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Unified QBF certification and its applications

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Abstract

Quantified Boolean formulae (QBF) allow compact encoding of many decision problems. Their importance motivated the development of fast QBF solvers. Certifying the results of a QBF solver not only ensures correctness, but also enables certain synthesis and verification tasks. To date the certificate of a true formula can be in the form of either a syntactic cube-resolution proof or a semantic Skolem-function model whereas that of a false formula is only in the form of a syntactic clause-resolution proof. The semantic certificate for a false QBF is missing, and the syntactic and semantic certificates are somewhat unrelated. This paper identifies the missing Herbrand-function countermodel for false QBF, and strengthens the connection between syntactic and semantic certificates by showing that, given a true QBF, its Skolem-function model is derivable from its cube-resolution proof of satisfiability as well as from its clause-resolution proof of unsatisfiability under formula negation. Consequently Skolem-function derivation can be decoupled from special Skolemization-based solvers and computed from standard search-based ones. Experimental results show strong benefits of the new method.

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Notes

  1. Since negating the QBFEVAL formulae using Tseitin’s conversion [20] may suffer from variable blow up, the Skolem functions are only derived with respect to the original formulae.

  2. Rewriting a proof involving long-distance resolution to one with only distance-1 resolution might potentially increase the proof size exponentially.

  3. We did not experiment with Ebddres [10] and yQuaffle [21] as the former tends to generate larger certificates for false QBF compared to squolem, and the latter has characteristics similar to QuBE-cert.

  4. In addition to sKizzo, in theory squolem can also compute Skolem-function countermodels of false QBF instances by formula negation. We only experimented with sKizzo, which can read in DNF formulae and thus requires no external DNF-to-CNF conversion, arising due to formula negation. Although squolem is not experimented in direct countermodel generation by formula negation, prior experience [10] suggested that it might be unlikely to cover much more cases than sKizzo.

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Acknowledgements

The authors are grateful to Roderick Bloem and Georg Hofferek for providing the relation determinization benchmarks.

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Authors and Affiliations

  1. Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Valeriy Balabanov

  2. Department of Electrical Engineering/Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Jie-Hong R. Jiang

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  1. Valeriy Balabanov
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  2. Jie-Hong R. Jiang
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Correspondence to Jie-Hong R. Jiang.

Additional information

This work is an extended version of [4] and was supported in part by the National Science Council under grants NSC 99-2221-E-002-214-MY3, 99-2923-E-002-005-MY3, and 100-2923-E-002-008. V. Balabanov has been partially supported by the Taiwan Scholarship Program.

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Balabanov, V., Jiang, JH.R. Unified QBF certification and its applications. Form Methods Syst Des 41, 45–65 (2012). https://doi.org/10.1007/s10703-012-0152-6

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