The Quest for Minimal Quotients for Probabilistic Automata

  • Christian Eisentraut
  • Holger Hermanns
  • Johann Schuster
  • Andrea Turrini
  • Lijun Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7795)


One of the prevailing ideas in applied concurrency theory and verification is the concept of automata minimization with respect to strong or weak bisimilarity. The minimal automata can be seen as canonical representations of the behaviour modulo the bisimilarity considered. Together with congruence results wrt. process algebraic operators, this can be exploited to alleviate the notorious state space explosion problem. In this paper, we aim at identifying minimal automata and canonical representations for concurrent probabilistic models. We present minimality and canonicity results for probabilistic automata wrt. strong and weak bisimilarity, together with polynomial time minimization algorithms.


Normal Form Canonical Representation Label Transition System Probabilistic Automaton Transitive Reduction 
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.
    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)CrossRefGoogle Scholar
  2. 2.
    Barbot, B., Chen, T., Han, T., Katoen, J.-P., Mereacre, A.: Efficient CTMC Model Checking of Linear Real-Time Objectives. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 128–142. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Cattani, S., Segala, R.: Decision Algorithms for Probabilistic Bisimulation. In: Brim, L., Jančar, P., Křetínský, M., Kučera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 371–386. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Chehaibar, G., Garavel, H., Mounier, L., Tawbi, N., Zulian, F.: Specification and Verification of the PowerScaleTM Bus Arbitration Protocol: An Industrial Experiment with LOTOS. In: FORTE, pp. 435–450 (1996)Google Scholar
  5. 5.
    Crouzen, P., Lang, F.: Smart Reduction. In: Giannakopoulou, D., Orejas, F. (eds.) FASE 2011. LNCS, vol. 6603, pp. 111–126. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Deng, Y., Hennessy, M.: On the Semantics of Markov Automata. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 307–318. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Eisentraut, C., Hermanns, H., Zhang, L.: On Probabilistic Automata in Continuous Time. Reports of SFB/TR 14 AVACS 62, SFB/TR 14 AVACS, long version of LICS 342–351 (2010)Google Scholar
  8. 8.
    Fernandez, J.-C., Mounier, L.: A Tool Set for Deciding Behavioral Equivalences. In: Groote, J.F., Baeten, J.C.M. (eds.) CONCUR 1991. LNCS, vol. 527, pp. 23–42. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  9. 9.
    Hermanns, H., Katoen, J.-P.: Automated Compositional Markov Chain Generation for a Plain-Old Telephone System. Science of Computer Programming 36(1), 97–127 (2000)zbMATHCrossRefGoogle Scholar
  10. 10.
    Hermanns, H., Turrini, A.: Deciding Probabilistic Automata weak Bisimulation in Polynomial Time. In: FSTTCS, pp. 435–447 (2012)Google Scholar
  11. 11.
    Kanellakis, P.C., Smolka, S.A.: CCS Expressions, Finite State Processes, and Three Problems of Equivalence. In: PODC, pp. 228–240 (1983)Google Scholar
  12. 12.
    Kwiatkowska, M.Z., Norman, G., Parker, D.: PRISM 4.0: Verification of Probabilistic Real-Time Systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Larsen, K.G., Skou, A.: Bisimulation through Probabilistic Testing (Preliminary Report). In: POPL, pp. 344–352 (1989)Google Scholar
  14. 14.
    Lynch, N.A., Segala, R., Vaandrager, F.W.: Observing Branching Structure through Probabilistic Contexts. SIAM Journal on Computing 37(4), 977–1013 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Milner, R.: Communication and Concurrency. Prentice-Hall International (1989)Google Scholar
  16. 16.
    Paige, R., Tarjan, R.E.: Three Partition Refinement Algorithms. SIAM Journal on Computing 16(6), 973–989 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, MIT (1995)Google Scholar
  18. 18.
    Segala, R., Lynch, N.A.: Probabilistic Simulations for Probabilistic Processes. Nordic Journal of Computing 2(2), 250–273 (1995)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christian Eisentraut
    • 1
  • Holger Hermanns
    • 1
  • Johann Schuster
    • 2
  • Andrea Turrini
    • 1
  • Lijun Zhang
    • 3
    • 1
  1. 1.Department of Computer ScienceSaarland UniversityGermany
  2. 2.Department of Computer ScienceUniversity of the Federal Armed Forces MunichGermany
  3. 3.DTU InformaticsTechnical University of DenmarkDenmark

Personalised recommendations