Structural Operational Semantics for Continuous State Probabilistic Processes

  • Giorgio Bacci
  • Marino Miculan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7399)


We consider the problem of modeling syntax and semantics of probabilistic processes with continuous states (e.g. with continuous data). Syntax and semantics of these systems can be defined as algebras and coalgebras of suitable endofunctors over Meas, the category of measurable spaces. In order to give a more concrete representation for these coalgebras, we present an SOS-like rule format which induces an abstract GSOS over Meas; this format is proved to yield a fully abstract universal semantics, for which behavioural equivalence is a congruence.

To this end, we solve several problems. In particular, the format has to specify how to compose the semantics of processes (which basically are continuous state Markov processes). This is achieved by defining a language of measure terms, i.e., expressions specifically designed for describing probabilistic measures. Thus, the transition relation associates processes with measure terms.

As an example application, we model a CCS-like calculus of processes placed in an Euclidean space. The approach we follow in this case can be readily adapted to other quantitative aspects, e.g. Quality of Service, physical and chemical parameters in biological systems, etc.


  1. 1.
    Abbott, E.A.: Flatland: A Romance of Many Dimensions. Blackwell (1884)Google Scholar
  2. 2.
    Adámek, J., Trnková, V.: Automata and Algebras in Categories, 1st edn. Kluwer Academic Publishers, Norwell (1990)MATHGoogle Scholar
  3. 3.
    Bacci, G., Miculan, M.: Measurable stochastics for Brane Calculus. Theoretical Comput. Sci. 431, 117–136 (2012), doi:10.1016/j.tcs.2011.12.055MATHCrossRefGoogle Scholar
  4. 4.
    Barbuti, R., Maggiolo-Schettini, A., Milazzo, P., Pardini, G.: Spatial calculus of looping sequences. Electr. Notes Theor. Comput. Sci. 229(1), 21–39 (2009)CrossRefGoogle Scholar
  5. 5.
    Barr, M.: Algebraically compact functors. Journal of Pure and Applied Algebra 82(3), 211–231 (1992)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bartels, F.: On Generalised Coinduction and Probabilistic Specification Formats: Distributive Laws in Coalgebraic Modelling. PhD thesis, CWI, Amsterdam (2004)Google Scholar
  7. 7.
    Bartels, F., Sokolova, A., de Vink, E.P.: A hierarchy of probabilistic system types. Electr. Notes Theor. Comput. Sci. 82(1) (2003)Google Scholar
  8. 8.
    Bloom, B., Istrail, S., Meyer, A.R.: Bisimulation can’t be traced. J. ACM 42(1), 232–268 (1995)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Brodo, L., Degano, P., Gilmore, S., Hillston, J., Priami, C.: Performance Evaluation for Global Computation. In: Priami, C. (ed.) GC 2003. LNCS, vol. 2874, pp. 229–253. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Cardelli, L., Gardner, P.: Processes in Space. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds.) CiE 2010. LNCS, vol. 6158, pp. 78–87. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Cardelli, L., Mardare, R.: The measurable space of stochastic processes. In: Proc. QEST, pp. 171–180. IEEE Computer Society (2010)Google Scholar
  12. 12.
    Danos, V., Desharnais, J., Laviolette, F., Panangaden, P.: Bisimulation and cocongruence for probabilistic systems. Inf. Comput. 204(4), 503–523 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    de Vink, E.P., Rutten, J.: Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 460–470. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  14. 14.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. Comput. 179(2), 163–193 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ding, J., Hillston, J.: Structural Analysis for Stochastic Process Algebra Models. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 1–27. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Doberkat, E.-E.: Stochastic relations: foundations for Markov transition systems. Chapman & Hall/CRC Studies in Informatics Series. Chapman & Hall/CRC (2007)Google Scholar
  17. 17.
    Giry, M.: A categorical approach to probability theory. In: Banaschewski (ed.) Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics, vol. 915, pp. 68–85. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  18. 18.
    Hermanns, H., Herzog, U., Katoen, J.-P.: Process algebra for performance evaluation. Theor. Comput. Sci. 274(1-2), 43–87 (2002)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Hillston, J.: Process algebras for quantitative analysis. In: Proc. LICS, pp. 239–248. IEEE Computer Society (2005)Google Scholar
  20. 20.
    Klin, B., Sassone, V.: Structural Operational Semantics for Stochastic Process Calculi. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 428–442. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Kurz, A.: Logics for Coalgebras and Applications to Computer Science. PhD thesis, Ludwig-Maximilians-Universität München (2000)Google Scholar
  22. 22.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94(1), 1–28 (1991)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Moss, L.S., Viglizzo, I.D.: Final coalgebras for functors on measurable spaces. Inf. Comput. 204(4), 610–636 (2006)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)Google Scholar
  25. 25.
    Plotkin, G.D.: A structural approach to operational semantics. DAIMI FN-19, Computer Science Department, Århus University, Århus (1981)Google Scholar
  26. 26.
    Smyth, M.B., Plotkin, G.D.: The category-theoretic solution of recursive domain equations. SIAM J. Comput. 11(4), 761–783 (1982)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Sokolova, A.: Probabilistic systems coalgebraically: A survey. Theoretical Comput. Sci. 412(38), 5095–5110 (2011)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Staton, S.: Relating coalgebraic notions of bisimulation. Logical Methods in Computer Science 7(1) (2011)Google Scholar
  29. 29.
    Turi, D., Plotkin, G.: Towards a mathematical operational semantics. In: Proc. 12th LICS Conf., pp. 280–291. IEEE Computer Society Press (1997)Google Scholar
  30. 30.
    Viglizzo, I.D.: Final Sequences and Final Coalgebras for Measurable Spaces. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 395–407. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Giorgio Bacci
    • 1
  • Marino Miculan
    • 1
  1. 1.Dept. of Mathematics and Computer ScienceUniversity of UdineItaly

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