We study \emph{partial-information} two-player turn-based games on graphs with omega-regular objectives, when the partial-information player has \emph{limited memory}. Such games are a natural formalization for reactive synthesis when the environment player is not genuinely adversarial to the system player. The environment player has goals of its own, but the exact goal of the environment player is unknown to the system player. We prove that the problem of determining the existence of a winning strategy for the system player is PSPACE-hard for reachability, safety, and parity objectives. Moreover, when the environment player is memoryless, the problem is PSPACE-complete. However, it is simpler to decide if the environment player has a winning strategy; it is only NP-complete. Additionally, we construct a game where the the partial-information player needs at least $\mathcal{O}(\sqrt{n})$ bits of memory to retain winning strategies in a game of size $\mathcal{O}(n)$.

中文翻译：

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与记忆力有限的对手比赛

当部分信息玩家的记忆有限时，我们在具有欧米茄规则目标的图上研究了 \emph{partial-information} 两人回合制游戏。当环境玩家不是真正对抗系统玩家时，这种游戏是反应合成的自然形式。环境玩家有自己的目标，但环境玩家的确切目标对系统玩家来说是未知的。我们证明了确定系统玩家是否存在获胜策略的问题对于可达性、安全性和均等性目标来说是 PSPACE 难的。而且，当环境播放器无记忆时，问题是PSPACE-complete。然而，判断环境玩家是否有获胜策略更简单；它只是NP完全的。此外，