Math. Log. Quart. 45 (1999) 3, 399 - 407

Mathematical Logic Quarterly

@ WILEY-VCH Verlag Berlin GmbH 1999

Two General Results on Intuitionistic Bounded Theories

Fernando Ferreira')

Departamento de MatemLtica, Universidade de Lisboa Rua Ernest0 de Vasconcelos, bloco C1-3, 1700 Lisboa, Portugal')

Abstract. We study, within the framework of intuitionistic logic, two well-known general results of (classical logic) bounded arithmetic. Firstly, Parikh's theorem on the existence of bounding terms for the provably total functions. Secondly, the result which states that adding the scheme of bounded collection to (suitable) bounded theories does not yield new I I 2 consequences.

Mathematics Subject Classification: 03F30, 03F55, 03C50.

Keywords: Bounded arithmetic, Bounding terms, Bounded collection, Intuitionism.

1 Introduction

The theory IAo is that subsystem of Peano Arithmetic in which the scheme of in- duction is restricted to bounded formulas. These are the formulas generated from atomic formulas via the successive applications of Boolean connectives and boun- ded quantifications, i. e., quantifications of the form (Vy 5 t(z)) (. . .) or of the form (3y 5 t(z)) (. . .), where t ( E ) is any term of the language in which the variable y does not occur (we are abbreviating a sequence of variables 2 1 , . . . , 2k by Z). In 1971, ROHIT PARIKH proved in [6] the following

T h e o r e m . If IAo I- Vz3yA('i, y), where A is a bounded formula, then there is a term t (Z) such that IAo I- E ( 3 y 5 t ) (Z) A @ , y).

The conclusion is false if A is not restricted to bounded formulas. For instance, IAo proves the sentence Vz3yVz(Exp(z1 z) + y = z), where Exp(z, z) is a bounded formula that suitably formalizes the relation 2" = z . Observe that there can be no bounding term for y. However, classical reasoning (in the form of a use of the excluded middle) seems to be needed to argue for the above sentence. As a matter of fact, it follows from Theorem I below that the above sentence is not provable intuitionistically in IAo (nor even in IAo supplemented with the law of excluded middle for bounded formulas).

In 1980, JEFF PARIS in [7] showed that the theory IAo plus the collection scheme BCo for bounded formulas is a IIa-conservative extension of IAo. In other words:

PRAXIS XXI. ')This research was partially supported by CMAF (Fundw%o da CiBncia e Tecnologia) and

2)e-mail: ferferrOptmat.Imc.fc.uI.pt

400 Fernando Ferreira

T h e o r e m . If IAo + BE0 I- VZ3yA(Z, y), where A is a bounded formula, then

Here, BCo is the scheme IAo I- VT 3~ A(Z, Y).

(VU 5 x) 3Y N u , Y) + 3w (VU 5 4 (3Y 5 4 4% Y), where A is any bounded formula that may contain additional free variables as param- eters.

A compactness argument yields a very simple proof of the earlier theorem. This argument readily generalizes to other bounded theories of arithmetic, such as those introduced by SAMUEL BUSS in his doctoral dissertation [2]. The original proof of the latter theorem does not extend to more general situations, although the theorem still holds for a wide class of bounded theories of arithmetic. This was stated and shown by Buss in [3].

In this paper, we envisage studying the above two theorems within the intuition- istic setting. We will place ourselves in a rather general framework, since our results (as well as the classical results) need not be framed in theories of arithmetic. Results such as the above two theorems - and their intuitionistic versions - depend solely on some general features of the binary relation 5 and of the provable structure of the terms (however, see the final section).

We shall be concerned with intuitionistic theories formulated in first-order re- cursively presented languages (this condition is here for convenience only, and can be discarded - we will comment on this in the final section) which include a distinguished binary relation symbol a. A quantification is bounded if it is of the form Vx (xat -, . . .) or of the form 32 (x a t A . . .), where t is a term of the language in which z does not occur. These quantifications are abbreviated by (Vxat) (. . .) and (3xat) (. . .), respec- tively. A formula is called 3n-free if all its existential quantifications are bounded. Note that bounded formulas are jn-free.

