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From DQBF to QBF by Dependency Elimination

  • Ralf WimmerEmail author
  • Andreas Karrenbauer
  • Ruben Becker
  • Christoph Scholl
  • Bernd Becker
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)

Abstract

In this paper, we propose the elimination of dependencies to convert a given dependency quantified Boolean formula (DQBF) to an equisatisfiable QBF. We show how to select a set of dependencies to eliminate such that we arrive at a smallest equisatisfiable QBF in terms of existential variables that is achievable using dependency elimination. This approach is improved by taking so-called don’t-care dependencies into account, which result from the application of dependency schemes to the formula and can be added to or removed from the formula at no cost. We have implemented this new method in the state-of-the-art DQBF solver HQS. Experiments show that dependency elimination is clearly superior to the previous method using variable elimination.

References

  1. 1.
    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
  2. 2.
    Beineke, L.W., Little, C.H.C.: Cycles in bipartite tournaments. J. Comb. Theor. Ser. B 32(2), 140–145 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    Bubeck, U.: Model-based transformations for quantified Boolean formulas. Ph.D. thesis, University of Paderborn (2010)Google Scholar
  6. 6.
    Bubeck, U., Büning, H.K.: 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). doi: 10.1007/11814948_21 CrossRefGoogle Scholar
  7. 7.
    Cai, M., Deng, X., Zang, W.: A min-max theorem on feedback vertex sets. Math. Oper. Res. 27(2), 361–371 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chatterjee, K., Henzinger, T.A., Otop, J., Pavlogiannis, A.: Distributed synthesis for LTL fragments. In: FMCAD 2013, pp. 18–25. IEEE, October 2013Google Scholar
  9. 9.
    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
  10. 10.
    Fröhlich, A., Kovásznai, G., Biere, A.: A DPLL algorithm for solving DQBF. In: International Workshop on Pragmatics of SAT (POS), Trento, Italy (2012)Google Scholar
  11. 11.
    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 2014), Vienna, Austria. EPiC Series, vol. 27, pp. 103–116. EasyChair, July 2014Google Scholar
  12. 12.
    Gitina, K., Reimer, S., Sauer, M., Wimmer, R., Scholl, C., Becker, B.: Equivalence checking of partial designs using dependency quantified Boolean formulae. In: ICCD 2013, Asheville, NC, USA, pp. 396–403. IEEE CS, October 2013Google Scholar
  13. 13.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: DATE 2015, Grenoble, France. IEEE, March 2015Google Scholar
  14. 14.
    Guo, J., Hüffner, F., Moser, H.: Feedback arc set in bipartite tournaments is NP-complete. Inf. Process. Lett. 102(2–3), 62–65 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Henkin, L.: Some remarks on infinitely long formulas. In: Infinitistic Methods: Proceedings of the 1959 Symposium on Foundations of Mathematics, Warsaw, pp. 167–183. Panstwowe Wydawnictwo Naukowe, Panstwowe, September 1961Google Scholar
  16. 16.
    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
  17. 17.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: Yang, Q., Wooldridge, M. (eds.) IJCAI 2015, Buenos Aires, Argentina, pp. 325–331. AAAI Press (2015). http://ijcai.org/Abstract/15/052
  18. 18.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Proceedings of the Symposium on the Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Plenum Press, New York, IBM Thomas J. Watson Research Center, Yorktown Heights (1972)Google Scholar
  19. 19.
    Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. J. Satisf. Boolean Model. Comput. 7(2–3), 71–76 (2010)Google Scholar
  20. 20.
    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). doi: 10.1007/978-3-662-44199-2_48 Google Scholar
  21. 21.
    Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time: preliminary report. In: STOC, pp. 1–9. ACM Press (1973)Google Scholar
  22. 22.
    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
  23. 23.
    Pigorsch, F., Scholl, C.: Exploiting structure in an AIG based QBF solver. In: DATE 2009, Nice, France, pp. 1596–1601. IEEE, April 2009Google Scholar
  24. 24.
    Pigorsch, F., Scholl, C.: An AIG-based QBF-solver using SAT for preprocessing. In: Sapatnekar, S.S. (ed.) DAC 2010, Anaheim, CA, USA, pp. 170–175. ACM Press, July 2010Google Scholar
  25. 25.
    Samer, M.: Variable dependencies of quantified CSPs. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS, vol. 5330, pp. 512–527. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-89439-1_49 CrossRefGoogle Scholar
  26. 26.
    Samer, M., Szeider, S.: Backdoor sets of quantified Boolean formulas. J. Autom. Reason. 42(1), 77–97 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Scholl, C., Becker, B.: Checking equivalence for partial implementations. In: DAC 2001, Las Vegas, NV, USA, pp. 238–243. ACM Press, June 2001Google Scholar
  28. 28.
    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). doi: 10.1007/978-3-642-31612-8_6 CrossRefGoogle Scholar
  29. 29.
    Slivovsky, F., Szeider, S.: Quantifier reordering for QBF. J. Autom. Reason. 56, 459–477 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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). doi: 10.1007/978-3-642-23786-7_59 CrossRefGoogle Scholar
  31. 31.
    Wimmer, K., Wimmer, R., Scholl, C., Becker, B.: Skolem functions for DQBF. In: Artho, C., Legay, A., Peled, D. (eds.) ATVA 2016. LNCS, vol. 9938, pp. 395–411. Springer, Cham (2016). doi: 10.1007/978-3-319-46520-3_25 CrossRefGoogle Scholar
  32. 32.
    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
  33. 33.
    Wimmer, R., Reimer, S., Marin, P., Becker, B.: HQSpre – an effective preprocessor for QBF and DQBF. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10205, pp. 373–390. Springer, Heidelberg (2017). doi: 10.1007/978-3-662-54577-5_21 CrossRefGoogle Scholar
  34. 34.
    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
  35. 35.
    Wimmer, R., Wimmer, K., Scholl, C., Becker, B.: Analysis of incomplete circuits using dependency quantified Boolean formulas. In: International Workshop on Logic and Synthesis (IWLS) (2016)Google Scholar
  36. 36.
    Wolsey, L.A.: Integer Programming. Wiley-Interscience, New York (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ralf Wimmer
    • 1
    Email author
  • Andreas Karrenbauer
    • 2
  • Ruben Becker
    • 2
  • Christoph Scholl
    • 1
  • Bernd Becker
    • 1
  1. 1.Albert-Ludwigs-Universität FreiburgFreiburg im BreisgauGermany
  2. 2.MPI for InformaticsSaarbrückenGermany

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