Contributions to the Theory of Practical Quantified Boolean Formula Solving

Abstract

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 depthmonotonic 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 depthmonotonic literals even when they are not at innermost scope.

Long-Distance Resolution: Proof Generation and Strategy Extraction in Search-Based QBF Solving

“…Tautological resolvents can entirely be avoided in clause learning [4]. However, this approach has an exponential worst case [14], in contrast to a more sophisticated polynomial-time procedure [10].…”

“…By Theorem 3.2 in [7], any Q-refutation of ϕ t for t ≥ 1 is exponential in t. The formula ϕ t has a polynomial size Q-resolution refutation if universal pivot variables are allowed [14]. In the following, we show how to obtain polynomial size LDQ-refutations in the form of a directed acyclic graph (DAG).…”

A Game Characterisation of Tree-like Q-resolution Size

“…QU-Resolution is a stronger system than Q-Resolution [42] (though this it is not known whether this also holds for the tree-like versions).…”

“…Here we use our new technique to show that these formulas require exponential-size proofs in tree-like QU-Resolution, which in contrast to the previous two examples provides a new hardness result. This also has the interesting consequence that the formulas of Kleine Büning et al exponentially separate tree-like and dag-like QU-Resolution, as they are known to have short proofs in dag-like QU-Resolution [42]. For the KBKF(t) formulas both the Delayer strategy as well as the scoring analysis are technically involved.…”

Towards the Semantics of QBF Clauses

“…[22,19,12,23,13], as well as various resolution-based, clausal calculi [21,29,3,20,7] which advance our understanding of the techniques and formalise the involved reasoning.…”