The following definition is an adaptation to our setting of a definition of BUSS in [3]:

D e f i n i t i o n . Let r be a theory in a first-order language as above. We say that r is a term-sufficient theory if there is a binary term maj, a unary term trans, and for each n-ary functional symbol f there is a n-ary term aj such that:

(1) r I-i VX (x a x); (2) r I-i VxVy(xamaj(x,y) A yamaj(x ,y) ) ; (3) r Fi Vx (Vy a x) (Vz a y) (z a trans(z)); (4) for any function symbol f(z) of the language, r I-i VZ(@ a z ) (f($ a oj(Z)),

where I-i denotes intuitionistic provability. For convenience, we also demand that there is at least a constant c in the language. We say that I' is a 3n-free term- sufficient theory if it is a term sufficient theory and if it is recursively axiomatized by a set of P- f ree formulas.

We are using some obvious abbreviations: for instance, (Vp a 5) abbreviates (Vy1 a ~ 1 ) . . . (Vyk a xk). Condition (3) is complicated because we are striving for generality. In many cases a will be a transitive relation and (3) comes automatically by taking trans(z) := x . Also, in many theories of bounded arithmetic, the function

Two General Results on Intuitionistic Bounded Theories 401

symbols of the language are (provably) monotonous and, thus, aj can be taken to be f itself. This is the case with the function symbols of the theories IA,, S;, Ti, etc.

In our discussion, we will need two simple facts concerning term-sufficient theories. Their proofs are simple and we omit them:

L e m m a . In a term-suficient theory r, for each positive natural number k there is an k-ary term majk(x l , . . . , xk) of the language such that

rt-i V q . . .V(zk(Vy1 a x 1 ) . ..(Vyk azk),&(yi amajk(z1 , . . .,zk)). L e m m a . Let I' be a term-suficient theory. For each term t(T) of the language

there is a term pi@) such that rt-i ' dz 'V (Ja?F) (Vwat (p ) ) (wap~(F) ) . The main aim of this paper is to prove the following two theorems: T h e o r e m I . If r i s a 3"-free term-suflcient theory and r t - i E 3 y A ( Z , y ) ,

where A is any (first-order) formula, then there is a term t(5) of the language such that r t-i E ( 3 y a t(Z)) A(5, y).

The scheme of 3n-free collection, denoted by Bg", consists of all formulas (Vu a 2) 3y A(u, y) -, 3w (Vu a z) (3y a w) A(u, y),

where A is a 3n-free formula, possibly with parameters. T h e o r e m 11. If I' is a 3n-free term-sufficient theory and r + B q n l-i E3yA(Z, y),

where A is 3"-free, then I' t-i V53yA(Z, y). The strategy for proving these theorems follows a technique introduced in [5].

This technique hinges on realizability arguments that take place within theories whose languages permit countable infinite disjunctions. For the first theorem, the theory is the same as the original theory. For the second theorem, we need an extra infinitary axiom schema. In the next section, we briefly describe these infinitary theories and mention the pertinent results concerning them. The third section introduces infinitary realizability notions convenient to proving the above theorems. The two theorems themselves are consequences of soundness theorems pertaining to the realizability notions.

2 The technical framework

In the present section, we shall introduce realizability notions framed in a language which permits countable infinite disjunctions. We deal with these countable infinite disjunctions semantically by adding an extra forcing clause in the Kripkean semantics ([9] is a good reference for the intuitionistic notions used in this paper). Given a Kripke model K: of a term-sufficient theory

Now it makes sense to say that I' forces a sentence A of the extended infinitary language. It means that the least node (hence, every node) of every Kripke model of I' forces the sentence A , and we write r II- A. With a view of proving Theorem 11, we need to consider a special class Rr of Kripke models of I?. This class is, by definition, the class of Kripke models of I' which force the following infinitary sentences (henceforth called the infinitary azioms):

and given a node (Y of Ic we let IF, Vn Fn := (3m E w ) IF, F,.

Vz(A(z) + Vn Fn(z)) + VntJz(A(z) -+ Vksn Fk(zC)),

402 Fernando Ferreira

where A(z) is a first-order formula and Fo(z), Fl(z), F z ( z ) , . . . is a recursive enu- meration of first-order formulas with only a finite number of parameters. Given a (first-order or infinitary) sentence A, we write I', II- A if every Kripke structure of fir forces A.

