.

Mathematics Subject Classification: 03F30, 03C62, 68415.

Keywords: Bounded arithmetic, IAo, Computational complexity, A P , Bounded collection.

1 Introduction

Consider a model of BUSS’ bounded arithmetic theory Si (see [2]). The elements of this model can be seen as finite (standard and non-standard) sequencea of zeroes and ones (strings). Its tally elements (i.e., its elements that are strings of ones) make up, in a natural way, a model of IAo (where, for instance, addition is given by the concatenation of two strings). In fact, the tally part of this model is even a model of BUSS’ stronger theory V,l. Furthermore, any model of V t is the tally part of a model of 5’;. A perfect matching, we may say. This correspondence is much more general than this particular case. For instance, the tally parts of models of are, exactly, the models of Vi. These results are essentially due to KRAJ~CEK [12]. They also appeared in the first version of this paper [5], and syntactic formulations of them were obtained by TAKEUTI and RAZBOROV (in [15] and [14], respectively). A question pops out: what is exactly the theory of which IAo is the tally part? That is, we want to solve the “equation”:

S: : V t = t : IAo.

Contrary to the case of the theories 5’; and ql, there is a (seemingly) unavoidable change of language between the language of arithmetic of IAo and the stringlanguage of the theory Th-FO that is the solution to the above “equation” (formulations of this theory were also independently given by ZAMBELLA [19] and CLOTE-TAKEUTI [3]).

2 Fernando Ferreira

The problem concerns the multiplication function. More specifically, the theory Th-FO has induction on notation for predicates in IMMERMAN’S class FO of first-order ezpressible properties (these are defined in terms of first-order definability in suitable finite structures with domain {(Ill,. . . ,n}; IMMERMAN’S original paper is [lo]). This class is included in ACO, the class of sets that can be decided by constant depth, polynomial size circuit families (one should view the class of FO-relations as a rather robust uniform version of AC”). Well, it stems from deep work of AJTAI [l] and, independently, of FURST-SAXE-SIPSER (see [9] for an exposition of this latter work) that the multiplication function is not an ACo notion (however, it is an open question whether the graph of multiplication is in AC”).

The language which we use for formulating Th-FO was introduced in [4], and an interesting feature of it is that although multiplication is conspicuous by its absence, i t is nevertheless possible to formulate exponential assumptions in Th-FO. More precisely, the binary length function A%(%), which gives (in binary notation) the length of a string, is an FO-function and provably satisfies in Th-FO some basic properties. This has some nice implications, namely that every model of IAo + l e x p sits, as a proper initial segment, in a model of Th-FO. This, in turn, implies that a modicum of bounded collection is provable in IAo + lexp. Moreover, this modicum is almost the best that we can, at present, hope for (some more of bounded collection entails a positive answering to the P # N P question). These results give insight into a question of WILKIE-PARIS, on whether the theory IAo + l e x p proves the scheme of collection for bounded formulas.

This paper is organized in the following way. In the next section we establish and give a precise meaning to the correspondence between Th-FO and IAo. While a t this, we define FO-relations and functions in arbitrary models of Th-FO. In the third section we expand the original language of Th-FO in order to have available a function symbol for each (description of a) FO-function. This expansion is technically useful in the final section and, incidentally, permits the characterization of the provably total functions (with appropriate graphs) of Th-FO. In the fourth section we show how to make exponential assumptions in models of Th-FO and prove that a modicum of bounded collection holds in models of IAo + Yexp. The last result of the paper separates the theories Th-FO and IAo by a IIY-sentence.

2 The correspondence between IAo and Th-FO

The first-order stringlanguage of the finite binary tree of the sequences of zeroes and ones consists of three constant symbols e, 0 and 1, two binary function symbols - (for concatenation, sometimes omitted) and x , and a binary relation symbol C_ (for initial subwordness). There are fourteen basic open axioms:

2-& = 2, t X & = e ,

z-(y-o) = (z-y)-o, 2 x (y-0) = (2 x y)^z ,

r ^ ( y ” l ) = (+-y) - l , 2 x (y-1) = (2 x y)^z , 2-0 = y-0 + 2 = y, z-1= y - 1 + 2 = y,

On End-Extensions of Models of Texp 3

bit(z, u) = 1

2 s E * z = El 2-0 # y-1, 2 s y-0 -2 E y V z = y-0, z C y-1 w z C y V z z y - 1 , 2-0 # E 2-1 # 6.

(3w C z) (w u A w l s z)

The interpretations of the symbols of the stringlanguage in the standard model are immediately clear, except (perhaps) for x . The language is simple and natural, but unfamiliarity takes a toll. For ease and convenience of reference, we present a small table:

symbol

6

0 1

X - Y

Z C Y

Z C Y Z C ’ Y tally( z)

Z l Y

z = y

Z X Y

formal definition primitive primitive primitive primitive primitive primitive

Z C Y A Z # Y

(32 s Y) ( 2 - 2 c Y) z = l x z

1 x z s 1 x y

t l Y A Y s =

standard int erprei ation empty word one bit word 0 one bit word 1 concatenation of z with y 2-2- . . . -z, length(y) times z is an initial subword of y z is a proper initial subword of y z is a subword of y z is a string of 1’s length(z) 5 length(y) z and y have the same length the (length(u) + 1)-bit of z is 1 the (length(u) + 1)-bit of z is 0

The class of swq-formulas (“subword quantification formulas”.) is the smallest class of formulas containing the atomic formulas and closed under the Boolean operations and subword quantification, i. e., quantifications of the form (Vz s* t ) (. . .) or (32 s’ t ) (. . .), where t is a term in which the variable z does not occur. Let q be a class of formulas of the stringlanguage. The theory \k‘-NIA (for Notation Induction Azioms) consists of the fourteen basic open axioms plus the induction scheme:

(1)

where F E Q, possibly with parameters. In the sequel, we will use at ease plenty of simple facts that can be deduced in the theory swq-NIA. Here is a sample of them:

= Y; x E z A y C z -+z C y v y c 2; z c y - z O C y V z l s y; and z C y h y c Z- z = y. A longer list of simple facts like these, together with deductions of them in swq-NIA, can be found in [4]. Two properties of swq-NIA are directly relevant for this paper. Firstly, the tally part of a model of swq-NIA is a model of 1 4 0 in a natural way: just interpret the constant 0 by E, the successor function S by “concatenation with l”, U+l> by l l - ) ? , L 1 . n by “x” and “5” by “C”. Secondly, the following holds:

F ( E ) A vz(F(z) ---* F(z0) A F(z1)) -$ vzF(Z),

0 # 1; ( z y ) ~ = z(yz); z C zz; zy C zw -$ y C W ; Z y A z s y -+

4 Fernando Ferreira

P r o p o s i t i o n .

swq-NIA I- VzVy(z i y A (Vu c 1 x z) (bit(z, u) = 1 - bit(y, u) = 1) + z = y).

