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Dependency Schemes for DQBF

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Theory and Applications of Satisfiability Testing – SAT 2016 (SAT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9710))

Abstract

Dependency schemes allow to identify variable independencies in QBFs or DQBFs. For QBF, several dependency schemes have been proposed, which differ in the number of independencies they are able to identify. In this paper, we analyze the spectrum of dependency schemes that were proposed for QBF. It turns out that only some of them are sound for DQBF. For the sound ones, we provide a correctness proof, for the others counter examples. Experiments show that a significant number of dependencies can either be added to or removed from a formula without changing its truth value, but with significantly increasing the flexibility for modifying the representation.

This work was partly supported by the German Research Council (DFG) as part of the project “Solving Dependency Quantified Boolean Formulas”.

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Notes

  1. 1.

    In [16] there is an additional constraint ‘the resolvent of \(C_i\) and \(C_{i+1}\) w.r.t. \(l_i\) is non-tautologous’. This constraint has to be removed according to [25, 26]. If it is not removed, resolution path dependencies are not sound.

  2. 2.

    The construction does not lead to \((Z'_{x_1} \setminus \{y_1\})\)-paths and thus \((x_1,y_1)\in {{\mathrm{{{\mathrm{qdep}}}^{\mathrm {rp}}}}}(\psi )\) cannot be proven (which is not surprising due to Theorem 2).

  3. 3.

    For notational convenience, we use here \(\mathrm {dep}(v_i) = D_{v_i}\) if \(v_i\) is existential and \(\mathrm {dep}(v_i) = \{v_i\}\) if \(v_i\) is universal.

  4. 4.

    Our tool and all benchmarks we used are available at https://projects.informatik.uni-freiburg.de/projects/dqbf.

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Acknowledgments

We thank the anonymous reviewers for their really helpful comments.

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Authors and Affiliations

  1. Albert-Ludwigs-Universität Freiburg im Breisgau, Freiburg im Breisgau, Germany

    Ralf Wimmer, Christoph Scholl, Karina Wimmer & Bernd Becker

  2. Dependable Systems and Software, Saarland University, Saarbrücken, Germany

    Ralf Wimmer

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  1. Ralf Wimmer
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Correspondence to Ralf Wimmer .

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  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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Wimmer, R., Scholl, C., Wimmer, K., Becker, B. (2016). Dependency Schemes for DQBF. In: Creignou, N., Le Berre, D. (eds) Theory and Applications of Satisfiability Testing – SAT 2016. SAT 2016. Lecture Notes in Computer Science(), vol 9710. Springer, Cham. https://doi.org/10.1007/978-3-319-40970-2_29

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  • DOI: https://doi.org/10.1007/978-3-319-40970-2_29

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