The following proposition is crucial: P r o p o s i t i o n (Conservativity). Let r be a term-suficient theory of arithmetic

and suppose that

(*) roo [I- va: V,(3Y a t ( 5 ) ) A@, Y), where A(5 ,y ) is a first-order formula with its free variables as shown and t ranges over all the appropriate terms. Then there is a t e rm t o f the language such that

In assertion (*), if r were instead of roo, a single compactness argument would suffice to prove the proposition. In the general case we need a compactness argument on top of multiple compactness arguments (in fact, on top of a recursive saturation argument). The framework for making these arguments consists in looking "at Kripke models from outside, as a complicated concoction of classical structures, and hence as a classical structure itself' (DIRK VAN DALEN [8, p. 2641). We will not describe the procedure of associating to each Kripke structure K its corresponding classical structure K". We refer the reader to the above mentioned work of VAN DALEN. For the present purposes, it is sufficient to say that the language L" supporting K" is a modification of the original language L tailored to describe the structure of the original Kripke structure K and, conversely, that we restrict ourselves to classical structures M which validate a number of (first-order) laws so conceived as to permit the extraction from M of a natural Kripke structure M ' . In actual fact, we have (Mi)" M M and (K")' M K.

By making a detour through the classical structures as described above, we can mimic the recursive saturation argument of the proof of the main lemma of [5, Sec- tion 31, and obtain

L e m m a. Suppose K is a (countable) Kripke model of I' for which the associated classical structure Ic" is recursively saturated. Then K E Rr .

We are now ready to prove the conservativity result. P r o o f of t h e P r o p o s i t i o n . Suppose the conclusion is false. Let t ( z ) be a

term of the language (to simplify notation, we work with a single variable z). Then r Fi Vx (3y a t ( z ) ) A(zJ y). By the Kripke completeness theorem, there is a (rooted) Kripke model Ic t such that Y O Vz (3y a t ( z ) ) A(z, y), where 0 is the root of K t . This means that there is a node CY of K t and an element d in the domain D(a) of the node CY such that for all d' E D ( a ) , if IF, d ' a t ( d ) , then W, A ( d , d ' ) . Consider the theory C consisting of I? plus the following sentences (one for each appropriate term t ) :

r ki E ( 3 y a t(a:)) A(Z, y).

E D ( K ) A (vy E D ( K ) ) ( I k K y a t ( k ) + Y K y))~ where k and K are new constant symbols. (We are abusing language in the above definition of C: in rigour, the sentences of C should have been stated in the language L" discussed earlier.) Using the first lemma after the definition of a term-sufficient theory, it is easy to argue that every finite subset of C has a (classical) model. Hence, by compactness, C has a countable (classical) model M . Let M , be a recursively

Two General Results on Intuitionistic Bounded Theories 403

saturated structure elementarily equivalent to M . From M , we can read-off a Kripke structure K, = M i . By the previous lemma, this structure is in the class Or . Thus, by hypothesis, the (infinitary) sentence Vz Vt(3yaI(z)) A ( z , y) is forced in K,. However, by construction, lc, has a node a and an element d E D ( a ) such that for every term t ,

(Vd‘ E D(ru)) ( I F a d’ at(d) +Fa A(d, d’)). This yields a contradiction. 0

3 Notions of infinitary realizabiliy

With the help of the conservativity result of the previous section, Theorems I and I1 are corollaries of soundness theorems for suitable notions of realizability. In the following definition we introduce the two notions of realizibility that we need.

D e f i n i t io n. Let L be the (first-order) language of a term-sufficient theory r. To each (first-order) formula A of L we associate a new formula zrA of the extended infinitary language according to the following clauses ( z is a new variable):

1. zrA is A, if A is an atomic formula; 2. zr(A A B ) is (3zo a z ) (321 a z ) (zor.4 A zlrB); 3. zr(A V B ) is (3w a z) (wrA V wrB); 4. zr(A + B ) is Vz (zrA 4 V,(3w a t (z , z)) wrB); 5. zrVx A(z) is Vx V,(3w a t(z, z)) wrA(x); 6. zr32 A(x) is (3x0 a z ) (321 a z ) zorA(zl),

where t ranges over appropriate terms. The notion of q-realizability has the same clauses as r-realizability, except for an extra requirement in the case of implication:

4*. zq(A + B ) is (A + B ) A Vx(xqA 4 Vt(3w a t ( z , s ) ) w q B ) .

In the above definition we omitted talk of parameters. However, it should be clear from the context what parameters are permitted in each term. For instance, in Clause 5, if 1 are the parameters of the matrix A , then t ranges over all terms of the form t ( z , E , E) , where all the variables are as displayed.

The realizability clauses for bounded quantifications simplify somewhat: L e m m a . In a term-suficient theory I‘ the following holds (intuitionistically):

5’. zr(Vzas(G))A(z,G) z$ (Vzas(’il))Vt(3w dt(z,C))wrA(z,’il); 6’. zr(3z a s(G)) A ( z , E) i$ (320 a z ) (3zL a 20) (3x1 a z ) (21 a t(G) A zhrA(?I, 2 1 ) ) .

P r o o f . We will prove the left-to-right implication of 5 ’ . The other three implica- tions are easy. Suppose zrVz (z a s(G) 4 A(z,C)) . Take z with z a s(?i). According to the definition of r-realizability, there is a term q and an element w’aq(z, 2, G) such that

Vv (wr(z as@)) ---f V,(3war(w’,v,~,E))wrA(zlC)). Taking v := c, we get a term T’ and an element w a ~ ’ ( w ‘ , z,G) such that wrA(z , a). It is not difficult to check that the term t(z,E) := er,(pn(z, s(G),G), s (G) ,E) does the job. 0

404 Fernando Ferreira

The following proposition is handy: P r o p o s i t i o n . Let I' be a term-suficient theory. For each 3"-free formula A@)

there as a term tA(Z) such that

(1) I? ki 3y(yrA(Z) ) -+ A(Z) and (ii) r I-i A ( Z ) -+ (3y a tA(Z)) yrA(:). The same holds for the notion of q-realizability.

P r o o f . The term t A is constructed by induction on the complexity of A:

(a) t A ( F ) := c , if A is an atomic formula;

(b) t A h B ( f ) := majZ( tA(Z) , tB( f ) ) ; ( c ) t A V B ( f ) := majZ(tA(F), tB(Z)); (d) tA-+B(Z) := C ;

(e) t V u A ( u ) ( F ) := c;

(f) t(3uas) A(" ) (5) := m a j 2 ( e t , (@, F), ~(5)).

We establish (i) and (ii) simultaneously by induction on the complexity of A. We will only check cases (d) and (f). The statements (i) corresponding to these cases are straightforward.

Let us argue for (ii) of case (d). Suppose that A@) -+ B ( z ) and assume that z r A ( z ) . By induction hypothesis and by Modus Ponens we get B(T). Again by induction hypothesis, we conclude that (3w a t B ( Z ) ) wrB(3). Thus, we have argued that rrA(T) -+ Vt(3w a t ( z ) ) w r B ( Z ) . This shows that cr(A(Z) -+ B ( Z ) ) .

Let us now check (ii) of case (f). Suppose that there is u a s(Z) such that A ( u , Z). By induction hypothesis there is yatA(u ,F) such that yrA(u ,Z ) . Clearly, both u and y are in the relation a with maja(et,(s(Z),Z), ~ ( 5 ) ) . Thus, our conclusion follows.

Similar (and sometimes more immediate) arguments work for the notion of q-realizability. 0

The notion of q-realizability is, as in usual realizability notions, r-realizability plus

L e m m a . Let r be a term-sufficient theory. If z does not occur in the formula A,

We have the following soundness theorem. S o u n d n e s s T h e o r e m I. If I' is a 3n-free term-sufficient theory and zf

r t-i A@) , where A @ ) as a (first-order) formula, then r IF V,(3w a t ( Z ) ) wrA(z ) , where t ranges over all terms whose variables are among the free variables Z of A@). The same is true for the notion of q-realitability.

P r o o f . The proof is by induction on the lengths of derivations in a Hilbert-type deduction system. For determinateness, we work with the deductive system described in [9, p. 681. By the above proposition, the mathematical axioms of I' pose no trouble since, by hypothesis, they are 3"-free. The equality axioms pose no trouble either. It would be tedious to go throught all the thirteen logical axioms of our deduction system. All these axioms are - in fact - realized by any element (by c , in particular), and we will see this for three typical axioms (to make the discussion more clear, we will usually avoid the consideration of parameters).

truth. The following mimics a standard result of the literature:

then r It- ( r q A ) -+ A .

Two General Results on Intuitionistic Bounded Theories 405

The logical axiom A -+ ( B -+ A) is r-realized by c if and only if Vz(zrA -+ Vt(3zo at(z))Vw (wrB -+ V4(3wo aq(z0,w))worB))).

Given z with zrA, take t ( z ) = z and let to = z. Then, given w with wrB, take q ( z 0 , w) = zo and let wo = q(z0 , w).

Now let us consider the axiom (A -+ C) -+ ( ( B -+ C) + (A V B -+ C)). We must see that zr(A -+ C) implies

Vr(3z’ a r(z))Vw (wr(B * C) -+ V,(3z”as(z’,w))Vy(yr(A V B ) + VP(3z”’ap(z’‘,y))z’’’re)).