P r o o f . We reason inside an arbitrary model of swq-NIA. Suppose that the an- tecedent of the implication holds. We show, by induction on notation on the variable z, that

Vz(z G z A (Vu C 1 x z)(bit(z, u) = 1 - bit(y, u) = 1) - z C_ y)).

Note that the case z = t yields z E y. Similarly, we can show that y C t and, hence, conclude that t = y.

The base case of the induction is clear. Assume that z0 E z and that

(Vu c 1 x zO)(bit(zO,u) = 1 - bit(y,u) = 1)

(the case for zl is handled similarly). By induction hypothesis, we can conclude that z y. Moreover, it is easy to argue that z # y. (Just this once we present the full argument: I = y + z 1 y 3 z 3 z 3 1 x z = 1 x t * (1 x z) l g 1 x z 3 1 x rO g 1 x I y. The second case leads to a contradiction. In effect, if zl y then bit(y, 1 x z) = 1 and, by assumption,

0

We caution the reader not to be misled by the apparent strength and elegance of the theory swq-NIA. In our opinion, this theory is uninteresting and artificial (see [7] for a discussion of this). The same cannot be said of the following theory:

D e f i n i t i o n. The theory Th-FO is the theory swq-NIA plus the following string- building principle:

(2) where F is a swq-formula, possibly with parameters.

In order to “grow” a model of Th-FO out of an arbitrary “IA~-tally”, it is convenient to work with a second-order (or relativized) version of IAo. This theory, called Ao-CAo (“CA” for comprehension axiom), is formulated in a two-sorted language with number variables, w , y, z, . . . and set variables W, Y, 2, . . .. A first-order formula is a formula that does not contain any quantifications over set variables; following [2], we say that a formula is bounded if it does not contain any unbounded quantification over number variables (it may contain quantifications over set variables or bounded quantifications over number variables). Equality between set variables “Y = 2 is not a primitive notion, but rather defined by Y w ( w E Y - w E 2)” and, hence, it is not a bounded formula. The axioms of Ao-CAo are the basic axioms of IAo (see, for instance, the fifteen axioms of PA- in Chapter 2 of [ll]), plus the induction axiom

z0 g z.) Hence, either z0 5 y or zl

bit(z0,l x z) = 1, which is a contradiction.

Vu(tally(u) + (32 E u)(Vu C u) (bit(z, u ) = 1 ~1 F ( u ) ) ) ,

vW(0 E w A v w ( w E W + w + 1 E W) +vw(w E w)), and the comprehension scheme

3WVw(w E W ++ A(w) ) ,

where A is a bounded first-order formula, possibly with number or set parameters. Remark that the combination of the axiom of induction with the comprehension scheme gives rise to the scheme of induction for bounded first-order formulas (i.e.,


IAo is a subtheory of Ao-CAo). A structure for the language of Ao-CAo is a pair M = (MIS), where M is a structure for the language of first-order arithmetic and S is a set of subsets of M (occasionally, we will use the notation SM to make visible the fact that S comes from the model M). The main thing to observe in the clauses for the truth definition is that the set variables range over S. This means that the logic of the theory Ao-CAo is a first-order logic.

P r o p o s i t i o n . For any model M of IAo, there is a sef S of subsets of M such fhat (MI S) is a model of Ao-CAo.

0 P r o o f . Just let S be the set of all Ao-definable subsets of M. C o r o 11 a r y. The theory Ao-CAo is conservafive over IAo. P r o o f . This is a consequence of the above proposition and the completeness

theorems. 0

A bounded formula of the theory Ao-CAo is called strictly bounded, if all its atomic subformulas of the form t E W occur in the context t E W h t < w (abbreviated, t E WW), with w not occuring in 2 . Given a model M = (MIS) of Ao-CAo, and given a E M and R E S, we denote by Ra the pair ( { P E M : M I= P E R A P < a},a) . The element a is called the M-length of the pair. (Note that the first entry of R" is in S.) Call s b the set of all these pairs.

D e f i n i t i o n. A description of a k-ary FO-relation is a first-order strictly bounded formula A(w1,. . . , Wk, , . . . , W,"'), where all the free variables are as shown.

Let A be as in the previous definition. Given a model M = (MIS) of Ao-CAo, there is a natural way of defining a k-ary relation [AIM in st,:

(nyl,. . . R:') E [AIM w M I= A ( a i , . . . I Qy' , . . . a:'). Let us give some examples/definitions:

1. 2. 3. 4.

5 .

equal(W"', YY): w = y A (Vz < w ) ( z E Ww - z E YY). initial(W" ,Yy): w 5 y A (Vz < w ) ( z E W"' w z E YY). part(Ww , YY): (32 5 y) (z + w 5 y A (Vz < w ) ( z E Ww - z + z E yy)). Given descriptions A and B of k-ary FO-relations, define neg(A) and and(A, B) by -A and A A B, respectively. Given a description A ( w , w l , . . . , W L , W"', Wr', . . . , W,"') of a (k + 1)-ary FO-relation, define the (k + 1)-ary FO-relation all(A) by

with z and u new variables. In the above, . . . , Wwu(I~t,,~, . . . modifies the formula at hand by replacing its subformulas of the form t E Ww by z + t E W". This definition is contrived so that, given any model M = (MIS) of Ao-CAo and given R", f ly1 , .. . , S-2:' in sb, then (na, ny' , . . . , n:') E [ a l l ( A ) ] ~ if and only if, for all Cp such that M I= part(C8, nu), ( Z p , Q y ' , . . . , n,"*) E [AIM.

(vz 5 ~ ) ( V U 5 W ) (2 5 tl + A(u - 2, wl , . . . , wk, WW(l[t,v], WF' . Wr'))

D e f i n i t i o n . A description of a k - a y FO-function is a pair consisting of a first-order strictly bounded formula A(u, w l , . . . , W L , Wr' , . . . , Wr') and a term t ( w 1 , . . . , w t ) , where all the free variables are as shown. The variable u is called the special variable of the description.

6 Fernando Ferreira

Let ( A , t ) be as in the previous definition. Given a model M = (MIS) of Ao-CAo, we want to define a k-ary total function [A, t ] M in Sb. Take elements a:' , . . . , n:' of sb. By (something equivalent to) the least number principle in Ao-CAo there is a unique a E M such that

M L[A,t](a~al,...,ak,n:'~...~R:'),

where L[A, t~ (u , wl, . . . , WL., W;',. . . , W;') is the following first-order strictly bounded formula:

[(VW 5 t(w1,. . . , wk) ) A(w, ~ 1 , . . . , WL., W?', . . . , Wr') A u = 01 v [U 5 t (W1, . . . , W k ) A -A(u, wl ,.. . , W ) , W;', . . . , W r h )

A (VW 5 t(w1,. . . , wk) ) (II < w + A(w, wi, . . . , wk, W?', . . . , Wf'))].

Now, define

[ A , ~ ] M ( Q : ' , . . . , 0;') = { p E M : M I= A(p,c*1,. . . , ak, Q:',. . . , 0;'))". The idea is that a, which is the greatest element not exceeding t for which A is false (otherwise, a is 0), operates as a mark for the M-length of [A, t ] M ( n : ' , . . . , Qp').

The following are some examples/definitions:

6. conc(W" , YY): (u E W"' V (w 5 u A u - w E YY), w + y). 7. prod(W", YY): ((36 < y)(3r < w ) (u = qw + r A r E W"'), wy). 8. tail(W",YY): (y + u E W" V y + u > w, w ) .

9. Constants (0-ary functions) are not ruled out in the above definition. The next three O-ary description will be of use later: (u = u, 0), ( u # u , l), and

10. Let ( A , t ) be a description of a k-ary FO-function such that, for any model M of AO-CAo, the M-length of any value in the range of [ A , ~ ] M is 1. We may consider this description as characteristic function of the k-ary FO-relation given by the description A(0, w l , . . . , w t , W;', . . . , Wr'). A suitable formulation of the converse statement also holds.

11. Given a description D = (B(u, yl, . . ., yk, Yf', . . ., Yf'), t(y1,. . . ,yk)) of a k-ary FO-function, and given k descriptions

of r-ary FO-functions, we define (the composition of these functions by) the r-ary function description comp(D; D1,. . . , Dt) whose first component is

(u = 0, 1).

Di = (Ai (u , ~ l , . . . wr, W?',.. . I W,!"r), t i ( w l , . . . , w , ) ) , 1 5 i 5 k,

(3Yl 5 tI(G))* * *(3Yk 5 tk(W'))(l\llil~.L[~,](Yi113,W;'i...,W~~.)

A ( B ( ~ , ~ , Y ~ ' I A ~ , . . . , Y I Y ' I A ~ ) V U > t(Yli..*,tk))) and whose second component is t(tl(G), . . . , t k ( G ) ) . In the above, . . . , Yyl,,.. . modifies the formulas at hand by replacing its subformulas of the form t E Yy by A(t , . . .) A t < y. This definition is contrived to make the following true: if M = (MI S) is a model of Ao-CAo then, for all 0:' , . . . ,n;' in s b l

[comp(D; ~ 1 , . . . , ~ k ) ] M ( ( n : l , . . . ,a(;') = [D]M([D1]M(R:',. . . ,0:')1* ..I[D~]M(R:'~...,R(;')).


12. Given a description D(w, w1, . . . , W k , W", W r ' , . . ., W;") of a (k + 1)-ary FO- relation, let witness(D) be the (k+ 1)-ary function description whose first component is

(VU 5 w)[D(u, w1,. . . , W k , WV, .., W;') A (VZ < U)+(Z, w1, . . . , w t , Wf, wr1,. . . , WY') 4 u E WV v u > u]

and whose second component is w . Consider a model M = ( M , S ) of Ao-CAo and take elements RQ, f l y 1 , . . . , R;' of s b for which there is /3 5 a such that (ap, Q y l , . . . ,a;') E [DIM. Then [witness(D)]~((SZ", nyL,. . . ,R;') = Q p for the least such p.

13. Given a description D(w1, . . . , W L , W r l , . . . , W,"') of a k-ary FO-relation, and given two descriptions Di = (A;(u, w l , . . . , W k , IVY',. . . , W;'), t i ( w l , . . . , W k ) ) ,

i E (0, l}, of k-ary functions, let case(D; D1, Da) be the k-ary function description whose first component is

[ D ( W l , . . . , w t , wy1,. . . , WF") A(Ai(U, W l , . . . , wk, W y ' , . . . , W;') v U > t l ( W 1 , . . . , Wk)]

V [ ~ D ( w l , . . . , wk,W,"', ...,W;') A(Az(u, wl,. . . , wk, Wr', . . . , W;') V U > tz(w1, . . .,wk))]

and whose second component is Il(w1 , . . . , wk) + tz(w1 , . . . , Wk). Consider a model M = (M, S) of Ao-CAo and take elements f lyi , . . . , Rp' of sb. Then

[ c ~ e ( D ; D 1 , ~ z ) ] ~ ( R ~ L , . . . , Q ~ k )

= { [DZ]M(Ry',. . .,a,"') otherwise. [D1]M(Ryl,. . . , fit') if D(nyl,.. . I 0;') E [DIM,

T h e o r e m . Every model ofIA0 i s ihe ially par t of a model ofTh-FO. P r o o f . Let M be a model of IAO and take S such that M = (M, S) is a model

of Ao-CAo. The model M is the basis for the construction of a model M o of Th-FO whose tally part is M. The universe of M o is s b ; the interpretations of the constant symbols e, 0 and 1 are Mo, 0l and {O}I, respectively; the interpretation of C_ is initial]^; and the interpretations of - and x are, respectively, [ c o n c ] ~ and prod]^. It is straightforward to show that N satisfies the fourteen basic axioms and that the map ;(a) := { p E M : p < a}O is an isomorphism between M and the tally part of MO. The following are also easy to check:

(i) for all R in S, M o I= Ro = E ;

(ii) for all a , p E M and R E S, P 5 a iff M o I= Ra C RQ; (iii) for all a, /3 E M and R, C E S, M I= part(Ra, Z p ) iff M o I= Ro 2. Ca; (iv) for all a E M and R E S, a E R (resp., a

(v) for all a,/3 E M and R E S such that /3 < a, M o CS bit(Ru, i ( p ) ) = 1 (resp., M o I= bit(Ra, i (p ) ) = 0).

