The Complexity of Partial-Observation Stochastic Parity Games with Finite-Memory Strategies

  • Krishnendu Chatterjee
  • Laurent Doyen
  • Sumit Nain
  • Moshe Y. Vardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8412)


We consider two-player partial-observation stochastic games on finitestate graphs where player 1 has partial observation and player 2 has perfect observation. The winning condition we study are ε-regular conditions specified as parity objectives. The qualitative-analysis problem given a partial-observation stochastic game and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). These qualitative-analysis problems are known to be undecidable. However in many applications the relevant question is the existence of finite-memory strategies, and the qualitative-analysis problems under finite-memory strategies was recently shown to be decidable in 2EXPTIME.We improve the complexity and show that the qualitative-analysis problems for partial-observation stochastic parity games under finite-memory strategies are EXPTIME-complete; and also establish optimal (exponential) memory bounds for finite-memory strategies required for qualitative analysis.


Stochastic Game Winning Strategy Input Tree Tree Automaton Game Graph 
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  1. 1.
    Full version of the paper. CoRR, abs/1401.3289 (2014)Google Scholar
  2. 2.
    Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. JACM 49, 672–713 (2002)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Baier, C., Bertrand, N., Größer, M.: On decision problems for probabilistic Büchi automata. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 287–301. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Cai, Y., Zhang, T.: Determinization complexities of ω automata, Technical report (2013),
  5. 5.
    Chatterjee, K.: Stochastic ω-regular Games. PhD thesis, UC Berkeley (2007)Google Scholar
  6. 6.
    Chatterjee, K., Chmelik, M., Tracol, M.: What is decidable about partially observable Markov decision processes with omega-regular objectives. In: CSL (2013)Google Scholar
  7. 7.
    Chatterjee, K., Doyen, L.: Partial-observation stochastic games: How to win when belief fails. In: LICS, pp. 175–184. IEEE (2012)Google Scholar
  8. 8.
    Chatterjee, K., Doyen, L., Gimbert, H., Henzinger, T.A.: Randomness for free. CoRR abs/1006.0673 (Full version) (2010); Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 246–257. Springer, Heidelberg (2010)Google Scholar
  9. 9.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Qualitative analysis of partially-observable Markov decision processes. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 258–269. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Chatterjee, K., Doyen, L., Henzinger, T.A., Raskin, J.-F.: Algorithms for omega-regular games of incomplete information. Logical Methods in Computer Science 3(3:4) (2007)Google Scholar
  11. 11.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Simple stochastic parity games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 100–113. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. JACM 42(4), 857–907 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford Univ. (1997)Google Scholar
  14. 14.
    Emerson, E.A., Jutla, C.: Tree automata, mu-calculus and determinacy. In: FOCS, pp. 368–377. IEEE (1991)Google Scholar
  15. 15.
    Gurevich, Y., Harrington, L.: Trees, automata, and games. In: STOC, pp. 60–65 (1982)Google Scholar
  16. 16.
    McNaughton, R.: Infinite games played on finite graphs. Annals of Pure and Applied Logic 65, 149–184 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Muller, D.E., Saoudi, A., Schupp, P.E.: Alternating automata, The weak monadic theory of the tree, and its complexity. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 275–283. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  18. 18.
    Muller, D.E., Schupp, P.E.: Alternating automata on infinite trees. TCS 54, 267–276 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Muller, D.E., Schupp, P.E.: Simulating alternating tree automata by nondeterministic automata: New results and new proofs of the theorems of Rabin, McNaughton and Safra. TCS 141(1&2), 69–107 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Nain, S., Vardi, M.Y.: Solving partial-information stochastic parity games. In: LICS, pp. 341–348. IEEE (2013)Google Scholar
  21. 21.
    Paz, A.: Introduction to probabilistic automata. Academic Press (1971)Google Scholar
  22. 22.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: POPL, pp. 179–190. ACM (1989)Google Scholar
  23. 23.
    Reif, J.H.: The complexity of two-player games of incomplete information. JCSS 29, 274–301 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages, Beyond Words, vol. 3, ch. 7, pp. 389–455. Springer (1997)Google Scholar
  25. 25.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state systems. In: Proc. of FOCS, pp. 327–338 (1985)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Laurent Doyen
    • 2
  • Sumit Nain
    • 3
  • Moshe Y. Vardi
    • 3
  1. 1.IST AustriaAustria
  2. 2.CNRS, LSVENS CachanFrance
  3. 3.Rice UniversityUSA

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