Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF

Abstract

Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called ‘full exhibition’ is central. We apply our approach to QBF proof systems from two distinct paradigms, termed ‘universal reduction’ and ‘universal expansion’. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky.

Building Strategies into QBF Proofs

“…As our final contribution we show that the new QBF system of M-Res naturally lifts to a sound and complete calculus for DQBF. As shown in [2], the lifting of Q-Res to DQBF is incomplete, whereas the combination of universal reduction and long-distance steps presents soundness issues, both in DQBF [13] as well as in the related framework of dependency schemes [7,8].…”

“…A DQBF can be written in one of two forms: Skolem-form (S-form) and Herbrand-form (H-form) [2]. To date, most of the DQBF literature has focused on S-form (whether in computational complexity [1,16], proof complexity [8,50], and solving [27,29,55,57,58]), whereas relatively little has been written about H-form [2]. The DQBF solver presented in [25] uses H-form DQBF to facilitate a reduction to QBF.…”

“…The second difference is the absence of universal reduction. The difficulty of dealing with universal reduction in the context of DQBF resolution is to some extent addressed in [7], where it is considered in the (closely related [8]) context of dependency schemes. There it is shown that the interplay between universal reduction and merging is problematic, and additional constraints must be placed on universal reduction to prevent unsound inferences.…”

Strong (D)QBF Dependency Schemes via Tautology-Free Resolution Paths

“…The performance gains are also underlined by theoretical findings. There is a sequence of results [7,29,35] that establish how and when dependency schemes are sound to use with a QBF proof system, such as the central QBF resolution systems Q-resolution [22] and long-distance Q-resolution [2]. In [6] it is demonstrated that using the reflexive resolution path dependency scheme (D rrs [35]) in Q-resolution can exponentially shorten proofs.…”

“…It seems natural that there should be a relationship between dependency schemes and DQBF, and indeed the paper [7] suggests that dependency schemes for QBF should be viewed as truth-preserving mappings from QBF to DQBF. Now, is there even a need for dependency schemes for DQBF?…”