On Unification of QBF Resolution-Based Calculi

Abstract

Abstract. Several calculi for quantified Boolean formulas (QBFs) exist, but relations between them are not yet fully understood. This paper defines a novel calculus, which is resolution-based and enables unification of the principal existing resolution-based QBF calculi, namely Qresolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res. All these calculi play an important role in QBF solving. This paper shows simulation results for the new calculus and some of its variants. Further, we demonstrate how to obtain winning strategies for the universal player from proofs in the calculus. We believe that this new proof system provides an underpinning necessary for formal analysis of modern QBF solvers.

Towards the Semantics of QBF Clauses

“…In this sense, a semantic method enables the notion of a sound inference, an inference that can be added to a calculus without affecting its soundness. In contrast, the currently known proof of soundness of IRM-calc [6] is global, manipulating the whole refutation monolithically under an arguably complex inductive invariant. Should one want to add another rule to IRM-calc, the whole proof might need to be redone from scratch.…”

“…[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.…”

“…A different notion of semantics can be provided to the expansion-derived calculi ∀Exp+Res [20] and IR-calc [6] via a translation from QBF to first-order logic [25], and so soundness of these calculi can be established with the help of first-order model theory [16,11]. Extending this approach to accommodate IRM-calc [6], a calculus which unifies the instantiation flavour inherent to the expansion-derived calculi with the essence of long-distance steps coming from LD-Q-Res (see Fig. 1), was one of the focal points behind this work.…”

Lower Bounds

“…This motivates our last general approach via instantiations, which unifies both reduction and expansion systems in a natural way. These general instantiation systems are inspired by a QBF resolution calculus recently introduced by Beyersdorff, Chew, and Janota (2014). Again the theoretical study of these new expansion and instantiation systems might exert a fruitful influence on QBF solving as modern solvers utilise these approaches.…”

Expansion-Based QBF Solving Without Recursion

“…In this paper, we present a novel solving algorithm based on non-recursive expansion for QBFs with arbitrary quantifier prefixes using only two SAT solvers. Our approach of non-recursive expansion is theoretically (i.e., from a proof complexity perspective) equivalent to approaches that apply recursive expansion since both non-recursive and recursive expansion rely on the ∀Exp+Res proof system [5]. However, the non-recursive expansion has practical implications such as a modified search strategy.…”