Bounded Arithmetic, Propositional Logic and Complexity Theory

Lecture Notes in Computer Science

Müller

2010

Abstract. The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70's and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper.

Section: Proof Systems With Advice and Bounded Arithmeticmentioning

confidence: 99%

Different Approaches to Proof Systems

Lecture Notes in Computer Science

Müller

2010

“…While it was already proven there that EF is simulated by SF , the converse simulation is considerably more involved and was shown independently by Dowd [10] and Krajíček and Pudlák [19]. For more detailed information on Frege systems and their extensions we refer to the monograph [18].…”

Section: Extensions Of Frege Systemsmentioning

confidence: 99%

“…These extensions are particularly important as every proof system P is p-simulated by a proof system of the form EF + Φ where the axioms Φ provide a propositional description of the reflection principle of P , expressing a strong form of the consistency of P (cf. [18] for details).…”

Section: Extensions Of Frege Systemsmentioning

confidence: 99%

The Deduction Theorem for Strong Propositional Proof Systems

FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science

2007

Abstract. This paper focuses on the deduction theorem for propositional logic. We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and even polynomially bounded proof systems. We also exhibit a similar, but apparently weaker condition that implies the existence of complete disjoint NP-pairs. In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NP-pairs.

“…A Π b 1 -formula ϕ(x) is translated into a sequence ϕ(x) n of propositional formulas containing one formula per input length for the number x, such that ϕ(x) is true, i.e., N |= (∀x)ϕ(x), if and only if ϕ(x) n is a tautology where n = |x| (cf. [16]). We use ϕ(x) to denote the set { ϕ(x) n | n ≥ 1}.…”

Section: Consistency Statementsmentioning

confidence: 99%

“…These schematic extensions enhance the extended Frege system by additional sets of polynomial-time decidable axiom schemes. Such systems are of particular importance: Firstly, because every propositional proof system is simulated by such an extension of EF , and secondly, because these systems admit a fruitful correspondence to theories of bounded arithmetic [8,14,16].…”

Section: Introductionmentioning

confidence: 99%

Logical Closure Properties of Propositional Proof Systems

Lecture Notes in Computer Science

Abstract. In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic.