In this paper we study the pairs (U; V ) of disjoint NP-sets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NP-pair which is complete for the class of disjoint NP-pairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [11]. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NP-pair by a (quasi)poly-time computable set. We remark that such a separation is obvious for the theory S(S 2 ) + S \Sigma 2 \Gamma PIND considered in [11], and this gives an alternative proof of the main result from that paper.

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Abstract We present a general method for introducing finitely axiomatizable "minimal " second-order theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the first-order theory \Delta b

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 define the notion of a combinatorics of a first order structure, and a relation of covering between first order structures and propositional proof systems. Namely, a first order structure M combinatorially satisfies an L-sentence \Phi iff \Phi holds in all L-structures definable in M. The combinatorics Comb(M) of M is the set of all sentences combinatorially satisfied in M. Structure M covers a propositional proof system P iff M combinatorially satisfies all \Phi for which the associated sequence of propositional formulas h\Phi i n, encoding that \Phi holds in Lstructures of size n, have polynomial size P-proofs. That is, Comb(M) contains all \Phi feasibly verifiable in P. Finding M that covers P but does not combinatorially satisfy \Phi thus gives a super polynomial lower bound for the size of P-proofs of h\Phi in. We show that any proof system admits a class of structures covering it; these structures are expansions of models of bounded arithmetic. We also give, using structures covering proof systems R \Lambda

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My research centers on the interface between computational complexity and logic, with an emphasis on proof complexity. Computational complexity aims at understanding the nature and limitations of efficient computation; logic, on the other hand, and in particular proof theory, studies what can formally be proved from a given set of axioms and deduction rules, in a given language. Accordingly, proof complexity aims at characterizing which statements can formally be proved with efficient (or feasible) proofs, in a given proof system. Here the term “efficient” typically stands for two separate (though interconnected) meanings: The first meaning refers to the size of the proofs, that is, the number of symbols (or bits) it takes to write down the proofs. The second meaning refers to the efficiency of the ‘concepts ’ one reasons with (the technical interpretation being the complexity class from which proof-lines are taken from). A large part of my work in proof complexity was devoted to the complexity-theoretic analysis of proof systems of an algebraic nature. My main contributions in that respect were: (i) in introducing and studying equational proofs of polynomial identities in Hrubeˇs & Tzameret (2009) and Tzameret (2008) – in relation to the Polynomial Identity Testing problem from algebraic complexity and computational derandomization theory; (ii) the development of multilinearproofs