R. J. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494-508, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for details. The usual formulation of WKL cannot be written down in G 2 A since it requires the coding functional Phi hi fx : hf0; f(x Gamma 1)i. In G 3 A one can show that WKL seq implies WKL. 12 Theorems 3.3,3. 5 can also be viewed as a vast extension of a result by Parikh [19]: Parikh considered a fragment PB of Peano arithmetic PA which contains the schema of induction only for bounded formulas. He shows that if a sentence 8x9yA(x; y) A(x; y) being a bounded formula) is provable in PB then there exists a polynomial p such that PB proves 8x9y p(x) A(x; y) So PB can ....

Parikh, R.J. Existence and feasibility in arithmetic. J. Symbolic Logic 36, pp.494--508 (1971).

....which may not reflect anything concrete or real . Extremely large numbers were troubling to some, and there was the idea that they should be treated differently from a small number like 37 which is closer to ordinary existence. The first mathematical treatment of feasible numbers was given in [36]. The philosophical discussions go back to Mannoury, Poincar e, and Wittgenstein. For this we start with the first order theory of arithmetic, and we add a unary predicate F . Roughly speaking F (x) is interpreted as meaning that x can be constructed in some feasible manner. We shall use the ....

....the cut rule one can make short proofs of feasibility which provide only implicit descriptions, as we shall soon see. There are effective methods for converting proofs with cuts into proofs without, at the cost of great expansion in the proofs. See [21, 45, 10] Let us mention one more point. In [36] an F : inequality rule is included in the axioms, to the effect that if y is feasible and x y then x is also feasible. For the historical concern about large numbers this is a reasonable requirement to consider, but we have omitted it intentionally. It does not fit as well with the idea of a ....

R. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494--508, 1971.

....[98] and Wainer [87] in this book. The work reported here is new and based largely on Constable and Crary [38] and Benzinger [14] as well as examples from Kreitz and Pientka [90] One interesting application of the resource indexed types is to define types like Parikh s feasible numbers [89], numbers that may be computed in a reasonable time. Benzinger [14] shows another application. Acknowledgements I would like to thank Juanita Heyerman for preparing the manuscript so carefully and for contributing numerous suggestions to improve its form. I also want to thank the excellent ....

R. Parikh. Existence and feasibility in arithmetic. Jour. Assoc. Symbolic Logic, 36:494--508, 1971.

....of the class of polytime functions P . He showed that the class of polytime functions is the smallest class closed under composition and limited recursion on notation; this allowed the treatment 1 Chapter 1. Introduction 2 of polynomial functions in the context of pure logic. In 1971, Parikh [Par71] proposed the rst system of bounded arithmetic, called I 0 , where the induction scheme was restricted to bounded ( 0 ) formulae. He showed that all functions that are 0 de nable in I 0 are polynomially bounded; i.e. if is a 0 formula and I 0 8 x9y ( x; y) then the value of y is ....

R. Parikh. Existence and feasibility of arithmetic. Journal of Symbolic Logic, 36:494-508, 1971.

....however nonconstructive. The true reason why we are interested in these systems is that they seem to be more amenable to logical analysis and for their close relation to complexity theory. The oldest considered system is I 0 which is Q plus induction axioms (1) for 0 formulas, introduced in [27]. If M is any model of I 0 , then any in nite initial segment closed under multiplication is a model of I 0 too. This implies that if I 0 proves that a function f(x) is de ned for all numbers, then the function is bounded by a polynomial in x. In particular it is not provable in I 0 that x ....

R. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36, (1971) pp. 494-508.

....One might try to think of a proof as being like an algorithm, in which terms evolve and a construction is in progress. This is an appealing idea, and indeed one can often see natural inputs for a proof. To make these ideas more concrete let us consider the concept of feasible numbers. As in [13], one works in arithmetic but allows an extra unary predicate F ( Delta) for which the intended meaning of F (n) is that n is feasible . All natural numbers are feasible, but one is interested in the sizes of proofs of feasibility (without allowing induction over formulas containing F ) A proof ....

R. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494--508, 1971.

....z of variables instead of x, y, z, if b AC is formulated for tuples. Furthermore instead of #w # A 0 we may also have #z # #z # A 0 where z # is of arbitrary type: It still is possible to bound #z # . Remark 3.2.5 Cor.3.2. 3 is a considerable generalization of a theorem due to Parikh ([27] ) Parikh shows for a subsystem (called PB) of the first order fragment of G 2 A # : If PB# #x#yA(x, y) where A contains only bounded quantifiers and only x, y as free variables) then there is a polynomial p such that PB# #x#y # p(x) A(x, y) Proof of thm.3.2.2 : For PA # the theorem is ....