(vi) for all Q E M and R E S, M o t= tally(Qa) iff Ru = ;(a).

R) iff M o I= RQ+l = Ro-1 (resp.,

P E R (resp., P f! R) iff

M o CS na+l = QO-0);

8 Fern an do Fer reir a

The following fact is crucial to showing that the schemes (1) and (2) hold in M o for swq-formulas:

( F l ) For euery swq-formula F(z1,. . . , z k ) of the language of Th-FO, there i s a first-order strictly bounded formula FO(w1,. . . , wk, wrl, . . . , W r k ) of the language of Ao-CAo such that, for all elements R:', . . . , in s b ,

M o I= F(R?' , . . . , Q F k ) M I= F o ( a l , . . . , a k I R y l , . . . I fitk). The map O is defined in two steps. Firstly, to each r-ary term t of the language

of Th-FO we associate a description to for a r-ary FO-function such that, for all $ 2 ° , R ~ 1 , . . . ,R;r in Sb, M o I= t (R? l , . . . ,a:.) = a" iff [ t o ] ~ ( R y l , . . . ,SIPr) = $2". This term-to-term map is easily defined by induction on the complexity o f t with the aid of the examples 6, 7, 9, and 11. Secondly, the map O is expanded to the swq-formulas with the aid of examples 1, 2, 4, 5, 10, and 11.

In order to show that (1) holds in MO, take an arbitrary swq-formula F ( z , . . .) and suppose that M o Pr F(Qo, . . .), for some R" in s b and for some parameters. By (Fl) , M b' F o ( a , . . . , R", . . .). Due to the fact that M is a model of Ao-CAo, either M o b' FO(0,. . . , no,. . .) or M o I= F o ( P , . . . , RD, . . . )AlFo(P+1 I " ' , @+I , . . .), where p is an element of M preceding a. In the first alternative, M o b' F(E , . . .); in the second alternative, M o i= F(Rfi, . . . ) A +'(RP+l,. . .) and, by the above property (iv), we get the right conclusion. To show that (2) holds in MO, take again a swq-formula F ( z , . . .), and let a E M. We need to prove that there is R in S such that, for all P < a, p E R if and only if M I= F 0 ( P , . . . , i ( P ) , . . .). The existence of such an 52 is a

0

Conversely, to a model N of Th-FO we can associate a model N* of Ao-CAo. The first-order part of N* is tally(N) := { a E N : N I= tally(a)} which, as we have already mentioned, is naturally a model of IAo. Given a E N , let

consequence of the comprehension scheme of Ao-CAo.

s(a) := { v E tally(N) : N I= u C 1 x a A bit(a, u ) = 1) .

Take the set SN of all subsets R of tally(N) satisfying the following condition: for all u E tally(N) there is a E N, a i 'u, such that s(a) = { u E tally(n') : u C u A u E a}. The structure N* is, by definition, (tally(N), SN).

T h e o r e m . Let N be a model of Th-FO. Then N* i s a model of Ao-CAo. P r o o f . We have to check that the induction axiom and the Ao-comprehension

scheme hold in N*. For the truth of the induction axiom, take 0 E SN, u E tally(N), and assume that 0 E R A u @ R. By definition of SN, there is a E N such that s(a) = { u E tally(N) : u c u-1 u E $2). Hence, N I= bit(a,e) = 1 A -bit(a, u) = 1. Now, the scheme of induction on notation for swq-formulas guarantees the existence

In order to so show that the Ao-comprehension scheme holds in N* we need a lemma and the following fact: (Bl) For euery first-order strictly bounded formula A ( w , . . . , wkI wr' I . . . , W r k ) of

the language of Ao-CAo, there i s a swq-formula A * ( z i , . . . , zt) of the language of Th-FO such that, for all elements a l , . . . ,ak in N ,

of u C u such that N I= bit(a, u ) = IA-bit(a, u-1) = 1, i.e., u E and u+" lN* e 52.

N* b A ( 1 X a l , . . . , I X ~ k , S ( O 1 ) i X ~ l , . . ~ , Q ( ~ k ) l x a k ) * N E A * ( a ~ , . . . , a k ) .


The map * is defined in three steps. Firstly, to each term t(y1,. . . , yr, wl, . . . , wk) of the language of Ao-CAo we associate a term t*(yl,. . . , yr, 2 1 , . . . , zk) of the language of Th-FO, obtained from t by replacing each occurence of 0 by E , of + by -, of . by X , and of each occurence of wi by 1 x zi. Secondly, we define ( t = a)* as t* = q*, (t 5 q)* as t* C q*, and (t E W:i)* as bit(zi,t*). Finally, let (-A)* be -A*, ( A A B)* be A* A PI and let ((Vy 5 t ) A(y, . . .))* be (Vy C t*) A*(y, . . .).

L e m m a . Let A(y1 , . . . , yk , Wl , . . . , Wr) be a bounded formula of the language of Ao-CAo, where all the free variables are as shown. Then the theory IAo, enlarged with the logical azioms for the second order language, proves the following sentence:

V Y ~ Z ( V ~ 2 z)vW1-..vWr(vy1 5 y)...(Vyk 5 Y ) ( A ( Y l ~ . . . ~ y ~ ~ ~ l ~ . . . , ~ r ) A(Y1, . . . I Ykl W,., . . . I W3).

