Advertisement

Tight Game Abstractions of Probabilistic Automata

  • Falak Sher Vira
  • Joost-Pieter Katoen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8704)

Abstract

We present a new game-based abstraction technique for probabilistic automata (PA). The key idea is to use distribution-based abstraction – preserving novel distribution-based (alternating) simulation relations – rather than classical state-based abstraction. These abstractions yield (simple) probabilistic game automata (PGA), turn-based 2 player stochastic games in which moves of both players – as opposed to classical stochastic games – yield distributions over states. As distribution-based (alternating) simulation relations are pre-congruences for composite PGA, abstraction can be done compositionally. Our abstraction yields tighter upper and lower bounds on (extremal) reachability probabilities than state-based abstraction. This shows the potential superiority over state-based abstraction of PA and Markov decision processes.

Keywords

Abstract Model Markov Decision Process Stochastic Game Concrete State Simulation Relation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology (1995)Google Scholar
  2. 2.
    Delahaye, B., Katoen, J.-P., Larsen, K.G., Legay, A., Pedersen, M.L., Sher, F., Wasowski, A.: Abstract probabilistic automata. Information and Computation 232, 66–116 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Delahaye, B., Katoen, J.-P., Larsen, K.G., Legay, A., Pedersen, M.L., Sher, F., Wąsowski, A.: Abstract probabilistic automata. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 324–339. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Sher, F., Katoen, J.-P.: Compositional abstraction techniques for probabilistic automata. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds.) TCS 2012. LNCS, vol. 7604, pp. 325–341. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  5. 5.
    Katoen, J.-P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for probabilistic systems. J. Log. Algebr. Program. 81(4), 356–389 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kattenbelt, M., Kwiatkowska, M.Z., Norman, G., Parker, D.: A game-based abstraction-refinement framework for Markov decision processes. Formal Methods in System Design 36(3), 246–280 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Shapley, L.S.: Stochastic games. Proceedings of the National Academy of Sciences of the United States of America 39(10), 1095–1100 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Condon, A.: The complexity of stochastic games. Information and Computation 96, 203–224 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Wachter, B., Zhang, L.: Best probabilistic transformers. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 362–379. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: LICS, pp. 266–277. IEEE CS Press (1991)Google Scholar
  11. 11.
    Condon, A., Ladner, R.E.: Probabilistic game automata. Journal of Computer and System Sciences 36(3), 452–489 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ash, R.B., Doléans-Dade, C.A.: Probability & Measure Theory, 2nd edn. Academic Press (2000)Google Scholar
  13. 13.
    Bertsekas, D.P., Tsitsiklis, J.N.: An analysis of stochastic shortest path problems. Mathematics of Operations Research 16, 580–595 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    de Alfaro, L.: Computing minimum and maximum reachability times in probabilistic systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 66–81. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  15. 15.
    Lynch, N.A., Segala, R., Vaandrager, F.W.: Observing branching structure through probabilistic contexts. SIAM J. Comput. 37(4), 977–1013 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Eisentraut, C., Hermanns, H., Zhang, L.: On probabilistic automata in continuous time. In: LICS, pp. 342–351. IEEE CS Press (2010)Google Scholar
  17. 17.
    Doyen, L., Henzinger, T.A., Raskin, J.F.: Equivalence of labeled Markov chains. Int. J. Found. Comput. Sci. 19(3), 549–563 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Alur, R., Henzinger, T.A., Kupferman, O., Vardi, M.Y.: Alternating refinement relations. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 163–178. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  19. 19.
    Kattenbelt, M.: Automated Quantitative Software Verification. PhD thesis, University of Oxford (2010)Google Scholar
  20. 20.
    Tarski, A., et al.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5(2), 285–309 (1955)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Falak Sher Vira
    • 1
  • Joost-Pieter Katoen
    • 1
  1. 1.Software Modeling and VerificationRWTH Aachen UniversityAachenGermany

Personalised recommendations