We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

Abstract. We present a novel third-order theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that from any configuration in the game of Hex, at least one player has a winning strategy. We then exhibit a translation of theorems of W 1 1 into families of propositional tautologies with polynomial-size proofs in BPLK (a recent propositional proof system for PSPACE and an alternative to G). This translation is clearer and more natural in several respects than the analogous ones for previous PSPACE theories. Keywords: Bounded arithmetic, propositional proof complexity, PSPACE, quantified propositional calculus 1

We give combinatorial principles GIk, based on k-turn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a non-logical question about multiparty communication protocols.

Let H be a proof system for the quantified propositional calculus (QPC). We j -witnessing problem for H to be: given a prenex S j -formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce

Abstract. 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 NPpairs 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. 1

. In this paper we prove that if there exists an optimal quantified propositional proof system then there exists a complete language for NP " co-NP. 1 Introduction There is a lot of activity around the complexity of propositional proof systems (see [7, 12]). Research into propositional proof systems is motivated by the open problem NP=co-NP? Research into quantified propositional proof systems is not so popular. The study of quantified propositional proof systems may be related to the problem NP=PSPACE? Some deep results about connections between quantified propositional proof systems and bounded arithmetic are contained in [8]. We propose to study the problem of the existence of an optimal quantified propositional proof system. The similar problem for propositional proof systems has been studied in [9]. It is not known whether complete languages exist for NP " co-NP and Sipser has shown in [10] that there are relativizations so that NP " co-NP has no complete languages (see also [4...

This paper proves exponential separations between depth d - LK and depth (d + 2 ) - LK for every d 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d - LK and depth (d+1) - LK for d N . We investigate the relationship between the sequence-size, tree-size and height of depth d - LK-derivations for d 2 N , and describe transformations between them.

We discuss the Paris-Wilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and non-relativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0-depth for PK-proofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1-definable functions of S12are polynomial time computable and that the \Sigma b1-definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma

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.

Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formula-based proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a non-parameter free variable. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of Σ B 0-rec into a family of tautologies that have polynomial size GL ∗ proofs. Σ B 0-rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ B 0-rec, and put Σ B 0-rec proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in Σ B 0-rec. This is done by giving a log space algorithm that witnesses GL ∗ proofs. 1