P r o o f of t h e L e m m a . The proof is by induction on the complexity of A. There are only three cases we need worrying about. When A is t(y1, . . . , yk) E W , put z = t(yl . . . , y). This z does the job because all terms of the language of IAo are provably monotonous. Consider now the case when A is 3W B(y1 , . . . , yk, W, WI, . . . , Wr): Given y there is, by induction hypothesis, an element z that works for the formula B. The very same z works for A. Lastly, suppose A is the formula

(vz 5 t(Y1,. . ., Yk)) B(+, Y l , . . . I Yk l Wll. * . I wr). Given y, let y' = max{y, t ( y , . . . , y)}. By induction hypothesis, take z that works for B and y'. Then z does the job for A and the original y. O( Lemma)

We are now in position to show that the Ao-comprehension scheme holds in N*. Given a first-order bounded formula A(y, yl , . . . , yk , WI, . . . , Wr) and given elements u1,. . . , uk in tally(N) and R1 , .. . , $2, in SN , we need to show that the set

{U E tally(N) : N* != A ( u , u ~ , . . . , ~ 1 . , n 1 , . . . , Q r ) }

is in SN. According to the definition of SN , this is equivalent to showing that for all u E tally(N) there is a E N such that s(a) is equal to the set

C = { u ~ t a l l y ( N ) : u ~ u a n d N*!=A(u,ui , . . . , ~ ~ . , n i , . . . , n r ) } . By the previous lemma, C is

{ u E tally(N) : u C u and N* I= A ( V , U I , . . . , Uk, Q : , . . . ,nF)), for some Q E tally(N). Take 0 1 , . . . I a, in N such that

C = { u E tally(N) : u < u and N* != A(u, u l , . . . , ut ,s (a l )u , . . . ,s(ar)u)}. According to property (Bl), there is a swq-formula F such that

N != F(u , u l , . . ., Uk, 0 1 , . . . , a,, a) N* != A(u, u l , . . . , uk, s (a l )p , . . . , s(ar)"). By the string-building principle (2), pick a satisfying the following property: For all u E tally(N) with u C u , N I= bit(a,u) w F(ulul , .. . , u t , a i , . . . , a , , a ) . This a does the job. O(Theorem)

Let N be a model of Th-FO. It is straightforward to argue that N * O is isomorphic to N in a natural way (and we write, N*O x N ) . Conversely, consider a model (MI S)

10 Fernando Ferreira

of Ao-CAo. It is easy to argue that MO* is (up to a natural isomorphism) of the form (MIS'), with Sb = SC, (we write MO* M). Hence, although MO* and M are not necessarily isomorphic, the above condition is sufficient to ensure that bounded formulas are absolute between MO* and M (the lemma inserted in the proof of the previous theorem is used to proving this). These considerations allow us to speak of natural pairs ( M , N ) , where M is a model of Ao-CAo, N is a model of Th-FO, N

Consider a natural pair (M,N), and let R" E Sf and a E N. We say that R" and a are in natural correspondence if Q = 1 x a and

Mo, and N* wb M.

{ u E M : u < a and u ER}= {u E tally(N) : u C Q and N k bit(a,u) = 1).

With this terminology, if Rpl , . . . , R;" and (11, . . . , ments, then the properties (Fl) and (Bl) can be rephrased as follows:

strictly bounded formula of the language of Ao-CAo such that

are pairwise corresponding ele-

(F2) Given a swq-formula F(z1,. . . , tk) ofthe language ofTh-FO, Fo i s o first-order

(B2) Given a first-order strictly bounded formula A(w1,. . . , wt, W r ' , . . . , W r k ) of the language of Ao-CAo, A * ( q , . . . , ti) is a swq-formula of the language of Th-FO such that

M A(a1, . . . , Qk, a:',.. . , a;') es> N k A*(al , .. . , at).

A consequence of the above discussion and of the completeness theorems is that the formulas F and FO* (resp., A and A*O) are equivalent in Th-FO (resp., in Ao-CAo).

3 The theory Th-FO expanded

In this section we define an expansion Th-FO+ of the theory Th-FO. The language of this new theory consists of the original language of Th-FO plus a k-ary function symbol fD for each description D of a k-ary FO-function. The axioms of Th-FO+ are those of Th-FO plus two function azioms for each such description D = (A, t ) :

(3) VZVZl .. . v Z t ( L ~ ] ( Z , Z l , .. . , Z t ) c* fD(Z1, . ..,Zt) Z),

Let D be a k-ary FO-description, let N be model of Th-FO+, and take a model M of Ao-CAo such that (MI N ) i s a natural pair. If a, a l l . . . , a k and Ra, a:', . . . , Qik are pairwise corresponding elements, then N k fD(a1,. . . , at) = a es> [D]M(R;ll,. . . , a;') = Ra.

The map O can be eztended to the open formulas of the language ofTh-FO+, with the equivalence (F2) still holding.


(c) Let D be a description of a k-ary function. Then there is a term q of the language ofTh-FO such that

Th-FO+ I- V t l . . . V Z ~ f ~ ( z 1 , . . . , ~ k ) 5 q(z1, . . . , ~ k ) .

(d) For each term t (z1, . . . , zk) ofthe language ofTh-FO+ there is a description D of a k-ary FO-function such that

(e) If F l ( z 1 , . . . ,zk), . . . , F r ( z 1 , . . . , t ~ ) are open formulas of the language of Th-FO+ and if D1,. . ., D,, Dr+l are descriptions of k-ary FO-function symbols, then there is a description D' of a k-ary FO-function such that

Th-FO' k V Z ~ , . . V t k t(z1 , .. . , z,) = f ~ ( t 1 , . . . , zk).

Th-FO' k (F1 A fDl = f ~ ] ) V (-F1 A Fz A fDl = f&) V . . . V (7F1 A . . . A l F r A fD8 = fD,+]).

(f) For any open formula F ( r , 2 1 , . . . , zk) of the language of Th-FO+ there is a ( k + 1)-ary description D of a FO-function such that

.Th-FO+ k ( 3 y Z) F ( y , 2 1 , . . . , ~ k )

+ f D ( z , z11. . . zk) 2 A F ( f D ( z 1 z 1 1 . s . I zk),z11 . . . I zk). The same is true (with a differed D ) if "initial subwordness" i s replaced b y "subwordness".

P r o o f . Let D = ( A , t ) and suppose that N I= f ~ ( a 1 , . .. , a k ) = a. Axiom (3) entails that

By (F2), M i= L ; ~ ( C Y , a l l . . . , ak, i(a), f l y ' , . . . , Q E k ) and, hence,

This shows that the second component of [D],(Q:l , . . . , Q Z k ) is, indeed, a. On the other hand, axiom (4) implies that

M I= (Vw < a) (w E Q" H A*O(w, a l l . . . , at, i (w) , Q:],.. . , a:.)). Hence,

M I= (VW < C Y ) ( W E a" ++ A(w, ~ 1 , . . . , ak, Q:',.. . In:')). From the above, we conclude that [D]M(R:', . . . , Q p k ) = Ra. The converse statement now follows from the bitwise characterization of the elements of N (see the opening proposition of this paper).

