A PSPACE Subclass of Dependency Quantified Boolean Formulas and Its Effective Solving

Scholl, Christoph; Jiang, Jie-Hong R.; Wimmer, Ralf; Ge-Ernst, Aile

doi:10.1609/aaai.v33i01.33011584

Cited by 5 publications

(7 citation statements)

References 24 publications

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“…Evaluating DQBF is NEXPTIME complete [3] in general, but some tractable subclasses have been identified in recent work [38,17].…”

Section: Related Workmentioning

confidence: 99%

Certified DQBF Solving by Definition Extraction

Reichl¹,

Slivovsky²,

Szeider³

2021

Preprint

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We propose a new decision procedure for dependency quantified Boolean formulas (DQBFs) that uses interpolation-based definition extraction to compute Skolem functions in a counter-example guided inductive synthesis (CEGIS) loop. In each iteration, a family of candidate Skolem functions is tested for correctness using a SAT solver, which either determines that a model has been found, or returns an assignment of the universal variables as a counterexample. Fixing a counterexample generally involves changing candidates of multiple existential variables with incomparable dependency sets. Our procedure introduces auxiliary variables-which we call arbiter variables-that each represent the value of an existential variable for a particular assignment of its dependency set. Possible repairs are expressed as clauses on these variables, and a SAT solver is invoked to find an assignment that deals with all previously seen counterexamples. Arbiter variables define the values of Skolem functions for assignments where they were previously undefined, and may lead to the detection of further Skolem functions by definition extraction.A key feature of the proposed procedure is that it is certifying by design: for true DQBF, models can be returned at minimal overhead. Towards certification of false formulas, we prove that clauses can be derived in an expansion-based proof system for DQBF. In an experimental evaluation on standard benchmark sets, a prototype implementation was able to match (and in some cases, surpass) the performance of state-of-the-art-solvers. Moreover, models could be extracted and validated for all true instances that were solved.

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“…Evaluating DQBF is NEXPTIME complete [3] in general, but some tractable subclasses have been identified in recent work [38,17].…”

Section: Related Workmentioning

confidence: 99%

Certified DQBF Solving by Definition Extraction

Reichl¹,

Slivovsky²,

Szeider³

2021

Preprint

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“…A DQBF with modular dependency sets can be rewritten as a QBF with a Σ 3 prefix (Scholl et al 2019). An inspection of the proof shows that the rewriting preserves primal treewidth.…”

Section: Modularity and Backdoorsmentioning

confidence: 99%

“…Hence, in order to achieve tractability, it is natural to combine primal treewidth with a measure that bounds the complexity of the interactions between the dependency sets. A suitable "base case" for such a parameterization is the previously studied class of DQBFs which require each pair of dependency sets to be either disjoint or equal (Scholl et al 2019); here, we refer to these as modular DQBFs. A natural parameter that suggests itself at this point is the total number of "omissions" from the dependency sets of a DQBF required to achieve modularity; this matches the notion of (deletion) backdoors (Williams, Gomes, and Selman 2003) whose size has been used as a parameter in the SAT, CSP, and QBF settings (Gaspers and Szeider 2012;Gaspers et al 2014;Samer and Szeider 2009a).…”

Section: Introductionmentioning

confidence: 99%

“…We show that the combinatorial explosion this operation introduces is capped by a function of our parameters, and moreover that the increase in treewidth is bounded as well, which effectively gives us an FPT-reduction to a modular DQBF. On this class, we can invoke a result of Scholl et al (2019), which shows that the formula can be reduced to a QBF with 2 quantifier alternations. We then prove that this translation is treewidthpreserving, which allows us to finally use fixed-parametertractability of bounded-alternation QBF to obtain our result.…”

Section: Introductionmentioning

confidence: 99%

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Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

Ganian

Peitl

Slivovsky

et al. 2020

Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning

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We study dependency quantified Boolean formulas (DQBF), an extension of QBF in which dependencies of existential variables are listed explicitly rather than being implicit in the order of quantifiers. DQBF evaluation is a canonical NEXPTIME-complete problem, a complexity class containing many prominent problems that arise in Knowledge Representation and Reasoning. One approach for solving such hard problems is to identify and exploit structural properties captured by numerical parameters such that bounding these parameters gives rise to an efficient algorithm. This idea is captured by the notion of fixed-parameter tractability (FPT). We initiate the study of DQBF through the lens of fixed-parameter tractability and show that the evaluation problem becomes FPT under two natural parameterizations: the treewidth of the primal graph of the DQBF instance combined with a restriction on the interactions between the dependency sets, and also the treedepth of the primal graph augmented by edges representing dependency sets.

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“…Unfortunately, the completeness part is incorrect. Nevertheless, fork extension is useful for deciding certain non-trivial subclasses of DQBF [66].…”

Section: Resolution and Universal Expansionmentioning

confidence: 99%

The (D)QBF Preprocessor HQSpre – Underlying Theory and Its Implementation1

Wimmer

Scholl

Becker

2019

SAT

Self Cite

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Preprocessing turned out to be an essential step for SAT, QBF, and DQBF solvers to reduce/modify the number of variables and clauses of the formula, before the formula is passed to the actual solving algorithm. These preprocessing techniques often reduce the computation time of the solver by orders of magnitude. In this paper, we present the preprocessor HQSpre that was developed for Dependency Quantified Boolean Formulas (DQBFs) and that generalizes different preprocessing techniques for SAT and QBF problems to DQBF. We give a presentation of the underlying theory together with detailed proofs as well as implementation details contributing to the efficiency of the preprocessor. HQSpre has been used with obvious success by the winners of the DQBF track, and, even more interestingly, the QBF tracks of QBFEVAL'18.

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A PSPACE Subclass of Dependency Quantified Boolean Formulas and Its Effective Solving

Cited by 5 publications

References 24 publications

Certified DQBF Solving by Definition Extraction

Certified DQBF Solving by Definition Extraction

Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

The (D)QBF Preprocessor HQSpre – Underlying Theory and Its Implementation1

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