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

TL;DR

This paper explores the complexity and completeness of certain propositional proof systems and bounded arithmetic theories, establishing reductions and completeness results for related NP search problems and propositional formulas.

Contribution

It introduces a new framework linking circuit evaluation, resolution proofs, and bounded arithmetic, proving completeness and reduction results for related NP search problems.

Findings

$ ext{Gamma}(0,s,k)$ is complete among relativized NP search problems in $V^1_1$.

The non-relativized NP search problem $i ext{Gamma}$ is complete in $V^1_2$.

Reductions are definable in $S^1_2$, connecting proof complexity and bounded arithmetic.

Abstract

We consider sets $\Gamma(n,s,k)$ 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 $\Gamma(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that $\Gamma(0,s,k)$ 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 to define from $\Gamma(0,s,k)$ a non-relativized NP search problem $i\Gamma$ 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 propositional 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.