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.