Finding Small Unsatisfiable Cores to Prove Unsatisfiability of QBFs

In the past few years, we have seen significant progress in the area of Boolean satisfiability (SAT) solving and its applications. More recently, new efforts have focused on developing solvers for Quantified Boolean Formulas (QBFs). Recent QBF evaluation results show that developing practical QBF solvers is more challenging than one might expect. Even relatively small QBF problems are sometimes beyond the reach of current QBF solvers. We present a new approach for solving unsatisfiable two-alternation QBFs. Our approach is able to solve hard random QBF formulas that current algorithms are not able to handle. Our solver WalkMinQBF combines the power of stochastic local search methods and complete SAT solvers. The solver is incomplete, in that it outputs unsat if a certificate for unsatisfiability is found, otherwise it outputs unknown. We test our solver on the model for random formulas introduced in [3] and the Models A and B introduced in [7]. We compare WalkMinQBF with the state-of-the-art QBF solvers Ssolve and QuBE-BJ. We show that WalkMinQBF outperforms Ssolve and QuBE-BJ in time and in the number of formulas solved. We believe our work provides new insights on strategies that should be useful in complete QBF solvers. As a side result we have developed a stochastic local search algorithm for the minimum unsatisfiability problem (MIN-SAT).