An Achilles’ Heel of Term-Resolution

  • Mikoláš Janota
  • Joao Marques-Silva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10423)


Term-resolution provides an elegant mechanism to prove that a quantified Boolean formula (QBF) is true. It is a dual to Q-resolution and is practically highly important as it enables certifying answers of DPLL-based QBF solvers. While term-resolution and Q-resolution are very similar, they are not completely symmetrical. In particular, Q-resolution operates on clauses and term-resolution operates on models of the matrix. This paper investigates the impact of this asymmetry. We will see that there is a large class of formulas (formulas with “big models”) whose term-resolution proofs are exponential. As a possible remedy, the paper suggests to prove true QBFs by refuting their negation (negate-refute), rather than proving them by term-resolution. The paper shows that from the theoretical perspective this is indeed a favorable approach. In particular, negation-refutation p-simulates term-resolution and there is an exponential separation between the two calculi. These observations further our understanding of proof systems for QBFs and provide a strong theoretical underpinning for the effort towards non-CNF QBF solvers.



This work was supported by national funds through Fundação para a Ciência e a Tecnologia (FCT) with reference UID/CEC/50021/2013 and by FCT funding of LASIGE Research Unit, reference UID/CEC/00408/2013.


  1. 1.
    Goultiaeva, A., Seidl, M., Biere, A.: Bridging the gap between dual propagation and CNF-based QBF solving. In. Proceedings of International Conference on Design, Automation and Test in Europe (DATE) (2013)Google Scholar
  2. 2.
    Ansótegui, C., Gomes, C.P., Selman, B.: The Achilles’ heel of QBF. In: Veloso, M.M., Kambhampati, S. (eds.) AAAI, pp. 275–281. AAAI Press/The MIT Press (2005)Google Scholar
  3. 3.
    Balabanov, V., Widl, M., Jiang, J.-H.R.: QBF resolution systems and their proof complexities. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 154–169. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_12 Google Scholar
  4. 4.
    Beyersdorf, O., Chew, L., Janota, M.: Extension variables in QBF resolution. In: Workshops at the Thirtieth AAAI Conference on Artificial Intelligence (2016)Google Scholar
  5. 5.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Proceedings of Symposium on Theoretical Aspects of Computer Science (STACS), pp. 76–89. LIPIcs Series (2015)Google Scholar
  6. 6.
    Brummayer, R., Lonsing, F., Biere, A.: Automated testing and debugging of SAT and QBF solvers. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 44–57. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14186-7_6 CrossRefGoogle Scholar
  7. 7.
    Cook, S.A.: A short proof of the pigeon hole principle using extended resolution. SIGACT News 8(4), 28–32 (1976)CrossRefGoogle Scholar
  8. 8.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 100–113. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31612-8_9 CrossRefGoogle Scholar
  10. 10.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Clause/term resolution and learning in the evaluation of quantified Boolean formulas. J. Artif. Intell. Res. 26(1), 371–416 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goultiaeva, A., Bacchus, F.: Exploiting QBF duality on a circuit representation. In: AAAI (2010)Google Scholar
  12. 12.
    Heule, M.J.H., Seidl, M., Biere, A.: A unified proof system for QBF preprocessing. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 91–106. Springer, Cham (2014). doi: 10.1007/978-3-319-08587-6_7 Google Scholar
  13. 13.
    Kleine Büning, H., Bubeck, U.: Theory of quantified Boolean formulas. In: Handbook of Satisfiability. IOS Press (2009)Google Scholar
  14. 14.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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). doi: 10.1007/978-3-642-14186-7_12 CrossRefGoogle Scholar
  16. 16.
    Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Math. Logic Q. 36(1), 29–46 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lonsing, F.: Dependency Schemes and Search-Based QBF Solving: Theory and Practice. Ph.D. thesis, Johannes Kepler Universität (2012).
  18. 18.
    McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45069-6_1 CrossRefGoogle Scholar
  19. 19.
    Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Silva, J.P.M., Lynce, I., Malik, S.: Conflict-driven clause learning sat solvers. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 131–153. IOS Press (2009)Google Scholar
  21. 21.
    Silva, J.P.M., Sakallah, K.A.: Conflict analysis in search algorithms for satisfiability. In: ICTAI, pp. 467–469 (1996)Google Scholar
  22. 22.
    Tseitin, G.S.: On the complexity of derivations in the propositional calculus. In: Studies in Constructive Mathematics and Mathematical Logic (1968)Google Scholar
  23. 23.
    Urquhart, A.: The complexity of propositional proofs. Bull. EATCS 64, 128–138 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Van Gelder, A.: Decision procedures should be able to produce (easily) checkable proofs. In: Workshop on Constraints in Formal Verification (in Conjunction with CP02) (2002)Google Scholar
  25. 25.
    Van Gelder, A.: Contributions to the theory of practical quantified Boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, pp. 647–663. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33558-7_47 CrossRefGoogle Scholar
  26. 26.
    Zhang, L.: Solving QBF by combining conjunctive and disjunctive normal forms. In: AAAI (2006)Google Scholar
  27. 27.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: ICCAD (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IST/INESC-IDLisbonPortugal
  2. 2.LaSIGE, Faculty of ScienceUniversity of LisbonLisbonPortugal

Personalised recommendations