Lifting QBF Resolution Calculi to DQBF

  • Olaf Beyersdorff
  • Leroy Chew
  • Renate A. Schmidt
  • Martin Suda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9710)


We examine existing resolution systems for quantified Boolean formulas (QBF) and answer the question which of these calculi can be lifted to the more powerful Dependency QBFs (DQBF). An interesting picture emerges: While for QBF we have the strict chain of proof systems \(\textsf {Q-Res}< \textsf {IR-calc} < \textsf {IRM-calc} \), the situation is quite different in DQBF. The obvious adaptations of Q-Res and likewise universal resolution are too weak: they are not complete. The obvious adaptation of IR-calc has the right strength: it is sound and complete. IRM-calc is too strong: it is not sound any more, and the same applies to long-distance resolution. Conceptually, we use the relation of DQBF to effectively propositional logic (EPR) and explain our new DQBF calculus based on IR-calc as a subsystem of first-order resolution.


Proof System Conjunctive Normal Form Predicate Symbol Partial Assignment Ground Instance 
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 research was supported by grant no. 48138 from the John Templeton Foundation, EPSRC grant EP/L024233/1, and a Doctoral Training Grant from the EPSRC (2nd author).

Martin Suda was supported by the EPSRC grant EP/K032674/1 and the ERC Starting Grant 2014 SYMCAR 639270.


  1. 1.
    Azhar, S., Peterson, G., Reif, J.: Lower bounds for multiplayer non-cooperative games of incomplete information. J. Comput. Math. Appl. 41, 957–992 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 19–99. Elsevier and MIT Press (2001)Google Scholar
  3. 3.
    Balabanov, V., Chiang, H.J.K., Jiang, J.H.R.: Henkin quantifiers and boolean formulae: a certification perspective of DQBF. Theor. Comput. Sci. 523, 86–100 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balabanov, V., Jiang, J.H.R.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    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, Heidelberg (2014)Google Scholar
  6. 6.
    Beyersdorff, O., Bonacina, I., Chew, L.: Lower bounds: from circuits to QBF proof systems. In: Proceedings of ACM Conference on Innovations in Theoretical Computer Science (ITCS 2016), pp. 249–260. ACM (2016)Google Scholar
  7. 7.
    Beyersdorff, O., Chew, L., Janota, M.: On unification of QBF resolution-based calculi. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 81–93. Springer, Heidelberg (2014)Google Scholar
  8. 8.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Proceedings of Symposium on Theoretical Aspects of Computer Science, pp. 76–89. LIPIcs series (2015)Google Scholar
  9. 9.
    Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Feasible interpolation for QBF resolution calculi. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 180–192. Springer, Heidelberg (2015)Google Scholar
  10. 10.
    Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Are short proofs narrow? QBF resolution is not simple. In: Proceedings of Symposium on Theoretical Aspects of Computer Science (STACS 2016) (2016)Google Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Finkbeiner, B., Tentrup, L.: Fast DQBF refutation. In: Sinz, C., Egly, U. (eds.) [26], pp. 243–251Google Scholar
  13. 13.
    Fröhlich, A., Kovásznai, G., Biere, A., Veith, H.: iDQ: Instantiation-based DQBF solving. In: Sinz, C., Egly, U. (eds.) [26], pp. 103–116Google Scholar
  14. 14.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: Nebel, W., Atienza, D. (eds.) Proceedings of the 2015 Design, Automation & Test in Europe Conference & Exhibition, DATE 2015, Grenoble, France, March 9–13, 2015. pp. 1617–1622. ACM (2015)Google Scholar
  15. 15.
    Giunchiglia, E., Marin, P., Narizzano, M.: Reasoning with 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. 761–780. IOS Press (2009)Google Scholar
  16. 16.
    Henkin, L.: Some remarks on infinitely long formulas. J. Symbolic Logic, pp. 167–183 (1961). Pergamon PressGoogle Scholar
  17. 17.
    Heule, M.J., Seidl, M., Biere, A.: Efficient extraction of skolem functions from QRAT proofs. In: Formal Methods in Computer-Aided Design (FMCAD), 2014, pp. 107–114. IEEE (2014)Google Scholar
  18. 18.
    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, Heidelberg (2014)Google Scholar
  19. 19.
    Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Janota, M., Marques-Silva, J.: On propositional QBF expansions and Q-resolution. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 67–82. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Büning, K.H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lewis, H.R.: Complexity results for classes of quantificational formulas. J. Comput. Syst. Sci. 21(3), 317–353 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Peterson, G.L., Reif, J.: Multiple-person alternation. In: 20th Annual Symposium on Foundations of Computer Science, 1979, pp. 348–363, October 1979Google Scholar
  24. 24.
    Segerlind, N.: The complexity of propositional proofs. Bull. Symbolic Logic 13(4), 417–481 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Seidl, M., Lonsing, F., Biere, A.: qbf2epr: a tool for generating EPR formulas from QBF. In: Fontaine, P., Schmidt, R.A., Schulz, S. (eds.) PAAR-2012. Third Workshop on Practical Aspects of Automated Reasoning. EPiC Series in Computing, vol. 21, pp. 139–148. EasyChair (2013)Google Scholar
  26. 26.
    Sinz, C., Egly, U. (eds.): SAT 2014. LNCS, vol. 8561. Springer, Heidelberg (2014)Google Scholar
  27. 27.
    Slivovsky, F., Szeider, S.: Variable dependencies and Q-resolution. In: Sinz, C., Egly, U. (eds.) [26], pp. 269–284Google Scholar
  28. 28.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time: preliminary report. In: Aho, A.V., Borodin, A., Constable, R.L., Floyd, R.W., Harrison, M.A., Karp, R.M., Strong, H.R. (eds.) Proceedings of the 5th Annual ACM Symposium on Theory of Computing, April 30 – May 2, 1973, Austin, Texas, USA, pp. 1–9. ACM (1973)Google Scholar
  29. 29.
    Urquhart, A.: The complexity of propositional proofs. Bull. Symbolic Logic 1, 425–467 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    Wimmer, R., Gitina, K., Nist, J., Scholl, C., Becker, B.: Preprocessing for DQBF. In: Heule, M., et al. (eds.) SAT 2015. LNCS, vol. 9340, pp. 173–190. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-24318-4_13 CrossRefGoogle Scholar
  32. 32.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: ICCAD, pp. 442–449 (2002)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Olaf Beyersdorff
    • 1
  • Leroy Chew
    • 1
  • Renate A. Schmidt
    • 2
  • Martin Suda
    • 2
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.School of Computer ScienceUniversity of ManchesterManchesterUK

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