Part (b) is a consequence of part (a) and of the examples given in the previous section. The proof of part (c) stems easily from the proof of part (a). The statements (d) and (e) and the first part of (f) follow from (a), the examples 6, 7, 9, 11 (for (d)), 13 (for (e)), 12 (for (f)) of the previous section, and the completeness theorems. The second part of (f) reduces to two applications of the first part. In effect, by the example 8 of the previous section, there is a description "tail" of a binary FO-function such that Th-FO+ I- VzVy(y z -+ y - f t a i l ( y , z ) = z). Now, observe that Th-FO+ proves the equivalence (3y C* z) F(y, . . .) c.) (32 C +)(3y C f t a i i ( 2 , z)) F ( y , . . .).


P r o p o s i t i o n . If N is a model ofTh-FO+ and R is a substructure of N (wi th respect to the language of Th-FO+), then R is also a model of Th-FO+.

P r o o f . The truth of the fourteen basic axioms is obviously inherited by R, since these axioms are universal. The scheme of induction on notation for swq-formulas can be reformulated by

F(E) A +(Y)

--* (3+ C y) ( F ( z ) A ( ( t o E Y A -F(zO)) V (21 G Y A i F ( ~ l ) ) ) ) ,

where F is a swq-formula. Hence, the inheritance by R of the truth of this scheme follows from the fact that swq-formulas are absolute between N and R. More specifically: if F(zl, . . . , tt) is a swq-formula and 01,. . . , at are elements of R , then R I= F(al,. . . , ak) if and only if N I= F(a1,. . . ,Q). This fact is proved by induction on the complexity of F , using property (f) of the above Lemma.

There remains to show that the string-building principle (2) is true in R. To see this, take a swq-formula F(t, z1,. . . , ZL), (11,. . . , Ok E R and Q E tally(N). Consider the following description D of a (k + 1)-ary FO-function:

(U < wt+i A Fo(u, wi,. . . , wk, i ( U ) , w;’, . . . I w,uI”), wk+i), where u is the special variable. Pick a model M of Ao-CAo such that ( M , N) is a natural pair, and let fly’, . . . , $2;” be the pairwise corresponding elements to all. . . , O k .

Then, for a E MI

[DIM (0:’ 1 . . . I ngk I i (o ) ) = { u E M : u < a A Fo(u, 01,. . .,at, i ( u ) , f ly ’ , . . . ,

According to (B2), the first component of this pair, which we designate by C, is { u E tally(N) : N I= u C o A F*(u, all . . . , at)}. Hence,

C = { u E tally(N) : N I= u c Q A F(u , 01,. . . , a t ) } .

By hypothesis, the element f~(a,a1,. . . a t ) is in R. By (a) of the above Lemma, this element is the corresponding element to Co. It follows from the definition of “corresponding element” that ~ D ( C Y , 01,. . . , a t ) is the value a such that

N k a = o A (Vu C a) (bit(a, u ) = 1 - F(u, al, . . . , a t ) ) .

Due to the absoluteness of the swq-formulas, this equivalence holds in R. 0

T h e o r e m . Suppose that Th-FOt I- Vz1 .. .Vzk3yF(zl,. . . ,tk, y), where F is an open formula of the language of Th-FO+. Then there is a description D of a k-ary FO-function such that

Th-FO’ I- V Z ~ . . .VttF(t1, . . . , t k , f~(t1,. . . , ~ k ) ) . P r o o f . By the previous proposition, Th-FO+ is a universal theory. Due to (d)

and (e) of the above Lemma, the result follows from Herbrand’s theorem for universal theories. 0

T h e o r e m . E u e y model of Th-FO is the reduct of a model of Th-FO’. P r o o f . Take an arbitrary model N of Th-FO and let D be a description of

a k-ary FO-function. We define the interpretation of fD in N in the obvious way:

On End-Extensions of Models of l e x p 13

Given elements 01 . . . , OL of N , consider f ly’ , . . . , fl;’, their pairwise corresponding elements in a model M - with ( M , N) a natural pair - and define f ~ ( z 1 , . . . , zk) as the corresponding element to [D]~( f ly l , . . . ,a,””). It is now a straightforward consequence of the properties (F2) and (B2) that the axioms (3) and (4) hold. 0

0 Define a swq+-formula exactly like a swq-formula, except for permitting the new

function symbols f~ in the language to start the recursive definition. By (b) and (f) of the above Lemma and by (B2), every such formula is equivalent (modulo ThFO+) to a swq-formula (we will say that this latter formula is obtained by “unwinding” the former formula). Hence, instantiations of the schemes (1) and (2) by swqt-formulas are still valid in Th-FO+. This is a very convenient fact. In effect, it enables the work with function symbols that witness the truth of V3swq-sentences which are provable in Th-FO in order to prove in Th-FO+ a sentence F of the original language and - finally - to conclude (by the above Corollary) that Th-FOt- F. We describe this process as “working with smoothly introduced function symbols”.

C o r o 11 a r y. The theory Th-FO+ i s conseruafiue over Th-FO.

4 Exponential assumptions and bounded collection

The formula “(z 5 yAz $ y)V(z y))”, abbreviated by “z <( y”, defines a total ordering in the standard model, first according to length and then, within the same length, lexicographically. With the help of some easy checkings, we can smoothly introduce a unary function symbol S for “the immediate successor of z with respect to <(”. That is,

yA(3w C z) (w0 s z A w l

Th-FO’ t- S(E) = 0 A Vz(S(z0) = z1 A S(z1) = S(z)O). L e m m a . There is a binary swq-formula L ( z , y) such f h a f

(a) Th-FO I- Vz3’yL(z, y),

(c) Th-FO+ I- VzVy(L(z, y) + L(z0, S(y)) A L(z1, S(y))). (b) Th-FO t- L(E,E) ,

P r o o f . By results of BENNETT and PARIS, we know that there is a Ao-formula B(w,y,z) which expresses the relation wy = z in the standard model of arithmetic and such that the usual defining recurrence relations of exponentiation are provable in IAo. Hence, it is possible to “speak in IAo” of binary expansions of numbers (see [16] for the details). We may also (smoothly) introduce in the language of IAo function symbols IwI and (w) , so that “if w = ~ ~ = o ( l + 6 u ) . 2 * - u , where 6, E (0, l}, then IwI = s+ 1 and ( w ) , = 6,”. As we know, the tally part of a model of Th-FO is a model of IAo. So, by the above discussion and the previous section, we can smoothly introduce in the language of Th-FO two function symbols 1.1 and (z), SO that, if tally(z) then “2 = Cu=o Is’-’( 1 + (z),) . 21r1-u-11 with all the arithmetic operations done in the tally part (and the understanding that 101 = 0 and that the empty sum is 0)”. Let L ( z , y) be the (unwinding of the) following formula,

y 11 x zl A (Vu C 11 x zI) (bit(y, u) = 1 ++ (1 x z), = 1).

