This paper presents a case study for the application of semiring semantics for fixed-point formulae to the analysis of strategies in B\"uchi games. Semiring semantics generalizes the classical Boolean semantics by permitting multiple truth values from certain semirings. Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how they win and which strategies they might use. This is well-understood for reachability games, where the winning region is definable as a least fixed point. The case of B\"chi games is of special interest, not only due to their practical importance, but also because it is the simplest case where the fixed-point definition involves a genuine alternation of a greatest and a least fixed point. We show that, in a precise sense, semiring semantics provide information about all absorption-dominant strategies - strategies that win with minimal effort, and we discuss how these relate to positional and the more general persistent strategies. This information enables applications such as game synthesis or determining minimal modifications to the game needed to change its outcome.

中文翻译：

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Büchi 游戏的半林起源：使用吸收多项式的策略分析

也因为它是最简单的情况，其中不动点定义涉及最大不动点和最小不动点的真正交替。我们表明，在精确的意义上，半环语义提供了有关所有吸收主导策略的信息 - 以最少的努力获胜的策略，我们讨论了这些与位置策略和更一般的持久策略之间的关系。此信息支持应用程序，例如游戏合成或确定更改其结果所需的游戏最小修改。