Modal Characterisations of Probabilistic and Fuzzy Bisimulations

  • Yuxin Deng
  • Hengyang Wu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8829)


This paper aims to investigate bisimulation on fuzzy systems. For that purpose we revisit bisimulation in the model of reactive probabilistic processes with countable state spaces and obtain two findings: (1) bisimilarity coincides with simulation equivalence, which generalises a result on finite-state processes originally established by Baier; (2) the modal characterisation of bisimilarity by Desharnais et al. admits a much simpler completeness proof. Furthermore, inspired by the work of Hermanns et al. on probabilistic systems, we provide a sound and complete modal characterisation of fuzzy bisimilarity.


Label Transition System Possibility Distribution Fuzzy Simulation Logical Characterization Characteristic Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulla, P.A., Legay, A., d’Orso, J., Rezine, A.: Tree regular model checking: A simulation-based approach. J. Logic. Algebr. Progr. 69(1/2), 93–121 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baier, C.: On Algorithmic Verification Methods for Probabilistic Systems. Habilitationsschrift zur Erlangung der venia legendi der Fakultät für Mathematik and Informatik, Universität Mannheim (1998)Google Scholar
  3. 3.
    Bailador, G., Triviño, G.: Pattern recognition using temporal fuzzy automata. Fuzzy Sets Syst. 161(1), 37–55 (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Bělohlávek, R.: Determinism and fuzzy automata. Inform. Sci. 142(1-4), 205–209 (2002)CrossRefGoogle Scholar
  5. 5.
    Bhattacharyya, M.: Fuzzy Markovian decision process. Fuzzy Sets Syst. 99(3), 273–282 (1998)CrossRefGoogle Scholar
  6. 6.
    Billingsley, P.: Probability and Measure. Wiley-Interscience, New York (1995)MATHGoogle Scholar
  7. 7.
    Cao, Y., Chen, G., Kerre, E.E.: Bisimulations for fuzzy transition systems. IEEE Trans. Fuzzy Syst. 19(3), 540–552 (2010)CrossRefGoogle Scholar
  8. 8.
    Cao, Y., Ezawa, Y.: Nondeterministic fuzzy automata. Inform. Sci. 191(1), 86–97 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ćirić, M., Ignjatović, J., Damljanović, N., Bašić, M.: Bisimulations for fuzzy automata. Fuzzy Sets Syst. 186(1), 100–139 (2012)CrossRefMATHGoogle Scholar
  10. 10.
    Deng, Y., Du, W.: Logical, Metric, and Algorithmic Characterisations of Probabilistic Bisimulation., Technical Report CMU-CS-11-145, Carnegie Mellon University (2011)Google Scholar
  11. 11.
    Deng, Y., van Glabbeek, R.J., Hennessy, M., Morgan, C.C.: Testing finitary probabilistic processes. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 274–288. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Desharnais, J.: Labelled Markov Processes. Ph.D. thesis, McGill University (1999)Google Scholar
  13. 13.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. Comput. 179(2), 163–193 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled Markov processes. Inf. Comput. 184(1), 160–200 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    D’Errico, L., Loreti, M.: A process algebra approach to fuzzy reasoning. In: Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, pp. 1136–1141 (2009)Google Scholar
  16. 16.
    Doberkat, E.-E.: Stochastic Coalgebraic Logic. Springer, Heidelberg (2010)MATHGoogle Scholar
  17. 17.
    Fisler, K., Vardi, M.Y.: Bisimulation minimization and symbolic model checking. Form. Method. Syst. Des. 21(1), 39–78 (2002)CrossRefMATHGoogle Scholar
  18. 18.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B., Tofts, C.M.N.: Reactive, generative, and stratified models of probabilistic processes. In: Proc. 5th Annu. IEEE Symp. Logic in Computer Science, pp. 130–141 (1990)Google Scholar
  19. 19.
    Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM. 32(1), 137–161 (1985)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hermanns, H., Parma, A., et al.: Probabilistic logical characterization. Inf. Comput. 209(2), 154–172 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ignjatović, J., Ćirić, M., Simović, V.: Fuzzy relation equations and subsystems of fuzzy transition systems. Knowl-Based Syst. 38(1), 48–61 (2013)CrossRefGoogle Scholar
  22. 22.
    Kupferman, O., Lustig, Y.: Latticed simulation relations and games. Int. J. Found. Comput. S. 21(2), 167–189 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Li, Y.M., Pedrycz, W.: Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids. Fuzzy Sets Syst. 156(1), 68–92 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lin, F., Ying, H.: Modeling and control of fuzzy discrete event systems. IEEE Trans. Syst., Man, Cybern., B, Cybern. 32(4), 408–415 (2002)CrossRefGoogle Scholar
  26. 26.
    Mordeson, J.N., Malik, D.S.: Fuzzy Automata and Languages:Theory and Applications. Chapman & Hall/CRC, Boca Raton (2002)CrossRefGoogle Scholar
  27. 27.
    Milner, R.: A Calculus of Communication Systems. LNCS, vol. 92. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  28. 28.
    Park, D.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104, pp. 167–183. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  29. 29.
    Parma, A., Segala, R.: Logical characterizations of bisimulations for discrete probabilistic systems. In: Seidl, H. (ed.) FOSSACS 2007. LNCS, vol. 4423, pp. 287–301. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  30. 30.
    Pedrycz, W., Gacek, A.: Learning of fuzzy automata. Int. J. Comput. Intell. Appl. 1(1), 19–33 (2001)CrossRefGoogle Scholar
  31. 31.
    Pedrycz, W., Gomide, F.: A generalized fuzzy Petri net model. IEEE Trans. Fuzzy Syst. 2(4), 295–301 (1994)CrossRefGoogle Scholar
  32. 32.
    Qiu, D.W.: Supervisory control of fuzzy discrete event systems: a formal approach. IEEE Trans. Syst., Man, Cybern., B, Cybern. 35(1), 72–88 (2005)CrossRefGoogle Scholar
  33. 33.
    Shen, V.R.L.: Knowledge representation using high-level fuzzy Petri nets. IEEE Trans. Syst., Man, Cybern. A, Syst., Humans 36(6), 1220–1227 (2006)CrossRefGoogle Scholar
  34. 34.
    Sack, J., Zhang, L.: A General Framework for Probabilistic Characterizing Formulae. In: Kuncak, V., Rybalchenko, A. (eds.) VMCAI 2012. LNCS, vol. 7148, pp. 396–411. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  35. 35.
    Sangiorgi, D., Rutten, J. (eds.): Advanced Topics in Bisimulation and Coinduction. Cambridge University Press (2011)Google Scholar
  36. 36.
    Segala, R., Lynch, N.A.: Probabilistic simulations for probabilistic process. Nord. J. Comput. 2(2), 250–273 (1995)MathSciNetMATHGoogle Scholar
  37. 37.
    Wee, W.G., Fu, K.S.: A formulation of fuzzy automata and its application as a model of learning systems. IEEE Trans. Syst. Sci. Cybern. SSC-5(3), 215–223 (1969)Google Scholar
  38. 38.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Zhang, L.: Decision Algorithms for Probabilistic Simulations. Ph.D. thesis, Saarland University (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yuxin Deng
    • 1
  • Hengyang Wu
    • 2
  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityChina
  2. 2.Information Engineer CollegeHangzhou Dianzi UniversityChina

Personalised recommendations