A First Step Towards a Unified Proof Checker for QBF

  • Toni Jussila
  • Armin Biere
  • Carsten Sinz
  • Daniel Kröning
  • Christoph M. Wintersteiger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4501)


Compared to SAT, there is no simple concept of what a solution to a QBF problem is. Furthermore, as the series of QBF evaluations shows, the QBF solvers that are available often disagree. Thus, proof generation for QBF seems to be even more important than for SAT. In this paper we propose a new uniform proof format, which captures refutations and witnesses for a variety of QBF solvers, and is based on a novel extended resolution rule for QBF. Our experiments show the flexibility of this new format. We also identify shortcomings of our format and conjecture that a purely resolution based proof calculus is not powerful enough to trace the most efficient solvers.


Conjunctive Normal Form Boolean Formula Satisfying Assignment Validation Time Unit Clause 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Stockmeyer, L.J.: The polynomial–time hierarchy. TCS 3, 1–22 (1976)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  3. 3.
    Rintanen, J.: Constructing conditional plans by a theorem-prover. Journal of Artificial Intelligence Research 10, 323–352 (1999)zbMATHGoogle Scholar
  4. 4.
    Otwell, C., Remshagen, A., Truemper, K.: An effective QBF solver for planning problems. In: MSV/AMCS, pp. 311–316. CSREA Press (2004)Google Scholar
  5. 5.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: QBF reasoning on real–world instances. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 105–121. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM Journal on Computing 6(3), 467–480 (1977),, doi:10.1137/0206033CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Biere, A., et al.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) ETAPS 1999 and TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Dershowitz, N., Hanna, Z., Katz, J.: Bounded model checking with QBF. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 408–414. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Jussila, T., Biere, A.: Compressing BMC encodings with QBF. In: Proc. 4th Intl. Work. on Bounded Model Checking (BMC). To be published in ENTCS, Elsevier, Amsterdam (2006)Google Scholar
  10. 10.
    Benedetti, M.: Experimenting with QBF-based formal verification. In: Proc. of the 3rd International Workshop on Constraints in Formal Verification (CFV). To be published in ENTCS, Elsevier, Amsterdam (2005)Google Scholar
  11. 11.
    Plaisted, D.A., Biere, A., Zhu, Y.: A satisfiability procedure for quantified boolean formulae. Discrete Appl. Math. 130(2), 291–328 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Zhang, L., Malik, S.: Towards a symmetric treatment of satisfaction and conflicts in quantified boolean formula evaluation. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 200–215. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Letz, R.: Lemma and model caching in decision procedures for quantified boolean formulas. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 160–175. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  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.
    Cadoli, M., Giovanardi, A., Schaerf, M.: An algorithm to evaluate quantified boolean formulae. In: Proc. of AAAI/IAAI, pp. 262–267. AAAI Press, Menlo Park (1998)Google Scholar
  16. 16.
    Samulowitz, H., Bacchus, F.: Using SAT in QBF. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 578–592. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005)Google Scholar
  18. 18.
    Pan, G., Vardi, M.Y.: Symbolic decision procedures for QBF. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 453–467. Springer, Heidelberg (2004)Google Scholar
  19. 19.
    Benedetti, M.: Evaluating QBFs via symbolic skolemization. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 285–300. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Samulowitz, H., Bacchus, F.: Binary clause reasoning in QBF. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Narizzano, M., Tacchella, A., Pulina, L.: Report of the third QBF solvers evaluation. JSAT 2, 145–164 (2006)zbMATHGoogle Scholar
  22. 22.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Studies in Constructive Mathematics and Mathematical Logic, vol. 2, pp. 115–125 (1968)Google Scholar
  23. 23.
    Benedetti, M.: sKizzo: A suite to evaluate and certify QBFs. In: Nieuwenhuis, R. (ed.) Automated Deduction – CADE-20. LNCS (LNAI), vol. 3632, pp. 369–376. Springer, Heidelberg (2005)Google Scholar
  24. 24.
    Yu, Y., Malik, S.: Validating the result of a quantified boolean formula (QBF) solver: theory and practice. In: Proc. of ASP-DAC, pp. 1047–1051. ACM Press, New York (2005)CrossRefGoogle Scholar
  25. 25.
    Kleine Büning, H., Karpinski, M., Flügel, A.: Resolution for quantified boolean formulas. Inf. Comput. 117(1), 12–18 (1995)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kleine Büning, H., Zhao, X.: On models for quantified boolean formulas. In: Lenski, W. (ed.) Logic versus Approximation. LNCS, vol. 3075, pp. 18–32. Springer, Heidelberg (2004)Google Scholar
  27. 27.
    Benedetti, M.: Extracting certificates from quantified boolean formulas. In: Proc. of IJCAI, pp. 47–53 (2005)Google Scholar
  28. 28.
    Büning, H.K., Subramani, K., Zhao, X.: On boolean models for quantified boolean horn formulas. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 93–104. Springer, Heidelberg (2004)Google Scholar
  29. 29.
    Sinz, C., Biere, A.: Extended resolution proofs for conjoining BDDs. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 600–611. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  30. 30.
    Jussila, T., Sinz, C., Biere, A.: Extended resolution proofs for symbolic SAT solving with quantification. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 54–60. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Toni Jussila
    • 1
  • Armin Biere
    • 1
  • Carsten Sinz
    • 2
  • Daniel Kröning
    • 3
  • Christoph M. Wintersteiger
    • 3
  1. 1.Formal Models and Verification, Johannes Kepler University, Linz 
  2. 2.Wilhelm-Schickard-Institute for Computer Science, University of Tübingen 
  3. 3.Computer Systems Institute, ETH Zürich 

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