Undecidability Results for Distributed Probabilistic Systems

  • Sergio Giro
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5902)

Abstract

In the verification of concurrent systems involving probabilities, the aim is to find out the maximum/minimum probability that a given event occurs (examples of such events being “the system reaches a failure state”,“a message is delivered”). Such extremal probabilities are obtained by quantifying over all the possible ways in which the processes may be interleaved. Interleaving choices are considered a particular case of nondeterministic behaviour. Such behaviour is dealt with by considering schedulers that resolve the nondeterministic choices. Each scheduler determines a Markov chain for which actual probabilities can be calculated. In the recent literature on distributed systems, particular attention has been paid to the fact that, in order to obtain accurate results, the analysis must rely on partial information schedulers, instead of full-history dependent schedulers used in the setting of Markov decision processes. In this paper, we present undecidability results for distributed schedulers. These schedulers were devised in previous works, and aim to capture the fact that each process has partial information about the actual state of the system. Some of the undecidability results we present are particularly impressive: in the setting of total information the same problems are inexpensive and, indeed, they are used as preprocessing steps in more general model checking algorithms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aumann, Y.: Efficient asynchronous consensus with the weak adversary scheduler. In: PODC, pp. 209–218 (1997)Google Scholar
  2. 2.
    Bianco, A., de Alfaro, L.: Model checking of probabalistic and nondeterministic systems. In: FSTTCS, pp. 499–513 (1995)Google Scholar
  3. 3.
    Canetti, R., Cheung, L., Kirli Kaynar, D., Lynch, N.A., Pereira, O.: Compositional security for Task-PIOAs. In: CSF, pp. 125–139. IEEE CS, Los Alamitos (2007)Google Scholar
  4. 4.
    Chatzikokolakis, K., Norman, G., Parker, D.: Bisimulation for demonic schedulers. In: FOSSACS, pp. 318–332 (2009)Google 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 Probabilistic PIOA: Parallel composition via distributed scheduling. Theor. Comput. Sci. 365(1-2), 83–108 (2006)MATHCrossRefMathSciNetGoogle 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.
    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
  9. 9.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University (1997), Technical report STAN-CS-TR-98-1601Google Scholar
  10. 10.
    PRISM development team. Prism case studies, http://www.prismmodelchecker.org/casestudies/index.php
  11. 11.
    Dijkstra, E.W.: Self-stabilizing systems in spite of distributed control. Commun. ACM 17(11), 643–644 (1974)MATHCrossRefGoogle Scholar
  12. 12.
    Giro, S., D’Argenio, P.R.: 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
  13. 13.
    Giro, S., D’Argenio, P.R.: On the expressive power of schedulers in distributed probabilistic systems. In: Proc. of QAPL 2009 (2009). Extended version to appear in ENTCS, cs.famaf.unc.edu.ar/~sgiro/QAPL09-ext.pdf
  14. 14.
    Giro, S., D’Argenio, P.R.: On the verification of probabilistic i/o automata with unspecified rates. In: SAC 2009: Proceedings of the 2009 ACM symposium on Applied Computing, pp. 582–586. ACM, New York (2009)CrossRefGoogle Scholar
  15. 15.
    Giro, S.: On the automatic verification of Distributed Probabilistic Automata with Partial Information. PhD thesis, Universidad Nacional de Córdoba (to appear)Google Scholar
  16. 16.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative, and stratified models of probabilistic processes. Information and Computation 121, 59–80 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for automatic verification of probabilistic systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Lynch, N.A., Tuttle, M.R.: An introduction to input/output automata. CWI Quarterly 2(3), 219–246 (1989)MATHMathSciNetGoogle Scholar
  19. 19.
    Madani, O., Hanks, S., Condon, A.: On the undecidability of probabilistic planning and related stochastic optimization problems. Artif. Intell. 147(1-2), 5–34 (2003)MATHMathSciNetGoogle Scholar
  20. 20.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Laboratory for Computer Science, MIT (1995)Google Scholar
  21. 21.
    Sipser, M.: Introduction to the Theory of Computation, 2nd edn., pp. 199–205. Thomson Course Technology (2005)Google Scholar
  22. 22.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite state programs. In: Procs. of 26th FOCS, pp. 327–338. IEEE Press, Los Alamitos (1985)Google Scholar
  23. 23.
    Wu, S.-H., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theor. Comput. Sci. 176(1-2), 1–38 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sergio Giro
    • 1
  1. 1.FaMAFUniversidad Nacional de Córdoba - CONICETCórdobaArgentina

Personalised recommendations