Advertisement

Playing Games with Boxes and Diamonds

  • Rajeev Alur
  • Salvatore La Torre
  • P. Madhusudan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2761)

Abstract

Deciding infinite two-player games on finite graphs with the winning condition specified by a linear temporal logic (Ltl) formula, is known to be 2Exptime-complete. The previously known hardness proofs encode Turing machine computations using the next and/or until operators. Furthermore, in the case of model checking, disallowing next and until, and retaining only the always and eventually operators, lowers the complexity from Pspace to Np. Whether such a reduction in complexity is possible for deciding games has been an open problem. In this paper, we provide a negative answer to this question. We introduce new techniques for encoding Turing machine computations using games, and show that deciding games for the Ltl fragment with only the always and eventually operators is 2Exptime-hard. We also prove- that if in this fragment we do not allow the eventually operator in the scope of the always operator and vice-versa, deciding games is Expspace-hard, matching the previously known upper bound. On the positive side, we show that if the winning condition is a Boolean combination of formulas of the form “eventually p” and “infinitely often p,” for a state-formula p, then the game can be decided in Pspace, and also establish a matching lower bound. Such conditions include safety and reachability specifications on game graphs augmented with fairness conditions for the two players.

Keywords

Temporal Logic Turing Machine Linear Temporal Logic Recursive Call Winning Strategy 
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.
    Abadi, M., Lamport, L., Wolper, P.: Realizable and unrealizable specifications of reactive systems. In: Ronchi Della Rocca, S., Ausiello, G., Dezani-Ciancaglini, M. (eds.) ICALP 1989. LNCS, vol. 372, pp. 1–17. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  2. 2.
    Alur, R., de Alfaro, L., Henzinger, T., Mang, F.: Automating modular verification. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 82–97. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Alur, R., Henzinger, T., Kupferman, O.: Alternating-time temporal logic. Journal of the ACM 49(5), 1–42 (2002)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Alur, R., La Torre, S.: Deterministic generators and games for LTL fragments. In: Proc. of LICS 2001, pp. 291–302 (2001)Google Scholar
  5. 5.
    Chandra, A., Kozen, D., Stockmeyer, L.: Alternation. Journal of the ACM 28(1), 114–133 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dill, D.: Trace Theory for Automatic Hierarchical Verification of Speed-independent Circuits. Distinguished Dissertation Series. MIT Press, Cambridge (1989)Google Scholar
  7. 7.
    Emerson, E., Jutla, C.: The complexity of tree automata and logics of programs. In: Proc. of FOCS 1988, pp. 328–337 (1988)Google Scholar
  8. 8.
    Holzmann, G.: The model checker SPIN. IEEE Transactions on Software Engineeri ng 23(5), 279–295 (1997)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  10. 10.
    Kupferman, O., Vardi, M.: Module checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 75–86. Springer, Heidelberg (1996)Google Scholar
  11. 11.
    Manna, Z., Pnueli, A.: The temporal logic of reactive and concurrent systems: Specification. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  12. 12.
    Marcinkowski, J., Truderung, T.: Optimal complexity bounds for positive ltl games. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 262–275. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Pnueli, A.: The temporal logic of programs. In: Proc. of FOCS 1977, pp. 46–77 (1977)Google Scholar
  14. 14.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Proc. of POPL 1989, pp. 179–190 (1989)Google Scholar
  15. 15.
    Sistla, A., Clarke, E.: The complexity of propositional linear temporal logics. The Journal of the ACM 32, 733–749 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Vardi, M.Y., Stockmeyer, L.: Improved upper and lower bounds for modal logics of programs. In: Proc. 17th Symp. on Theory of Computing, pp. 240–251 (1985)Google Scholar
  17. 17.
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoretical Computer Science 200, 135–183 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rajeev Alur
    • 1
  • Salvatore La Torre
    • 2
  • P. Madhusudan
    • 1
  1. 1.University of Pennsylvania 
  2. 2.Università degli Studi di Salerno 

Personalised recommendations