Suppose that zr(A -+ C), wr(B + C) and yr(A V B). Take yo a y such that yor A V yorB. Suppose that yorA. Then there is a term t and there is zo a t ( z , yo) such that zorC. On the other hand, if yorB, then there are a term q and a woaq(w, yo) such that worC. It is a matter of simple checking to see that we can take r ( z ) = z , z’ = z , s (z’ , w) = majZ(z‘, w), z” = s (z ’ , w) and p(z”, y) = majz(et(z”, y), eq(z” , y)).

The third axiom that we consider is Vx(A(x) -+ B ) -+ (3yA(y) -+ I?), where z is not free in B. We must see that

zrVx(A(z) -+ B) -+ Vr(3z’ ar(z))Vw (wr3yA(y) -+ Vp(3w’ ap(z’, w)) w‘rB). Suppose that zrVx(A(z) -+ B ) and wr3yA(y). Then, for some wo Q w and for some w1 aw, we have worA(w1). Hence, there is a term t ( z , wl) and an element zo a t ( r , wl) such that zor(A(w1) + B). Thus there exists a term s and an element yo a s(z0, wo) such that yorB. It is not difficult to check that we can take r ( z ) = z , z’ = z and

Let us now consider the two rules of our deduction system, Modus Ponens and Generalization.

For Modus Ponens suppose that we have both l? IF V,(3w at(z))wrA(x) and I? IF Vr(3yar(x))Vz(zrA(x) -+ V (3zoaq(z, y, x)) zorB(z)). We reason (intuitionisti- cally) inside I’. Take t and w a t t z ) with wrA(z). We know that there is a term T

and an element y a ~ ( x ) such that for some term q and element zo a q(w, y, x) we have zorB(x). Let s(x) := eq(t(z), r ( z ) , x). It is clear that there exists zo a s ( x ) with zor B( x) .

Finally, we consider the rule of Generalization, A(z, 1) / VzA(z, T i ) . By induction hypothesis, I’ IF V,(3w at(z,G))wrA(z,E). Inside I’, we must see that

Just take q ( E ) = majk(u1,. . . ,uk), y = q ( E ) and s(y,z) = pt(z,y, . . .y) (k many y’s), where u1, . . . , uk is the sequence of variables 1.

Although we gave an argument for the result pertaining to the notion of r-realizability, it should be clear that a similar argument yields the result for the

~ ( z ’ , w) = eS(et(z’, w), w).

Vq(3Y a q(Ei>) vx V,(3z a S(Y, zrA(x, 1).

notion of q-realizability. 0

We are now ready to prove Theorem I: P r o o f of T h e o r e m I: Suppose that r is a 3”-free term-sufficient theory and

that r I-i 3yA(Z, y), where A is a (first-order) formula. By the above soundness theo- rem for the notion of q-realizability, I’ IF V t ( 3 w a t @ ) ) (3w0 a w) (3y a w) woqA(f, y). Since q-realizability implies truth, we get I’ I t - V,(3y a trans(t(F))) A(f , y). The con-

0 clusion of the theorem follows from the conservativity result.

406 Fernando Ferreira

S o u n d n e s s T h e o r e m 11. If I7 is a 3n-free term-suficient theory and if r+BTn k, A@), where A(Z) is a first-order formula, then roo IF Vt(3wat(5)) wrA(c), where t ranges over all terms whose variables are among the free variables 5 of A@).

P r o o f . We just have to supplement the argument given in proving Soundness Theorem I with the proof that each instance of the BJO-scheme is r-realized by c within roo. Let (Vu a z) 3yA(u, y) -+ 3w (Vu a z) (3y a w) A(u, y) be an instance of the B3 -scheme, where A is a 3n-free formula. We must see that

rM I ~ V z ( t r ( V u a z ) 3 y A ( u , y ) -+ V t ( 3 v at(r,z))vr3w(Vuat)(3yaw)A(u,y)). We reason (intuitionistically) inside roo. Suppose that zr(Vu a z) 3yA(u, y). Thus, (Vu a z) v , ( 3 y a Q(Z, z)) (3y0 a y) (3yl Q y) yorA(u, yl) . Since A is an 3n-free formula, the (possibly infinitary) formula “yOrA(u,y1)” can be replaced in the above by the first-order formula “A(u, ~ 1 ) ” . Using the infinitary axioms (and the first lemma after the definition of a term-sufficient theory), we may then conclude that there exists a term Q such that (Vu a x ) (3y a ~ ( z , z)) (3yl a y) A(u, y l ) . Using again the fact that A is a 3n-free formula, we get (Vu a z) (3y a t ( z , z)) (3yo ay ) (3yl a y) yorA(u, yl) , where t ( z , x ) is the term m a j z ( q ( t , x ) , et,(z, trans(q(z, z)))). A small amount of detailed checking shows that this implies t ( r , z)r3w (Vu a x) (3y a w) A(u, y). This yields our result. 0

