Skolem Functions for DQBF

  • Karina WimmerEmail author
  • Ralf Wimmer
  • Christoph Scholl
  • Bernd Becker
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9938)


We consider the problem of computing Skolem functions for satisfied dependency quantified Boolean formulas (DQBFs). We show how Skolem functions can be obtained from an elimination-based DQBF solver and how to take preprocessing steps into account. The size of the Skolem functions is optimized by don’t-care minimization using Craig interpolants and rewriting techniques. Experiments with our DQBF solver HQS show that we are able to effectively compute Skolem functions with very little overhead compared to the mere solution of the formula.


Boolean Function Conjunctive Normal Form Winning Strategy Boolean Formula Satisfying Assignment 
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  1. 1.
    Ashar, P., Ganai, M.K., Gupta, A., Ivancic, F., Yang, Z.: Efficient SAT-based bounded model checking for software verification. In: Proceedings of ISoLA. Technical report, vol. TR-2004-6, pp. 157–164. University of Cyprus (2004)Google Scholar
  2. 2.
    Balabanov, V., Chiang, H.K., Jiang, J.R.: Henkin quantifiers and Boolean formulae: a certification perspective of DQBF. Theor. Comput. Sci. 523, 86–100 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balabanov, V., Jiang, J.R.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Balabanov, V., Jiang, J.H.R.: Reducing satisfiability and reachability to DQBF (2015). Talk at the International Workshop on Quantified Boolean Formulas (QBF)Google Scholar
  5. 5.
    Benedetti, M.: Evaluating QBFs via symbolic Skolemization. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 285–300. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. Adv. Comput. 58, 117–148 (2003)CrossRefGoogle Scholar
  7. 7.
    Bloem, R., Könighofer, R., Seidl, M.: SAT-based synthesis methods for safety specs. In: McMillan, K.L., Rival, X. (eds.) VMCAI 2014. LNCS, vol. 8318, pp. 1–20. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  8. 8.
    Brayton, R., Mishchenko, A.: ABC: an academic industrial-strength verification tool. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 24–40. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Bubeck, U., Kleine Büning, H.: Dependency quantified horn formulas: models and complexity. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 198–211. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of STOC, pp. 151–158. ACM Press (1971)Google Scholar
  11. 11.
    Craig, W.: Linear reasoning. a new form of the Herbrand-Gentzen theorem. J. Symbolic Logic 22(3), 250–268 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Czutro, A., Polian, I., Lewis, M.D.T., Engelke, P., Reddy, S.M., Becker, B.: Thread-parallel integrated test pattern generator utilizing satisfiability analysis. Int. J. Parallel Programm. 38(3–4), 185–202 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Eggersglüß, S., Drechsler, R.: A highly fault-efficient SAT-based ATPG flow. IEEE Des. Test Comput. 29(4), 63–70 (2012)CrossRefGoogle Scholar
  15. 15.
    Finkbeiner, B., Tentrup, L.: Fast DQBF refutation. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 243–251. Springer, Heidelberg (2014)Google Scholar
  16. 16.
    Fröhlich, A., Kovásznai, G., Biere, A.: A DPLL algorithm for solving DQBF. In: International Workshop on Pragmatics of SAT (POS) (2012)Google Scholar
  17. 17.
    Fröhlich, A., Kovásznai, G., Biere, A., Veith, H.: iDQ: instantiation-based DQBF solving. In: International Workshop on Pragmatics of SAT (POS). EPiC Series, vol. 27, pp. 103–116. EasyChair (2014)Google Scholar
  18. 18.
    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. IEEE CS (2013)Google Scholar
  19. 19.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: Proceedings of DATE. IEEE (2015)Google Scholar
  20. 20.
    Goultiaeva, A., Van Gelder, A., Bacchus, F.: A uniform approach for generating proofs and strategies for both true and false QBF formulas. In: Proceedings of IJCAI, pp. 546–553. IJCAI/AAAI (2011)Google Scholar
  21. 21.
    Henkin, L.: Some remarks on infinitely long formulas. In: Infinitistic Methods: Proceedings of the 1959 Symposium on Foundations of Mathematics, pp. 167–183. Pergamon Press, Warsaw (1961)Google Scholar
