可数MDP中的均等目标
Parity Objectives in Countable MDPs
摘要：
我们研究可数无限MDPs奇偶目标，和特殊情况下，有界数量的颜色在Mostowski层次结构（包括可达性，安全性，Buchi和合作Buchi）。在有限的MDP总是存在最佳的无记忆确定性（MD）的奇偶校验目标的策略，但这通常不适用于可数无限的MDP。特别是，最优策略不一定存在。对于可数无限MDP，我们提供了一个完整的图片的内存需求的最佳（分别为，$ε$ -最优）策略。特别是，两种不同类型的目标之间存在着强烈的二分法。对于第一种类型，最优策略，如果它们存在，可以选择MD，而对于第二种类型的最优策略需要无限的记忆。(I.e.,对于Mostowski层次中的所有目标，如果有限记忆随机化策略足够，那么MD策略也足够。类似地，某些目标允许$ε$ -最优MD-策略，而对于其他目标，$ε$ -最优策略需要无限内存。这样的二分法也适用于可数无限的MDPs的子类，这些MDPs是分枝的，尽管这里有更多的目标允许MD策略。
英文摘要：
We study countably infinite MDPs with parity objectives, and special cases with a bounded number of colors in the Mostowski hierarchy (including reachability, safety, Buchi and co-Buchi). In finite MDPs there always exist optimal memoryless deterministic (MD) strategies for parity objectives, but this does not generally hold for countably infinite MDPs. In particular, optimal strategies need not exist. For countable infinite MDPs, we provide a complete picture of the memory requirements of optimal (resp., $ε$-optimal) strategies for all objectives in the Mostowski hierarchy. In particular, there is a strong dichotomy between two different types of objectives. For the first type, optimal strategies, if they exist, can be chosen MD, while for the second type optimal strategies require infinite memory. (I.e., for all objectives in the Mostowski hierarchy, if finite-memory randomized strategies suffice then also MD strategies suffice.) Similarly, some objectives admit $ε$-optimal MD-strategies, while for others $ε$-optimal strategies require infinite memory. Such a dichotomy also holds for the subclass of countably infinite MDPs that are finitely branching, though more objectives admit MD-strategies here.
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