Acta Informatica

, Volume 49, Issue 4, pp 203–224 | Cite as

Stochastic game logic

  • Christel Baier
  • Tomáš Brázdil
  • Marcus Größer
  • Antonín Kučera
Original Article


Stochastic game logic (SGL) is a new temporal logic for multi-agent systems modeled by turn-based multi-player games with discrete transition probabilities. It combines features of alternating-time temporal logic (ATL), probabilistic computation tree logic and extended temporal logic. SGL contains an ATL-like modality to specify the individual cooperation and reaction facilities of agents in the multi-player game to enforce a certain winning objective. While the standard ATL modality states the existence of a strategy for a certain coalition of agents without restricting the range of strategies for the semantics of inner SGL formulae, we deal with a more general modality. It also requires the existence of a strategy for some coalition, but imposes some kind of strategy binding to inner SGL formulae. This paper presents the syntax and semantics of SGL and discusses its model checking problem for different types of strategies. The model checking problem of SGL turns out to be undecidable when dealing with the full class of history-dependent strategies. We show that the SGL model checking problem for memoryless deterministic strategies as well as the model checking problem of the qualitative fragment of SGL for memoryless randomized strategies is PSPACE-complete, and we establish a close link between natural syntactic fragments of SGL and the polynomial hierarchy. Further, we give a reduction from the SGL model checking problem under memoryless randomized strategies into the Tarski algebra which proves the problem to be in EXPSPACE.


Model Check Markov Decision Process Linear Temporal Logic Atomic Proposition Label Transition System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ågotnes, T., Goranka, V., Jamroga, W.: Alternating-time temporal logic with irrevocable strategies. In: 11th Confernce on Theoretical Aspects of Rationality and Knowledge (TARK’07), pp. 15–24 (2007)Google Scholar
  2. 2.
    Alur R., Henzinger T.A., Kupferman O.: Alternating-time temporal logic. J. ACM 49, 672–713 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baier, C., Brázdil, T., Größer, M., Kučera, A.: Stocahstic game logic. In: Fourth International Conference on Quantitative Evaluation of Systems (QEST’07), pp. 227–236. IEEE Computer Society Press, Los Alamos (2007)Google Scholar
  4. 4.
    Baier, C., Größer, M., Leucker, M., Ciesinski, F., Bollig, B.: Controller synthesis for probabilistic systems. In: IFIP Worldcongress, Theoretical Computer Science (2004)Google Scholar
  5. 5.
    Baier C., Katoen J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)MATHGoogle Scholar
  6. 6.
    Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) Proceedings of Foundations of Software Technology and Theoretical Computer Science, 15th Conference, Bangalore, India, December 18–20, Lecture Notes in Computer Science 1026, pp. 499–513 (1995). ISBN 3-540-60692-0Google Scholar
  7. 7.
    Brázdil, T., Forejt, V., Kučera, A.: Controller Synthesis and Verification for Markov Decision Processes with Qualitative Branching Time Objectives. In: Proceedings of the ICALP 2008, Lecture Notes in Computer Science, vol. 5126, pp. 148–159 (2008)Google Scholar
  8. 8.
    Brázdil, T., Brožek, V., Forejt, V., Kučera, A.: Stochastic games with branching-time winning objectives. In: Proceedings if the LICS 2006, pp. 349–358 (2006)Google Scholar
  9. 9.
    Brihaye, T., Da Costa Lopes, A., Laroussinie, F., Markey, N.: ATL with strategy contexts and bounded memory. In: International Symposium on Logical Foundations of Computer Science (LFCS), Lecture Notes in Computer Science, vol. 5407, pp. 92–106 (2009)Google Scholar
  10. 10.
    Bulling N., Jamroga W.: What agents can probably enforce. Fundam. Inform. 93(1–3), 81–96 (2009)MathSciNetMATHGoogle Scholar
  11. 11.
    Canny, J.: Some algebraic and geometric computations in PSPACE. In: Proceedings of the STOC’88, pp. 460–467 (1988)Google Scholar
  12. 12.
    Chatterjee K., Henzinger T.A., Piterman N.: Strategy logic. Inf. Comput. 208(6), 677–693 (2010)MathSciNetMATHGoogle Scholar
  13. 13.
    Clarke E.M., Grumberg O., Kurshan R.P.: A synthesis of two approaches for verifying finite state concurrent systems. J. Log. Comput. 2(5), 605–618 (1992)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Clarke E.M., Peled D., Grumberg O.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  15. 15.
    Cook, S.A.: The complexity of theorem-proving procedures. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 151–158 (1971)Google Scholar
  16. 16.
    Courcoubetis C., Yannakakis M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    De Alfaro, L.: Formal Verification of Probabilistic Systems. Ph.D.Thesis. Department of Computer Science, Stanford University (1997)Google Scholar
  18. 18.
    de Alfaro, L., Henzinger, T.A.: Concurrent omega-regular games. In: Proceedings of the LICS, pp. 141–154 (2000)Google Scholar
  19. 19.
    de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 564–575 (1998)Google Scholar
  20. 20.
    Grigoriev D.: Complexity of deciding Tarski algebra. J. Symb. Comput. 5(1–2), 65–108 (1988)CrossRefGoogle Scholar
  21. 21.
    Hopcroft J.E., Ullman J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  22. 22.
    Kučera, A., Stražovský, O.: On the controller synthesis for finite-state Markov decision processes. In: Proceedings of the FSTTCS, Lecture Notes in Computer Science, vol. 3821, pp. 541–552 (2005)Google Scholar
  23. 23.
    Meyer, A.R., Stockmeyer, L.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, pp. 125–129 (1972)Google Scholar
  24. 24.
    Papadimitriou C.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  25. 25.
    Safra, S.: On the complexity of ω-automata. In: 29th Annual IEEE Symposium on Foundations of Computer Science, pp. 319–327. White Plains, New York (1988)Google Scholar
  26. 26.
    Tarski A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5(2), 285–309 (1955)MathSciNetMATHGoogle Scholar
  27. 27.
    Thomas, W.: Computation tree logic and regular omega-languages. In: Lecture Notes in Computer Science, vol. 354, pp. 690–713 (1988)Google Scholar
  28. 28.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, chapt. 4, pp. 133–191. Elsevier Science Publishers B.V., Amsterdam (1990)Google Scholar
  29. 29.
    Vardi, M.Y.: Probabilistic linear-time model checking: an overview of the automata-theoretic approach Lecture Notes in Computer Science, vol. 1601, pp. 265–276 (1999)Google Scholar
  30. 30.
    Vardi, M.Y., Wolper, P.:Yet another process logic (preliminary version). In: Lecture Notes in Computer Science, vol. 164, pp. 501–512 (1983)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Christel Baier
    • 1
  • Tomáš Brázdil
    • 2
  • Marcus Größer
    • 1
  • Antonín Kučera
    • 2
  1. 1.Faculty for Computer Science, Institute for Theoretical Computer ScienceTechnische Universität DresdenDresdenGermany
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations