Skip to main content
SpringerLink
Log in
Book cover

International Joint Conference on Automated Reasoning

IJCAR 2018: Automated Reasoning pp 1–18Cite as

An Assumption-Based Approach for Solving the Minimal S5-Satisfiability Problem

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10900))

Abstract

Recent work on the practical aspects on the modal logic S5 satisfiability problem showed that using a SAT-based approach outperforms other existing approaches. In this work, we go one step further and study the related minimal S5 satisfiability problem (MinS5-SAT), the problem of finding an S5 model, a Kripke structure, with the smallest number of worlds. Finding a small S5 model is crucial as soon as the model should be presented to a user, displayed on a screen for instance. SAT-based approaches tend to produce S5-models with a large number of worlds, thus the need to minimize them. That optimization problem can obviously be solved as a pseudo-Boolean optimization problem. We show in this paper that it is also equivalent to the extraction of a maximal satisfiable set (MSS). It can thus be solved using a standard pseudo-Boolean or solver, or a MSS-extractor. We show that a new incremental, SAT-based approach can be proposed by taking into account the equivalence relation between the possible worlds on S5 models. That specialized approach presented the best performance on our experiments conducted on a wide range of benchmarks from the modal logic community and a wide range of pseudo-Boolean and solvers. Our results demonstrate once again that domain knowledge is key to build efficient SAT-based tools.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    https://fai.cs.uni-saarland.de/hoffmann/ff/cff-tests.tgz.

  2. 2.

    http://www.cril.fr/~montmirail/planning-to-s5/.

  3. 3.

    http://www.cril.fr/~montmirail/s52SAT.

References

  1. Fairtlough, M., Mendler, M.: An intuitionistic modal logic with applications to the formal verification of hardware. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 354–368. Springer, Heidelberg (1995). https://doi.org/10.1007/BFb0022268

    Chapter  MATH  Google Scholar 

  2. Fitting, M.: Modality and databases. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS (LNAI), vol. 1847, pp. 19–39. Springer, Heidelberg (2000). https://doi.org/10.1007/10722086_2

    Chapter  Google Scholar 

  3. Murphy VII, T., Crary, K., Harper, R.: Distributed control flow with classical modal logic. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 51–69. Springer, Heidelberg (2005). https://doi.org/10.1007/11538363_6

    Chapter  MATH  Google Scholar 

  4. Bienvenu, M., Fargier, H., Marquis, P.: Knowledge compilation in the modal logic S5. In: Proceedings of AAAI 2010 (2010)

    Google Scholar 

  5. Niveau, A., Zanuttini, B.: Efficient representations for the modal logic S5. In: Proceedings of IJCAI 2016, pp. 1223–1229 (2016)

    Google Scholar 

  6. Hustadt, U., Schmidt, R.A., Weidenbach, C.: MSPASS: subsumption testing with SPASS. In: Proceedings of DL 1999. CEUR Workshop Proceedings, vol. 22. CEUR (1999)

    Google Scholar 

  7. Tacchella, A.: *SAT system description. In: Proceedings of DL 1999, vol. 22. CEUR (1999)

    Google Scholar 

  8. Sebastiani, R., Vescovi, M.: Automated reasoning in modal and description logics via SAT encoding: the case study of K(m)/ALC-satisfiability. J. Artif. Intell. Res. 35, 343–389 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Nalon, C., Hustadt, U., Dixon, C.: K\(_S\)P: a resolution-based prover for multimodal K. In: Proceedings of IJCAR 2016, pp. 406–415 (2016)

    Google Scholar 

  10. Caridroit, T., Lagniez, J.M., Le Berre, D., de Lima, T., Montmirail, V.: A SAT-based approach for solving the modal logic S5-satisfiability problem. In: Proceedings of AAAI 2017 (2017)

    Google Scholar 

  11. Lagniez, J.M., Le Berre, D., de Lima, T., Montmirail, V.: A recursive shortcut for CEGAR: application to the modal logic K satisfiability problem. In: Proceedings of IJCAI 2017 (2017)

    Google Scholar 

  12. Simon, L., Le Berre, D., Hirsch, E.A.: The SAT2002 competition. Ann. Math. Artif. Intell. 43(1), 307–342 (2005)

    Article  Google Scholar 

  13. Marques-Silva, J., Janota, M.: On the query complexity of selecting few minimal sets. Electron. Colloq. Comput. Complex. (ECCC) 21, 31 (2014)

