Preprocessing for DQBF

  • Ralf WimmerEmail author
  • Karina Gitina
  • Jennifer Nist
  • Christoph Scholl
  • Bernd Becker
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9340)


For SAT and QBF formulas many techniques are applied in order to reduce/modify the number of variables and clauses of the formula, before the formula is passed to the actual solving algorithm. It is well known that these preprocessing techniques often reduce the computation time of the solver by orders of magnitude. In this paper we generalize different preprocessing techniques for SAT and QBF problems to dependency quantified Boolean formulas (DQBF) and describe how they need to be adapted to work with a DQBF solver core. We demonstrate their effectiveness both for CNF- and non-CNF-based DQBF algorithms.


Conjunctive Normal Form Boolean Formula Universal Expansion Universal Variable Conjunctive Normal Form Formula 
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.


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  1. 1.
    Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. Advances in Computers 58, 117–148 (2003)CrossRefGoogle Scholar
  2. 2.
    Czutro, A., Polian, I., Lewis, M.D.T., Engelke, P., Reddy, S.M., Becker, B.: TIGUAN: thread-parallel integrated test pattern generator utilizing satisfiability analysis. In: International Conference on VLSI Design, pp. 227–232. IEEE Computer Society, New Delhi, India (2009)Google Scholar
  3. 3.
    Rintanen, J.: Constructing conditional plans by a theorem-prover. Journal of Artificial Intelligence Research 10, 323–352 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Sinz, C., Kaiser, A., Küchlin, W.: Formal methods for the validation of automotive product configuration data. AI EDAM 17(1), 75–97 (2003)Google Scholar
  5. 5.
    Mironov, I., Zhang, L.: Applications of SAT solvers to cryptanalysis of hash functions. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 102–115. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  6. 6.
    Cook, S.A.: The complexity of theorem-proving procedures. In: Annual ACM Symposium on Theory of Computing (STOC), ACM Press, pp. 151–158 (1971)Google Scholar
  7. 7.
    Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time: Preliminary report. In: Annual ACM Symposium on Theory of Computing (STOC), pp. 1–9. ACM Press (1973)Google Scholar
  8. 8.
    Peterson, G., Reif, J., Azhar, S.: Lower bounds for multiplayer non-cooperative games of incomplete information. Computers and Mathematics with Applications 41(7–8), 957–992 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gitina, K., Reimer, S., Sauer, M., Wimmer, R., Scholl, C., Becker, B.: Equivalence checking of partial designs using dependency quantified Boolean formulae. In: IEEE Int’l Conf. on Computer Design (ICCD), Asheville, NC, USA, IEEE Computer Society, pp. 396–403 (2013)Google Scholar
  10. 10.
    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
  11. 11.
    Fröhlich, A., Kovásznai, G., Biere, A., Veith, H.: iDQ: Instantiation-based DQBF solving. In: Berre, D.L. (ed.) Int’l Workshop on Pragmatics of SAT (POS). EPiC Series, vol. 27, pp. 103–116. Vienna, Austria, EasyChair (2014)Google Scholar
  12. 12.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: Int’l Conf. on Design, Automation and Test in Europe (DATE), Grenoble, France, IEEE (2015)Google Scholar
  13. 13.
    Jr., R.J.B., Schrag, R.: Using CSP look-back techniques to solve real-world SAT instances. In: Kuipers, B., Webber, B.L. (eds.): National Conference on Artificial Intelligence / Innovative Applications of Artificial Intelligence Conference (AAAI/IAAI), Providence, Rhode Island, USA, AAAI Press / The MIT Press, pp. 203–208 (1997)Google Scholar
  14. 14.
    Silva, J.P.M., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Transactions on Computers 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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) CrossRefGoogle Scholar
  16. 16.
    Manthey, N.: Coprocessor 2.0 – a flexible CNF simplifier. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 436–441. Springer, Heidelberg (2012) CrossRefGoogle 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) CrossRefGoogle Scholar
  18. 18.
    Biere, A., Lonsing, F., Seidl, M.: Blocked clause elimination for QBF. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 101–115. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  19. 19.
    Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Veloso, M.M., Kambhampati, S. (eds.): National Conference on Artificial Intelligence / Int’l Conf. on Innovative Applications of Artificial Intelligence (IAAI), Pittsburgh, Pennsylvania, USA, AAAI Press / The MIT Press, pp. 1368–1373 (2005)Google Scholar
  20. 20.
    Janota, M., Lynce, I., Marques-Silva, J.: Algorithms for computing backbones of propositional formulae. AI Communications 28(2), 161–177 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    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
  22. 22.
    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) CrossRefGoogle Scholar
  23. 23.
