刻画无限图博弈的有限记忆决定性中的Omega正则性
Characterizing Omega-Regularity through Finite-Memory Determinacy of Games on Infinite Graphs
AI总结：
本文通过无限图博弈的有限记忆决定性刻画了ω-正则目标，证明双方玩家在色度有限记忆下最优时目标必为ω-正则，并提供了一对二玩家提升及折扣和目标的实例。
摘要：
我们考虑无限图上的零和博弈，其目标被指定为某种颜色字母表上的无限字集合。一个被广泛研究的目标类别是$\omega$-正则目标，因为它与理论计算机科学中的许多自然问题相关。我们关注策略复杂度问题：给定一个目标，每个玩家需要多少记忆才能尽可能好地发挥？一个经典结果是，当目标是$\omega$-正则时，有限记忆策略对双方玩家都足够。我们展示了该陈述的逆命题：当双方玩家在所有无限博弈图中都能使用色度有限记忆结构（即其更新只能观察颜色的结构）进行最优博弈时，那么该目标必须是$\omega$-正则的。这提供了$\omega$-正则目标的博弈论刻画，并且该刻画有助于获得记忆界限。此外，我们刻画的一个副产品是一个新的一对二玩家提升：为了证明色度有限记忆结构足以在无限图上的双人博弈中实现最优博弈，只需证明在无限图上的单人博弈的简单情形下成立即可。我们用折扣和目标的家族来说明我们的结果，对于这些目标，$\omega$-正则性取决于某些参数的值。
英文摘要：
We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of $ω$-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is $ω$-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be $ω$-regular. This provides a game-theoretic characterization of $ω$-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which $ω$-regularity depends on the value of some parameters.
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