Playing Games with Boxes and Diamonds
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.
KeywordsTemporal Logic Turing Machine Linear Temporal Logic Recursive Call Winning Strategy
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- 4.Alur, R., La Torre, S.: Deterministic generators and games for LTL fragments. In: Proc. of LICS 2001, pp. 291–302 (2001)Google Scholar
- 6.Dill, D.: Trace Theory for Automatic Hierarchical Verification of Speed-independent Circuits. Distinguished Dissertation Series. MIT Press, Cambridge (1989)Google Scholar
- 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
- 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
- 13.Pnueli, A.: The temporal logic of programs. In: Proc. of FOCS 1977, pp. 46–77 (1977)Google Scholar
- 14.Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Proc. of POPL 1989, pp. 179–190 (1989)Google Scholar
- 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