The provably total search problems of bounded arithmetic

Abstract

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 Tk2 of the bounded arithmetic hierarchy and hence characterize the ∀ Σ ^ 1 b consequences of Tk2. 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 ∀ Σ ^ 1 b ( α ) conservativity of T 2 i + 1 ( α ) over T 2 i ( α ) already implies ∀ Σ ^ 1 b ( α ) conservativity of T2(α) over T 2 i ( α ). We translate this into propositional form and give a polylogarithmic width CNF GI ¯ 3 such that if GI ¯ 3 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 GI ¯ 3 . We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T 2 i ( α ) in terms of a non-logical question about multiparty communication protocols.