  22. 22.
    Heule, M., Järvisalo, M., Lonsing, F., Seidl, M., Biere, A.: Clause elimination for SAT and QSAT. J. Artif. Intell. Res. 53, 127–168 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Heule, M., Seidl, M., Biere, A.: Efficient extraction of Skolem functions from QRAT proofs. In: Proceedings of FMCAD, pp. 107–114. IEEE (2014)Google Scholar
  24. 24.
    Jiang, J.-H.R.: Quantifier elimination via functional composition. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 383–397. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  25. 25.
    Jussila, T., Biere, A., Sinz, C., Kroning, D., Wintersteiger, C.M.: A first step towards a unified proof checker for QBF. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 201–214. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  26. 26.
    Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Proceedings of NAI/IAAI, pp. 1368–1373. AAAI Press/The MIT Press (2005)Google Scholar
  27. 27.
    Kuehlmann, A., Paruthi, V., Krohm, F., Ganai, M.K.: Robust Boolean reasoning for equivalence checking and functional property verification. IEEE Trans. CAD Integr. Circ. Syst. 21(12), 1377–1394 (2002)CrossRefGoogle Scholar
  28. 28.
    Lonsing, F., Bacchus, F., Biere, A., Egly, U., Seidl, M.: Enhancing search-basedQBF solving by dynamic blocked clause elimination. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR-20 2015. LNCS, vol. 9450, pp. 418–433. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  29. 29.
    Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. J. Satisfiability Boolean Modell. Comput. 7(2–3), 71–76 (2010)Google Scholar
  30. 30.
    McMillan, K.L.: Applications of Craig interpolants in model checking. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 1–12. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  31. 31.
    Niemetz, A., Preiner, M., Lonsing, F., Seidl, M., Biere, A.: Resolution-based certificate extraction for QBF. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 430–435. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  32. 32.
    Peterson, G., Reif, J., Azhar, S.: Lower bounds for multiplayer non-cooperative games of incomplete information. Comput. Math. Appl. 41(7–8), 957–992 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Pigorsch, F., Scholl, C.: Exploiting structure in an AIG based QBF solver. In: Proceedings of DATE, pp. 1596–1601. IEEE (2009)Google Scholar
  34. 34.
    Pigorsch, F., Scholl, C.: An AIG-based QBF-solver using SAT for preprocessing. In: Proceedings of DAC, pp. 170–175. ACM Press (2010)Google Scholar
  35. 35.
    Pudlák, P.: Lower bounds for resolution and cutting planes proofs and monotone computations. J. Symbolic Logic 62(3), 981–998 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rintanen, J., Heljanko, K., Niemelä, I.: Planning as satisfiability: parallel plans and algorithms for plan search. Artif. Intell. 170(12–13), 1031–1080 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Scholl, C., Becker, B.: Checking equivalence for partial implementations. In: Proceedings of DAC, pp. 238–243. ACM Press (2001)Google Scholar
  38. 38.
    Tentrup, L., Rabe, M.N.: CAQE: a certifying QBF solver. In: Proceedings of FMCAD, pp. 136–143. IEEE (2015)Google Scholar
  39. 39.
    Wegener, I.: Branching Programs and Binary Decision Diagrams. Discrete Mathematics and Applications. SIAM, SIAM Monographs on Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  40. 40.
    Wimmer, K., Wimmer, R., Scholl, C., Becker, B.: Skolem functions for DQBF (extended version). Technical report, FreiDok plus, Universitätsbibliothek Freiburg, Freiburg im Breisgau, Germany, June 2016.
  41. 41.
    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

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Karina Wimmer
    • 1
    Email author
  • Ralf Wimmer
    • 1
    • 2
  • Christoph Scholl
    • 1
  • Bernd Becker
    • 1
  1. 1.Albert-Ludwigs-Universität FreiburgFreiburg im BreisgauGermany
  2. 2.Dependable Systems and SoftwareSaarland UniversitySaarbrückenGermany

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