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 witness-oracle Turing machines.

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.

2 Background and Past Work The primary motivation for studying propositional proof systems is the theorem of Cook and Reckhow[8, 11] that NP=co-NP 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 co-NP: 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.

a formalized no-gap theorem

Johnson, Papadimitriou and Yannakakis introduce the class PLS consisting of optimization problems for which ecient localsearch heuristics exist. We formulate a type-2 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.

We study the long-standing 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.