Parikh, R.J. Existence and feasibility in arithmetic. J. Symbolic Logic 36, pp.494--508 (1971).

....proved. These questions influenced several directions of research. We shall mention some results related to the consistency problem. The consistency problem was an initiating idea for studying inconsistent theories. The basic result about useful inconsistent theories is due to Parikh [37], who constructed theories which are inconsistent, but where any proof of contradiction is extremely long, and showed some nice properties of such theories. For further results see [11, 16, 34] Another problem connected with the result of Godel was the question, for how weak theories the second ....

....called oracle results, which means that we can show independence of a certain combinatorial principle for 2 I have presented it on an invited talk at Logic Colloquium 84 in Manchester, but nobody told me about Friedman s paper there. 3 Parikh had proposed to study such systems already in 1971 [37], but no new results had been published till late seventies. 4 The sentences of Paris and Harrington are Pi 2 , the only explicit Pi 1 sentences that we know of are ConT . an extra uninterpreted predicate (see e.g. 31] It is still possible that interesting independence results will be ....

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R. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494--508, 1971.

....problem is Problem 7.10. Describe all =F q on pseudo nite elds, or at least on ultraproducts of nite elds. This problem is related to [9, Thm.7.3] see remarks there) 14 JAN KRAJ I CEK AND THOMAS SCANLON 8. Examples from bounded arithmetic Bounded arithmetic I 0 , de ned by Parikh [14], is a subtheory of Peano arithmetic with induction for bounded formulas only (the language is f0; 1; g) see also [8] for a general reference on bounded arithmetic) One of the oldest and most interesting open problems about bounded arithmetic was posed by A. Macintyre some twenty years ....

R. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36, (1971), pp.494-508.

....extended this result to composite p and q. The preceding results are significant not just from the point of view of propositional complexity theory, but also as providing independence results in systems of bounded arithmetic. The system I # 0 of first order bounded arithmetic introduced by Parikh [47] has been extensively studied; in it, the induction scheme is restricted to formulas containing only bounded quantifiers. Let I # 0 (f) be the system obtained from I # 0 by adding a new function symbol f that is allowed to appear in the induction scheme. Let PHP(f) be the formula in the expanded ....

Rohit Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, vol. 36 (1971), pp. 494--508.

....extended this result to composite p and q. The preceding results are significant not just from the point of view of propositional complexity theory, but also as providing independence results in systems of bounded arithmetic. The system I # 0 of first order bounded arithmetic introduced by Parikh [47] has been extensively studied; in it, the induction scheme is restricted to formulas containing only bounded quantifiers. Let I # 0 (f) be the system obtained from I # 0 by adding a new function symbol f that is allowed to appear in the induction scheme. Let PHP(f) be the formula in the expanded ....

Rohit Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, vol. 36 (1971), pp. 494--508.

....of the function and relation symbols. BASIC properly contains Robinson s Q since it has to be used with weaker induction axioms. 2) The Sigma b i IND axioms. T Gamma1 2 has no induction axioms. T 2 is the union of the T i 2 s. T 2 is equivalent to I Delta 0 Omega 1 (see Parikh [40] and Wilkie and Paris [50] modulo differences in the nonlogical language. Definition Let i 0 . S i 2 is the first order theory with language 0 , S , Delta , b 1 2 xc , jxj , # and and axioms: 8 (1) The BASIC axioms, and (2) The Sigma b i PIND axioms. S Gamma1 2 = T Gamma1 ....

....i 1 . Let A be a Sigma b i formula. Suppose S i 2 (8 x) 9y)A( x; y) Then there is a Sigma b i formula B and a function f 2 p i and a term t so that (1) S i 2 (8 x; y) B( x; y) oe A( x; y) 2) S i 2 (8 x) 9 y)B( x; y) 3) S i 2 (8 x) 9y t)B( x; y) see Parikh [40]) 4) For all n , N j= B( n; f( n) Theorem 7 If f 2 p i then there is a formula B 2 Sigma b i and a term t so that (2) 3) and (4) hold. Corollary 8 (i 1) The Sigma b i definable functions of S i 2 are precisely the functions in p i . The most interesting case of the above ....

