Dependency Schemes in QBF Calculi: Semantics and Soundness

Abstract

Abstract. We study the parametrisation of QBF resolution calculi by dependency schemes. One of the main problems in this area is to understand for which dependency schemes the resulting calculi are sound. Towards this end we propose a semantic framework for variable independence based on 'exhibition' by QBF models, and use it to express a property of dependency schemes called full exhibition that is known to be sufficient for soundness in Q-resolution. Introducing a generalised form of the long-distance resolution rule, we propose a complete parametrisation of classical long-distance Q-resolution, and show that full exhibition remains sufficient for soundness. We demonstrate that our approach applies to the current research frontiers by proving that the reflexive resolution path dependency scheme is fully exhibited.

Long Distance Q-Resolution with Dependency Schemes

“…To gauge the potential of clause learning based on LDQ(D std ), we ran experiments with the search-based solver DepQBF. 6 By default, DepQBF supports proof generation only in combination with the trivial dependency scheme-in that case, it generates Q-resolution or long-distance Q-resolution proofs (depending on whether long-distance resolution is enabled). However, by uncommenting a few lines in the source code, proof generation can also be enabled with the standard dependency scheme, and this option can even be combined with long-distance resolution.…”

“…Recently and independently of this work, Beyersdorff and Blinkhorn investigated the soundness of Q-resolution proof systems parameterized by dependency schemes [6]. They define a property of dependency schemes D-full exhibition-which ensures that a certain version of long-distance Q(D)-resolution is sound, and show that the reflexive resolution-path dependency scheme has that property.…”

“…The resolution rule as defined here is more general than the one considered in an earlier version of this paper[34], in that we admit complementary universal literals to be "merged" as long as the pivot is independent according to D (rather than D trv ). This definition-which we think is required to capture proofs generated by DepQBF-was proposed in (independent) work by Beyersdorff and Blinkhorn[6].…”

Towards the Semantics of QBF Clauses

“…One of most important properties of a calculus is its soundness and one of the most common methods for showing soundness is relating the inferences of the calculus and the semantics of the manipulated clauses by a notion of logical entailment. We know how to show soundness of Q-Res using semantical methods [24,27] and I will argue that this technique is getting implicitly extended to LD-Q-Res via the notion of a shadow clause [4] introduced for the purpose of strategy extraction [2]. 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].…”

DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL

“…[3]). Recently, the soundness of the combination of LDQ-resolution of clauses and dependency schemes in QRES has been proved [4,37], leaving the soundness of cube resolutions as an open problem. Therefore, the combination of LDQ-resolution and dependency schemes is not supported in DepQBF 6.0.…”