Temporal graphs are a popular modelling mechanism for dynamic complex systems that extend ordinary graphs with discrete time. Simply put, time progresses one unit per step and the availability of edges can change with time. We consider the complexity of solving $\omega$-regular games played on temporal graphs where the edge availability is ultimately periodic and fixed a priori. We show that solving parity games on temporal graphs is decidable in PSPACE, only assuming the edge predicate itself is in PSPACE. A matching lower bound already holds for what we call punctual reachability games on static graphs, where one player wants to reach the target at a given, binary encoded, point in time. We further study syntactic restrictions that imply more efficient procedures. In particular, if the edge predicate is in $P$ and is monotonically increasing for one player and decreasing for the other, then the complexity of solving games is only polynomially increased compared to static graphs.

翻译：时间图是一种流行的动态复杂系统建模机制，它在普通图的基础上扩展了离散时间。简言之，时间每步推进一个单位，边的可用性随时间变化。我们考虑在边可用性最终周期且固定先验的时间图上求解 $\omega$-正则博弈的复杂性。结果表明，仅假设边谓词本身属于PSPACE，时间图上的奇偶博弈的可判定性在PSPACE内。一个匹配的下界已存在于我们称为静态图上的准时可达性博弈中，其中一方玩家希望在给定二进制编码的时间点到达目标。我们进一步研究了暗示更高效程序句法限制。特别地，如果边谓词属于$P$，且对一方玩家单调递增而对另一方单调递减，则博弈求解的复杂性相较于静态图仅多项式增加。