R. J. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36 (1971), pp. 494--508.

....modulo q. Thus we also need to consider binomial equations over Z q . In the case we work over Z the argument only has a chance to take place in models of Bounded Arithmetic if c 1 ; c 2 ; c u are integers bound by a term in the underlying language. This follows by Parikh s theorem [Parikh 71] In general both u : u(r) and k : k(r) can be functions of r. Constraint on their growth rate is closely linked to the systems of Bounded Arithmetic we have fixed. For instance the argument has only a chance to take place in systems where that Gamma r u Delta is bound by a term t(r) in ....

....of r. Constraint on their growth rate is closely linked to the systems of Bounded Arithmetic we have fixed. For instance the argument has only a chance to take place in systems where that Gamma r u Delta is bound by a term t(r) in the underlying language (again because of Parikhs theorem [Parikh 71] So in the case of I Delta 0 (ff) where all terms are polynomials, we need only to consider arguments where u 2 O(1) Summarizing we only consider the question whether c 1 ; c 2 ; c u can be chosen bounded by a fixed polynomial in r such that equation (Eq 3) has solutions for ....

R. Parikh; Existence and feasibility in arithmetic, J. Symbolic Logic, 36 (1971) pp 494-508.

....the only Extensionality Axiom relative to Delta 0 formulas. The class of basic operations and also the Basic Set Theory were extended in [30] 39] to more reach versions called there Delta language and Delta set theory or Bounded Set Theory (by some analogy with Bounded Arithmetic; cf. e.g. [5, 25, 28]) Bounded Set Theory and its class of Delta definable or, equivalently, provably computable operations over sets exactly correspond to PTIME computability over HF. In this paper we consider the basic language of R.Gandy under the same name Delta and several its extensions corresponding to ....

Parikh, R.: Existence and feasibility in arithmetic. JSL, 36 No 3 (1971) 494--508

....into it. However, such a construction is quite complicated even for the restricted cases considered above, see [26, Chpt.13] 3 Bounded Arithmetic Bounded arithmetic is a subsystem of Peano Arithmetic in which the induction axioms are restricted to bounded formulas only. It was proposed by [38]. Several other systems, differing in the language and in the particular class of bounded formulas for which the induction is assumed, were defined and studied since then ( 16, 39, 40, 41, 10] Here we shall consider only the theory S 1 2 of [10] perhaps the most important one in the present ....

Parikh, R. (1971) Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36, pp.494-508.

....The goal of this paper is to outline the present state of affairs in investigating various versions of Bounded Set Theory (BST) These are analogues of S. Cook s theory PV [8] of Bounded Primitive Recursive Functions of A. Cobham [7] over natural numbers and to Bounded Arithmetic (cf. e.g. [5, 33, 30]) Also, they are similar to the predicative systems of S. Feferman [13] and G. Jager [24] however essentially more weak, even weaker than the Primitive Recursive Set Theory of R. Jensen and C. Karp [26] BST s reasonably extend the fragment KP 0 of Kripke Platek Set Theory KP (cf. 3] obtained ....

....a Feasible Mathematics we must rather conceptually identify N with f1g , the set of finite unary strings, or with a set of sufficiently short binary strings. cf. also the corresponding discussion and some formalizations of feasibility and constructibility concepts for finite objects in [30, 33, 38, 42] and in the end of this paper. There are many different kinds of finite objects such as unary and binary strings, hereditarily finite sets, etc. whose interrelations must be investigated from the point of view of feasibility of these objects and also of corresponding computations. There are ....

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Parikh, R.: 1971, `Existence and feasibility in arithmetic', Journal of Symbolic Logic 36, (3), pp. 494--508.

....theorem remains in force. From a classical predicativistic viewpoint, E. Nelson ( Nel86] attacked full first order induction, arguing that we have no good reason to assume that the tally numbers are closed under exponentiation; a point made before in a somewhat different way by Parikh ([Par71]) Within a constructivistic setting one can also criticize unrestrained induction. Already with the acceptation of full first order induction in intuitionistic mathematics the idealization, or if you want schematization sets in. For, on the one hand, one pictures natural numbers as obtained by ....

R. Parikh. Existence and feasibility in arithmetic. The Journal of Symbolic Logic, 36:494--508, 1971.

....is that when the existence of an object is proven, that object may be constructed. However, there never been a guarantee that such a construction will be feasible to perform. By proving the existence of an object in resource bounded logic, we guarantee that the object may be constructed feasibly [35]. 2 Partial Object Type Theory In this section we lay out the underlying type theory for the results of this paper. Our work builds on the type theory of Nuprl [9] a type theory in the style of Martin Lof [26, 27] We briefly summarize Nuprl in Sections 2.1. This starting theory is essentially a ....

