The continuous modal mu-calculus is a fragment of the modal mu-calculus, where the application of fixpoint operators is restricted to formulas whose functional interpretation is Scott-continuous, rather than merely monotone. By game-theoretic means, we show that this relatively expressive fragment still allows two important techniques of basic modal logic, which notoriously fail for the full modal mu-calculus: filtration and canonical models. In particular, we show that the Filtration Theorem holds for formulas in the language of the continuous modal mu-calculus. As a consequence we obtain the finite model property over a wide range of model classes. Moreover, we show that if a basic modal logic L is canonical and the class of L-frames admits filtration, then the logic obtained by adding continuous fixpoint operators to L is sound and complete with respect to the class of L-frames. This generalises recent results on a strictly weaker fragment of the modal mu-calculus, viz. PDL.

中文翻译：

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连续模态 mu 演算的过滤和典型完备性

连续模态 mu 演算是模态 mu 演算的一个片段，其中不动点算子的应用仅限于函数解释是 Scott 连续的，而不仅仅是单调的公式。通过博弈论的手段，我们表明这个相对表达的片段仍然允许基本模态逻辑的两种重要技术，众所周知，这些技术对于完整的模态 mu 演算是失败的：过滤和规范模型。特别是，我们证明了过滤定理适用于连续模态 mu 演算语言中的公式。因此，我们获得了范围广泛的模型类的有限模型属性。此外，我们证明，如果基本模态逻辑 L 是规范的并且 L 框架类允许过滤，那么通过将连续的定点运算符添加到 L 获得的逻辑就 L 帧的类而言是健全和完整的。这概括了最近对模态 mu 演算的一个严格较弱的片段的结果，即。PDL。