Disjoint NP-pairs from propositional proof systems

For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist

A Tight Karp-Lipton Collapse Result in Bounded Arithmetic

Cook and Krajíček [9] have obtained the following Karp-Lipton result in bounded arithmetic: if the theory proves , then collapses to , and this collapse is provable in . Here we show the converse implication, thus answering an open question from [9]. We obtain this result by formalizing in a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček [9]. In particular, we obtain several optimal and even p-optimal proof systems using advice. We further show that these p-optimal systems are equivalent to natural extensions of Frege systems

The deduction theorem for strong propositional proof systems

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

Does Advice Help to Prove Propositional Tautologies?

One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [6], where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajíček [5] have recently started the investigation of proof systems which are computed by poly-time functions using advice. While this yields a more powerful model, it is also less directly applicable in practice. In this note we investigate the question whether the usage of advice in propositional proof systems can be simplified or even eliminated. While in principle, the advice can be very complex, we show that proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. In addition, we show that if advice is ”not very helpful” for proving tautologies, then there exists an optimal propositional proof system without advice. In our main result, we prove that advice can be transferred from the proof to the formula, leading to an easier computational model. We obtain this result by employing a recent technique by Buhrman and Hitchcock [4]

Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem:Extended Abstract

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propo-sitional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for Ω(n1.5) many clauses [13]. We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly autom-atizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [25]). This reduces the problem of refuting random 3CNF with n vari-ables and Ω(n1.4) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin)

Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes.

In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class C? For concrete languages L (such as TAUT or SAT) and concrete promise classes C (such as NP∩coNP, UP, BPP, disjoint NP-pairs etc.), these ques-tions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new character-izations for Q1 and Q2 that apply to almost all promise classes C and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount on advice is avail-able in the underlying machine model. This continues a recent line of research on proof systems with advice started by Cook and Kraj́ıček [6]

The Deduction Theorem for Strong Propositional Proof Systems

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 NPUnknown control sequence '\mathsf' -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 NPUnknown control sequence '\mathsf' -pairs

Nondeterministic Instance Complexity and Proof Systems with Advice

Motivated by strong Karp-Lipton collapse results in bounded arithmetic, Cook and Krajíček [1] have recently introduced the notion of propositional proof systems with advice. In this paper we investigate the following question: Given a language L , do there exist polynomially bounded proof systems with advice for L ? Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that the above question is tightly linked with the question whether L has small nondeterministic instance complexity

Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

A Note on Universal Measures for Weak Implicit Computational Complexity

Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate “univer-sal measures ” by “dynamic ordinals”, and “weak implicit computational complexity ” by “bounded arithmetic”. Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing ma-chines for bounded arithmetic theories