We introduce an extension of Strategy Logic for the imperfect-information setting, called SL ii and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of “hierarchical instances” for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SL ii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises the decidability of distributed synthesis for systems with hierarchical information. It allows us to easily derive new decidability results concerning strategic problems under imperfect information such as the existence of Nash equilibria or rational synthesis. To establish this result, we go through an intermediary, “low-level” logic much more adapted to automata techniques. QCTL * is an extension of CTL * with second-order quantification over atomic propositions that has been used to study strategic logics with perfect information. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. The simple syntax of the resulting logic, QCTL * ii , allows us to provide a conceptually neat reduction of SL ii to QCTL * ii that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of QCTL * ii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable.

中文翻译：

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不完全信息的战略逻辑

我们为不完全信息设置引入了策略逻辑的扩展，称为 SLii并研究其模型检验问题。由于这种逻辑自然会捕捉到信息不完整的多人游戏，所以这个问题是不可判定的；但是我们引入了一个句法类的“分层实例”，直观地说，当一个沿着公式的句法树向下移动时，策略量化与模型的更精细的观察有关，我们证明了模型检查 SLii仅限于分层实例是可判定的。这个结果，因为它允许对策略进行存在和普遍量化的复杂模式，极大地概括了具有分层信息的系统的分布式综合的可判定性。它使我们能够很容易地推导出关于不完全信息下的战略问题的新的可判定性结果，例如纳什均衡或理性综合的存在。为了确定这个结果，我们通过一个更适合自动机技术的中间“低级”逻辑。QCTL*是 CTL 的扩展*对原子命题进行二阶量化，已用于研究具有完美信息的战略逻辑。我们通过用观察参数化二阶量词将其扩展到不完美信息设置。结果逻辑的简单语法 QCTL* ii, 允许我们提供一个概念上简洁的 SL 缩减ii到 QCTL* ii这将关注点分开，让人们忘记策略和参与者，只关注二阶量化。而QCTL的模型检验问题* ii是，一般来说，不可判定，我们识别层次公式的句法片段，并使用自动机理论方法证明它是可判定的。