Confluence Reduction for Probabilistic Systems

  • Mark Timmer
  • Mariëlle Stoelinga
  • Jaco van de Pol
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6605)


This paper presents a novel technique for state space reduction of probabilistic specifications, based on a newly developed notion of confluence for probabilistic automata. We prove that this reduction preserves branching probabilistic bisimulation and can be applied on-the-fly. To support the technique, we introduce a method for detecting confluent transitions in the context of a probabilistic process algebra with data, facilitated by an earlier defined linear format. A case study demonstrates that significant reductions can be obtained.


  1. 1.
    Baier, C., D’Argenio, P.R., Größer, M.: Partial order reduction for probabilistic branching time. In: Proc. of the 3rd Workshop on Quantitative Aspects of Programming Languages (QAPL). ENTCS, vol. 153(2), pp. 97–116 (2006)Google Scholar
  2. 2.
    Baier, C., Größer, M., Ciesinski, F.: Partial order reduction for probabilistic systems. In: Proc. of the 1st International Conference on Quantitative Evaluation of Systems (QEST), pp. 230–239. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  3. 3.
    Blom, S.C.C.: Partial τ-confluence for efficient state space generation. Technical Report SEN-R0123, CWI, Amsterdam (2001)Google Scholar
  4. 4.
    Blom, S.C.C., van de Pol, J.C.: State space reduction by proving confluence. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 596–609. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    D’Argenio, P.R., Niebert, P.: Partial order reduction on concurrent probabilistic programs. In: Proc. of the 1st International Conference on Quantitative Evaluation of Systems (QEST), pp. 240–249. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  6. 6.
    Fokkink, W., Pang, J.: Simplifying Itai-Rodeh leader election for anonymous rings. In: Proc. of the 4th International Workshop on Automated Verification of Critical Systems (AVoCS). ENTCS, vol. 128(6), pp. 53–68 (2005)Google Scholar
  7. 7.
    Giro, S., D’Argenio, P.R., Ferrer Fioriti, L.M.: Partial order reduction for probabilistic systems: A revision for distributed schedulers. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 338–353. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Groote, J.F., Sellink, M.P.A.: Confluence for process verification. Theoretical Computer Science 170(1-2), 47–81 (1996)CrossRefMATHGoogle Scholar
  9. 9.
    Größer, M.: Reduction Methods for Probabilistic Model Checking. PhD thesis, Technische Universität Dresden (2008)Google Scholar
  10. 10.
    Katoen, J.-P., van de Pol, J.C., Stoelinga, M.I.A., Timmer, M.: A linear process-algebraic format for probabilistic systems with data. In: Proc. of the 10th International Conference on Application of Concurrency to System Design (ACSD), pp. 213–222. IEEE Computer Society, Los Alamitos (2010)Google Scholar
  11. 11.
    De Nicola, R., Vaandrager, F.W.: Action versus state based logics for transition systems. In: Guessarian, I. (ed.) LITP 1990. LNCS, vol. 469, pp. 407–419. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  12. 12.
    Pace, G.J., Lang, F., Mateescu, R.: Calculating τ-confluence compositionally. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 446–459. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Peled, D.: All from one, one for all: on model checking using representatives. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 409–423. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  14. 14.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology (1995)Google Scholar
  15. 15.
    Segala, R., Lynch, N.A.: Probabilistic simulations for probabilistic processes. Nordic Journal of Computation 2(2), 250–273 (1995)MATHGoogle Scholar
  16. 16.
    Stoelinga, M.I.A.: Alea jacta est: verification of probabilistic, real-time and parametric systems. PhD thesis, University of Nijmegen (2002)Google Scholar
  17. 17.
    Timmer, M., Stoelinga, M.I.A., van de Pol, J.C.: Confluence reduction for probabilistic systems (extended version). Technical Report 1011.2314, ArXiv e-prints (2010)Google Scholar
  18. 18.
    van de Pol, J.C., Timmer, M.: State space reduction of linear processes using control flow reconstruction. In: Liu, Z., Ravn, A.P. (eds.) ATVA 2009. LNCS, vol. 5799, pp. 54–68. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    van Glabbeek, R.J., Weijland, W.P.: Branching time and abstraction in bisimulation semantics. Journal of the ACM 43(3), 555–600 (1996)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mark Timmer
    • 1
  • Mariëlle Stoelinga
    • 1
  • Jaco van de Pol
    • 1
  1. 1.Formal Methods and Tools, Faculty of EEMCSUniversity of TwenteThe Netherlands

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