A Resolution-Style Proof System for DQBF

  • Markus N. Rabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)


This paper presents a sound and complete proof system for Dependency Quantified Boolean Formulas (DQBF) using resolution, universal reduction, and a new proof rule that we call fork extension. This opens new avenues for the development of efficient algorithms for DQBF.



The author expresses his gratitude to Armin Biere, Benjamin Caulfield, Daniel J. Fremont, Martina Seidl, Sanjit A. Seshia, Martin Suda, Leander Tentrup, and Ralf Wimmer for supportive comments and detailed discussions on this work.

This work was supported in part by NSF grants CCF-1139138, CNS-1528108, and CNS-1646208, and by TerraSwarm, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.


  1. 1.
    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
  2. 2.
    Beyersdorff, O., Chew, L., Schmidt, R.A., Suda, M.: Lifting QBF resolution calculi to DQBF. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 490–499. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_30 Google Scholar
  3. 3.
    Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005). doi: 10.1007/11527695_5 CrossRefGoogle Scholar
  4. 4.
    Biere, A., Lonsing, F., Seidl, M.: Blocked clause elimination for QBF. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 101–115. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22438-6_10 CrossRefGoogle Scholar
  5. 5.
    Bruttomesso, R., Cimatti, A., Franzén, A., Griggio, A., Santuari, A., Sebastiani, R.: To Ackermann-ize or not to Ackermann-ize? On efficiently handling uninterpreted function symbols in \(\mathit{SMT}(\cal{EUF} \cup \cal{T})\). In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS, vol. 4246, pp. 557–571. Springer, Heidelberg (2006). doi: 10.1007/11916277_38 CrossRefGoogle Scholar
  6. 6.
    Buning, H.K., Karpinski, M., Flogel, A.: Resolution for quantified boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM (JACM) 7(3), 201–215 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Faymonville, P., Finkbeiner, B., Rabe, M.N., Tentrup, L.: 3 encodings of reactive synthesis. In: Proceedings of QUANTIFY, pp. 20–22 (2015)Google Scholar
  9. 9.
    Faymonville, P., Finkbeiner, B., Rabe, M.N., Tentrup, L.: Encodings of bounded synthesis. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 354–370. Springer, Heidelberg (2017). doi: 10.1007/978-3-662-54577-5_20 CrossRefGoogle Scholar
  10. 10.
    Finkbeiner, B., Schewe, S.: Uniform distributed synthesis. In: Proceedings of LICS, Washington, DC, USA, pp. 321–330. IEEE Computer Society (2005)Google Scholar
  11. 11.
    Finkbeiner, B., Tentrup, L.: Fast DQBF refutation. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 243–251. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_19 Google Scholar
  12. 12.
    Fröhlich, A., Kovásznai, G., Biere, A.: A DPLL algorithm for solving DQBF. In: Proceedings of Pragmatics of SAT 2012 (2012)Google Scholar
  13. 13.
    Fröhlich, A., Kovásznai, G., Biere, A., Veith, H.: iDQ: instantiation-based DQBF solving. In: Proceedings of Pragmatics of SAT, pp. 103–116 (2014)Google Scholar
  14. 14.
    Gitina, K., Reimer, S., Sauer, M., Wimmer, R., Scholl, C., Becker, B.: Equivalence checking of partial designs using dependency quantified Boolean formulae. In: Proceedings of ICCD, pp. 396–403, October 2013Google Scholar
  15. 15.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: Proceedings of DATE (2015)Google Scholar
  16. 16.
    Giunchiglia, E., Narizzano, M., Pulina, L., Tacchella, A.: Quantified Boolean formulas satisfiability library (QBFLIB) (2005).
  17. 17.
    Henkin, L.: Some remarks on infinitely long formulas. J. Symb. Logic 30, 167–183 (1961)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: Proceedings of IJCAI, pp. 325–331. AAAI Press (2015)Google Scholar
  19. 19.
    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). doi: 10.1007/978-3-642-21581-0_19 CrossRefGoogle Scholar
  20. 20.
    Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. JSAT 7(2–3), 71–76 (2010)Google Scholar
  21. 21.
    Peterson, G., Reif, J., Azhar, S.: Lower bounds for multiplayer noncooperative games of incomplete information. Comput. Math. Appl. 41(7), 957–992 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Peterson, G.L., Reif, J.H.: Multiple-person alternation. In: Proceedings of FOCS, pp. 348–363. IEEE (1979)Google Scholar
  23. 23.
    Rabe, M.N., Seshia, S.A.: Incremental determinization. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 375–392. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_23 Google Scholar
  24. 24.
    Rabe, M.N., Leander Tentrup, C.: A certifying QBF solver. In: Proceedings of FMCAD, pp. 136–143 (2015)Google Scholar
  25. 25.
    Siekmann, J., Wrightson, G.: Automation of Reasoning: 2: Classical Papers on Computational Logic 1967–1970. Springer, Heidelberg (1983). doi: 10.1007/978-3-642-81955-1 zbMATHGoogle Scholar
  26. 26.
    Silva, J.P.M., Sakallah, K.A.: GRASP - a new search algorithm for satisfiability. In: Proceedings of CAD, pp. 220–227. IEEE (1997)Google Scholar
  27. 27.
    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). doi: 10.1007/978-3-319-40970-2_24 Google Scholar
  28. 28.
    Tentrup, L.: On expansion and resolution in CEGAR based QBF solving. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 475–494. Springer, Cham (2017)Google Scholar
  29. 29.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. Stud. Constr. Math. Math. Logic 2, 115–125 (1968). Reprinted in [2]: 10–13Google Scholar
  30. 30.
    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
  31. 31.
    Wimmer, R., Gitina, K., Nist, J., Scholl, C., Becker, B.: Preprocessing for DQBF. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 173–190. Springer, Cham (2015). doi: 10.1007/978-3-319-24318-4_13 CrossRefGoogle Scholar
  32. 32.
    Wimmer, R., Reimer, S., Marin, P., Becker, B.: HQSpre-an effective preprocessor for QBF and DQBF. In: Proceedings of TACAS (2017)Google Scholar
  33. 33.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: Proceedings of ICCAD, pp. 442–449, November 2002Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA

Personalised recommendations