A Resolution-Based Interactive Proof System for UNSAT

Abstract

Modern SAT or QBF solvers are expected to produce correctness certificates. However, certificates have worst-case exponential size (unless $$\textsf{NP}=\textsf{coNP}$$ NP = coNP ), and at recent SAT competitions the largest certificates of unsatisfiability are starting to reach terabyte size.Recently, Couillard, Czerner, Esparza, and Majumdar have suggested to replace certificates with interactive proof systems based on the $$\textsf {IP}=\textsf {PSPACE}$$ IP = PSPACE theorem. They have presented an interactive protocol between a prover and a verifier for an extension of QBF. The overall running time of the protocol is linear in the time needed by a standard BDD-based algorithm, and the time invested by the verifier is polynomial in the size of the formula. (So, in particular, the verifier never has to read or process exponentially long certificates). We call such an interactive protocol competitive with the BDD algorithm for solving QBF.While BDD-algorithms are state-of-the-art for certain classes of QBF instances, no modern (UN)SAT solver is based on BDDs. For this reason, we initiate the study of interactive certification for more practical SAT algorithms. In particular, we address the question whether interactive protocols can be competitive with some variant of resolution. We present two contributions. First, we prove a theorem that reduces the problem of finding competitive interactive protocols to finding an arithmetisation of formulas satisfying certain commutativity properties. (Arithmetisation is the fundamental technique underlying the $$\textsf {IP}=\textsf {PSPACE}$$ IP = PSPACE theorem.) Then, we apply the theorem to give the first interactive protocol for the Davis-Putnam resolution procedure.