Consistency of circuit evaluation, extended resolution and total NP search problems

We consider sets Γ(n, s, k) of narrow clauses expressing that no definition of a size s circuit with n inputs is refutable in resolution R in k steps. We show that every CNF shortly refutable in Extended R, ER, can be easily reduced to an instance of Γ(0, s, k) (with s, k depending on the size of the ER-refutation) and, in particular, that Γ(0, s, k) when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory V 1 1. We use the ideas of implicit proofs from [9, 10] to define from Γ(0, s, k) a non-relativized NP search problem iΓ and we show that it is complete among all such problems provably total in bounded arithmetic theory V 1 2. The reductions are definable in S 1 2. We indicate how similar results can be proved for some other propo-sitional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them. The value of y s is the output value of C and is denoted also as C(x). Let Def n,s C (x, y) be the canonical 3CNF formula expressing the conjunction of all instructions. For example, instruction y i := 0 is represented by one clause 1