Blocked Clause Elimination for QBF

  • Armin Biere
  • Florian Lonsing
  • Martina Seidl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6803)


Quantified Boolean formulas (QBF) provide a powerful framework for encoding problems from various application domains, not least because efficient QBF solvers are available. Despite sophisticated evaluation techniques, the performance of such a solver usually depends on the way a problem is represented. However, the translation to processable QBF encodings is in general not unique and may either introduce variables and clauses not relevant for the solving process or blur information which could be beneficial for the solving process. To deal with both of these issues, preprocessors have been introduced which rewrite a given QBF before it is passed to a solver.

In this paper, we present novel preprocessing methods for QBF based on blocked clause elimination (BCE), a technique successfully applied in SAT. Quantified blocked clause elimination (QBCE) allows to simulate various structural preprocessing techniques as BCE in SAT. We have implemented QBCE and extensions of QBCE in the preprocessor bloqqer. In our experiments we show that preprocessing with QBCE reduces formulas substantially and allows us to solve considerable more instances than the previous state-of-the-art.


Related Clause Boolean Formula Propositional Formula Satisfying Assignment 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    Le Berre, D.: Exploiting the Real Power of Unit Propagation Lookahead. Electronic Notes in Discrete Mathematics 9, 59–80 (2001)CrossRefzbMATHGoogle 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)CrossRefGoogle Scholar
  4. 4.
    Bubeck, U., Kleine Büning, H.: Bounded Universal Expansion for Preprocessing QBF. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 244–257. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Büning, H.K., Karpinski, M., Flögel, A.: Resolution for Quantified Boolean Formulas. Information and Computation 117(1), 12–18 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    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
  7. 7.
    Giunchiglia, E., Marin, P., Narizzano, M.: QuBE 7.0. JSAT 7, 83–88 (2010)Google Scholar
  8. 8.
    Giunchiglia, E., Marin, P., Narizzano, M.: sQueezeBF: An Effective Preprocessor for QBFs Based on Equivalence Reasoning. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 85–98. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Heule, M., Järvisalo, M., Biere, A.: Clause elimination procedures for cnf formulas. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 357–371. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Heule, M., Järvisalo, M., Biere, A.: Covered Clause Elimination. CoRR, abs/1011.5202 (2010); Short paper proceedings LPAR-17Google Scholar
  11. 11.
    Järvisalo, M., Biere, A., Heule, M.: Blocked Clause Elimination. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 129–144. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Kullmann, O.: On a Generalization of Extended Resolution. Discrete Applied Mathematics 96, 149–176 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    Lonsing, F., Biere, A.: DepQBF: A Dependency-Aware QBF Solver (System Description). JSAT 7, 71–76 (2010)Google Scholar
  15. 15.
    Mangassarian, H., Le, B., Goultiaeva, A., Veneris, A.G., Bacchus, F.: Leveraging Dominators for Preprocessing QBF. In: Design, Automation and Test in Europe (DATE 2010), pp. 1695–1700. IEEE, Los Alamitos (2010)Google Scholar
  16. 16.
    Ostrowski, R., Grgoire, E., Mazure, B., Saïs, L.: Recovering and Exploiting Structural Knowledge from CNF Formulas. In: Proc. of the 8th Int. Conf. on Princ. and Pract. of Constraint Prog (CP 2002), pp. 199–206. Springer, Heidelberg (2006)Google Scholar
  17. 17.
    Pigorsch, F., Scholl, C.: An AIG-Based QBF-Solver Using SAT for Preprocessing. In: Proc. of the 47th Design Aut. Conf (DAC 2010), pp. 170–175 (2010)Google Scholar
  18. 18.
    Plaisted, D.A., Greenbaum, S.: A Structure-Preserving Clause Form Translation. Journal of Symbolic Computation 2(3), 293–304 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Samulowitz, H., Davies, J., Bacchus, F.: Preprocessing QBF. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 514–529. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Tseitin, G.S.: On the Complexity of Derivation in Propositional Calculus. Studies in Constructive Mathematics and Mathematical Logic 2(115-125), 10–13 (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Armin Biere
    • 1
  • Florian Lonsing
    • 1
  • Martina Seidl
    • 1
  1. 1.Institute for Formal Models and VerificationJohannes Kepler UniversityLinzAustria

Personalised recommendations