Abstract

In this paper we investigate the use of preprocessing when solving Quantified Boolean Formulas (QBF). Many different problems can be efficiently encoded as QBF instances, and there has been a great deal of recent interest and progress in solving such instances efficiently. Ideas from QBF have also started to migrate to CSP with the exploration of Quantified CSPs which offer an intriguing increase in representational power over traditional CSPs. Here we show that QBF instances can be simplified using techniques related to those used for preprocessing SAT. These simplifications can be performed in polynomial time, and are used to preprocess the instance prior to invoking a worst case exponential algorithm to solve it. We develop a method for preprocessing QBF instances that is empirically very effective. That is, the preprocessed formulas can be solved significantly faster, even when we account for the time required to perform the preprocessing. Our method significantly improves the efficiency of a range of state-of-the-art QBF solvers. Furthermore, our method is able to completely solve some instances just by preprocessing, including some instances that to our knowledge have never been solved before by any QBF solver.

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References

  1. 1.
    Aspvall, B., Plass, M., Tarjan, R.: A linear-time algorithms for testing the truth of certain quantified boolean formulas. Information Processing Letters 8, 121–123 (1979)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bacchus, F.: Enhancing davis putnam with extended binary clause reasoning. In: Eighteenth national conference on Artificial intelligence, pp. 613–619 (2002)Google Scholar
  3. 3.
    Bacchus, F., Winter, J.: Effective preprocessing with hyper-resolution and equality reduction. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 341–355. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Benedetti, M.: Skizzo: a QBF decision procedure based on propositional skolemization and symbolic reasoning. Technical Report TR04-11-03 (2004)Google Scholar
  5. 5.
    Benedetti, M.: Evaluating QBFs via Symbolic Skolemization. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS, vol. 3452, pp. 285–300. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Benedetti, M.: Extracting Certificates from Quantified Boolean Formulas. In: Proc. of 9th International Joint Conference on Artificial Intelligence (IJCAI 2005) (2005)Google Scholar
  7. 7.
    Benedetti, M.: Quantifier Trees for QBFs. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 378–385. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 238–246. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Büning, H.K., Karpinski, M., Flügel, A.: Resolution for quantified boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MATHCrossRefGoogle Scholar
  10. 10.
    Eén, N., Biere, A.: Effective Preprocessing in SAT Through Variable and Clause Elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Egly, U., Eiter, T., Tompits, H., Woltran, S.: Solving advanced reasoning tasks using quantified boolean formulas. In: AAAI/IAAI, pp. 417–422 (2000)Google Scholar
  12. 12.
    Gent, I.P., Nightingale, P., Stergiou, K.: Qcsp-solve: A solver for quantified constraint satisfaction problems. In: Proceedings of the International Joint Conference on Artifical Intelligence (IJCAI), pp. 138–143 (2005)Google Scholar
  13. 13.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Quantified Boolean Formulas satisfiability library (QBFLIB) (2001), http://www.qbflib.org/
  14. 14.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: QUBE: A system for deciding quantified boolean formulas satisfiability. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 364–369. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Narizzano, M., Tacchella, A.: QBF evaluation (2005), http://www.qbflib.org/qbfeval/2005
  16. 16.
    Rintanen, J.: Constructing conditional plans by a theorem-prover. Journal of Artificial Intelligence Research 10, 323–352 (1999)MATHGoogle Scholar
  17. 17.
    Samulowitz, H., Bacchus, F.: Using SAT in QBF. In: Principles and Practice of Constraint Programming, pp. 578–592. Springer, New York (2005), Available at: http://www.cs.toronto.edu/~fbacchus/sat.html CrossRefGoogle Scholar
  18. 18.
    Samulowitz, H., Bacchus, F.: Binary clause reasoning in QBF. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 353–367. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Samulowitz, H., Davies, J., Bacchus, F.: QBF Preprocessor Prequel (2006), Available at: http://www.cs.toronto.edu/~fbacchus/sat.html
  20. 20.
    Stergiou, K.: Repair-based methods for quantified csps. In: Principles and Practice of Constraint Programming, pp. 652–666 (2005)Google Scholar
  21. 21.
    Zhang, L., Malik, S.: Towards symmetric treatment of conflicts and satisfaction in quantified boolean satisfiability solver. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 185–199. Springer, Heidelberg (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Horst Samulowitz
    • 1
  • Jessica Davies
    • 1
  • Fahiem Bacchus
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoCanada

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