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Approximating Labelled Markov Processes Again!

  • Philippe Chaput
  • Vincent Danos
  • Prakash Panangaden
  • Gordon Plotkin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5728)

Abstract

Labelled Markov processes are continuous-state fully probabilistic labelled transition systems. They can be seen as co-algebras of a suitable monad on the category of measurable space. The theory as developed so far included a treatment of bisimulation, logical characterization of bisimulation, weak bisimulation, metrics, universal domains for LMPs and approximations. Much of the theory involved delicate properties of analytic spaces.

Recently a new kind of averaging procedure was used to construct approximations. Remarkably, this version of the theory uses a dual view of LMPs and greatly simplifies the theory eliminating the need to consider aanlytic spaces. In this talk I will survey some of the ideas that led to this work.

Keywords

Markov Process Conditional Expectation Markov Decision Process Ultrametric Space Markov Kernel 
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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References

  1. 1.
    Sontag, E.: Mathematical Control Theory. Texts in Applied Mathematics, vol. 6. Springer, Heidelberg (1990)zbMATHGoogle Scholar
  2. 2.
    Blute, R., Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. In: Proceedings of the Twelfth IEEE Symposium On Logic In Computer Science, Warsaw, Poland (1997)Google Scholar
  3. 3.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labeled Markov processes. Information and Computation 179(2), 163–193 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  5. 5.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, Chichester (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Larsen, K.G., Skou, A.: Bisimulation through probablistic testing. Information and Computation 94, 1–28 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Vink, E., Rutten, J.J.M.M.: Bisimulation for probabilistic transition systems: A coalgebraic approach. In: Proceedings of the 24th International Colloquium On Automata Languages And Programming (1997)Google Scholar
  8. 8.
    Kozen, D.: A probabilistic PDL. Journal of Computer and Systems Sciences 30(2), 162–178 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Danos, V., Desharnais, J., Panangaden, P.: Conditional expectation and the approximation of labelled markov processes. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 477–491. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Feller, W.: An Introduction to Probability Theory and its Applications II, 2nd edn. John Wiley and Sons, Chichester (1971)zbMATHGoogle Scholar
  11. 11.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: The metric analogue of weak bisimulation for labelled Markov processes. In: Proceedings of the Seventeenth Annual IEEE Symposium On Logic In Computer Science, pp. 413–422 (2002)Google Scholar
  12. 12.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Van Nostrand (1960)Google Scholar
  13. 13.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labeled Markov processes. Information and Computation 184(1), 160–200 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: A metric for labelled Markov processes. Theoretical Computer Science 318(3), 323–354 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hopf, E.: The general temporally discrete Markoff process. J. Rational Math. Mech. Anal. 3, 13–45 (1954)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Danos, V., Desharnais, J., Laviolette, F., Panangaden, P.: Bisimulation and cocongruence for probabilistic systems. Information and Computation 204(4), 503–523 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bartels, F., Sokolova, A., de Vink, E.: A hierarchy of probabilistic system types. Theoretical Computer Science 327, 3–22 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Billingsley, P.: Probability and Measure. Wiley Interscience, Hoboken (1995)zbMATHGoogle Scholar
  19. 19.
    Choksi, J.: Inverse limits on measure spaces. Proc. London Math. Soc. 8(3), 321–342 (1958)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Philippe Chaput
    • 1
  • Vincent Danos
    • 2
  • Prakash Panangaden
    • 1
  • Gordon Plotkin
    • 2
  1. 1.School of Computer ScienceMcGill UniversityCanada
  2. 2.School of InformaticsUniversity of EdinburghUK

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