A Non-prenex, Non-clausal QBF Solver with Game-State Learning

  • William Klieber
  • Samir Sapra
  • Sicun Gao
  • Edmund Clarke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6175)


We describe a DPLL-based solver for the problem of quantified boolean formulas (QBF) in non-prenex, non-CNF form. We make two contributions. First, we reformulate clause/cube learning, extending it to non-prenex instances. We call the resulting technique game-state learning. Second, we introduce a propagation technique using ghost literals that exploits the structure of a non-CNF instance in a manner that is symmetric between the universal and existential variables. Experimental results on the QBFLIB benchmarks indicate our approach outperforms other state-of-the-art solvers on certain benchmark families, including the tipfixpoint and tipdiam families of model checking problems.


QBF DPLL non-clausal non-prenex clause learning 


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  1. 1.
    Ansótegui, C., Gomes, C.P., Selman, B.: The Achilles’ Heel of QBF. In: AAAI (2005)Google Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Biere, A.: Resolve and Expand. In: Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Egly, U., Seidl, M., Woltran, S.: A Solver for QBFs in Nonprenex Form. In: ECAI (2006)Google Scholar
  5. 5.
    Ganai, M.K., Ashar, P., Gupta, A., Zhang, L., Malik, S.: Combining strengths of circuit-based and CNF-based algorithms for a high-performance SAT solver. In: DAC (2002)Google Scholar
  6. 6.
    Gent, I.P., Giunchiglia, E., Narizzano, M., Rowley, A.G.D., Tacchella, A.: Watched Data Structures for QBF Solvers. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 25–36. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Quantifier structure in search based procedures for QBFs. In: DATE 2006 (2006)Google Scholar
  8. 8.
    Goultiaeva, A., Iverson, V., Bacchus, F.: Beyond CNF: A Circuit-Based QBF Solver. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 412–426. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Jain, H., Bartzis, C., Clarke, E.: Satisfiability Checking of Non-clausal Formulas Using General Matings. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 75–89. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an Efficient SAT Solver. In: DAC 2001 (2001)Google Scholar
  12. 12.
    Narizzano, M., Pulina, L., Tacchella, A.: QBFEVAL,
  13. 13.
    Pigorsch, F., Scholl, C.: Exploiting structure in an AIG based QBF solver. In: DATE 2009 (2009)Google Scholar
  14. 14.
    Sabharwal, A., Ansótegui, C., Gomes, C.P., Hart, J.W., Selman, B.: QBF Modeling: Exploiting Player Symmetry for Simplicity and Efficiency. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 382–395. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Samulowitz, H., Bacchus, F.: Dynamically Partitioning for Solving QBF. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 215–229. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Thiffault, C., Bacchus, F., Walsh, T.: Solving Non-clausal Formulas with DPLL Search. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 663–678. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Tseitin, G.S.: On the complexity of derivations in the propositional calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic, Part-II (1968)Google Scholar
  18. 18.
    Zhang, L.: Solving QBF by Combining Conjunctive and Disjunctive Normal Forms. In: AAAI 2006 (2006)Google Scholar
  19. 19.
    Zhang, L., Malik, S.: Conflict Driven Learning in a Quantified Boolean Satisfiability Solver. In: ICCAD 2002 (2002)Google Scholar
  20. 20.
    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, p. 200. Springer, Heidelberg (2002)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • William Klieber
    • 1
  • Samir Sapra
    • 1
  • Sicun Gao
    • 1
  • Edmund Clarke
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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