Dependency Learning for QBF

  • Tomáš Peitl
  • Friedrich Slivovsky
  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)

Abstract

Quantified Boolean Formulas (QBFs) can be used to succinctly encode problems from domains such as formal verification, planning, and synthesis. One of the main approaches to QBF solving is Quantified Conflict Driven Clause Learning (QCDCL). By default, QCDCL assigns variables in the order of their appearance in the quantifier prefix so as to account for dependencies among variables. Dependency schemes can be used to relax this restriction and exploit independence among variables in certain cases, but only at the cost of nontrivial interferences with the proof system underlying QCDCL. We propose a new technique for exploiting variable independence within QCDCL that allows solvers to learn variable dependencies on the fly. The resulting version of QCDCL enjoys improved propagation and increased flexibility in choosing variables for branching while retaining ordinary (long-distance) Q-resolution as its underlying proof system. In experiments on standard benchmark sets, an implementation of this algorithm shows performance comparable to state-of-the-art QBF solvers.

References

  1. 1.
    Balabanov, V., Jiang, J.R.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)CrossRefMATHGoogle Scholar
  2. 2.
    Balabanov, V., Jiang, J.R., Janota, M., Widl, M.: Efficient extraction of QBF (counter)models from long-distance resolution proofs. In: Bonet, B., Koenig, S. (eds.) Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, Austin, Texas, USA, 25–30 January 2015, pp. 3694–3701. AAAI Press (2015)Google Scholar
  3. 3.
    Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005). doi:10.1007/11527695_5 CrossRefGoogle Scholar
  4. 4.
    Lonsing, F., Biere, A.: A compact representation for syntactic dependencies in QBFs. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 398–411. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02777-2_37 CrossRefGoogle Scholar
  5. 5.
    Lonsing, F., Biere, A.: Integrating dependency schemes in search-based QBF solvers. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 158–171. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14186-7_14 CrossRefGoogle Scholar
  6. 6.
    Biere, A., Lonsing, F., Seidl, M.: Blocked clause elimination for QBF. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 101–115. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22438-6_10 CrossRefGoogle Scholar
  7. 7.
    Cadoli, M., Schaerf, M., Giovanardi, A., Giovanardi, M.: An algorithm to evaluate Quantified Boolean Formulae and its experimental evaluation. J. Autom. Reason. 28(2), 101–142 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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 2013. LNCS, vol. 8312, pp. 291–308. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45221-5_21 CrossRefGoogle Scholar
  9. 9.
    Faymonville, P., Finkbeiner, B., Rabe, M.N., Tentrup, L.: Encodings of bounded synthesis. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 354–370. Springer, Heidelberg (2017). doi:10.1007/978-3-662-54577-5_20 CrossRefGoogle Scholar
  10. 10.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Clause/term resolution and learning in the evaluation of Quantified Boolean Formulas. J. Artif. Intell. Res. 26, 371–416 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    Goultiaeva, A., Bacchus, F.: Recovering and utilizing partial duality in QBF. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 83–99. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39071-5_8 CrossRefGoogle Scholar
  12. 12.
    Heule, M.J.H., Kullmann, O., Marek, V.W.: Solving and verifying the boolean pythagorean triples problem via cube-and-conquer. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 228–245. Springer, Cham (2016). doi:10.1007/978-3-319-40970-2_15 Google Scholar
  13. 13.
    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
  14. 14.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.: Solving QBF with counterexample guided refinement. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 114–128. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31612-8_10 CrossRefGoogle Scholar
  15. 15.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: Yang, Q., Wooldridge, M. (eds.) Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, pp. 325–331. AAAI Press (2015)Google Scholar
