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Long Distance Q-Resolution with Dependency Schemes

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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))

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Abstract

Resolution proof systems for quantified Boolean formulas (QBFs) provide a formal model for studying the limitations of state-of-the-art search-based QBF solvers, which use these systems to generate proofs. In this paper, we define a new proof system that combines two such proof systems: Q-resolution with generalized universal reduction according to a dependency scheme and long distance Q-resolution. We show that the resulting proof system is sound for the reflexive resolution-path dependency scheme—in fact, we prove that it admits strategy extraction in polynomial time. As a special case, we obtain soundness and polynomial-time strategy extraction for long distance Q-resolution with universal reduction according to the standard dependency scheme. We report on experiments with a configuration of DepQBF that generates proofs in this system.

This research was partially supported by FWF grants P27721 and W1255-N23.

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Notes

  1. 1.

    We consider QBFs in prenex normal form.

  2. 2.

    The original definition of dependency schemes [34] is more restrictive than the one given here, but the additional requirements are irrelevant for the purposes of this paper.

  3. 3.

    Strictly speaking, it uses a refined version of the standard dependency scheme [28, p. 49].

  4. 4.

    Our definition slightly differs from the original for the resolution rule: if restriction removes the pivot variable from both premises, we simply pick the (restriction of the) first premise as the result (rather than the clause containing fewer literals). This simplifies the certificate extraction argument given below.

  5. 5.

    http://lonsing.github.io/depqbf/

  6. 6.

    As a sanity check, we verified that all configurations that were able to solve a particular instance returned the same result.

  7. 7.

    http://fmv.jku.at/bloqqer/

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Acknowledgments

We would like to thank Florian Lonsing for helpful discussions and for pointing out how to modify DepQBF so that it generates LDQ(D\(^\text {std}\)) proofs.

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Authors and Affiliations

  1. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Tomáš Peitl, Friedrich Slivovsky & Stefan Szeider

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  1. Tomáš Peitl
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Correspondence to Stefan Szeider .

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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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Peitl, T., Slivovsky, F., Szeider, S. (2016). Long Distance Q-Resolution with Dependency Schemes. 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_31

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