Skip to main content

Symbolic and Parametric Model Checking of Discrete-Time Markov Chains

  • Conference paper
Theoretical Aspects of Computing - ICTAC 2004 (ICTAC 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3407))

Included in the following conference series:


We present a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains. The probability with which a path formula is satisfied is represented by a regular expression. A recursive evaluation of the regular expression yields an exact rational value when transition probabilities are rational, and rational functions when some probabilities are left unspecified as parameters of the system. This allows for parametric model checking by evaluating the regular expression for different parameter values, for instance, to study the influence of a lossy channel in the overall reliability of a randomized protocol.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others


  1. Aziz, A., Singhal, V., Balarin, F., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: 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 

  2. Bergstra, J., Fokkink, W., Ponse, A.: Process algebra with recursive operations. In: Bergstra, J., Ponse, A., Smolka, S. (eds.) Handbook of Process Algebrea. Elsevier Science, Amsterdam (2001)

    Google Scholar 

  3. Bergstra, J.A., Bethke, I., Ponse, A.: Process algebra with iteration and nesting. The Computer Journal 37(4), 243–258 (1994)

    Article  Google Scholar 

  4. Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs in Computer Science. Springer, Heidelberg (1988)

    Google Scholar 

  5. Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)

    Google Scholar 

  6. Bohnenkamp, H., van der Stok, P., Hermanns, H., Vaandrager, F.: Cost-optimization of the IPv4 zeroconf protocol. In: Proceedings of the 2003 International Conference on Dependable Systems and Networks (DSN 2003), June 2003, pp. 531–540. IEEE Computer Society, Los Alamitos (2003)

    Chapter  Google Scholar 

  7. Brzozowsky, J.: Derivatives of regular expressions. Journal of ACM 11(4) (1964)

    Google Scholar 

  8. Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. D’Argenio, P., Jeannet, B., Jensen, H., Larsen, K.: Reachability analysis of probabilistic systems by successive refinements. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 29–56. Springer, Heidelberg (2001)

    Google Scholar 

  10. D’Argenio, P., Jeannet, B., Jensen, H., Larsen, K.: Reduction and refinement strategies for probabilistic analysis. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, p. 57. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  11. Gordon, M.J.C.: HOL: A proof generating system for higher-order logic. In: Birtwistle, G., Subrahmanyam, P.A. (eds.) VLSI Specification, Verification and Synthesis, pp. 73–128. Kluwer Academic Publishers, Boston (1988)

    Google Scholar 

  12. Gordon, M.J.C., Melham, T.F.: Introduction to HOL (A theorem-proving environment for higher order logic). Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  13. Gramond, Rodger: Using JFLAP to interact with theorems in automata theory. SIGCSEB: SIGCSE Bulletin (ACM Special Interest Group on Computer Science Education) 31 (1999)

    Google Scholar 

  14. Hansson, H., Jonsson, B.: A logic for reasoning about time and probability. Formal Aspects of Computing 6(5), 512–535 (1994)

    Article  MATH  Google Scholar 

  15. Hopcroft, J.E., Motwani, R.R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Longman, Reading (2000)

    Google Scholar 

  16. Hurd, J.: Formal Verification of Probabilistic Algorithms. PhD thesis, University of Cambridge (2002)

    Google Scholar 

  17. Jeannet, B., D’Argenio, P., Larsen, K.: Rapture: A tool for verifying Markov Decision Processes. In: I. Cerna, editor, Tools Day 2002, Brno, Czech Republic, Technical Report. Faculty of Informatics, Masaryk University Brno (2002)

    Google Scholar 

  18. JFLAP (java formal languages and automata package) web page,

  19. Kemeny, J., Snell, J., Knapp, A.: Denumerable Markov Chains. Graduate Texts in Mathematics, 2nd edn. Springer, Heidelberg (1976)

    Google Scholar 

  20. Knuth, D.E., Yao, A.C.: The complexity of nonuniform random number generation. In: Traub, J.F. (ed.) Algorithms and Complexity: New Directions and Recent Results. Academic Press, New York (1976)

    Google Scholar 

  21. Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, pp. 200–204. Springer, Heidelberg (2002)

    Google Scholar 

  22. PRISM web page,

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Daws, C. (2005). Symbolic and Parametric Model Checking of Discrete-Time Markov Chains. In: Liu, Z., Araki, K. (eds) Theoretical Aspects of Computing - ICTAC 2004. ICTAC 2004. Lecture Notes in Computer Science, vol 3407. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25304-4

  • Online ISBN: 978-3-540-31862-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics