Hyperfinite Approximations to Labeled Markov Transition Systems

  • Ernst-Erich Doberkat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4019)


The problem of finding an approximation to a labeled Markov transition system through hyperfinite transition systems is addressed. It is shown that we can find for each countable family of stochastic relations on Polish spaces a family of relations defined on a hyperfinite set that is infinitely close. This is applied to Kripke models for a simple modal logic in the tradition of Larsen and Skou. It follows that we can find for each Kripke model a hyperfinite one which is infinitely close.


Modal Logic Transition System Weak Topology Polish Space Atomic Proposition 
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.


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  1. 1.
    Anderson, R.A.: Star-finite representations of measure spaces. Trans. Amer. Math. Soc. 271(2), 667–687 (1982)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Anderson, R.M., Rashid, S.: A nonstandard characterization of weak convergence. Proc. Amer. Math. Soc. 69(2), 327–332 (1978)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blackburn, P., de Rjike, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  4. 4.
    Cutland, N.: Nonstandard measure theory and its applications. Bull. London Math. Soc. 15, 529–589 (1983)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Danos, V., Desharnais, J., Panangaden, P.: Labelled markov processes: Stronger and faster approximations. Electr. Notes Theor. Comp. Sci. 87, 44 (2004)Google Scholar
  6. 6.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation of labelled Markov-processes. Information and Computation 179(2), 163–193 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Desharnais, J., Jagadeesan, R., Gupta, V., Panangaden, P.: Approximating labeled Markov processes. In: Proc. 15th Ann. IEEE Symp. on Logic in Computer Science, pp. 95–106. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  8. 8.
    Doberkat, E.-E.: Semi-pullbacks and bisimulations in categories of stochastic relations. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 996–1007. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Doberkat, E.-E.: Semi-pullbacks for stochastic relations over analytic spaces. Math. Struct. Comp. Sci. 15, 647–670 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Doberkat, E.-E.: Notes on stochastic relations. Research report, Department of Computer Science, University of Dortmund (January 2006)Google Scholar
  11. 11.
    Doberkat, E.-E.: Stochastic relations: congruences, bisimulations and the Hennessy-Milner theorem. SIAM J. Computing 35(3), 590–626 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fajardo, S., Keisler, H.J.: Model Theory of Stochastic Processes, Lecture Notes in Logic. Association for Symbolic Logic, Natick, Mass., vol. 14 (2002)Google Scholar
  13. 13.
    Giry, M.: A categorical approach to probability theory. In: Categorical Aspects of Topology and Analysis. Lect. Notes Math., vol. 915, pp. 68–85. Springer, Berlin (1981)CrossRefGoogle Scholar
  14. 14.
    Keisler, H.J.: Infinitesimals in probability theory. In: Cutland, N. (ed.) Nonstandard analysis and its applications. London Mathematical Society Student Texts, vol. 10, pp. 106–139. Cambridge University Press, Cambridge (1988)Google Scholar
  15. 15.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94, 1–28 (1991)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lindstrøm, T.: An invitation to nonstandard analysis. In: Cutland, N. (ed.) Nonstandard analysis and its applications. London Mathematical Society Student Texts, vol. 10, pp. 1–105. Cambridge University Press, Cambridge (1988)Google Scholar
  17. 17.
    Loeb, P.A.: Conversion to nonstandard measure spaces and applications to probability theory. Trans. Amer. Math. Soc. 211, 113–122 (1975)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Srivastava, S.M.: A Course on Borel Sets. Graduate Texts in Mathematics. Springer, Berlin (1998)MATHGoogle Scholar
  19. 19.
    Stroyan, K.D., Luxemburg, W.A.J.: Introduction to the theory of Infinitesimals. In: Pure and Applied Mathematics, Academic Press, New York (1976)Google Scholar
  20. 20.
    Sun, Y.: Economics and nonstandard analysis. In: Loeb, P.A., Wolff, M. (eds.) Nonstandard analysis for the Working Mathematician, Mathematics and its Applications, pp. 259–305. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  21. 21.
    van Breugel, F., Mislove, M., Ouaknine, J., Worrell, J.: Domain theory, testing and simulation for labelled Markov processes. Theoret. Comp. Sci. 333, 171–197 (2005)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

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

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