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Lifting QBF Resolution Calculi to 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))

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Abstract

We examine existing resolution systems for quantified Boolean formulas (QBF) and answer the question which of these calculi can be lifted to the more powerful Dependency QBFs (DQBF). An interesting picture emerges: While for QBF we have the strict chain of proof systems \(\textsf {Q-Res}< \textsf {IR-calc} < \textsf {IRM-calc} \), the situation is quite different in DQBF. The obvious adaptations of Q-Res and likewise universal resolution are too weak: they are not complete. The obvious adaptation of IR-calc has the right strength: it is sound and complete. IRM-calc is too strong: it is not sound any more, and the same applies to long-distance resolution. Conceptually, we use the relation of DQBF to effectively propositional logic (EPR) and explain our new DQBF calculus based on IR-calc as a subsystem of first-order resolution.

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References

  1. Azhar, S., Peterson, G., Reif, J.: Lower bounds for multiplayer non-cooperative games of incomplete information. J. Comput. Math. Appl. 41, 957–992 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 19–99. Elsevier and MIT Press (2001)

    Google Scholar 

  3. Balabanov, V., Chiang, H.J.K., Jiang, J.H.R.: Henkin quantifiers and boolean formulae: a certification perspective of DQBF. Theor. Comput. Sci. 523, 86–100 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balabanov, V., Jiang, J.H.R.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)

    Article  MATH  Google Scholar 

  5. Balabanov, V., Widl, M., Jiang, J.-H.R.: QBF resolution systems and their proof complexities. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 154–169. Springer, Heidelberg (2014)

    Google Scholar 

  6. Beyersdorff, O., Bonacina, I., Chew, L.: Lower bounds: from circuits to QBF proof systems. In: Proceedings of ACM Conference on Innovations in Theoretical Computer Science (ITCS 2016), pp. 249–260. ACM (2016)

    Google Scholar 

  7. Beyersdorff, O., Chew, L., Janota, M.: On unification of QBF resolution-based calculi. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 81–93. Springer, Heidelberg (2014)

    Google Scholar 

  8. Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Proceedings of Symposium on Theoretical Aspects of Computer Science, pp. 76–89. LIPIcs series (2015)

    Google Scholar 

  9. Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Feasible interpolation for QBF resolution calculi. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 180–192. Springer, Heidelberg (2015)

    Google Scholar 

  10. Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Are short proofs narrow? QBF resolution is not simple. In: Proceedings of Symposium on Theoretical Aspects of Computer Science (STACS 2016) (2016)

    Google Scholar 

  11. Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 100–113. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  12. Finkbeiner, B., Tentrup, L.: Fast DQBF refutation. In: Sinz, C., Egly, U. (eds.) [26], pp. 243–251

    Google Scholar 

  13. Fröhlich, A., Kovásznai, G., Biere, A., Veith, H.: iDQ: Instantiation-based DQBF solving. In: Sinz, C., Egly, U. (eds.) [26], pp. 103–116

    Google Scholar 

  14. Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: Nebel, W., Atienza, D. (eds.) Proceedings of the 2015 Design, Automation & Test in Europe Conference & Exhibition, DATE 2015, Grenoble, France, March 9–13, 2015. pp. 1617–1622. ACM (2015)

    Google Scholar 

  15. Giunchiglia, E., Marin, P., Narizzano, M.: Reasoning with quantified boolean formulas. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 761–780. IOS Press (2009)

    Google Scholar 

  16. Henkin, L.: Some remarks on infinitely long formulas. J. Symbolic Logic, pp. 167–183 (1961). Pergamon Press

    Google Scholar 

  17. Heule, M.J., Seidl, M., Biere, A.: Efficient extraction of skolem functions from QRAT proofs. In: Formal Methods in Computer-Aided Design (FMCAD), 2014, pp. 107–114. IEEE (2014)

    Google Scholar 

  18. Heule, M.J.H., Seidl, M., Biere, A.: A unified proof system for QBF preprocessing. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 91–106. Springer, Heidelberg (2014)

    Google Scholar 

  19. Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Janota, M., Marques-Silva, J.: On propositional QBF expansions and Q-resolution. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 67–82. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  21. Büning, K.H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lewis, H.R.: Complexity results for classes of quantificational formulas. J. Comput. Syst. Sci. 21(3), 317–353 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Peterson, G.L., Reif, J.: Multiple-person alternation. In: 20th Annual Symposium on Foundations of Computer Science, 1979, pp. 348–363, October 1979

    Google Scholar 

  24. Segerlind, N.: The complexity of propositional proofs. Bull. Symbolic Logic 13(4), 417–481 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Seidl, M., Lonsing, F., Biere, A.: qbf2epr: a tool for generating EPR formulas from QBF. In: Fontaine, P., Schmidt, R.A., Schulz, S. (eds.) PAAR-2012. Third Workshop on Practical Aspects of Automated Reasoning. EPiC Series in Computing, vol. 21, pp. 139–148. EasyChair (2013)

    Google Scholar 

  26. Sinz, C., Egly, U. (eds.): SAT 2014. LNCS, vol. 8561. Springer, Heidelberg (2014)

    Google Scholar 

  27. Slivovsky, F., Szeider, S.: Variable dependencies and Q-resolution. In: Sinz, C., Egly, U. (eds.) [26], pp. 269–284

    Google Scholar 

  28. Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time: preliminary report. In: Aho, A.V., Borodin, A., Constable, R.L., Floyd, R.W., Harrison, M.A., Karp, R.M., Strong, H.R. (eds.) Proceedings of the 5th Annual ACM Symposium on Theory of Computing, April 30 – May 2, 1973, Austin, Texas, USA, pp. 1–9. ACM (1973)

    Google Scholar 

  29. Urquhart, A.: The complexity of propositional proofs. Bull. Symbolic Logic 1, 425–467 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Van Gelder, A.: Contributions to the theory of practical quantified boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 647–663. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  31. Wimmer, R., Gitina, K., Nist, J., Scholl, C., Becker, B.: Preprocessing for DQBF. In: Heule, M., et al. (eds.) SAT 2015. LNCS, vol. 9340, pp. 173–190. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24318-4_13

    Chapter  Google Scholar 

  32. Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: ICCAD, pp. 442–449 (2002)

    Google Scholar 

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Acknowledgments

This research was supported by grant no. 48138 from the John Templeton Foundation, EPSRC grant EP/L024233/1, and a Doctoral Training Grant from the EPSRC (2nd author).

Martin Suda was supported by the EPSRC grant EP/K032674/1 and the ERC Starting Grant 2014 SYMCAR 639270.

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Authors and Affiliations

  1. School of Computing, University of Leeds, Leeds, UK

    Olaf Beyersdorff & Leroy Chew

  2. School of Computer Science, University of Manchester, Manchester, UK

    Renate A. Schmidt & Martin Suda

Authors
  1. Olaf Beyersdorff
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  2. Leroy Chew
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  3. Renate A. Schmidt
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  4. Martin Suda
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Corresponding author

Correspondence to Leroy Chew .

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Editors and Affiliations

  1. Aix-Marseille Université, Marseille, France

    Nadia Creignou

  2. Université d'Artois, Lens, France

    Daniel Le Berre

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Beyersdorff, O., Chew, L., Schmidt, R.A., Suda, M. (2016). Lifting QBF Resolution Calculi to 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_30

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  • DOI: https://doi.org/10.1007/978-3-319-40970-2_30

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