A (Biased) Proof Complexity Survey for SAT Practitioners

Abstract

Abstract. This talk is intended as a selective survey of proof complexity, focusing on some comparatively weak proof systems that are of particular interest in connection with SAT solving. We will review resolution, polynomial calculus, and cutting planes (related to conflict-driven clause learning, Gröbner basis computations, and pseudo-Boolean solvers, respectively) and some proof complexity measures that have been studied for these proof systems. We will also briefly discuss if and how these proof complexity measures could provide insights into SAT solver performance.Proof complexity studies how hard it is to find succinct certificates for the unsatisfiability of formulas in conjunctive normal form (CNF), i.e., proofs that formulas always evaluate to false under any truth value assignment, where these proofs should be efficiently verifiable. It is generally believed that there cannot exist a proof system where such proofs can always be chosen of size at most polynomial in the formula size. If this belief could be proven correct, it would follow that NP = coNP, and hence P = NP, and this was the original reason research in proof complexity was initiated by Cook and Reckhow [18]. However, the goal of separating P and NP in this way remains very distant.Another, perhaps more recent, motivation for proof complexity is the connection to applied SAT solving. Any algorithm for deciding SAT defines a proof system in the sense that the execution trace on an unsatisfiable instance is itself a polynomial-time verifiable witness (often referred to as a refutation rather than a proof ). In the other direction, most SAT solvers in effect search for proofs in systems studied in proof complexity, and upper and lower bounds for these proof systems hence give information about the potential and limitations of such SAT solvers.In addition to running time, an important concern in SAT solving is memory consumption. In proof complexity, time and memory are modelled by proof size and proof space. It therefore seems interesting to understand these two complexity measures and how they are related to each other, and such a study reveals intriguing connections that are also of intrinsic interest to proof complexity. In this context, it is natural to concentrate on comparatively weak proof systems that are, or could plausibly be, used as a basis for SAT solvers. This talk will focus on such proof systems, and the purpose of these notes is to summarize the main points. Readers interested in more details can refer to, e.g, the survey [31].