Filtration and canonical completeness for continuous modal µ-calculi

The continuous modal µ-calculus is a fragment of the modal µ-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 µ-calculus: filtration and canonical models. In particular, we show that the Filtration Theorem holds for formulas in the language of the continuous modal µ-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 µ-calculus, viz. PDL.