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Classification of Search Problems and Their Definability in Bounded Arithmetic
, 2001
"... Classication of Search Problems and Their Denability in Bounded Arithmetic Tsuyoshi Morioka Master of Science Graduate Department of Computer Science University of Toronto 2001 We present a new framework for the study of search problems and their denability in bounded arithmetic. We identify t ..."
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Classication of Search Problems and Their Denability in Bounded Arithmetic Tsuyoshi Morioka Master of Science Graduate Department of Computer Science University of Toronto 2001 We present a new framework for the study of search problems and their denability in bounded arithmetic. We identify two notions of complexity of search problems: veri cation complexity and computational complexity. Notions of exact solvability and exact reducibility are developed, and exact b i denability of search problems in bounded arithmetic is introduced. We specify a new machine model called the oblivious witnessoracle Turing machines.
Consistency and Gamesin Search of New Combinatorial Principles
, 2004
"... We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences tha ..."
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We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences that characterize all 8 1 sentences provable in S 2 . We use the concept of a two player game to describe these sentences.
Fragments of Approximate Counting
, 2012
"... We study the longstanding open problem of giving ∀Σ b 1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeˇrábek’s theories for approximate counting and their subtheories. ..."
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We study the longstanding open problem of giving ∀Σ b 1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeˇrábek’s theories for approximate counting and their subtheories. We show that the ∀Σ b 1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP NP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeonhole principle for polynomial time functions.
Higher complexity search problems for bounded arithmetic and
, 2010
"... a formalized nogap theorem ..."
Research Statement
, 2006
"... 2 Background and Past Work The primary motivation for studying propositional proof systems is the theorem of Cook and Reckhow[8, 11] that NP=coNP iff there exists a polynomially bounded proof system for propositional tautologies. Many proof systems are studied and there are some notable successes i ..."
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2 Background and Past Work The primary motivation for studying propositional proof systems is the theorem of Cook and Reckhow[8, 11] that NP=coNP iff there exists a polynomially bounded proof system for propositional tautologies. Many proof systems are studied and there are some notable successes in the search for lower bounds, e.g.[13, 1], but this problem is very hard in general; nevertheless, there are many much more accessible problems than NP vs coNP: at one end of the scale, the detailed study of weak proof systems and their interrelations,and at the other, capturing different forms of reasoning with stronger proof systems.
The Relative Complexity of Local Search Heuristics And The Iteration Principle
, 2003
"... Johnson, Papadimitriou and Yannakakis introduce the class PLS consisting of optimization problems for which ecient localsearch heuristics exist. We formulate a type2 problem ITER that characterizes PLS in style of Beame et al., and prove a criterion for type2 problems to be nonreducible to ITER. As ..."
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Johnson, Papadimitriou and Yannakakis introduce the class PLS consisting of optimization problems for which ecient localsearch heuristics exist. We formulate a type2 problem ITER that characterizes PLS in style of Beame et al., and prove a criterion for type2 problems to be nonreducible to ITER. As a corollary, we obtain the rst relative separation of PLS from Papadimitriou's classes PPA, PPAD, PPADS, and PPP. Based on the criterion, we derive a special case of Riis's independence criterion for the Bounded Arithmetic 2 (L). We also prove that PLS is closed under Turing reducibility.
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