Efficient Clause Learning for Quantified Boolean Formulas via QBF Pseudo Unit Propagation

  • Florian Lonsing
  • Uwe Egly
  • Allen Van Gelder
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7962)


Recent solvers for quantified boolean formulas (QBF) use a clause learning method based on a procedure proposed by Giunchiglia et al. (JAIR 2006), which avoids creating tautological clauses. Recently, an exponential worst case for this procedure has been shown by Van Gelder (CP 2012). That paper introduced QBF Pseudo Unit Propagation (QPUP) for non-tautological clause learning in a limited setting and showed that its worst case is theoretically polynomial, although it might be impractical in a high-performance QBF solver, due to excessive time and space consumption. No implementation was reported.

We describe an enhanced version of QPUP learning that is practical to incorporate into high-performance QBF solvers, being compatible with pure-literal rules and dependency schemes. It can be used for proving in a concise format that a QBF formula is either unsatisfiable or satisfiable (working on both proofs in tandem). A lazy version of QPUP permits non-tautological clauses to be learned without actually carrying out resolutions, but this version is unable to produce proofs.

Experimental results show that QPUP is somewhat faster than the previous non-tautological clause learning procedure on benchmarks from QBFEVAL-12-SR.


Decision Level Partial Assignment Original Formula Unit Clause Dependency Scheme 
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.
    Balabanov, V., Jiang, J.R.: Unified QBF certification and its applications. Formal Methods in System Design 41, 45–65 (2012)CrossRefGoogle Scholar
  2. 2.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Clause/term resolution and learning in the evaluation of quantified boolean formulas. JAIR 26, 371–416 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Goultiaeva, A., Bacchus, F.: Exploiting QBF duality on a circuit representation. In: AAAI (2010)Google Scholar
  5. 5.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified boolean formulas. Information and Computation 117, 12–18 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kleine Büning, H., Lettmann, T.: Propositional Logic: Deduction and Algorithms. Cambridge University Press (1999)Google Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Lonsing, F., Biere, A.: DepQBF: A dependency-aware QBF solver: System description. JSAT 7, 71–76 (2010)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP–a search algorithm for propositional satisfiability. IEEE Transactions on Computers 48, 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: 39th Design Automation Conference (June 2001)Google Scholar
  13. 13.
    Niemetz, A., Preiner, M., Lonsing, F., Seidl, M., Biere, A.: Resolution-based certificate extraction for QBF (tool presentation). In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 430–435. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Samer, M.: Variable dependencies of quantified CSPs. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 512–527. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Samer, M., Szeider, S.: Backdoor sets of quantified boolean formulas. JAR 42, 77–97 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Van Gelder, A.: Generalized conflict-clause strengthening for satisfiability solvers. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 329–342. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Van Gelder, A.: Contributions to the theory of practical quantified boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 647–663. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Van Gelder, A.: Producing and verifying extremely large propositional refutations: Have your cake and eat it too. AMAI 65(4), 329–372 (2012)zbMATHGoogle Scholar
  19. 19.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified boolean satisfiability solver. In: Proc. ICCAD, pp. 442–449 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Florian Lonsing
    • 1
  • Uwe Egly
    • 1
  • Allen Van Gelder
    • 2
  1. 1.Technische Universität WienAustria
  2. 2.University of CaliforniaSanta CruzUSA

Personalised recommendations