HQSpre – An Effective Preprocessor for QBF and DQBF

  • Ralf WimmerEmail author
  • Sven Reimer
  • Paolo Marin
  • Bernd Becker
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10205)


We present our new preprocessor HQSpre, a state-of-the-art tool for simplifying quantified Boolean formulas (QBFs) and the first available preprocessor for dependency quantified Boolean formulas (DQBFs). The latter are a generalization of QBFs, resulting from adding so-called Henkin-quantifiers to QBFs. HQSpre applies most of the preprocessing techniques that have been proposed in the literature. It can be used both as a standalone tool and as a library. It is possible to tailor it towards different solver back-ends, e. g., to preserve the circuit structure of the formula when a non-CNF solver back-end is used. Extensive experiments show that HQSpre allows QBF solvers to solve more benchmark instances and is able to decide more instances on its own than state-of-the-art tools. The same impact can be observed in the DQBF domain as well.


Boolean Formula Universal Expansion Universal Variable Variable Elimination Existential Variable 
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.


  1. 1.
    Balabanov, V., Chiang, H.K., Jiang, J.R.: Henkin quantifiers and Boolean formulae: a certification perspective of DQBF. Theoret. Comput. Sci. 523, 86–100 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Biere, A.: Aiger format (2007).
  4. 4.
    Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. Adv. Comput. 58, 117–148 (2003)CrossRefGoogle Scholar
  5. 5.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2008)Google Scholar
  6. 6.
    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
  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). doi: 10.1007/978-3-642-54013-4_1 CrossRefGoogle Scholar
  8. 8.
    Bubeck, U., Kleine Büning, H.: Bounded universal expansion for preprocessing QBF. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 244–257. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72788-0_24 CrossRefGoogle Scholar
  9. 9.
    del Val, A.: Simplifying binary propositional theories into connected components twice as fast. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 392–406. Springer, Heidelberg (2001). doi: 10.1007/3-540-45653-8_27 CrossRefGoogle Scholar
  10. 10.
    Eén, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005). doi: 10.1007/11499107_5 CrossRefGoogle Scholar
  11. 11.
    Eggersglüß, S., Drechsler, R.: A highly fault-efficient SAT-based ATPG flow. IEEE Des. Test Comput. 29(4), 63–70 (2012)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Fröhlich, A., Kovásznai, G., Biere, A., Veith, H.: iDQ: instantiation-based DQBF solving. In: Le Berre, D. (ed.) International Workshop on Pragmatics of SAT (POS). EPiC Series, vol. 27, pp. 103–116. EasyChair (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. IEEE CS (2013)Google Scholar
  15. 15.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: Proceedings of DATE, pp. 1617–1622. IEEE (2015)Google Scholar
  16. 16.
    Giunchiglia, E., Marin, P., Narizzano, M.: QuBE7.0. J. Satisf. Boolean Model. Comput. 7(2–3), 83–88 (2010)Google Scholar
  17. 17.
    Giunchiglia, E., Marin, P., Narizzano, M.: sQueezeBF: an effective preprocessor for QBFs based on equivalence reasoning. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 85–98. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14186-7_9 CrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    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). doi: 10.1007/978-3-642-31612-8_10 CrossRefGoogle Scholar
  20. 20.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: Proceedings of IJCAI, pp. 325–331. AAAI Press (2015)Google Scholar
  21. 21.
    Järvisalo, M., Biere, A., Heule, M.: Blocked clause elimination. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 129–144. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-12002-2_10 CrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Kupferschmid, S., Lewis, M., Schubert, T., Becker, B.: Incremental preprocessing methods for use in BMC. Form. Methods Syst. Des. 39(2), 185–204 (2011)CrossRefzbMATHGoogle Scholar
  24. 24.
    Li, C.M., Anbulagan, A.: Heuristics based on unit propagation for satisfiability problems. In: Proceedings of IJCAI, vol. 1, pp. 366–371. Morgan Kaufmann Publishers Inc. (1997)Google Scholar
  25. 25.
    Lonsing, F., Bacchus, F., Biere, A., Egly, U., Seidl, M.: Enhancing search-based QBF solving by dynamic blocked clause elimination. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 418–433. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48899-7_29 CrossRefGoogle Scholar
  26. 26.
    Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time: preliminary report. In: Proceedings of STOC, pp. 1–9. ACM Press (1973)Google Scholar
  27. 27.
    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
  28. 28.
    Piette, C., Hamadi, Y., Sais, L.: Vivifying propositional clausal formulae. In: Ghallab, M., Spyropoulos, C.D., Fakotakis, N., Avouris, N.M. (eds.) Proceedings of ECAI. Frontiers in Artificial Intelligence and Applications, vol. 178, pp. 525–529. IOS Press (2008)Google Scholar
  29. 29.
    Pigorsch, F., Scholl, C.: Exploiting structure in an AIG based QBF solver. In: Proceedings of DATE, pp. 1596–1601. IEEE (2009)Google Scholar
  30. 30.
    Pigorsch, F., Scholl, C.: An AIG-based QBF-solver using SAT for preprocessing. In: Sapatnekar, S.S. (ed.) Proceedings of DAC, pp. 170–175. ACM Press (2010)Google Scholar
  31. 31.
    Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
  33. 33.
    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
  34. 34.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. J. Autom. Reason. 42(1), 77–97 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Scholl, C., Becker, B.: Checking equivalence for partial implementations. In: Proceedings of DAC, pp. 238–243. ACM Press (2001)Google Scholar
  36. 36.
    Schubert, T., Reimer, S.: antom (2016). In:
  37. 37.
    Slivovsky, F., Szeider, S.: Soundness of Q-resolution with dependency schemes. Theoret. Comput. Sci. 612, 83–101 (2016)MathSciNetCrossRefzbMATHGoogle 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.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. Stud. Constr. Math. Math. Log. Part 2, 115–125 (1970)CrossRefGoogle Scholar
  40. 40.
    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
  41. 41.
    Wimmer, R., Scholl, C., Wimmer, K., Becker, B.: Dependency schemes for DQBF. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 473–489. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_29 Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Ralf Wimmer
    • 1
    Email author
  • Sven Reimer
    • 1
  • Paolo Marin
    • 1
  • Bernd Becker
    • 1
  1. 1.Albert-Ludwigs-Universität FreiburgFreiburg im BreisgauGermany

Personalised recommendations