Part (a) is a consequence of (2) and the bitwise characterization of the elements in models of Th-FO. Part (b) is trivial, and part (c) holds because the theory IAo proves


that both the indexes and the length of the binary expansion of a number modify in 0

According to the previous section, we can smoothly introduce in Th-FO a unary the right way when a unit is added to it.

function symbol t h such that

Th-FO' I- th(&) = E and Th-FO' I- l h ( z 0 ) = lh(z1) = S( lh ( z ) ) .

The existence of this function symbol permits the formulation of exponential assumptions in models of Th-FO without having to go through the definition of multiplication.

Lemma.

(a) Th-FO' I- z 5 y A z f y -+ l h ( z ) <t lh(y), (b) Th-FO' I- y <( lh (2 ) -+ (32 Z) l h ( z ) = y.

P r o o f . To show (a), fix z and prove by induction on notation on z that

Vz(z # & -+ l h ( z ) <( l h ( z - 2 ) ) .

To argue for (b), pick y and z with y <t lh ( z ) and, to obtain a contradiction, assume that (Vz E z) lh ( z ) # y. Argue by induction on notation on z that Vz(z z 4

0

If we view each element z of an arbitrary model of IAo as the string of zeroes and ones (z)O(z)l. . . ( z ) l Z ~ - l , it is clear that all the primitive symbols of the stringlanguage have a Ao-rendering in the language of arithmetic, except for the binary function symbol x (a model of IAo is closed under x if and only if it satisfies the so-called axiom ill). A swq"'-formula of the stringlanguage is defined exactly as a swq-formula except for the x-function symbol, which is replaced by a ternary relational graph counterpart. Clearly, the swqm-formulas have natural Ao-renderings. In the sequel, when we speak of swqm-formulas in the language of arithmetic, we mean their natural Ao-renderings.

With the above observations in mind, consider an arbitrary model M of IAo. As we have seen, M is the tally part of a model N of Th-FO. Let l h ( N ) be the set {z E N : (3y E tally(N)) Ch(y) = z}. By (b) of the previous lemma, l h ( N ) is an initial segment of N, i.e., if z , y E N, z <( y and y E l h ( N ) , then z E . fh (N) . On the other hand, l h ( N ) can be seen as a copy of M by viewing it as inheriting the Ao-structure of M via the binary length function (which is injective in the tally part of N). Due to the way the things are set up, the relations of l h ( N ) that come from swqm-relations in M via the function l h coincide with the swqm-relations that come from the fact that l h ( N ) is a substructure of N. Furthermore, it is clear that the order < of M translates (via l h ) into the order <(, and that the successor operation in M translates into the S operation.

The discussion in the previous paragraph is summarized by the first part of the following result:

M a i n T h e o r e m . Every model M of IAo is the initial segment of a model N of Th-FO such that t h ( N ) = M . Moreover, if M is not a model of exp, then M i s a proper initial segment of N.

P r o o f . In order to prove the second part of this result, we need to make a preliminary observation. In the observation, as well as in the remaining of the proof,

&(z) <( y). The case z = z originates a contradiction.

On End-Extensions of Models of i exp 15

we identify M with tally(N) (this is not so in the statement of the theorem). Suppose that y E tally(N) is an element such that z = Ih(y) E tally(N). Since z is a string of ones in N , we conclude, by the definition of Ih , that M I="y = C"-'2z-u", u=o

i.e., M k ' ' ~ = 2=+' - 2". (The converse also holds, i.e., if z , y belong to M and M I="y = 2'+' - 2", then N I= Ih(y) = z.)

Now, assume that I h ( N ) = N and take an arbitrary element z E tally(N). By assumption, there is y E tally(N) such that Ph(y) = z. Hence, M I="y = 2't1 - 2".

0

We have the folowing interesting situation. Every model M of IAo + l exp is a proper initial segment of a model of Th-FO but, in the meantime, while going from the original model to its end-extension, the language changes from the language of arithmetic to the stringlanguage of the binary tree. Moreover, this is a genuine change, since it does not seem possible to "smoothly" define the multiplication function in the end-extension.

There is a canonical way of associating a model NM of Th-FO to a model M of IAo satisfying the specifications of the previous theorem. Just let NM be ( M , S ) O , where S is the set of all Ao-definable subsets of M . This permits to give a meaning to the concept of FO-relation in an arbitrary model of IAo. More precisely, we say that X C M' defines an r-ary FO-relation in M if there is a swq-formula F(Z, y') and parameters F i n M such that X = {Z E M' : NM I= F(Z,p')} . Obviously, there is a direct way to define FO-relations in models of IAo + 0 1 . The reader should convince himself that, in this case, the two notions coincide.

The next three theorems are variations on a single theme, that of "a modicum of bounded collection holds in models of IAo + iexp" .

T h e o r e m (variation I). Let M be a model of IAo + Yexp and suppose that X

This entails that M I= 3y"y = 2'".

M x M is a binary FO-relation in M . Then, for all a E M ,

( 5 ) M I= (vz I a)3y(T(y) A (I, y) E X ) -+ 34Vz I . I ( ~ Y 5 2) (2, Y) E XI where T(y) stands for 3 ~ ~ ~ 2 " ~ ' - 2 = y".

P r o o f . First of all note that for any a E M , M I= T(a) e NM I= tally(a).

(This was argued for in the proof of the Main Theorem.) By definition of FO-relation in M I there is a swq-formula F ( z , y, a) and elements F i n M such that

X = {(z,y) E M x M : NM k F(z ,y ,F) ) .

Assume, by hypothesis, that M t= (Vz I a)3y(T(y) A (2, y) E X). We may rephrase this by

NM I= (Vz It a)(3y E M ) (tally(y) A F ( z , Y, $1). Hence, for every element yo E tally(NM) \ M I we have

Let b E tally(NM) such that N M I= Ih(b) = a. Using the fact that

{z E M : z < f a} = {Ih(u) : u C b},


we get

(6) for every yo in tally(NM) \ M. The formula to the right of the symbol "F' is a swq+-formula. By an underspill argument, we may conclude that there is yo in tally(NM)nM for which (6) holds. This entails M I= ( V t 5 a)(3y I yo) (2, y) E X. 0

The next variation is a weaker, purely syntactical corollary of the previous variation.