P r o of of T h e o r e m 11: Suppose that r is a 3’-free term-sufficient theory and that I’ + Bgn t i 3yA(Z, y), where A is a 3n-free formula. By the above soundness theorem, roo IF Vt(3w a t(5)) (3wo a w) (3y a w) worA(5, y). Since A is a 3”-free formula, r-realizability implies truth. Thus, roo IF V,(3y Q trans(t(5))) A@, y). The

0

-51

conclusion of the theorem follows from the conservativity result.

4 Remarks and questions

The restriction of the above theorems to recursively presented languages is not neces- sary. The same arguments hold for any countable language by relativizing everything to the Turing degree of the presentation. Even the restriction to countable languages is not necessary because the arguments can be localized to the pertinent countable fragment of the language.

The proofs of this paper are infinitary in nature, since they use the semantic appa- ratus of Kripke models cum languages which permit infinitary disjunctions. However, if we weaken the statement of Theorem I to bounded term-sufficient theories and to bounded matrices A, then the proof-theoretic argument in [2] at the end of Chapter 4 readily adapts to the intuitionistic case. The question is: can we also give a finitistic argument for the general case ?

A similar sort of question can be asked for Theorem 11. Moreover, in [4] we pre- sented a very simple model-theoretic proof of the classical version of Theorem I1 for theories in which the structure of terms does not play a prominent role (contrary to Theorem I, the statement of Theorem I1 does not mention terms anywhere). [4] re- places the second and third clauses of the definition of a term-sufficient theory by

(2’) I’ t- VxVy3w (z a w A y a w ) ; (3’) r k Vx 3w (Vy a x)(Vt a y) (z a w),

Two General Results on Intuitionistic Bounded Theories 407

and also permits axioms of the form ‘ ‘VElF(VZ 4 Z)(3-2: 4 E ) A(E, 2)’’ , where A @ , F) comes from the bounded formula A@, Z) by substituting the variables and E by the new variables and TI respectively. In this generality, is an intuitionistic counterpart of the theorem in [4] also t rue? If so, can a finitistic proof be provided?

Recently and independently, WOLFGANG BURR produced some work related to the issues of this paper using (an adaptation of) Godel’s functional interpretation. As of now, it is not clear to us how BURR’S work compares logically to our work. That notwithstanding, we want to draw the attention of the reader to the last section of [I].

References

[l] BURR, W., Fragments of Heyting-arithmetic. Manuscript 1998. [2] BUSS, S., Bounded Arithmetic. PhD thesis, Princeton University, June 1985. A revision

of this thesis was published by Bibliopolis, Naples, in 1986. [3] BUSS, S., A conservation result concerning bounded theories and the collection scheme.

Proc. Amer. Math. SOC. 100 (1987), 109 - 116. [4] FERREIRA, F., A note on a result of Buss concerning bounded theories and the collection

scheme. Portugaliae Math. 52 (1995), 331 - 336. [5] FERREIRA, F., and A. MARQUES, Extracting algorithms from intuitionistic proofs. Math.

Logic Quarterly 44 (1998), 143 - 160. [6] PARIKH, R., Existence and feasibility in arithmetic. J. Symb. Logic 36 (1971), 494 - 508. [7] PARIS, J . , Some conservation results for fragments of arithmetic. In: Model Theory and

Arithmetic ( K . MCALOON, C. BERLINE and J.-P. RESSAYRE, eds.), Lecture Notes in Mathematics 890, Springer-Verlag, Berlin-Heidelberg-New York 1980, pp. 251 - 262.

[8 ] VAN DALEN, D., Intuitionistic logic. In: Handbook of Philosophical Logic, Chapter 111.4, (D. GABBAY and F. GUENTHNER, eds.), D. Reidel Publ. Comp., Dordrecht 1986.

[9] VAN DALEN, D., and A. S. TROELSTRA, Constructive Mathematics: An Introduction. Volume 1. North-Holland Publ. Comp., Amsterdam 1988.

(Received: December 14, 1997; Revised: November 1, 1998)