    Pigorsch, F., Scholl, C.: Exploiting structure in an AIG based QBF solver. In: Conf, I. (ed.) on Design, Automation and Test in Europe (DATE), pp. 1596–1601. IEEE, Nice, France (2009)Google Scholar
  24. 24.
    Biere, A.: Resolve and expand. In: H. Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  25. 25.
    Wimmer, R., Gitina, K., Nist, J., Scholl, C., Becker, B.: Preprocessing for DQBF (extended version). Reports of SFB/TR 14 AVACS number 110 (2015).
  26. 26.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. Studies in Constructive Mathematics and Mathematical Logic Part 2, 115–125 (1970)CrossRefGoogle Scholar
  27. 27.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. vol. 185 of Frontiers in Artificial Intelligence and Applications. IOS Press (2008)Google Scholar
  28. 28.
    Fröhlich, A., Kovásznai, G., Biere, A.: A DPLL algorithm for solving DQBF. In: Int’l Workshop on Pragmatics of SAT (POS), Trento, Italy (2012)Google Scholar
  29. 29.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. Reports of SFB/TR 14 AVACS 107 (2015).
  30. 30.
    Pigorsch, F., Scholl, C.: An AIG-based QBF-solver using SAT for preprocessing. In: Sapatnekar, S.S. (ed.) ACM/IEEE Design Automation Conference (DAC), pp. 170–175. ACM Press, Anaheim, CA, USA (2010)Google Scholar
  31. 31.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Brafman, R.I.: A simplifier for propositional formulas with many binary clauses. IEEE Transactions on Systems, Man, and Cybernetics, Part B 34(1), 52–59 (2004)CrossRefGoogle Scholar
  33. 33.
    Gelder, A.V.: Toward leaner binary-clause reasoning in a satisfiability solver. Ann. Math. Artif. Intell. 43(1), 239–253 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gershman, R., Strichman, O.: Cost-effective hyper-resolution for preprocessing CNF formulas. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 423–429. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  35. 35.
    Heule, M.J.H., Järvisalo, M., Biere, A.: Efficient CNF simplification based on binary implication graphs. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 201–215. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  36. 36.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. Journal of Automated Reasoning 42(1), 77–97 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Samer, M.: Variable dependencies of quantified CSPs. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 512–527. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  38. 38.
    Lonsing, F., Biere, A.: Efficiently representing existential dependency sets for expansion-based QBF solvers. Electronic Notes in Theoretical Computer Science 251, 83–95 (2009)CrossRefzbMATHGoogle Scholar
  39. 39.
    Van Gelder, A.: Variable independence and resolution paths for quantified boolean formulas. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 789–803. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  40. 40.
    Slivovsky, F., Szeider, S.: Computing resolution-path dependencies in linear time. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 58–71. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  41. 41.
    Balabanov, V., Chiang, H.K., Jiang, J.R.: Henkin quantifiers and Boolean formulae: A certification perspective of DQBF. Theoretical Computer Science 523, 86–100 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    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
  43. 43.
    Bubeck, U.: Model-based transformations for quantified Boolean formulas. Ph.D. thesis, University of Paderborn (2010)Google Scholar
  44. 44.
    Heule, M., Järvisalo, M., Biere, A.: Clause elimination procedures for CNF formulas. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 357–371. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  45. 45.
    Heule, M., Järvisalo, M., Biere, A.: Covered clause elimination. In: Voronkov, A., Sutcliffe, G., Baaz, M., Fermüller, C.G. (eds.): Int’l Conf. on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR) (Short papers). vol. 13 of EPiC Series, Yogyakarta, Indonesia, EasyChair, pp. 41–46 (2010)Google Scholar
  46. 46.
    Heule, M.J.H., Seidl, M., Biere, A.: Blocked literals are universal. In: Havelund, K., Holzmann, G., Joshi, R. (eds.) NFM 2015. LNCS, vol. 9058, pp. 436–442. Springer, Heidelberg (2015) Google Scholar
  47. 47.
    Lonsing, F., Egly, U.: Incremental QBF solving by DepQBF. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 307–314. Springer, Heidelberg (2014) Google Scholar
  48. 48.
    Lonsing, F., Biere, A.: DepQBF: A dependency-aware QBF solver. Journal on Satisfiability, Boolean Modelling and Computation 7(2–3), 71–76 (2010)Google Scholar
  49. 49.
    Kuehlmann, A., Paruthi, V., Krohm, F., Ganai, M.K.: Robust Boolean reasoning for equivalence checking and functional property verification. IEEE Transactions on CAD of Integrated Circuits and Systems 21(12), 1377–1394 (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ralf Wimmer
    • 1
    Email author
  • Karina Gitina
    • 1
  • Jennifer Nist
    • 1
  • Christoph Scholl
    • 1
  • Bernd Becker
    • 1
  1. 1.Albert-Ludwigs-Universität FreiburgFreiburg im BreisgauGermany

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