Advertisement

Duality for Labelled Markov Processes

  • Michael Mislove
  • Joël Ouaknine
  • Dusko Pavlovic
  • James Worrell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2987)

Abstract

Labelled Markov processes (LMPs) are automata whose transitions are given by probability distributions. In this paper we present a ‘universal’ LMP as the spectrum of a commutative C *-algebra consisting of formal linear combinations of labelled trees. We characterize the state space of the universal LMP as the set of homomorphims from an ordered commutative monoid of labelled trees into the multiplicative unit interval. This yields a simple semantics for LMPs which is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with entry points and exit points in the setting of iteration theories. We define an iteration theory of LMPs by specifying its categorical dual: a certain category of C *-algebras. We find that the basic operations for composing LMPs have simple definitions in the dual category.

Keywords

Commutative Ring Probabilistic Choice Exit Point Label Transition System Ring Homomorphism 
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.

References

  1. 1.
    Abramsky, S.: A Domain Equation for Bisimulation. Information and Computation 92, 161–218 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abramsky, S., Vickers, S.: Quantales, observational logic and process semantics. Mathematical Structures in Computer Science 3, 161–227 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Abramsky, S., Blute, R., Panangaden, P.: Nuclear and Trace Ideals in Tensor-∗- categories. Journal of Pure and Applied Algebra 143, 3–47 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Aceto, L., Ésik, Z., Ingólfsdóttir, A.: Equational Axioms for Probabilistic Bisimilarity. In: Kirchner, H., Ringeissen, C. (eds.) AMAST 2002. LNCS, vol. 2422, pp. 239–253. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Bloom, S., Esik, Z.: Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  6. 6.
    Bloom, S., Esik, Z.: The Equational Logic of Fixed Points. Theoretical Computer Science 179, 1–60Google Scholar
  7. 7.
    van Breugel, F., Worrell, J.B.: An algorithm for quantitative verification of probabilistic transition systems. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, p. 336. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    van Breugel, F., Mislove, M., Ouaknine, J., Worrell, J.: Domains, Testing and Similarity for Labelled Markov Processes. To appear in Proceedings of FOSSACS 2003, Theoret. Comp. Sci. (2003)Google Scholar
  9. 9.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for Labelled Markov Processes. Information and Computation 179(2), 163–193 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labeled markov systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, p. 258. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating Labeled Markov Processes. Information and Computation 184(1), 160–200 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Esik, Z., Kuich, W.: Formal Tree Series. Journal of Automata Languages and Combinatorics 8(2), 145–185 (2003)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Naimark, M.A.: Normed Rings, 2nd edn., Nauka, Moscow (1968), reprint of the revised English edition, Wolters-Noordhoff, Groningen (1970)Google Scholar
  14. 14.
    van Glabbeek, R., Smolka, S., Steffen, B.: Reactive, generative and stratified models of probabilistic processes. Information and Computation 121(1), 59–80 (1996)CrossRefGoogle Scholar
  15. 15.
    Kozen, D.: The Semantics of Probabilistic Programs. Journal of Computer and System Science 22, 328–350 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Johnstone, P.: Stone Spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  17. 17.
    Jonsson, B., Larsen, K., Yi, W.: Probabilistic Extensions of Process Algebras. In: Bergstra, J.A., Ponse, A., Smolka, S. (eds.) Handbook of Process Algebra, pp. 685–710. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  18. 18.
    Larsen, K.G., Skou, A.: Bisimulation through Probabilistic Testing. Information and Computation 94(1), 1–28 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mislove, M., Ouaknine, J., Worrell, J.: Axioms for Probability and Nondeterminism. In: Proc. EXPRESS 2003. ENTCS, vol. 91(3) (2003)Google Scholar
  20. 20.
    Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, London (1967)zbMATHGoogle Scholar
  21. 21.
    Di Pierro, A., Hankin, C., Wiklicky, H.: Quantitative Relations and Approximate Process Equivalences. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 508–522. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Stark, E.W., Smolka, S.A.: A complete axiom system for finite-state probabilistic processes. In: Proof, Language, and Interaction: Essays in Honour of Robin Milner, MIT Press, Cambridge (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michael Mislove
    • 1
  • Joël Ouaknine
    • 2
  • Dusko Pavlovic
    • 3
  • James Worrell
    • 1
  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA
  2. 2.Computer Science DepartmentCarnegie Mellon UniversityUSA
  3. 3.Kestrel InstitutePalo AltoUSA

Personalised recommendations