Proofs of Unsatisfiability Via Semidefinite Programming

Abstract

The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable X is maximized (or minimized) subject to linear constraints on the elements of X and the additional constraint that X be positive semidefinite. We focus on the application of SDP to obtain proofs of unsatisfiability. Using a new SDP relaxation for SAT, we obtain proofs of unsatisfiability for some hard instances with up to 260 variables and over 400 clauses. In particular, we can prove the unsatisfiability of the smallest unsatisfiable instance that remained unsolved during the SAT Competition 2003. This shows that the SDP relaxation is competitive with the top solvers in the competition, and that this technique has the potential to complement existing techniques for SAT.