T h e o r e m (variation 11). For any swf'-fomula A ( z , y) of the language of arithmetic, possibly with parameters,

NM t= (VU C b ) ( 3 ~ 5 YO) F(th(u), Y,S),

(7) IAo + -xp I- (V+ I a)gy(T(y) A A ( z , Y)) - 3z(V+ I a)(3y I A(+ , Y),

where T(y) stands for 3u"2"t1 - 2 = y". 0

Ever since the seminal work of PARIS and KIRBY in [13], it is well known that the theory IAo does not prove the scheme of collection for bounded arithmetic formulas, i. e., does not prove every instance of

(8) ( V Z I a)3y A ( s , Y) - W't I a)(3y I 2 ) A(+, Y)I where A is a bounded formulas of the language of arithmetic (possibly with parameters). More recently, a t the end of [17], WILKIE and PARIS asked the "intriguing" (sic) question of whether the above scheme (8) is provable in IAo + iexp. The previous two variations say that a modicum of bounded collection is, indeed, a consequence of this theory. Remark that this modicum is non-trivial since, in the presence of exp, it implies the full scheme of bounded collection (note that when exponentiation holds, bounded quantifications can be replaced by subword quantifications).

There are two natural ways to attempt to generalize variation I. One way is to remove, or to weaken, condition T(y). The other way is to allow more general relations X in statement (5). As a matter of fact, there is a balance between these two possibilities: simply removing condition T(y) is (easily seen to be) equivalent to permitting, instead, Cf-relations X. And this latter is beyond our present reach because of work in [6] showing that this entails P # N P . So, either we try to weaken (without removing) condition T(y) or, else, we try to consider binary relations X with complexity falling between FO and NP. We explore the first possibility.

D e f i n i t i o n . A swq-predicate P(t) is provably sparse in Th-FO if

(9) Th-FO I- V U ~ ~ V Z ( Z I u A P ( z ) -+ z C* y).

The computer scientist says that a set S 2 {0,1}' is sparse if there is an integer polynomial p ( n ) such that, for every n E w , card{% E S : length(z) 5 n} I p(n). We want to remark that a provably sparse (in Th-FO) swq-predicate P(t) does, indeed, define a sparse set in the standard model. This is a consequence of a Parikh type result (see, for instance, Chapter V.l of [8]), to the effect that the statement (9) implies the existence of a term t ( u ) of the language of Th-FO such that v u ( 3 y 5 t (u ) )Vz ( z 5 u A P ( z ) + z c* y).

Let M be a model of IAo. We say that a set S C M is FO-sparse if there is a provably sparse swq-predicate P ( z ) such that S = {t E M : NM P P(t)}.


T h e o r e m (variation 111). Let M be a model of IAo + Texp and suppose that S C M is a FO-sparse set in M and that X E M x M is a binary FO-relaiion in M . Then, for all a E M ,

M I= (Vz I a)3y(y E S A (2, y) E X) 4 3z(Vz I a)(3y 5 r) (2, y) E X. P r o o f . Let P ( z ) be the provably sparse swq-predicate defining S and fix an

element 6 E tally(NM) \ M. According to condition (9), there is yo E NM such that

N M I= (Vr _< ii) (P(2) + 2 C’ yo).

Assume, by hypothesis, that M I= (Vz 5 a)3y(y E S A ( z , y ) E X) and take an arbitrary element u E N M \ M such that u ii. By the first condition on u,

NM I= (Vz If 0 ) ( 3 Y I .) (P(Y) A F ( z , Y I i j ) ) , where F and p’ are, respectively, the swq-formula and the parameters that define the FO-relation X. Now, the second condition on u plus the considerations on the previous paragraph yield

(10)

(11)

NM I= (Vz If a)(3y S’ Yo) (Y I u P(Y) A F ( z , Y, $1).

N M I= ( V V E b)(3Y E’ Yo) (Y I A P(Y) A F(Jh(v ) , Y,F)). Let b E tally(NM) such that NM I= Ph(b) = a. We can rephrase (10) by

By an underspill argument, we may conclude that there is u E tally(NM)nM for which (ll), and hence ( lo ) , holds. We conclude that M I= (Vz 0 a)(3y < u) (z, y) E X.

We finish with the following separation result: T h e o r e m . There i s a swf‘-fonnula A(z ) such that

IAo I- VzA(2) but Th-FO )L V+A(z).

P r o o f . In order to obtain a contradition, assume that no such swqm-formula exists. Take a model Mo of IAo + R1 in which there is a semantic tableaux proof d of 0 = 1 from IAo + R1. This means that MO I= 3 z F ( z , d ) for a certain swq- formula F . (This rests on the fact that NP-relations can be defined by formulas of the form “(32 5 2(. . . ) ) A ( z , . . .)”, with A a swq-formula - see [4].) Clearly, there exists a swqm-formula G so that the theory Th-FO (and, a fortiori, IAo + R1) proves Vy(3zF(z, y) H 3zG(r, y)). This is done by absorbing existential commit- ments into the quantifier “ 3 ~ ” . By assumption and easy model theory, the model N M ~ of the theory Th-FO can be embedded into a model Mi of IAo such that swqm- formulas are absolute between N M ~ and MI. Moreover, we may assume that Mi also satisfies the axiom R1 (if the original model does not satisfy this axiom, consider its initial segment cofinal in N M ~ ) . Iterate this process w-times to obtain a chain Mo C N M ~ S M I S N M ~ ..., where the E-signs mean that each left entry is a substructure of the right entry with respect to the modified language of Th-FO (obtained by replacing the binary x-function symbol by a ternary relational graph counterpart), and that the truth of swqm-formulas is preserved between the entries. Clearly UnEw N M , is a model of Th-FO+{Vz3yLh(y) = z}, i.e., it is a model of IAo + exp. By results of WILKIE and PARIS in [18], UnEw NM. I= v + v y ~ F ( + , y). Hence, by absoluteness (via the formula G) Mo I= V z ~ F ( z , d ) . This is a contradiction. 0


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(Received: August 19, 1994; Revised: April 13, 1995)

Logic 48 (1990), 261 - 276.

On End-Extensions of Models of ¬exp

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