.... Using these, we may define the resource indexed type [T ] t s of terms that evaluate (to a member of T ) within time t and space s: T ] t s def = fe : T ] j Time(e; t) Space(e; s)g One interesting application of the resource indexed types is to define types like Parikh s feasible numbers [35], numbers that may be computed in a reasonable time. 5.1 Complexity Classes With time complexity measures defined above, we may define complexity classes of functions. Complexity classes are expressed as function types whose members are required to fit within complexity constraints. We call such ....

R. Parikh. Existence and feasibility in arithmetic. Jour. Assoc. Symbolic Logic, 36:494--508, 1971.

....n Omega Gamma37 . q.e.d. Note that by a suitable choice of we can get a lower bound of the form 2 Omega Gamma n 1 3 Gammaffl ) for arbitrary small ffl 0. 8 An independence result for the bounded arithmetic theory S 2 2 (ff) The first bounded arithmetic theory was introduced in [34]. Current research is centered around the theories defined in [5] In this section we give a new presentation of the proof of the independence result for the theory S 2 2 (ff) obtained in [43] For the definition of the theory as well as for the details of bounded arithmetic the reader should ....

Parikh, R. (1971) Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36: 494-508.

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R. J. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36 (1971), pp. 494--508.

....are consistent. This gives us a 500 consistent, inconsistent collection of axioms, which we nd reasonable to work with in practice. Sorites Paradoxes Consider the following set of axioms: I. 0 is small. II. For all n, if n is small, then n 1 is small. III. 10 10 10 is not small. Parikh [28] has shown that while the system Peano Arithmetic with axioms I III above is inconsistent, all theorems proved in it whose proofs are short (and are purely about numbers) are true. The B structures model provides a similar plausible treatment of Sorites type paradoxes as seen in the bald person ....

....that among desirable systems with certain features, ours is maximal so that a system with a more generous notion of inference would run into problems. A second issue is pragmatic. Since we are dealing with inconsistent systems we can hardly claim that all the consequences will be true. However, [28] shows that an inconsistent extension of Peano arithmetic proves only true formulas provided that we concentrate on formulas which are solely about numbers and whose proofs are short. It should happen also with B structures used in practice that in some sense they usually work. The wily airline ....

Rohit Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 1971.

....cycles into spirals A. Carbone Math ematiques Informatique Universit e de Paris XII 61 Avenue du G en eral de Gaulle 94010 Cr eteil, France In [13] Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noticing that very large numbers can be actually constructed through very short proofs. ....

R. Parikh. Existence and feasibility in arithmetic. In Journal of Symbolic Logic, 36:494--508, 1971.

....some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. 1 Introduction This article discusses two papers of Rohit Parikh on feasibility and bounded arithmetic and on the complexity of proofs: the first is the 1971 paper Existence and Feasibility in Arithmetic [30] and the second is the 1973 paper Some Results on Length of Proofs [31] Both papers were seminal and influential and led to large research areas which are still active and fruitful 25 years later. The first paper addressed the intuitive concept of feasibility, discussed the infeasibility of ....

....shown to hold for exponentiation. ii) The most important contribution of the feasibility paper was probably the definition of the theory of bounded arithmetic, denoted PB in that paper, but now usually denoted I Delta 0 . We ll use the modern notation in this paper. Definition 1 (Parikh [30]) I Delta 0 (or PB) is the first order theory with nonlogical symbols 0, S, and Delta and containing the axioms (1) 0 6= S(x) 5) x S(y) S(x y) 2) S(x) S(y) x = y (6) x Delta 0 = 0 (3) x = 0 9y(x = S(y) 7) x Delta S(y) x Delta y x (4) x 0 = x (8 n ) A(0) 8x(A(x) ....

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R. J. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36 (1971), pp. 494--508.

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R. J. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494-508, 1971.

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R. J. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36 (1971), pp. 494--508.

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R. J. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36 (1971), pp. 494--508.

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R. J. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494--508, 1971.

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R. J. Parikh, Existence and feasibility in arithmetic, Journal of Symbolic Logic, 36 (1971), pp. 494--508.

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R. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36(3):494--508, 1971.

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R. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494--508, 1971.

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R. Parikh. Existence and feasibility in arithmetic. Journal of Symbolic Logic, 36:494--508, 1971.

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Parikh, R. (1971) Existence and feasibility in arithmetic, JSL, 36, (3), 494--508.

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