  16. 16.
    Jordan, C., Klieber, W., Seidl, M.: Non-cnf QBF solving with QCIR. In: Darwiche, A. (ed.) Beyond NP, Papers from the 2016 AAAI Workshop. AAAI Workshops, vol. WS-16-05. AAAI Press (2016)Google Scholar
  17. 17.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Klieber, W., Sapra, S., Gao, S., Clarke, E.: A non-prenex, non-clausal QBF solver with game-state learning. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 128–142. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14186-7_12 CrossRefGoogle Scholar
  19. 19.
    Lonsing, F., Schemes, D., Solving, Search-Based QBF: Theory and Practice. PhD thesis, Johannes Kepler University, Linz, Austria, April 2012Google Scholar
  20. 20.
    Lonsing, F., Bacchus, F., Biere, A., Egly, U., Seidl, M.: Enhancing search-based QBF solving by dynamic blocked clause elimination. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 418–433. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48899-7_29 CrossRefGoogle Scholar
  21. 21.
    Lonsing, F., Biere, A.: Nenofex: expanding NNF for QBF solving. In: Kleine Büning, H., Zhao, X. (eds.) SAT 2008. LNCS, vol. 4996, pp. 196–210. Springer, Heidelberg (2008). doi:10.1007/978-3-540-79719-7_19 CrossRefGoogle Scholar
  22. 22.
    Malik, S., Zhang, L.: Boolean satisfiability from theoretical hardness to practical success. Commun. ACM 52(8), 76–82 (2009)CrossRefGoogle Scholar
  23. 23.
    Marques-Silva, J.P., Lynce, I., Malik, S.: Conflict-driven clause learning sat solvers. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 131–153. IOS Press (2009)Google Scholar
  24. 24.
    Meel, K.S., Vardi, M.Y., Chakraborty, S., Fremont, D.J., Seshia, S.A., Fried, D., Ivrii, A., Malik, S.: Constrained sampling and counting: universal hashing meets SAT solving. In Darwiche, A. (ed.) Beyond NP, Papers from the 2016 AAAI Workshop, Phoenix, Arizona, USA. AAAI Workshops, February 12, 2016, vol. WS-16-05. AAAI Press (2016)Google Scholar
  25. 25.
    Peitl, T., Slivovsky, F., Szeider, S.: Long distance Q-resolution with dependency schemes. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 500–518. Springer, Cham (2016). doi:10.1007/978-3-319-40970-2_31 Google Scholar
  26. 26.
    Pulina, L.: The ninth QBF solvers evaluation - preliminary report. In: Lonsing, F., Seidl, M. (eds.) Proceedings of the 4th International Workshop on Quantified Boolean Formulas (QBF 2016). CEUR Workshop Proceedings, vol. 1719, pp. 1–13. CEUR-WS.org (2016)Google Scholar
  27. 27.
    Rabe, M.N., Tentrup, L.: CAQE: a certifying QBF solver. In: Kaivola, R., Wahl, T. (eds.) Formal Methods in Computer-Aided Design - FMCAD 2015, pp. 136–143. IEEE Computer Soc. (2015)Google Scholar
  28. 28.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. J. Autom. Reason. 42(1), 77–97 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Scholl, C., Pigorsch, F.: The QBF solver AIGSolve. In: Lonsing, F., Seidl, M. (eds.) Proceedings of the 4th International Workshop on Quantified Boolean Formulas (QBF 2016). CEUR Workshop Proceedings, vol. 1719, pp. 55–62. CEUR-WS.org (2016)Google Scholar
  30. 30.
    Slivovsky, F., Szeider, S.: Soundness of Q-resolution with dependency schemes. Theoret. Comput. Sci. 612, 83–101 (2016)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Tentrup, L.: Non-prenex QBF solving using abstraction. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 393–401. Springer, Cham (2016). doi:10.1007/978-3-319-40970-2_24 Google Scholar
  32. 32.
    Gelder, A.: Variable independence and resolution paths for quantified boolean formulas. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 789–803. Springer, Heidelberg (2011). doi:10.1007/978-3-642-23786-7_59 CrossRefGoogle Scholar
  33. 33.
    Vizel, Y., Weissenbacher, G., Malik, S.: Boolean satisfiability solvers and their applications in model checking. Proc. IEEE 103(11), 2021–2035 (2015)CrossRefGoogle Scholar
  34. 34.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: Pileggi, L.T., Kuehlmann, A. (eds.) Proceedings of the 2002 IEEE/ACM International Conference on Computer-Aided Design, ICCAD 2002, San Jose, California, USA, 10–14 November 2002, pp. 442–449. ACM/IEEE Computer Society (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Tomáš Peitl
    • 1
  • Friedrich Slivovsky
    • 1
  • Stefan Szeider
    • 1
  1. 1.Algorithms and Complexity GroupTU WienViennaAustria

Personalised recommendations