    Google Scholar 

  14. Marques-Silva, J., Lynce, I.: On Improving MUS extraction algorithms. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 159–173. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21581-0_14

    Chapter  Google Scholar 

  15. Belov, A., Marques-Silva, J.: Accelerating MUS extraction with recursive model rotation. In: Proceedings of FMCAD 2011, pp. 37–40 (2011)

    Google Scholar 

  16. Iser, M., Sinz, C., Taghdiri, M.: Minimizing models for tseitin-encoded SAT instances. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 224–232. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39071-5_17

    Chapter  MATH  Google Scholar 

  17. Soh, T., Inoue, K.: Identifying necessary reactions in metabolic pathways by minimal model generation. In: Proceedings of ECAI 2010, vol. 215, pp. 277–282. IOS Press (2010)

    Google Scholar 

  18. Koshimura, M., Nabeshima, H., Fujita, H., Hasegawa, R.: Minimal model generation with respect to an atom set. In: Proceedings of FTP 2009, pp. 49–59 (2009)

    Google Scholar 

  19. Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)

    Article  MathSciNet  Google Scholar 

  20. Cook, S.A.: Characterizations of pushdown machines in terms of time-bounded computers. J. ACM 18(1), 4–18 (1971)

    Article  MathSciNet  Google Scholar 

  21. Li, C.M., Manyà, F.: MaxSAT, hard and soft constraints. In: Handbook of Satisfiability, pp. 613–631. IOS Press (2009)

    Google Scholar 

  22. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995)

    MATH  Google Scholar 

  23. Eiter, T., Faber, W., Leone, N., Pfeifer, G., Polleres, A.: Planning under incomplete knowledge. In: Lloyd, J., Dahl, V., Furbach, U., Kerber, M., Lau, K.-K., Palamidessi, C., Pereira, L.M., Sagiv, Y., Stuckey, P.J. (eds.) CL 2000. LNCS (LNAI), vol. 1861, pp. 807–821. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44957-4_54

    Chapter  Google Scholar 

  24. Kripke, S.A.: Semantical analysis of modal logic I. Normal propositional calculi. Z. M. L. G. M. 9(56), 67–96 (1963)

    MathSciNet  MATH  Google Scholar 

  25. Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24605-3_37

    Chapter  Google Scholar 

  26. Grégoire, É., Lagniez, J., Mazure, B.: An experimentally efficient method for (MSS, CoMSS) partitioning. In: Proceedings of AAAI 2014, pp. 2666–2673 (2014)

    Google Scholar 

  27. Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25, 97–116 (2012)

    MathSciNet  MATH  Google Scholar 

  28. O’Sullivan, B., Papadopoulos, A., Faltings, B., Pu, P.: Representative explanations for over-constrained problems. In: Proceedings of AAAI 2007, pp. 323–328 (2007)

    Google Scholar 

  29. Audemard, G., Lagniez, J.-M., Simon, L.: Improving glucose for incremental SAT solving with assumptions: application to MUS extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 309–317. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39071-5_23

    Chapter  MATH  Google Scholar 

  30. Roussel, O., Manquinho, V.M.: Pseudo-Boolean and cardinality constraints. In: Handbook of Satisfiability, pp. 695–733. IOS Press (2009)

    Google Scholar 

  31. Davies, J., Bacchus, F.: Exploiting the power of mip solvers in maxsat. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 166–181. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39071-5_13

    Chapter  MATH  Google Scholar 

  32. dos Reis Morgado, A.J., Ignatiev, A.S., Silva, J.M.: MSCG: robust core-guided MaxSAT solving. J. Satisfiability Boolean Model. Comput. 9, 129–134 (2014)

    MathSciNet  Google Scholar 

  33. Heras, F., Morgado, A., Marques-Silva, J.: Core-guided binary search algorithms for maximum satisfiability. In: Proceedings of AAAI 2011 (2011)

    Google Scholar 

  34. Sakai, M., Nabeshima, H.: Construction of an ROBDD for a PB-constraint in band form and related techniques for PB-solvers. IEICE Trans. 98–D(6), 1121–1127 (2015)

