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Optimal acceptors and optimal proof systems
"... Abstract. Unless we resolve the P vs NP question, we are unable to say whether there is an algorithm (acceptor) that accepts Boolean tautologies in polynomial time and does not accept nontautologies (with no time restriction). Unless we resolve the coNP vs NP question, we are unable to say whether ..."
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Abstract. Unless we resolve the P vs NP question, we are unable to say whether there is an algorithm (acceptor) that accepts Boolean tautologies in polynomial time and does not accept nontautologies (with no time restriction). Unless we resolve the coNP vs NP question, we are unable to say whether there is a proof system that has a polynomialsize proof for every tautology. In such a situation, it is typical for complexity theorists to search for “universal ” objects; here, it could be the “fastest ” acceptor (called optimal acceptor) and a proof system that has the “shortest ” proof (called optimal proof system) for every tautology. Neither of these objects is known to the date. In this survey we review the connections between these questions and generalizations of acceptors and proof systems that lead or may lead to universal objects. 1 Introduction and basic definitions
Does Advice Help to Prove Propositional Tautologies?
"... Abstract. 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 polytime computable functions which have all propositional tautologies as their range. Motivated by provability consequences in boun ..."
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Abstract. 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 polytime 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 polytime 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 polytime with access to a sparse NPoracle. 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]. 1
Towards NP−P via Proof Complexity and Search
, 2009
"... This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP. ..."
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This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP.
Logical Closure Properties of Propositional Proof Systems
"... 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 E ..."
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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. 1
Tuples of Disjoint NPSets ⋆ Olaf Beyersdorff ⋆⋆
"... Abstract. Disjoint NPpairs are a well studied complexitytheoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NPpairs to disjoint ktuples of NPsets for k ≥ 2. We define subc ..."
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Abstract. Disjoint NPpairs are a well studied complexitytheoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NPpairs to disjoint ktuples of NPsets for k ≥ 2. We define subclasses of the class of all disjoint ktuples of NPsets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint NPpairs exist if and only if complete disjoint ktuples of NPsets exist for all k ≥ 2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all ktuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems. 1
ON THE CORRESPONDENCE BETWEEN ARITHMETIC THEORIES AND PROPOSITIONAL PROOF SYSTEMS OLAF BEYERSDORFF
"... 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 an ..."
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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 [42]. 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
Olaf Beyersdorff
"... Abstract. Disjoint NPpairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NPpairs to disjoint ktuples of NPsets for k ≥ 2. We define subc ..."
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Abstract. Disjoint NPpairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NPpairs to disjoint ktuples of NPsets for k ≥ 2. We define subclasses of the class of all disjoint ktuples of NPsets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint NPpairs exist if and only if complete disjoint ktuples of NPsets exist for all k ≥ 2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all ktuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems. 1
Proof Systems that Take Advice ⋆
"... Abstract. One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [13], where they defined propositional proof systems as polytime computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bou ..."
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Abstract. One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [13], where they defined propositional proof systems as polytime computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajíček [12] have recently started the investigation of proof systems which are computed by polytime functions using advice. In this paper we concentrate on three fundamental questions regarding this new model. First, we investigate whether a given language L admits a polynomially bounded proof system with advice. 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 this question is tightly linked with the question whether L has small nondeterministic instance complexity. The second question concerns the existence of optimal proof systems with advice. For propositional proof systems, Cook and Krajíček [12] gave a surprising positive answer which we extend to all languages. These results show that providing proof systems with advice yields a more powerful model, but this model is also less directly applicable in practice. Our third question therefore asks 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 propositional proof systems with logarithmic advice are also computable in polytime with access to a sparse NPoracle. Employing a recent technique of Buhrman and Hitchcock [10] we also manage to transfer the advice from the proof to the proven formula, which leads to a more practical computational model. 1
Classes of Representable Disjoint NPPairs 1 Olaf Beyersdorff 2
"... For a propositional proof system P we introduce the complexity class DNPP(P) of all disjoint NPpairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical NPpairs associated with these proof syst ..."
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For a propositional proof system P we introduce the complexity class DNPP(P) of all disjoint NPpairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical NPpairs associated with these proof systems hard or complete for DNPP(P). Moreover, we demonstrate that nonequivalent 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. Key words: disjoint NPpairs, propositional proof systems 1