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

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Theory and Applications of Satisfiability Testing – SAT 2018 (SAT 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10929))

Abstract

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.

This work was partly supported by the German Research Council (DFG) as part of the project “Solving Dependency Quantified Boolean Formulas”.

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References

  1. Abramovici, M., Breuer, M.A., Friedman, A.D.: Digital Systems Testing and Testable Design. Computer Science Press, New York (1990)

    Google Scholar 

  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). https://doi.org/10.1007/3-540-36126-X_12

    Chapter  Google Scholar 

  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). https://doi.org/10.1016/j.tcs.2013.12.020

    Article  MathSciNet  MATH  Google Scholar 

  4. Beineke, L.W., Little, C.H.C.: Cycles in bipartite tournaments. J. Comb. Theory Ser. B 32(2), 140–145 (1982). https://doi.org/10.1016/0095-8956(82)90029-6

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-40970-2_30

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/11527695_5

    Chapter  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-22438-6_10

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-54013-4_1

    Chapter  Google Scholar 

  9. Bubeck, U.: Model-based transformations for quantified Boolean formulas. Ph.D. thesis, University of Paderborn, Germany (2010)

    Google Scholar 

  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). https://doi.org/10.1007/11814948_21

    Chapter  MATH  Google Scholar 

  11. Cai, M., Deng, X., Zang, W.: A min-max theorem on feedback vertex sets. Math. Oper. Res. 27(2), 361–371 (2002). https://doi.org/10.1287/moor.27.2.361.328

    Article  MathSciNet  MATH  Google Scholar 

  12. Chatterjee, K., Henzinger, T.A., Otop, J., Pavlogiannis, A.: Distributed synthesis for LTL fragments. In: FMCAD 2013, pp. 18–25. IEEE, October 2013. https://doi.org/10.1109/FMCAD.2013.6679386

  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). https://doi.org/10.1007/11499107_5

    Chapter  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-09284-3_19

    Chapter  Google Scholar 

  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. 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. 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. https://doi.org/10.1109/ICCD.2013.6657071

  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. https://doi.org/10.7873/date.2015.0098

  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). https://doi.org/10.1007/978-3-642-14186-7_9

    Chapter  Google Scholar 

  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). https://doi.org/10.1016/j.ipl.2006.11.016

    Article  MathSciNet  MATH  Google Scholar 

  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 1961

    Google Scholar 

  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). https://doi.org/10.1109/VTEST.2000.843854

  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). https://doi.org/10.1007/978-3-642-31612-8_10

    Chapter  Google Scholar 

  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). http://ijcai.org/Abstract/15/052

  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. 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). https://doi.org/10.1007/978-3-662-44199-2_48

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-31612-8_34

    Chapter  Google Scholar 

  28. Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time: preliminary report. In: STOC 1973, pp. 1–9. ACM Press (1973). https://doi.org/10.1145/800125.804029

  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). https://doi.org/10.1016/S0898-1221(00)00333-3

    Article  MathSciNet  MATH  Google Scholar 

  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. https://doi.org/10.1109/SFCS.1979.25

  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. https://doi.org/10.1109/FSCS.1990.89597

  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). https://doi.org/10.1007/978-3-319-66263-3_20

    Chapter  Google Scholar 

  33. Scholl, C., Becker, B.: Checking equivalence for partial implementations. In: DAC 2001, Las Vegas, NV, USA, pp. 238–243. ACM Press, June 2001. https://doi.org/10.1145/378239.378471

  34. Silva, J.P.M., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999). https://doi.org/10.1109/12.769433

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-46520-3_25

    Chapter  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-24318-4_13

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-66263-3_21

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-40970-2_29

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-67295-3_7

    Chapter  Google Scholar 

  40. Wolsey, L.A.: Integer Programming. Wiley-Interscience, New York (1998)

    MATH  Google Scholar 

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Acknowledgment

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.

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Correspondence to Christoph Scholl .

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Scholl, C., Wimmer, R. (2018). Dependency Quantified Boolean Formulas: An Overview of Solution Methods and Applications. In: Beyersdorff, O., Wintersteiger, C. (eds) Theory and Applications of Satisfiability Testing – SAT 2018. SAT 2018. Lecture Notes in Computer Science(), vol 10929. Springer, Cham. https://doi.org/10.1007/978-3-319-94144-8_1

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