Journal of Automated Reasoning

, Volume 56, Issue 4, pp 459–477 | Cite as

Quantifier Reordering for QBF

  • Friedrich Slivovsky
  • Stefan Szeider


State-of-the-art procedures for evaluating quantified Boolean formulas often expect input formulas in prenex conjunctive normal form (PCNF). We study dependency schemes as a means of reordering the quantifier prefix of a PCNF formula while preserving its truth value. Dependency schemes associate each formula with a binary relation on its variables (the dependency relation) that imposes constraints on certain operations manipulating the formula’s quantifier prefix. We prove that known dependency schemes support a stronger reordering operation than was previously known. We present an algorithm that, given a formula and its dependency relation, computes a compatible reordering with a minimum number of quantifier alternations. In combination with a dependency scheme that can be computed in polynomial time, this yields a polynomial time heuristic for reducing the number of quantifier alternations of an input formula. The resolution-path dependency scheme is the most general dependency scheme introduced so far. Using an interpretation of resolution paths as directed paths in a formula’s implication graph, we prove that the resolution-path dependency relation can be computed in polynomial time.


Quantified Boolean formulas Quantifier reordering  Variable dependencies Q-resolution 



This research was supported by the European Research Council (ERC), Project Complex Reason 239962.


  1. 1.
    Atserias, A., Oliva, S.: Bounded-width QBF is PSPACE-complete. In: Portier, N., Wilke, T. (eds.) 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27–March 2, Kiel, Germany, Volume 20 of LIPIcs, pp. 44–54. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  2. 2.
    Biere, A., Cimatti, A., Clarke, E.M., Zhu, Y.: Symbolic model checking without bdds. In: Cleaveland, R. (eds.) 5th International Conference in Tools and Algorithms for Construction and Analysis of Systems (TACAS ’99), Held as Part of the European Joint Conferences on the Theory and Practice of Software (ETAPS ’99), Amsterdam, The Netherlands, March 22–28, Proceedings, Volume 1579 of Lecture Notes in Computer Science, pp. 193–207. Springer, Berlin (1999)Google Scholar
  3. 3.
    Biere, A., Lonsing, F.: Integrating dependency schemes in search-based QBF solvers. In: Strichman, O., Szeider, S. (eds.) Theory and Applications of Satisfiability Testing—(SAT 2010), Volume 6175 of Lecture Notes in Computer Science, pp. 158–171. Springer, Berlin (2010)Google Scholar
  4. 4.
    Biere, A., Lonsing, F., Seidl, M.: Blocked clause elimination for QBF. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) International Conference on Automated Deduction—(CADE 23), Volume 6803 of Lecture Notes in Computer Science, pp. 101–115. Springer, Berlin (2011)Google Scholar
  5. 5.
    Bjesse, P., Leonard, T., Mokkedem, A.: Finding bugs in an alpha microprocessor using satisfiability solvers. In: Berry, G., Comon, H., Finkel, A. (eds.) Computer Aided Verification: Proceedings of the 13th International Conference (CAV 2001), Paris, France, July 18–22, pp. 454–464 (2001)Google Scholar
  6. 6.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kautz, H., Selman, B.: Pushing the envelope: planning, propositional logic, and stochastic search. In: Proceedings of the Thirteenth AAAI Conference on Artificial Intelligence (AAAI ’96), pp. 1194–1201. AAAI Press, Palo Alto (1996)Google Scholar
  8. 8.
    Kleine Büning, H., Lettman, T.: Propositional Logic: Deduction and Algorithms. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  9. 9.
    Lonsing, F.: Dependency Schemes and Search-Based QBF Solving: Theory and Practice. Ph.D. thesis, Johannes Kepler University, Linz, Austria (2012)Google Scholar
  10. 10.
    Prasad, A.G.M., Biere, A.: A survey of recent advances in SAT-based formal verification. Softw. Tools Technol. Transf. 7(2), 156–173 (2005)CrossRefGoogle Scholar
  11. 11.
    Pan, G., Vardi, M.Y.: Fixed-parameter hierarchies inside PSPACE. In: Proceedings of the 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12–15 August, Seattle, WA, USA, pp. 27–36. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  12. 12.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. J. Autom. Reason. 42(1), 77–97 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Slivovsky, F., Szeider, S.: Computing resolution-path dependencies in linear time. In: Cimatti, A., Sebastiani, R. (eds.) Theory and Applications of Satisfiability Testing—SAT 2012, Volume 7317 of Lecture Notes in Computer Science, pp. 58–71. Springer, Berlin (2012)Google Scholar
  14. 14.
    Slivovsky, F., Szeider, S.: Variable dependencies and Q-resolution. In: Sinz, C., Egly, U. (eds.) Theory and Applications of Satisfiability Testing—SAT 2014, Volume 8561 of Lecture Notes in Computer Science, pp. 269–284. Springer, Berlin (2014)Google Scholar
  15. 15.
    Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Van Gelder, A.: Variable independence and resolution paths for quantified Boolean formulas. In: Lee, J. (eds.) Principles and Practice of Constraint Programming—CP 2011, Volume 6876 of Lecture Notes in Computer Science, pp. 789–803. Springer, Berlin (2011)Google Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institute of Computer Graphics and AlgorithmsTU WienViennaAustria

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