The Converse of a Stochastic Relation

  • Ernst-Erich Doberkat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2620)


Transition probabilities are proposed as the stochastic counterparts to set-based relations. We propose the construction of the converse of a stochastic relation. It is shown that two of the most useful properties carry over: the converse is idempotent as well as anticommutative. The nondeterminism associated with a stochastic relation is defined and briefly investigated. We define a bisimulation relation, and indicate conditions under which this relation is transitive; moreover it is shown that bisimulation and converse are compatible.


Stochastic relations concurrency bisimulation converse relational calculi nondeterminism 


  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]
    P. Aczel and N. Mendler. A final coalgebra theorem. In H. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 357–365, 1989.CrossRefGoogle Scholar
  3. [3]
    P. Billingsley. Probability and Measure. JohnWiley and Sons, New York, 3 edition, 1995.zbMATHGoogle Scholar
  4. [4]
    C. Brink, W. Kahl, and G. Schmidt, editors. Relational Methods in Computer Science. Advances in Computing Science. Springer-Verlag, Wien, New York, 1997.Google Scholar
  5. [5]
    D. Cantone, E. G. Omodeo, and A. Policriti. Set Theory for Computing. Springer-Verlag, 2001. In print.Google Scholar
  6. [6]
    J. Desharnais, A. Edalat, and P. Panangaden. Bisimulation of labelled Markovprocesses. Technical report, School of Computer Science, McGill University, Montreal, 1998.Google Scholar
  7. [7]
    E.-E. Doberkat. Stochastic Automata — Nondeterminism, Stability, and Prediction, volume 113 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1981.Google Scholar
  8. [8]
    E.-E. Doberkat. The demonic product of probabilistic relations. InMogens 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
  9. [9]
    E.-E. Doberkat. Semi-pullbacks and bisimulations in categories of stochastic relations. Technical Report 130, Chair for Software Technology, University of Dortmund, November 2002.Google Scholar
  10. [10]
    A. Edalat. Semi-pullbacks and bisimulation in categories of Markov processes. Math. Struct. in Comp. Science, 9(5):523–543, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    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.Google Scholar
  12. [12]
    S. Mac Lane. Categories for the Working Mathematician. Number 5 in Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2 edition, 1997.Google Scholar
  13. [13]
    E. Michael. Topologies on spaces of subsets. Trans. Am. Math. Soc., 71(2):152–182, 1951.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    P. Panangaden. Probabilistic relations. In C. Baier, M. Huth, M. Kwiatkowska, and M. Ryan, editors, Proc. PROBMIV, pages 59–74, 1998. Also available from the School of Computer Science, McGill University, Montreal.Google Scholar
  15. [15]
    P. Panangaden. Does combining nondeterminism and probability make sense? Bulletin of the EATCS, (75):182–189, Oct. 2001.Google Scholar
  16. [16]
    K. R. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, New York, 1967.zbMATHGoogle Scholar
  17. [17]
    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
  18. [18]
    A. Tarski and S. Givant. A Formalization of Set-Theory Without Variables. Number 42 in Colloquium Publications. American Mathematical Society, Providence, R. I., 1987.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ernst-Erich Doberkat
    • 1
  1. 1.Chair for Software TechnologyUniversity of DortmundDortmund

Personalised recommendations