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We give combinatorial principles GI k , 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 ∀Σ b 1 consequences of T k 2 . 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 bWe 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.

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