Advertisement

Semi-pullbacks and Bisimulations in Categories of Stochastic Relations

  • Ernst-Erich Doberkat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2719)

Abstract

The problem of constructing a semi-pullback in a category is intimately connected to the problem of establishing the transitivity of bisimulations. Edalat shows that a semi-pullback can be constructed in the category of Markov processes on Polish spaces, when the underlying transition probability functions are universally measurable, and the morphisms are measure preserving continuous maps. We demonstrate that the simpler assumption of Borel measurability suffices. Markov processes are in fact a special case: we consider the category of stochastic relations over Standard Borel spaces. At the core of the present solution lies a selection argument from stochastic dynamic optimization. An example demonstrates that (weak) pullbacks do not exist in the category of Markov processes.

Keywords

Bisimulation semi-pullback stochastic relations labelled Markov processes Hennessy-Milner logic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Abramsky, R. Blute, and P. Panangaden. Nuclear and trace ideal in tensored *-categories. Journal of Pure and Applied Algebra, 143(1–3):3–47, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Desharnais, A. Edalat, and P. Panangaden. Bisimulation of labelled markov-processes. Information and Computation, 179(2):163–193, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Approximating labeled Markov processes. In Proc. 15th Symposium on Logic in Computer Science, pages 95–106. IEEE, 2000.Google Scholar
  4. [4]
    E.-E. Doberkat. The demonic product of probabilistic relations. In Mogens Nielsen and Uffe Engberg, editors, Proc. Foundations of Software Science and Computation Structures, volume 2303 of Lecture Notes in Computer Science, pages 113–127, Berlin, 2002. Springer-Verlag.Google Scholar
  5. [5]
    E.-E. Doberkat. A remark on A. Edalat’s paper Semi-Pullbacks and Bisimulations in Categories of Markov-Processes. Technical Report 125, Chair for Software Technology, University of Dortmund, July 2002.Google Scholar
  6. [6]
    A. Edalat. Semi-pullbacks and bisimulation in categories of Markov processes. Math. Struct. in Comp. Science, 9(5):523–543, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    M. Giry. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis, volume 915 of Lecture Notes in Mathematics, pages 68–85, Berlin, 1981. Springer-Verlag.CrossRefMathSciNetGoogle Scholar
  8. [8]
    C. J. Himmelberg and F. Van Vleck. Some selection theorems for measurable functions. Can. J. Math., 21:394–399, 1969.zbMATHGoogle Scholar
  9. [9]
    K. R. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, New York, 1967.zbMATHGoogle Scholar
  10. [10]
    J. J. M. M. Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science, 249(1):3–80, 2000. Special issue on modern algebra and its applications.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    S. M. Srivastava. A Course on Borel Sets. Number 180 in Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1998.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ernst-Erich Doberkat
    • 1
  1. 1.University of DortmundDortmund

Personalised recommendations