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Solving Parameterised Boolean Equation Systems with Infinite Data Through Quotienting

  • Thomas NeeleEmail author
  • Tim A. C. Willemse
  • Jan Friso Groote
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11222)

Abstract

Parameterised Boolean Equation Systems (PBESs) can be used to represent many different kinds of decision problems. Most notably, model checking and equivalence problems can be encoded in a PBES. Traditional instantiation techniques cannot deal with PBESs with an infinite data domain. We propose an approach that can solve PBESs with infinite data by computing the bisimulation quotient of the underlying graph structure. Furthermore, we show how this technique can be improved by repeatedly searching for finite proofs. Unlike existing approaches, our technique is not restricted to subfragments of PBESs. Experimental results show that our ideas work well in practice and support a wider range of models and properties than state-of-the-art techniques.

Notes

Acknowledgements

We would like to thank the anonymous reviewers for their constructive feedback. Their suggestions helped us to improve the paper before publication.

References

  1. 1.
    Alur, R., Courcoubetis, C., Halbwachs, N., Dill, D., Wong-Toi, H.: Minimization of timed transition systems. In: Cleaveland, W.R. (ed.) CONCUR 1992. LNCS, vol. 630, pp. 340–354. Springer, Heidelberg (1992).  https://doi.org/10.1007/BFb0084802CrossRefGoogle Scholar
  2. 2.
    Behrmann, G., David, A., Larsen, K.G.: A tutorial on Uppaal. In: Bernardo, M., Corradini, F. (eds.) SFM-RT 2004. LNCS, vol. 3185, pp. 200–236. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-30080-9_7CrossRefGoogle Scholar
  3. 3.
    Bouajjani, A., Fernandez, J.-C., Halbwachs, N., Raymond, P., Ratel, C.: Minimal state graph generation. Sci. Comput. Programm. 18(3), 247–269 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, T., Ploeger, B., van de Pol, J., Willemse, T.A.C.: Equivalence checking for infinite systems using parameterized boolean equation systems. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 120–135. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-74407-8_9CrossRefGoogle Scholar
  5. 5.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000).  https://doi.org/10.1007/10722167_15CrossRefGoogle Scholar
  6. 6.
    Cranen, S., Groote, J.F., Keiren, J.J.A., Stappers, F.P.M., de Vink, E.P., Wesselink, W., Willemse, T.A.C.: An overview of the mCRL2 toolset and its recent advances. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 199–213. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36742-7_15CrossRefzbMATHGoogle Scholar
  7. 7.
    Cranen, S., Keiren, J.J.A., Willemse, T.A.C.: A cure for stuttering parity games. In: Roychoudhury, A., D’Souza, M. (eds.) ICTAC 2012. LNCS, vol. 7521, pp. 198–212. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32943-2_16CrossRefGoogle Scholar
  8. 8.
    Cranen, S., Luttik, B., Willemse, T.A.C.: Proof graphs for parameterised boolean equation systems. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013. LNCS, vol. 8052, pp. 470–484. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40184-8_33CrossRefGoogle Scholar
  9. 9.
    Fisler, K., Vardi, M.Y.: Bisimulation and Model Checking. In: Pierre, L., Kropf, T. (eds.) CHARME 1999. LNCS, vol. 1703, pp. 338–342. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48153-2_29Google Scholar
  10. 10.
    Fontana, P., Cleaveland, R.: The power of proofs: new algorithms for timed automata model checking. In: Legay, A., Bozga, M. (eds.) FORMATS 2014. LNCS, vol. 8711, pp. 115–129. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-10512-3_9CrossRefzbMATHGoogle Scholar
  11. 11.
    Garavel, H., Lang, F., Mateescu, R., Serwe, W.: CADP 2011: a toolbox for the construction and analysis of distributed processes. STTT 15(2), 89–107 (2013)CrossRefGoogle Scholar
  12. 12.
    Groote, J.F., Willemse, T.A.C.: Parameterised boolean equation systems. Theor. Comput. Sci. 343(3), 332–369 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hesselink, W.H.: Invariants for the construction of a handshake register. Inf. Process. Lett. 68(4), 173–177 (1998)CrossRefGoogle Scholar
  14. 14.
    Kant, G., van de Pol, J.: Efficient instantiation of parameterised boolean equation systems to parity games. In: GRAPHITE 2012, volume 99 of EPTCS, pp. 50–65 (2012)CrossRefGoogle Scholar
  15. 15.
    Keiren, J.J.A., Wesselink, W., Willemse, T.A.C.: Liveness analysis for parameterised boolean equation systems. In: Cassez, F., Raskin, J.-F. (eds.) ATVA 2014. LNCS, vol. 8837, pp. 219–234. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-11936-6_16CrossRefzbMATHGoogle Scholar
  16. 16.
    Keiren, J.J.A., Willemse, T.A.C.: Bisimulation minimisations for boolean equation systems. In: Namjoshi, K., Zeller, A., Ziv, A. (eds.) HVC 2009. LNCS, vol. 6405, pp. 102–116. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19237-1_12CrossRefGoogle Scholar
  17. 17.
    Knuth, D.E.: Textbook examples of recursion. Artif. Math. Theory Comput. 91, 207–229 (1991)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Koolen, R.P.J., Willemse, T.A.C., Zantema, H.: Using SMT for solving fragments of parameterised boolean equation systems. In: Finkbeiner, B., Pu, G., Zhang, L. (eds.) ATVA 2015. LNCS, vol. 9364, pp. 14–30. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24953-7_3CrossRefzbMATHGoogle Scholar
  19. 19.
    Lamport, L.: A new solution of Dijkstra’s concurrent programming problem. Commun. ACM 17(8), 453–455 (1974)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lee, D., Yannakakis, M.: Online minimization of transition systems (extended abstract). In: STOC 1992, pp. 264–274 (1992)Google Scholar
  21. 21.
    Nagae, Y., Sakai, M.: Reduced dependency spaces for existential parameterised boolean equation systems. In: WPTE 2017, volume 265 of EPTCS, pp. 67–81 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nagae, Y., Sakai, M., Seki, H.: An extension of proof graphs for disjunctive parameterised boolean equation systems. In: WPTE 2016, volume 235 of EPTCS, pp. 46–61 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Neele, T., Willemse, T.A.C., Groote, J.F.: Solving Parameterised Boolean Equation Systems with Infinite Data Through Quotienting (Technical Report). Technical report, Eindhoven University of Technology (2018)Google Scholar
  24. 24.
    Park, D.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104, pp. 167–183. Springer, Heidelberg (1981).  https://doi.org/10.1007/BFb0017309CrossRefGoogle Scholar
  25. 25.
    Tripakis, S., Yovine, S.: Analysis of timed systems using time-abstracting bisimulations. FMSD 18(1), 25–68 (2001)zbMATHGoogle Scholar
  26. 26.
    Willemse, T.A.C.: Consistent correlations for parameterised boolean equation systems with applications in correctness proofs for manipulations. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 584–598. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15375-4_40CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Thomas Neele
    • 1
    Email author
  • Tim A. C. Willemse
    • 1
  • Jan Friso Groote
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

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