Taking It to the Limit: Approximate Reasoning for Markov Processes

  • Kim Guldstrand Larsen
  • Radu Mardare
  • Prakash Panangaden
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


We develop a fusion of logical and metrical principles for reasoning about Markov processes. More precisely, we lift metrics from processes to sets of processes satisfying a formula and explore how the satisfaction relation behaves as sequences of processes and sequences of formulas approach limits. A key new concept is dynamically-continuous metric bisimulation which is a property of (pseudo)metrics. We prove theorems about satisfaction in the limit, robustness theorems as well as giving a topological characterization of various classes of formulas. This work is aimed at providing approximate reasoning principles for Markov processes.


Markov Process Convergent Sequence Logical Formula Approximate Reasoning 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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  1. 1.
    Arveson, W.: An Invitation to C *-Algebra. Springer (1976)Google Scholar
  2. 2.
    Aumann, R.: Interactive epistemology I: knowledge. International Journal of Game Theory 28, 263–300 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ballarini, P., Mardare, R., Mura, I.: Analysing Biochemical Oscillations through Probabilistic Model Checking. In: FBTC 2008. ENTCS, vol. 229(1), pp. 3–19 (2009)Google Scholar
  4. 4.
    Cardelli, L., Larsen, K.G., Mardare, R.: Continuous Markovian logic - from complete axiomatization to the metric space of formulas. In: CSL, pp. 144–158 (2011)Google Scholar
  5. 5.
    Cardelli, L., Larsen, K.G., Mardare, R.: Modular Markovian Logic. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 380–391. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labeled Markov processes. I&C 179(2), 163–193 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labeled Markov processes. I&C 184(1), 160–200 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: A metric for labelled Markov processes. TCS 318(3), 323–354 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Desharnais, J., Panangaden, P.: Continuous stochastic logic characterizes bisimulation for continuous-time Markov processes. JLAP 56, 99–115 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Doberkat, E.-E.: Stochastic Relations. Foundations for Markov Transition Systems. Chapman and Hall, New York (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brookes/Cole (1989)Google Scholar
  12. 12.
    de Vink, E., Rutten, J.J.M.M.: Bisimulation for probabilistic transition systems: A coalgebraic approach. TCS 221(1/2), 271–293 (1999)zbMATHCrossRefGoogle Scholar
  13. 13.
    Fagin, R., Halpern, J.Y.: Reasoning about knowledge and probability. JACM 41(2), 340–367 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Giacalone, A., Jou, C.-C., Smolka, S.A.: Algebraic Reasoning for Probabilistic Concurrent Systems. In: IFIP WG 2.2/2.3 Working Conference on Programming Concepts and Methods (1990)Google Scholar
  15. 15.
    Larsen, K.G., Skou, A.: Bisimulation through probablistic testing. Information and Computation 94, 1–28 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Moss, L.S., Viglizzo, I.D.: Harsanyi type spaces and final coalgebras constructed from satisfied theories. ENTCS 106, 279–295 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)Google Scholar
  18. 18.
    van Breugel, F., Mislove, M.W., Ouaknine, J., Worrell, J.B.: An Intrinsic Characterization of Approximate Probabilistic Bisimilarity. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 200–215. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    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, pp. 336–350. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Zhou, C.: A complete deductive system for probability logic with application to Harsanyi type spaces. PhD thesis, Indiana University (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kim Guldstrand Larsen
    • 1
  • Radu Mardare
    • 1
  • Prakash Panangaden
    • 2
  1. 1.Aalborg UniversityDenmark
  2. 2.McGill UniversityCanada

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