Comparative Branching-Time Semantics for Markov Chains

  • Christel Baier
  • Holger Hermanns
  • Joost-Pieter Katoen
  • Verena Wolf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2761)


This paper presents various semantics in the branching-time spectrum of discrete-time and continuous-time Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation pre-orders are covered and are logically characterised in terms of the temporal logics PCTL and CSL. Apart from presenting various existing branching-time relations in a uniform manner, our contributions are: (i) weak simulation for DTMCs is defined, (ii) weak bisimulation equivalence is shown to coincide with weak simulation equivalence, (iii) logical characterisation of weak (bi)simulations are provided, and (iv) a classification of branching-time relations is presented, elucidating the semantics of DTMCs, CTMCs and their interrelation.


Markov Chain Exit Rate Reachability Condition Atomic Proposition Simulation Relation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    de Alfaro, L.: Temporal logics for the specification of performance and reliability. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 165–176. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Andova, S., Baeten, J.: Abstracion in probabilistic process algebra. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 204–219. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Aziz, A., Singhal, V., Balarin, F., Brayton, R., Sangiovanni-Vincentelli, A.: It usually works: the temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)Google Scholar
  4. 4.
    Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996)Google Scholar
  5. 5.
    Baier, C.: On algorithmic verification methods for probabilistic systems. Habilitation thesis, University of Mannheim (1998)Google Scholar
  6. 6.
    Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: Model checking continuous-time Markov chains by transient analysis. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 358–372. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Baier, C., Hermanns, H.: Weak bisimulation for fully probabilistic processes. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 119–130. Springer, Heidelberg (1997)Google Scholar
  8. 8.
    Baier, C., Katoen, J.-P., Hermanns, H., Haverkort, B.: Simulation for continuous-time Markov chains. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 338–354. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Baier, C., Hermanns, H., Katoen, J.-P., Wolf, V.: Comparative branching-time semantics for Markov chains. Tech. Rep., Univ. of Twente (2003)Google Scholar
  10. 10.
    Baier, C., Katoen, J.-P., Hermanns, H.: Approximate symbolic model checking of continuoustime Markov chains. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–162. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Bernardo, M., Gorrieri, R.: Extended Markovian process algebra. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 315–330. Springer, Heidelberg (1996)Google Scholar
  12. 12.
    Bravetti, M.: Revisiting interactive Markov chains. In: 3rd Workshop on Models for Time-Critical Systems. BRICS Notes NP-02-3, pp. 68–88 (2002)Google Scholar
  13. 13.
    Brown, M., Clarke, E., Grumberg, O.: Characterizing finite Kripke structures in propositional temporal logic. Th. Comp. Sc. 59, 115–131 (1988)CrossRefGoogle Scholar
  14. 14.
    Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. J. of Appl. Prob. 31, 59–75 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Clarke, E., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Tr. on Progr. Lang. and Sys. 16(5), 1512–1542 (1994)CrossRefGoogle Scholar
  16. 16.
    Desharnais, J.: Labelled Markov Processes. PhD Thesis, McGill University (1999)Google Scholar
  17. 17.
    Desharnais, J.: Logical characterisation of simulation for Markov chains. Workshop on Probabilistic Methods in Verification, Tech. Rep. CSR-99-8, Univ. of Birmingham, pp. 33–48 (1999)Google Scholar
  18. 18.
    Desharnais, J., Edalat, A., Panangaden, P.: Alogical characterisation of bisimulation for labeled Markov processes. In: IEEE Symp. on Logic in Comp. Sc., pp. 478–487 (1998)Google Scholar
  19. 19.
    Desharnais, J., Panangaden, P.: Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes. J. of Logic and Alg. Progr. 56, 99–115 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled Markov processes. In: IEEE Symp. on Logic in Comp. Sc., pp. 95–106 (2000)Google Scholar
  21. 21.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Weak bisimulation is sound and complete for PCTL∗. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 355–370. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    van Glabbeek, R.J.: The linear time – branching time spectrum I. The semantics of concrete, sequential processes. Ch. 1 in Handbook of Process Algebra, pp. 3–100 (2001)Google Scholar
  23. 23.
    van Glabbeek, R.J.: The linear time – branching time spectrum II. The semantics of sequential processes with silent moves. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 66–81. Springer, Heidelberg (1993)Google Scholar
  24. 24.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative, and stratified models of probabilistic processes. Inf. & Comp. 121, 59–80 (1995)zbMATHCrossRefGoogle Scholar
  25. 25.
    Feller, W.: An Introduction to Probability Theory and its Applications. John Wiley, Chichester (1968)zbMATHGoogle Scholar
  26. 26.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Form. Asp. Of Comp. 6, 512–535 (1994)zbMATHCrossRefGoogle Scholar
  27. 27.
    Hermanns, H. (ed.): Interactive Markov Chains. LNCS, vol. 2428. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  28. 28.
    Hillston, J.: A Compositional Approach to Performance Modelling. Cambr. Univ. Press, Cambridge (1996)CrossRefGoogle Scholar
  29. 29.
    Jones, C., Plotkin, G.: A probabilistic powerdomain of evaluations. In: IEEE Symp. on Logic in Computer Science, pp. 186–195 (1989)Google Scholar
  30. 30.
    Jonsson, B.: Simulations between specifications of distributed systems. In: Groote, J.F., Baeten, J.C.M. (eds.) CONCUR 1991. LNCS, vol. 527, pp. 346–360. Springer, Heidelberg (1991)Google Scholar
  31. 31.
    Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: IEEE Symp. on Logic in Comp. Sc., pp. 266–277 (1991)Google Scholar
  32. 32.
    Jou, C.-C., Smolka, S.A.: Equivalences, congruences, and complete axiomatizations for probabilistic processes. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 367–383. Springer, Heidelberg (1990)Google Scholar
  33. 33.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Van Nostrand (1960)Google Scholar
  34. 34.
    Kulkarni, V.G.: Modeling and Analysis of Stochastic Systems. Chapman & Hall, Boca Raton (1995)zbMATHGoogle Scholar
  35. 35.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. and Comp. 94(1), 1–28 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  37. 37.
    De Nicola, R., Vaandrager, F.: Three logics for branching bisimulation (extended abstract). In: IEEE Symp. on Logic in Comp. Sc., pp. 118–129 (1992)Google Scholar
  38. 38.
    Philippou, A., Lee, I., Sokolsky, O.: Weak bisimulation for probabilistic systems. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 334–349. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  39. 39.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, Chichester (1994)zbMATHGoogle Scholar
  40. 40.
    Segala, R., Lynch, N.A.: Probabilistic simulations for probabilistic processes. Nordic J. of Computing 2(2), 250–273 (1995)zbMATHMathSciNetGoogle Scholar
  41. 41.
    Stoelinga, M.I.A.: Verification of Probabilistic, Real-Time and Parametric Systems. PhD Thesis, University of Nijmegen (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Christel Baier
    • 1
  • Holger Hermanns
    • 2
    • 3
  • Joost-Pieter Katoen
    • 2
  • Verena Wolf
    • 1
  1. 1.Institut für Informatik IUniversity of BonnBonnGermany
  2. 2.Department of Computer ScienceUniversity of TwenteEnschedeThe Netherlands
  3. 3.Department of Computer ScienceSaarland UniversitySaarbrückenGermany

Personalised recommendations