On Propositional QBF Expansions and Q-Resolution

  • Mikoláš Janota
  • Joao Marques-Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7962)


Over the years, proof systems for propositional satisfiability (SAT) have been extensively studied. Recently, proof systems for quantified Boolean formulas (QBFs) have also been gaining attention. Q-resolution is a calculus enabling producing proofs from DPLL-based QBF solvers. While DPLL has become a dominating technique for SAT, QBF has been tackled by other complementary and competitive approaches. One of these approaches is based on expanding variables until the formula contains only one type of quantifier; upon which a SAT solver is invoked. This approach motivates the theoretical analysis carried out in this paper. We focus on a two phase proof system, which expands the formula in the first phase and applies propositional resolution in the second. Fragments of this proof system are defined and compared to Q-resolution.


Proof System Conjunctive Normal Form Universal Variable Resolution Step Conjunctive Normal Form Formula 
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.
    Ayari, A., Basin, D.: QUBOS: Deciding quantified Boolean logic using propositional satisfiability solvers. In: Aagaard, M.D., O’Leary, J.W. (eds.) FMCAD 2002. LNCS, vol. 2517, pp. 187–201. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Balabanov, V., Jiang, J.H.R.: Unified QBF certification and its applications. Formal Methods in System Design 41(1), 45–65 (2012)CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    Bubeck, U.: Model-based transformations for quantified Boolean formulas. Ph.D. thesis, University of Paderborn (2010)Google Scholar
  6. 6.
    Bubeck, U., Büning, H.K.: 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
  7. 7.
    Büning, H.K., Bubeck, U.: Theory of quantified boolean formulas. In: Handbook of Satisfiability. IOS Press (2009)Google Scholar
  8. 8.
    Büning, H.K., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1) (1995)Google Scholar
  9. 9.
    Cadoli, M., Schaerf, M., Giovanardi, A., Giovanardi, M.: An algorithm to evaluate quantified Boolean formulae and its experimental evaluation. J. Autom. Reasoning 28(2), 101–142 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cimatti, A., Sebastiani, R. (eds.): SAT 2012. LNCS, vol. 7317. Springer, Heidelberg (2012)Google Scholar
  11. 11.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti, Sebastiani (eds.) [10], pp. 100–113Google Scholar
  13. 13.
    Giunchiglia, E., Marin, P., Narizzano, M.: QuBE 7.0 system description. Journal on Satisfiability, Boolean Modeling and Computation 7 (2010)Google Scholar
  14. 14.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Clause/term resolution and learning in the evaluation of quantified Boolean formulas. Journal of Artificial Intelligence Research 26(1), 371–416 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. In: Cimatti, Sebastiani (eds.) [10], pp. 114–128Google Scholar
  16. 16.
    Janota, M., Marques-Silva, J.: Abstraction-based algorithm for 2QBF. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 230–244. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Mathematical Logic Quarterly 36(1), 29–46 (1990)zbMATHCrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Lonsing, F., Biere, A.: DepQBF: A dependency-aware QBF solver. JSAT (2010)Google Scholar
  21. 21.
    Rintanen, J.: Improvements to the evaluation of quantified Boolean formulae. In: Dean, T. (ed.) IJCAI, pp. 1192–1197. Morgan Kaufmann (1999)Google Scholar
  22. 22.
    Urquhart, A.: The complexity of propositional proofs. Bulletin of the EATCS 64 (1998)Google Scholar
  23. 23.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: ICCAD (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mikoláš Janota
    • 1
  • Joao Marques-Silva
    • 2
  1. 1.IST/INESC-IDLisbonPortugal
  2. 2.University College DublinIreland

Personalised recommendations