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

Abstract

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 ...

Hard Instances of Algorithms and Proof Systems

“…We prove the deterministic case, the nondeterministic case is obtained by the obvious modifications. So assume that the algorithm A decides Q and has a hard sequence (x s ) s∈N ; in particular, (1) t A (x s ) is not polynomially bounded in s.…”

“…A proof system P for Q is p-optimal or polynomially optimal if for every proof system P for Q there is a polynomial translation from P into P. A proof system P for Q is optimal if for every proof system P for Q and every w ∈ Σ * there is a w ∈ Σ * such that P(w) = P (w ) and |w| ≤ |w | O (1) . Clearly, every p-optimal proof system is optimal.…”

“…A sequence (x s ) s∈N of strings in Q is hard for A if it is computable in polynomial time and the sequence (t A (x s ) s∈N ) is not polynomially bounded in s. 1 Here, t A (x) denotes the number of steps the algorithm A takes on input x. Clearly, if A is polynomial time, then A has no hard sequences.…”

Lectures on Logic and Computation

“…In one of these papers, Raymond Reiter defined what is now called Reiter's default logic [97], which is still one of the most popular systems under investigation in this branch of logic. 11 In a nutshell, non-monotonic logics are a family of knowledge representation formalisms mostly targeted at modelling common-sense reasoning. Unlike in classical logic, the characterising feature of such logics is that an increase in information may lead to the withdrawal of previously accepted information or may blocks previously possible inferences.…”

On the Existence of Complete Disjoint NP-Pairs

“…A direct construction of such P -proofs would be quite tedious, but we can use the correspondence of extensions of EF to first-order arithmetic theories (cf. [33], [34] for background information).…”