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A Little Blocked Literal Goes a Long Way

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)

Abstract

Q-resolution is a generalization of propositional resolution that provides the theoretical foundation for search-based solvers of quantified Boolean formulas (QBFs). Recently, it has been shown that an extension of Q-resolution, called long-distance resolution, is remarkably powerful both in theory and in practice. However, it was unknown how long-distance resolution is related to \(\mathsf {QRAT}\), a proof system introduced for certifying the correctness of QBF-preprocessing techniques. We show that \(\mathsf {QRAT}\) polynomially simulates long-distance resolution. Two simple rules of \(\mathsf {QRAT}\) are crucial for our simulation—blocked-literal addition and blocked-literal elimination. Based on the simulation, we implemented a tool that transforms long-distance-resolution proofs into \(\mathsf {QRAT}\) proofs. In a case study, we compare long-distance-resolution proofs of the well-known Kleine Büning formulas with corresponding \(\mathsf {QRAT}\) proofs.

References

  1. 1.
    Balabanov, V., Jiang, J.R.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)CrossRefzbMATHGoogle Scholar
  2. 2.
    Balabanov, V., Widl, M., Jiang, J.R.: QBF resolution systems and their proof complexities. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 154–169. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_12 Google Scholar
  3. 3.
    Benedetti, M., Mangassarian, H.: QBF-based formal verification: experience and perspectives. J. Satisf. Boolean Model. Comput. (JSAT ) 5(1–4), 133–191 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beyersdorff, O., Bonacina, I., Chew, L.: Lower bounds: from circuits to QBF proof systems. In: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science (ITCS 2016), pp. 249–260. ACM (2016)Google Scholar
  5. 5.
    Beyersdorff, O., Chew, L., Janota, M.: On unification of QBF resolution-based calculi. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 81–93. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44465-8_8 Google Scholar
  6. 6.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Proceedings of the 32nd Internation Symposium on Theoretical Aspects of Computer Science (STACS 2015). LIPIcs, vol. 30, pp. 76–89. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)Google Scholar
  7. 7.
    Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Are short proofs narrow? QBF resolution is not simple. In: Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). LIPIcs, vol. 47, pp. 15:1–15:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  8. 8.
    Beyersdorff, O., Pich, J.: Understanding Gentzen and Frege systems for QBF. In: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016), pp. 146–155. ACM (2016)Google Scholar
  9. 9.
    Chen, H.: Proof complexity modulo the polynomial hierarchy: understanding alternation as a source of hardness. In: Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). LIPIcs, vol. 55, pp. 94:1–94:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  10. 10.
    Egly, U.: On stronger calculi for QBFs. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 419–434. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_26 Google Scholar
  11. 11.
    Egly, U., Lonsing, F., Widl, M.: Long-distance resolution: proof generation and strategy extraction in search-based QBF solving. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19. LNCS, vol. 8312, pp. 291–308. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-45221-5_21 CrossRefGoogle Scholar
  12. 12.
    Haken, A.: The intractability of resolution. Theor. Comput. Sci. 39, 297–308 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.D.: Expressing symmetry breaking in DRAT proofs. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 591–606. Springer, Cham (2015). doi: 10.1007/978-3-319-21401-6_40 CrossRefGoogle Scholar
  14. 14.
    Heule, M.J.H., Seidl, M., Biere, A.: Blocked literals are universal. In: Havelund, K., Holzmann, G., Joshi, R. (eds.) NFM 2015. LNCS, vol. 9058, pp. 436–442. Springer, Cham (2015). doi: 10.1007/978-3-319-17524-9_33 Google Scholar
  15. 15.
    Heule, M.J.H., Seidl, M., Biere, A.: Solution validation and extraction for QBF preprocessing. J. Autom. Reason. 58(1), 97–125 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Janota, M.: On Q-resolution and CDCL QBF solving. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 402–418. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_25 Google Scholar
  17. 17.
    Janota, M., Grigore, R., Marques-Silva, J.P.: On QBF proofs and preprocessing. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19. LNCS, vol. 8312, pp. 473–489. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-45221-5_32 CrossRefGoogle Scholar
  18. 18.
    Janota, M., Klieber, W., Marques-Silva, J.P., Clarke, E.M.: Solving QBF with counterexample guided refinement. Artif. Intell. 234, 1–25 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kleine Büning, H., Bubeck, U.: Theory of quantified Boolean formulas. In: Handbook of Satisfiability, pp. 735–760. IOS Press (2009)Google Scholar
  20. 20.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kullmann, O.: On a generalization of extended resolution. Discrete Appl. Math. 96–97, 149–176 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lonsing, F., Egly, U.: DepQBF 6.0: a search-based QBF solver beyond traditional QCDCL. CoRR abs/1702.08256 (2017)Google Scholar
  23. 23.
    Slivovsky, F., Szeider, S.: Variable dependencies and Q-resolution. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 269–284. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_21 Google Scholar
  24. 24.
    Van Gelder, A.: Contributions to the theory of practical quantified Boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, pp. 647–663. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33558-7_47 CrossRefGoogle Scholar
  25. 25.
    Wetzler, N.D., Heule, M.J.H., Hunt Jr., W.A.: DRAT-trim: efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 422–429. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_31 Google Scholar
  26. 26.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: Proceedings of the 2002 IEEE/ACM International Conference on Computer-Aided Design (ICCAD 2002), pp. 442–449. ACM/IEEE Computer Society (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria
  2. 2.Department of Computer ScienceThe University of Texas at AustinAustinUSA
  3. 3.Institute for Formal Models and VerificationJKU LinzLinzAustria

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