Dependency Quantified Boolean Formulas: An Overview of Solution Methods and Applications

Extended Abstract
  • Christoph SchollEmail author
  • Ralf Wimmer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10929)


Dependency quantified Boolean formulas (DQBFs) as a generalization of quantified Boolean formulas (QBFs) have received considerable attention in research during the last years. Here we give an overview of the solution methods developed for DQBF so far. The exposition is complemented with the discussion of various applications that can be handled with DQBF solving.



We are grateful to Bernd Becker, Ruben Becker, Andreas Karrenbauer, Jennifer Nist, Sven Reimer, Matthias Sauer, and Karina Wimmer for heavily contributing to the contents summarized in this paper.


  1. 1.
    Abramovici, M., Breuer, M.A., Friedman, A.D.: Digital Systems Testing and Testable Design. Computer Science Press, New York (1990)Google Scholar
  2. 2.
    Ayari, A., Basin, D.: Qubos: deciding quantified Boolean logic using propositional satisfiability solvers. In: Aagaard, M.D., O’Leary, J.W. (eds.) FMCAD 2002. LNCS, vol. 2517, pp. 187–201. Springer, Heidelberg (2002). Scholar
  3. 3.
    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). Scholar
  4. 4.
    Beineke, L.W., Little, C.H.C.: Cycles in bipartite tournaments. J. Comb. Theory Ser. B 32(2), 140–145 (1982). Scholar
  5. 5.
    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). Scholar
  6. 6.
    Biere, A.: Resolve and expand. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 59–70. Springer, Heidelberg (2005). Scholar
  7. 7.
    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). Scholar
  8. 8.
    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). Scholar
  9. 9.
    Bubeck, U.: Model-based transformations for quantified Boolean formulas. Ph.D. thesis, University of Paderborn, Germany (2010)Google Scholar
  10. 10.
    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). Scholar
  11. 11.
    Cai, M., Deng, X., Zang, W.: A min-max theorem on feedback vertex sets. Math. Oper. Res. 27(2), 361–371 (2002). Scholar
  12. 12.
    Chatterjee, K., Henzinger, T.A., Otop, J., Pavlogiannis, A.: Distributed synthesis for LTL fragments. In: FMCAD 2013, pp. 18–25. IEEE, October 2013.
  13. 13.
    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). Scholar
  14. 14.
    Finkbeiner, B., Tentrup, L.: Fast DQBF refutation. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 243–251. Springer, Cham (2014). Scholar
  15. 15.
    Fröhlich, A., Kovásznai, G., Biere, A.: A DPLL algorithm for solving DQBF. In: International Workshop on Pragmatics of SAT (POS) 2012, Trento, Italy (2012)Google Scholar
  16. 16.
    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. EPiC Series, vol. 27, pp. 103–116. EasyChair, Vienna (2014)Google Scholar
  17. 17.
    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 2013.
  18. 18.
    Gitina, K., Wimmer, R., Reimer, S., Sauer, M., Scholl, C., Becker, B.: Solving DQBF through quantifier elimination. In: DATE 2015, Grenoble, France. IEEE, March 2015.
  19. 19.
    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). Scholar
  20. 20.
    Guo, J., Hüffner, F., Moser, H.: Feedback arc set in bipartite tournaments is NP-complete. Inf. Process. Lett. 102(2–3), 62–65 (2007). Scholar
  21. 21.
    Henkin, L.: Some remarks on infinitely long formulas. In: Infinitistic Methods: Proceedings of the 1959 Symposium on Foundations of Mathematics, Warsaw, Panstwowe, pp. 167–183. Pergamon Press, September 1961Google Scholar
  22. 22.
    Jain, A., Boppana, V., Mukherjee, R., Jain, J., Fujita, M., Hsiao, M.S.: Testing, verification, and diagnosis in the presence of unknowns. In: IEEE VLSI Test Symposium (VTS) 2000, Montreal, Canada, pp. 263–270. IEEE Computer Society (2000).
  23. 23.
    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). Scholar
  24. 24.
    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).
  25. 25.
    Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. J. Satisf. Boolean Model. Comput. 7(2–3), 71–76 (2010)Google Scholar
  26. 26.
    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). Scholar
  27. 27.
    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). Scholar
  28. 28.
    Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time: preliminary report. In: STOC 1973, pp. 1–9. ACM Press (1973).
  29. 29.
    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). Scholar
  30. 30.
    Peterson, G.L., Reif, J.H.: Multiple-person alternation. In: Annual Symposium on Foundations of Computer Science (FOCS), San Juan, Puerto Rico, pp. 348–363. IEEE Computer Society, October 1979.
  31. 31.
    Pnueli, A., Rosner, R.: Distributed reactive systems are hard to synthesize. In: Annual Symposium on Foundations of Computer Science 1990, St. Louis, Missouri, USA, pp. 746–757. IEEE Computer Society, October 1990.
  32. 32.
    Rabe, M.N.: A resolution-style proof system for DQBF. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 314–325. Springer, Cham (2017). Scholar
  33. 33.
    Scholl, C., Becker, B.: Checking equivalence for partial implementations. In: DAC 2001, Las Vegas, NV, USA, pp. 238–243. ACM Press, June 2001.
  34. 34.
    Silva, J.P.M., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999). Scholar
  35. 35.
    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). Scholar
  36. 36.
    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). Scholar
  37. 37.
    Wimmer, R., Karrenbauer, A., Becker, R., Scholl, C., Becker, B.: From DQBF to QBF by dependency elimination. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 326–343. Springer, Cham (2017). Scholar
  38. 38.
    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). Scholar
  39. 39.
    Wimmer, R., Wimmer, K., Scholl, C., Becker, B.: Analysis of incomplete circuits using dependency quantified Boolean formulas. In: Reis, A.I., Drechsler, R. (eds.) Advanced Logic Synthesis, pp. 151–168. Springer, Cham (2018). Scholar
  40. 40.
    Wolsey, L.A.: Integer Programming. Wiley-Interscience, New York (1998)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Albert-Ludwigs-Universität FreiburgFreiburg im BreisgauGermany

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