Circuit-Based Search Space Pruning in QBF

  • Mikoláš JanotaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10929)


This paper describes the algorithm implemented in the QBF solver CQESTO, which has placed second in the non-CNF track of the last year’s QBF competition. The algorithm is inspired by the CNF-based solver QESTO. Just as QESTO, CQESTO invokes a SAT solver in a black-box fashion. However, it directly operates on the circuit representation of the formula. The paper analyzes the individual operations that the solver performs.


Quantified Boolean Formulas (QBF) RAReQS Quabbe Subformula Prenex Form 
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.



This work was supported by national funds through Fundação para a Ciência e a Tecnologia (FCT) with reference UID/CEC/50021/2013. The author would like to thank Nikolaj Bjørner and João Marques-Silva for the helpful discussions on the topic.


  1. 1.
    Ansótegui, C., Gomes, C.P., Selman, B.: The Achilles’ heel of QBF. In: National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference (AAAI), pp. 275–281 (2005)Google Scholar
  2. 2.
    Balabanov, V., Jiang, J.-H.R., Scholl, C., Mishchenko, A., Brayton, R.K.: 2QBF: challenges and solutions. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 453–469. Springer, Cham (2016). Scholar
  3. 3.
    Benedetti, M., Mangassarian, H.: QBF-based formal verification: experience and perspectives. J. Satisfiability Bool. Model. Comput. (JSAT) 5(1–4), 133–191 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bjørner, N., Janota, M.: Playing with quantified satisfaction. In: International Conferences on Logic for Programming LPAR-20, Short Presentations, vol. 35, pp. 15–27. EasyChair (2015)Google Scholar
  5. 5.
  6. 6.
    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). Scholar
  7. 7.
    Farzan, A., Kincaid, Z.: Strategy synthesis for linear arithmetic games. In: Proceedings of the ACM on Programming Languages 2 (POPL), pp. 61:1–61:30, December 2017. Scholar
  8. 8.
    Goultiaeva, A., Seidl, M., Biere, A.: Bridging the gap between dual propagation and CNF-based QBF solving. In: Design, Automation & Test in Europe (DATE), pp. 811–814 (2013)Google Scholar
  9. 9.
    Goultiaeva, A., Van Gelder, A., Bacchus, F.: A uniform approach for generating proofs and strategies for both true and false QBF formulas. In: International Joint Conference on Artificial Intelligence (IJCAI), pp. 546–553 (2011)Google Scholar
  10. 10.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.: Solving QBF with counterexample guided refinement. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 114–128. Springer, Heidelberg (2012). Scholar
  11. 11.
    Janota, M.: Towards generalization in QBF solving via machine learning. In: AAAI Conference on Artificial Intelligence (2018)Google Scholar
  12. 12.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.: Solving QBF with counterexample guided refinement. Artif. Intell. 234, 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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). Scholar
  14. 14.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: International Joint Conference on Artificial Intelligence (IJCAI) (2015)Google Scholar
  15. 15.
    Janota, M., Marques-Silva, J.: An Achilles’ heel of term-resolution. In: Conference on Artificial Intelligence (EPIA), pp. 670–680 (2017)CrossRefGoogle Scholar
  16. 16.
    Jordan, C., Klieber, W., Seidl, M.: Non-CNF QBF solving with QCIR. In: Proceedings of BNP (Workshop) (2016)Google Scholar
  17. 17.
    Kleine Büning, H., Bubeck, U.: Theory of quantified Boolean formulas. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 735–760. IOS Press (2009)Google Scholar
  18. 18.
    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). Scholar
  19. 19.
    Manna, Z., Waldinger, R.: The Logical Basis for Computer Programming, vol. 2. Addison-Wesley, Reading (1985)Google Scholar
  20. 20.
    Marques Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Peitl, T., Slivovsky, F., Szeider, S.: Dependency learning for QBF. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 298–313. Springer, Cham (2017). Scholar
  22. 22.
  23. 23.
    Rabe, M.N., Tentrup, L.: CAQE: a certifying QBF solver. In: Formal Methods in Computer-Aided Design, FMCAD, pp. 136–143 (2015)Google Scholar
  24. 24.
    Reynolds, A., King, T., Kuncak, V.: Solving quantified linear arithmetic by counterexample-guided instantiation. Formal Methods Syst. Des. 51(3), 500–532 (2017). Scholar
  25. 25.
    Tentrup, L.: Non-prenex QBF solving using abstraction. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 393–401. Springer, Cham (2016). Scholar
  26. 26.
    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
  27. 27.
    Tu, K.-H., Hsu, T.-C., Jiang, J.-H.R.: QELL: QBF reasoning with extended clause learning and levelized SAT solving. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 343–359. Springer, Cham (2015). Scholar
  28. 28.
    Zhang, L.: Solving QBF by combining conjunctive and disjunctive normal forms. In: National Conference on Artificial Intelligence and the Eighteenth Innovative Applications of Artificial Intelligence Conference (AAAI) (2006)Google Scholar
  29. 29.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: International Conference On Computer Aided Design (ICCAD), pp. 442–449 (2002)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IST/INESC-IDUniversity of LisbonLisbonPortugal

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