Contributions to the theory of practical quantified boolean formula solving

Abstract

Recent solvers for quantified boolean formulas (QBFs) use a clause learning method based on a procedure proposed by Giunchiglia et al. (JAIR 2006), which avoids creating tautological clauses. The underlying proof system is Q-resolution. This paper shows an exponential worst case for the clause-learning procedure. This finding confirms empirical observations that some formulas take mysteriously long times to solve, compared to other apparently similar formulas. Q-resolution is known to be refutation complete for QBF, but not all logically implied clauses can be derived with it. A stronger proof system called QU-resolution is introduced, and shown to be complete in this stronger sense. A new procedure called QPUP for clause learning without tautologies is also described. A generalization of pure literals is introduced, called effectively depth-monotonic literals. In general, the variable-elimination resolution operation, as used by Quantor, sQueezeBF, and Bloqqer is unsound if the existential variable being eliminated is not at innermost scope. It is shown that variable-elimination resolution is sound for effectively depth-monotonic literals even when they are not at innermost scope. © 2012 Springer-Verlag.