Multiset Bisimulations as a Common Framework for Ordinary and Probabilistic Bisimulations

  • David de Frutos Escrig
  • Miguel Palomino
  • Ignacio Fábregas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5048)


Our concrete objective is to present both ordinary bisimulations and probabilistic bisimulations in a common coalgebraic framework based on multiset bisimulations. For that we show how to relate the underlying powerset and probabilistic distributions functors with the multiset functor by means of adequate natural transformations. This leads us to the general topic that we investigate in the paper: a natural transformation from a functor F to another G transforms F-bisimulations into G-bisimulations but, in general, it is not possible to express G-bisimulations in terms of F-bisimulations. However, they can be characterized by considering Hughes and Jacobs’ notion of simulation, taking as the order on the functor F the equivalence induced by the epi-mono decomposition of the natural transformation relating F and G. We also consider the case of alternating probabilistic systems where non-deterministic and probabilistic choices are mixed, although only in a partial way, and extend all these results to categorical simulations.


Transition System Probabilistic System Natural Transformation Probabilistic Choice Common Framework 
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.


  1. 1.
    Aczel, P., Mendler, N.P.: A final coalgebra theorem. In: Pitt, D.H., Rydeheard, D.E., Dybjer, P., Pitts, A.M., Poigné, A. (eds.) Category Theory and Computer Science. LNCS, vol. 389, pp. 357–365. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  2. 2.
    Bartels, F., Sokolova, A., de Vink, E.P.: A hierarchy of probabilistic system types. Theoretical Computer Science 327(1-2), 3–22 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Frutos-Escrig, D., Rosa-Velardo, F., Gregorio-Rodríguez, C.: New Bisimulation Semantics for Distributed Systems. In: Derrick, J., Vain, J. (eds.) FORTE 2007. LNCS, vol. 4574, pp. 143–159. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, Generative and Stratified Models of Probabilistic Processes. Information and Computation 121(1), 59–80 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hasuo, I.: Generic forward and backward simulations. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 406–420. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Hughes, J., Jacobs, B.: Simulations in coalgebra. Theoretical Computer Science 327(1-2), 71–108 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jacobs, B.: Introduction to coalgebra. towards mathematics of states and observations. Book in preparation, Available at:
  8. 8.
    Jacobs, B., Rutten, J.: A tutorial on (co)algebras and (co)induction. Bulletin of the European Association for Theoretical Computer Science 62, 222–259 (1997)zbMATHGoogle Scholar
  9. 9.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94(1), 1–28 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mislove, M.W.: Nondeterminism and Probabilistic Choice: Obeying the Laws. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 350–364. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249(1), 3–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Segala, R., Lynch, N.A.: Probabilistic Simulations for Probabilistic Processes. Nordic Journal on Computing 2(2), 250–273 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Tix, R., Keimel, K., Plotkin, G.: Semantic Domains for Combining Probability and Non-Determinism. ENTCS, vol. 129, pp. 1–104. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
  15. 15.
    Varacca, D.: The powerdomain of indexed valuations. In: LICS 2002: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, pp. 299–310. IEEE Computer Society, Los Alamitos (2002)Google Scholar
  16. 16.
    Varacca, D., Winskel, G.: Distributing Probabililty over Nondeterminism. Mathematical Structures in Computer Science 16(1), 87–113 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    de Vink, E.P., Rutten, J.J.M.M.: Bisimulation for probabilistic transition systems: A coalgebraic approach. Theoretical Computer Science 221(1-2), 271–293 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2008

Authors and Affiliations

  • David de Frutos Escrig
    • 1
  • Miguel Palomino
    • 1
  • Ignacio Fábregas
    • 1
  1. 1.Departamento de Sistemas Informáticos y ComputaciónUniversidad Complutense de MadridSpain

Personalised recommendations