Partial Order Reduction for Probabilistic Systems: A Revision for Distributed Schedulers

  • Sergio Giro
  • Pedro R. D’Argenio
  • Luis María Ferrer Fioriti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5710)


The technique of partial order reduction (POR) for probabilistic model checking prunes the state space of the model so that a maximizing scheduler and a minimizing one persist in the reduced system. This technique extends Peled’s original restrictions with a new one specially tailored to deal with probabilities. It has been argued that not all schedulers provide appropriate resolutions of nondeterminism and they yield overly safe answers on systems of distributed nature or that partially hide information. In this setting, maximum and minimum probabilities are obtained considering only the subset of so-called distributed or partial information schedulers. In this article we revise the technique of partial order reduction (POR) for LTL properties applied to probabilistic model checking. Our reduction ensures that distributed schedulers are preserved. We focus on two classes of distributed schedulers and show that Peled’s restrictions are valid whenever schedulers use only local information. We show experimental results in which the elimination of the extra restriction leads to significant improvements.


Model Check Generative Transition Probabilistic System Markov Decision Process Reactive Transition 
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.
    Alur, R., Brayton, R.A., Henzinger, T.A., Qadeer, S., Rajamani, S.K.: Partial-order reduction in symbolic state-space exploration. Formal Methods in System Design 18(2), 97–116 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Baier, C., Größer, M., Ciesinski, F.: Partial order reduction for probabilistic systems. In: QEST 2004, Washington, DC, USA, pp. 230–239. IEEE CS, Los Alamitos (2004)Google Scholar
  3. 3.
    Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 288–299. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. 4.
    Chaum, D.: The dining cryptographers problem: Unconditional sender and recipient untraceability. J. Cryptology 1(1), 65–75 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheung, L.: Reconciling Nondeterministic and Probabilistic Choices. PhD thesis, Radboud Universiteit Nijmegen (2006)Google Scholar
  6. 6.
    Cheung, L., Lynch, N., Segala, R., Vaandrager, F.: Switched PIOA: Parallel composition via distributed scheduling. Theor. Comput. Sci. 365(1-2), 83–108 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ciesinski, F., Baier, C.: LiQuor: A tool for qualitative and quantitative linear time analysis of reactive systems. In: QEST 2006, pp. 131–132. IEEE CS, Los Alamitos (2006)Google Scholar
  8. 8.
    Ciesinski, F., Baier, C., Größer, M., Klein, J.: Reduction techniques for model checking markov decision processes. In: QEST 2008, pp. 45–54 (2008)Google Scholar
  9. 9.
    Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  10. 10.
    D’Argenio, P., Niebert, P.: Partial order reduction on concurrent probabilistic programs. In: QEST 2004, Washington, DC, USA, pp. 240–249. IEEE CS, Los Alamitos (2004)Google Scholar
  11. 11.
    de Alfaro, L.: The verification of probabilistic systems under memoryless partial-information policies is hard. In: PROBMIV 1999. TR CSR-99-8, University of Birmingham, pp. 19–32 (1999)Google Scholar
  12. 12.
    de Alfaro, L., Henzinger, T.A., Jhala, R.: Compositional methods for probabilistic systems. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 351–365. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Giro, S., D’Argenio, P.: Quantitative model checking revisited: neither decidable nor approximable. In: Raskin, J.-F., Thiagarajan, P.S. (eds.) FORMATS 2007. LNCS, vol. 4763, pp. 179–194. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Giro, S., D’Argenio, P.: On the expressive power of schedulers in distributed probabilistic systems. In: Proc. of QAPL 2009, York, UK, March 28-29 (2009), Extended version,
  15. 15.
    Giro, S., D’Argenio, P.: On the verification of probabilistic I/O automata with unspecified rates. In: Proc. of 24th SAC, pp. 582–586. ACM Press, New York (2009)Google Scholar
  16. 16.
    Giro, S., D’Argenio, P.: Partial order reduction for probabilistic systems assuming distributed schedulers. Technical Report Serie A, Inf. 2009/02, FaMAF, UNC (2009),
  17. 17.
    Glabbeek, R.v., Smolka, S., Steffen, B.: Reactive, generative, and stratified models of probabilistic processes. Information and Computation 121, 59–80 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Godefroid, P.: Partial-Order Methods for the Verification of Concurrent Systems. LNCS, vol. 1032. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  19. 19.
    Jeannet, B., D’Argenio, P., Larsen, K.: Rapture: A tool for verifying Markov Decision Processes. In: Cerna, I. (ed.) Tools Day 2002, Brno, Czech Republic, Technical Report, Faculty of Informatics, Masaryk University Brno (2002)Google Scholar
  20. 20.
    Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. International Journal on Software Tools for Technology Transfer (STTT) 6(2), 128–142 (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Laboratory for Computer Science, MIT (1995)Google Scholar
  22. 22.
    Vardi, M.: Automatic verification of probabilistic concurrent finite state programs. In: Procs. of 26th FOCS, pp. 327–338. IEEE Press, Los Alamitos (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sergio Giro
    • 1
  • Pedro R. D’Argenio
    • 1
  • Luis María Ferrer Fioriti
    • 1
  1. 1.FaMAFUniversidad Nacional de Córdoba - CONICET, Ciudad UniversitariaCórdobaArgentina

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