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]

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

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

Hard instances of algorithms and proof systems

"Vegeu el resum a l'inici del document del fitxer adjunt"

Nondeterministic functions and the existence of optimal proof systems

We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system

On an optimal propositional proof system and the structure of easy subsets of TAUT

. In this paper we develop a connection between optimal propositional proof systems and structural complexity theory - specifically, there exists an optimal propositional proof system if and only if there is a suitable recursive presentation of the class of all easy (polynomial time recognizable) subsets of TAUT . As a corollary we obtain the result that if there does not exist an optimal propositional proof system, then for every theory T there exists an easy subset of TAUT which is not T-provably easy. 1 Introduction The first classification of propositional proof systems by their relative efficiency was done by S. Cook and R. Reckhow [4] in 1979. The key tool for comparing the relative strength of proof systems is psimulation. Intuitively a proof system h p-simulates a second one g if there is a polynomial time computable function translating proofs in g into proofs in h. A propositional proof system is called p-optimal if it p-simulates any propositional proof system. The ..