Bisimulation for probabilistic transition systems: A coalgebraic approach

  • E. P. de Vink
  • J. J. M. M. Rutten
Session 12:Process Equivalences
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)


The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendier in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation.


Bisimulation probabilistic transition system coalgebra ultrametric space Borel measure final coalgebra 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Acz88]
    P. Aczel. Non-Well-Founded Sets. CSLI Lecture Notes 14. Center for the Study of Languages and Information, Stanford, 1988.Google Scholar
  2. [AM89]
    P. Aczel and N. Mendier. A final coalgebra theorem. In D.H. Pitt et al., editors, Proc. Category Theory and Computer Science, pages 357–365. LNCS 389, 1989.Google Scholar
  3. [AR89]
    P. America and J.J.M.M. Rutten. Solving reflexive domain equations in a category of complete metric spaces. Journal of Computer Systems and Sciences, 39:343–375, 1989.CrossRefGoogle Scholar
  4. [Bar93]
    M. Barr. Terminal coalgebras in well-founded set theory. Theoretical Computer Science, 114:299–315, 1993. See also the addendum in Theoretical Computer Science, 124:189-192, 1994.CrossRefGoogle Scholar
  5. [BDEP97]
    R. Blute, J. Desharnais, A. Edalat, and P. Panangaden. Bisimulation for labelled Markov processes. In Proc. LICS'97. Warzaw, 1997.Google Scholar
  6. [BV96]
    J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.Google Scholar
  7. [Eda94]
    A. Edalat. Domain theory and integration. In Proc. LICS'94, pages 115–124. Paris, 1994.Google Scholar
  8. [GJS90]
    A. Giacalone, C. Jou, and S.A. Smolka. Algebraic reasoning for probabilisitic concurrent systems. In Proc. Working Conference on Programming Concepts and Methods. IFIP TC2, Sea of Gallilee, 1990.Google Scholar
  9. [GSS95]
    R.J. van Glabbeek, S.A. Smolka, and B. Steffen. Reactive, generative and stratified models of probabilistic processes. Information and Computation, 121:59–80, 1995.Google Scholar
  10. [Hen95]
    T.A. Henzinger. Hybrid automata with finite bisimulations. In Z. Fülöp and F. Gécseg, editors, Proc. ICALP'95, pages 324–335. LNCS 944, 1995.Google Scholar
  11. [JL91]
    B. Jonsson and K.G-. Larsen. Specification and refinement of probabilistic processes. In Proc. LICS'91, pages 266–277. Amsterdam, 1991.Google Scholar
  12. [JP89]
    C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proc. LICS'89, pages 186–195. Asilomar, 1989.Google Scholar
  13. [LS91]
    K.G. Larsen and A. Skou. Bisimulation through probabilistic testing. Information and Computation, 94:1–28, 1991.CrossRefGoogle Scholar
  14. [RT93]
    J.J.M.M. Rutten and D. Turi. On the foundations of final semantics: non-standard sets, metric spaces, partial orders. In J.W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, Proc. REX Workshop on Semantics: Foundations and Applications, pages 477–530. LNCS 666, 1993.Google Scholar
  15. [RT94]
    J.J.M.M. Rutten and D. Turi. Initial algebra and final coalgebra semantics for concurrency. In J.W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, Proc. REX School/Symposium ‘A Decade of Concurrency', pages 530–582. LNCS 803, 1994.Google Scholar
  16. [Rud66]
    W. Rudin. Real and Complex Analysis. McGraw-Hill, 1966.Google Scholar
  17. [Rut96]
    J.J.M.M. Rutten. Universal coalgebra: a theory of systems. Report CSR9652, CWI, 1996. Ftp-available at as pub/CWIreports/AP/ Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • E. P. de Vink
    • 1
  • J. J. M. M. Rutten
    • 2
  1. 1.Faculty of Mathematics and Computer ScienceVrije UniversiteitHV AmsterdamThe Netherlands
  2. 2.Department of Software TechnologyCWIGB AmsterdamThe Netherlands

Personalised recommendations