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Clausal Abstraction for DQBF

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Theory and Applications of Satisfiability Testing – SAT 2019 (SAT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11628))

Abstract

Dependency quantified Boolean formulas (DQBF) is a logic admitting existential quantification over Boolean functions, which allows us to elegantly state synthesis problems in verification such as the search for invariants, programs, or winning regions of games. In this paper, we lift the clausal abstraction algorithm for quantified Boolean formulas (QBF) to DQBF. Clausal abstraction for QBF is an abstraction refinement algorithm that operates on a sequence of abstractions that represent the different quantifier levels. For DQBF we need to generalize this principle to partial orders of abstractions. The two challenges to overcome are: (1) Clauses may contain literals with incomparable dependencies, which we address by the recently proposed proof rule called Fork Extension, and (2) existential variables may have spurious dependencies, which we prevent by tracking consistency requirements during the execution. Our implementation \(\textsc {dCAQE}\) solves significantly more formulas than the existing DQBF algorithms.

M. N. Rabe—Work partially done while at University of California, Berkeley.

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Notes

  1. 1.

    Fork Extension as introduced in [20] is incomplete for general DQBF. However, it is complete for a normal form of DQBF. We refer to the full version [24] for details.

  2. 2.

    A formal correctness proof is given in the full version [24].

  3. 3.

    Available at https://github.com/ltentrup/caqe.

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Acknowledgments

We thank Bernd Finkbeiner for his valuable feedback on earlier versions of this paper. This work was partially supported by the German Research Foundation (DFG) as part of the Collaborative Research Center “Foundations of Perspicuous Software Systems” (TRR 248, 389792660) and by the European Research Council (ERC) Grant OSARES (No. 683300).

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Authors and Affiliations

  1. Reactive Systems Group, Saarland University, Saarbrücken, Germany

    Leander Tentrup

  2. Google Research, Mountain View, CA, USA

    Markus N. Rabe

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  1. Leander Tentrup
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Correspondence to Leander Tentrup .

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Editors and Affiliations

  1. University of Lisbon, Lisbon, Portugal

    Mikoláš Janota

  2. University of Lisbon, Lisbon, Portugal

    Inês Lynce

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Tentrup, L., Rabe, M.N. (2019). Clausal Abstraction for DQBF. In: Janota, M., Lynce, I. (eds) Theory and Applications of Satisfiability Testing – SAT 2019. SAT 2019. Lecture Notes in Computer Science(), vol 11628. Springer, Cham. https://doi.org/10.1007/978-3-030-24258-9_27

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  • DOI: https://doi.org/10.1007/978-3-030-24258-9_27

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