Rudimentary Predicates and Relative Computation

Wrathall, Celia

doi:10.1137/0207018

Cited by 100 publications

(28 citation statements)

References 13 publications

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“…Because the existential quantifier ∃y ≤ t(x) can be thought of as a suitable polynomial-size witness corresponding to the input x, a Σ b 1 -formula describes an NP-set of natural numbers. But also all NP-sets can be defined by Σ b 1 -formulas, as the next theorem which is a variant of a result of Wrathall [56] (see e.g. [34]) shows.…”

Section: Supported By Dfg Grant Ko 1053/5-1 This Is the Pre-peer Revimentioning

confidence: 83%

On the correspondence between arithmetic theories and propositional proof systems – a survey

Beyersdorff¹

2009

Mathematical Logic Qtrly

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Abstract. Bounded arithmetic is closely related to propositional proof systems, and this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and Pudlák [41]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties.Using logical methods has a rich tradition in complexity theory. In particular, there are very close relations between computational complexity, propositional proof complexity, and bounded arithmetic, and the central tasks in these areas, i.e., separating complexity classes, proving lower bounds to the length of propositional proofs, and separating arithmetic theories, can be understood as different approaches towards the same problem. While each of these fields supplies its own techniques to address these problems, many exciting results have been obtained that decisively use the interplay of combinatorial and logical methods (e.g. [1, ??, 48, 49]), and it is expected that this exchange of ideas will continue to exert substantial influence on the development of theoretical computer science in general. 1Nevertheless, complexity theorists and logicians quite often seem to have different traditions, regarding notation and prerequisites that can be assumed without explanation, and these "cultural" differences sometimes make logic-oriented research difficult to access for a wider complexity-theoretic audience. These observations particularly apply, in my opinion, to the field of propositional proof complexity, that can be addressed both from a completely combinatorial perspective as well as by utilizing the correspondence to bounded arithmetic. 2Supported by DFG grant KO 1053/5-1 THIS IS THE PRE-PEER REVIEWED VERSION OF THE FOLLOWING ARTI-CLE: FULL CITE 1 In [34] Jan Krajíček formulates "It is to be expected that a nontrivial combinatorial or algebraic argument will be required for the solution of the P versus NP problem. However, I believe that the close relations of this problem to bounded arithmetic and propositional logic indicate that such a solution should also require a nontrivial insight into logic."2 This opinion has been confirmed in many conversations and was also reiterated by some of the referees, when I used bounded arithmetic in a complexity-theoretic context (e.g. in [9]). 1 2 OLAF BEYERSDORFFThis relation works for a number of diverse pairs of proof systems and corresponding arithmetic theories, each of which presents a number of specific nontrivial technical problems. A unifying approach for a general correspondence was suggested by Krajíček and Pudlák [41]. It is the aim of this paper to explain ...

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Section: Supported By Dfg Grant Ko 1053/5-1 This Is the Pre-peer Revimentioning

confidence: 83%

On the correspondence between arithmetic theories and propositional proof systems – a survey

Beyersdorff¹

2009

Mathematical Logic Qtrly

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“…Thus, Appendices C and D of [67] explained how the first three of these predicates can receive ∆ * 0 encodings when one applies the theory of LinH functions [20,25,73]. In such a context, Equation (7) illustrates one possible ∆ * 0 encoding for SubstPrf…”

Section: IVmentioning

confidence: 99%

An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency

Willard¹

2005

J. symb. log.

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Abstract. This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: 1) α treats multiplication as a 3-way relation (rather than as a total function), and that 2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae.Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions.

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“…The linear-time hierarchy LinH is k Σ k -TIME(O(n)). A relation on the natural numbers is in LinH exactly if it can be defined by a ∆ 0 formula [Wra78].…”

Section: Preliminariesmentioning

confidence: 99%

End-Extensions of Models of Weak Arithmetic From Complexity-Theoretic Containments

Kołodziejczyk¹

2016

J. symb. log.

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We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of Π 1 (N) + ¬Ω 1 has a proper end-extension to a model of Π 1 (N), and so Π 1 (N) + ¬Ω 1 ⊢ BΣ 1 . Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, Π 1 (N) + ¬Exp ⊢ BΣ 1 . Both assumptions can be modified to versions which make it possible to replace Π 1 (N) by I∆ 0 as the base theory.We also show that any proof that I∆ 0 + ¬Exp does not prove a given finite fragment of BΣ 1 has to be "non-relativizing", in the sense that it will not work in the presence of an arbitrary oracle.The work presented below aims at a better understanding of the following notoriously hard open problem about moderately weak theories of arithmetic:Here I∆ 0 is induction for bounded formulas in the language of ordered rings, Exp is the axiom ∀x ∃y (y = 2 x ) with y = 2 x expressed by an appropriate ∆ 0 formula, and BΣ 1 is the Σ 1 collection scheme,where ϕ is ∆ 0 and may contain parameters. It is well-known and easy to show that I∆ 0 ⊢ BΣ 1 [Par70], but all known proofs of this (e.g. [Par70, PK78, Ada88, Bek98]) use objects of at least exponential size, often in the form of a definition of satisfaction for Σ 1 formulas.Problem (⋆) was first explicitly stated in [WP89]. The expected answer is negative, and there have been a number of results of the form "the answer to (⋆) is negative under some complexity-theoretic assumptions" [WP89,Fer94,

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Rudimentary Predicates and Relative Computation

Cited by 100 publications

References 13 publications

On the correspondence between arithmetic theories and propositional proof systems – a survey

On the correspondence between arithmetic theories and propositional proof systems – a survey

An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency

End-Extensions of Models of Weak Arithmetic From Complexity-Theoretic Containments

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