Dependency Quantified Horn Formulas: Models and Complexity

Bubeck, Uwe; Büning, Hans Kleine

doi:10.1007/11814948_21

Cited by 20 publications

(12 citation statements)

References 16 publications

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“…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.…”

Section: S-form Versus H-formmentioning

confidence: 99%

Building Strategies into QBF Proofs

2020

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Strategy extraction is of great importance for quantified Boolean formulas (QBF), both in solving and proof complexity. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several advantages: (1) lines of our calculus have a clear semantic meaning as they are accompanied by semantic objects; (2) partial strategies are represented succinctly (in contrast to some previous approaches); (3) our calculus has strategy extraction by design; and (4) the partial strategies allow new sound inference steps which are disallowed in previous central QBF calculi such as Q-Resolution and long-distance Q-Resolution. The last item (4) allows us to show an exponential separation between our new system and the previously studied reductionless long-distance resolution calculus. Our approach also naturally lifts to dependency QBFs (DQBF), where it yields the first sound and complete CDCL-style calculus for DQBF, thus opening future avenues into CDCL-based DQBF solving.

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Section: S-form Versus H-formmentioning

confidence: 99%

Building Strategies into QBF Proofs

2020

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“…We give an overview on relevant techniques that are used in the literature in the context of DQBF solving. Universal expansion, which has been defined for QBF, can be easily generalized to DQBF as observed, e. g., in (Bubeck and Kleine Büning 2006;Bubeck 2010;Balabanov, Chiang, and Jiang 2014;Gitina et al 2013).…”

Section: Universal Expansion Resolution Universal Reduction and Fomentioning

confidence: 99%

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

Scholl

Jiang

Wimmer

et al. 2019

AAAI

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Dependency quantified Boolean formulas (DQBFs) are a powerful formalism, which subsumes quantified Boolean formulas (QBFs) and allows an explicit specification of dependencies of existential variables on universal variables. This enables a succinct encoding of decision problems in the NEXPTIME complexity class. As solving general DQBFs is NEXPTIME complete, in contrast to the PSPACE completeness of QBF solving, characterizing DQBF subclasses of lower computational complexity allows their effective solving and is of practical importance. Recently a DQBF proof calculus based on a notion of fork extension, in addition to resolution and universal reduction, was proposed by Rabe in 2017. We show that this calculus is in fact incomplete for general DQBFs, but complete for a subclass of DQBFs, where any two existential variables have either identical or disjoint dependency sets over the universal variables. We further characterize this DQBF subclass to be Σ P 3 complete in the polynomial time hierarchy. Essentially using fork extension, a DQBF in this subclass can be converted to an equisatisfiable 3QBF with only a linear increase in formula size. We exploit this conversion for effective solving of this DQBF subclass and point out its potential as a general strategy for DQBF quantifier localization. Experimental results show that the method outperforms state-of-the-art DQBF solvers on a number of benchmarks, including the 2018 DQBF evaluation benchmarks.

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“…Universal expansion [10,9,1,27] is the corresponding method for eliminating universal variables. Universal expansion of a universal variable x allows to remove x by introducing a copy y for every existential variable y depending on x, which has to depend on the same variables as y.…”

Section: Resolution and Universal Expansionmentioning

confidence: 99%

“…vivify(ψ); simplify(ψ) (10) variableEliminationByResolution(ψ); simplify(ψ) (11) syntacticConstants(ψ); simplify(ψ) (12) if first iteration then (13) semanticConstants(ψ); simplify(ψ) (14) trivialMatrixChecks(ψ) (15) end if (16) universalExpansion(ψ); simplify(ψ) (17) iteration ← iteration + 1 (18) until ψ has not been changed anymore or ψ is decided (19) return ψ First of all, we always apply tautology checks, universal reduction, and backward subsumption checks when we insert a clause C into the clause database or when we strengthen a clause C as a result of some preprocessing step. Backward subsumption looks for clauses C with C ⊆ C that are already included in the clause database.…”

Section: Data Structuresmentioning

confidence: 99%

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

Wimmer

Scholl

Becker

2019

SAT

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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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Dependency Quantified Horn Formulas: Models and Complexity

Cited by 20 publications

References 16 publications

Building Strategies into QBF Proofs

Building Strategies into QBF Proofs

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

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

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