    Article  Google Scholar 

  35. Le Berre, D., Parrain, A.: The SAT4J library, release 2.2. JSAT 7(2–3), 59–64 (2010)

    Google Scholar 

  36. Maher, S.J., Fischer, T., Gally, T., Gamrath, G., Gleixner, A., Gottwald, R.L., Hendel, G., Koch, T., Lübbecke, M.E., Miltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schenker, S., Schwarz, R., Serrano, F., Shinano, Y., Weninger, D., Witt, J.T., Witzig, J.: The SCIP optimization suite 4.0. Technical report 17–12, ZIB, Takustr. 7, 14195 Berlin (2017)

    Google Scholar 

  37. Mencía, C., Previti, A., Marques-Silva, J.: Literal-based MCS extraction. In: Proceedings of IJCAI 2015, pp. 1973–1979 (2015)

    Google Scholar 

  38. Argelich, J., Min Li, C., Manyà, F., Planes, J.: Max-SAT 2016: eleventh Max-SAT evaluation (2016). http://www.maxsat.udl.cat/16/

  39. Saikko, P., Berg, J., Järvisalo, M.: LMHS: a SAT-IP hybrid MaxSAT solver. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 539–546. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_34

    Chapter  MATH  Google Scholar 

  40. Massacci, F., Donini, F.M.: Design and results of TANCS-2000 non-classical (modal) systems comparison. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS (LNAI), vol. 1847, pp. 52–56. Springer, Heidelberg (2000). https://doi.org/10.1007/10722086_4

    Chapter  MATH  Google Scholar 

  41. Kaminski, M., Tebbi, T.: InKreSAT: modal reasoning via incremental reduction to SAT. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 436–442. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_31

    Chapter  Google Scholar 

  42. Balsiger, P., Heuerding, A., Schwendimann, S.: A benchmark method for the propositional modal logics K, KT, S4. J. Autom. Reason. 24(3), 297–317 (2000)

    Article  MathSciNet  Google Scholar 

  43. Patel-Schneider, P.F., Sebastiani, R.: A new general method to generate random modal formulae for testing decision procedures. J. Artif. Intell. Res. 18, 351–389 (2003)

    MathSciNet  MATH  Google Scholar 

  44. Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)

    Article  MathSciNet  Google Scholar 

  45. Rintanen, J.: Planning and SAT. In: Handbook of Satisfiability, pp. 483–504. IOS Press (2009)

    Google Scholar 

  46. Petrick, R.P.A., Bacchus, F.: A knowledge-based approach to planning with incomplete information and sensing. In: Proceedings of AIPS 2002, pp. 212–222 (2002)

    Google Scholar 

  47. Hoffmann, J., Brafman, R.I.: Conformant planning via heuristic forward search: a new approach. Artif. Intell. 170(6–7), 507–541 (2006)

    Article  MathSciNet  Google Scholar 

  48. Götzmann, D., Kaminski, M., Smolka, G.: Spartacus: a tableau prover for hybrid logic. Electron. Notes Theor. Comput. Sci. 262, 127–139 (2010)

    Article  MathSciNet  Google Scholar 

  49. Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Proceedings of CADE 2009, pp. 140–145 (2009)

    Chapter  Google Scholar 

  50. Lagniez, J.M., Le Berre, D., de Lima, T., Montmirail, V.: On checking Kripke models for modal logic K. In: Proceedings of PAAR@IJCAR 2016, pp.69–81 (2016)

    Google Scholar 

Download references

Acknowledgments

We thank the anonymous reviewers for their insightful comments. Part of this work was supported by the French Ministry for Higher Education and Research, the Nord-Pas de Calais Regional Council through the “Contrat de Plan État Région (CPER) DATA” and an EC FEDER grant.

Author information

Authors and Affiliations

  1. CRIL, Artois University and CNRS, 62300, Lens, France

    Jean-Marie Lagniez, Daniel Le Berre, Tiago de Lima & Valentin Montmirail

Authors
  1. Jean-Marie Lagniez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Le Berre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tiago de Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Valentin Montmirail
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valentin Montmirail .

Editor information

Editors and Affiliations

  1. Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Didier Galmiche

  2. Baden-Wuerttemberg Cooperative State University, Stuttgart, Germany

    Stephan Schulz

  3. University of Trento, Trento, Italy

    Roberto Sebastiani

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lagniez, JM., Le Berre, D., de Lima, T., Montmirail, V. (2018). An Assumption-Based Approach for Solving the Minimal S5-Satisfiability Problem. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-94205-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94204-9

  • Online ISBN: 978-3-319-94205-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation