This paper studies two-player zero-sum games played on graphs and makes contributions toward the following question: given an objective, how much memory is required to play optimally for that objective? We study regular objectives, where the goal of one of the two players is that eventually the sequence of colors along the play belongs to some regular language of finite words. We obtain different characterizations of the chromatic memory requirements for such objectives for both players, from which we derive complexity-theoretic statements: deciding whether there exist small memory structures sufficient to play optimally is NP-complete for both players. Some of our characterization results apply to a more general class of objectives: topologically closed and topologically open sets.

翻译：本文研究图上的两人零和博弈，旨在回答以下问题：给定一个目标，实现该目标的最优策略需要多少记忆？我们研究正则目标，其中一方玩家的目标是：博弈过程中颜色序列最终属于某个有限词的正则语言。我们得到了双方玩家针对此类目标的不同色记忆需求特征描述，并由此推导出复杂度理论结论：判断是否存在足以实现最优策略的小型记忆结构，对双方玩家而言都是NP完全的。我们部分特征描述结果适用于更一般的目标类别：拓扑闭集与拓扑开集。