Completeness of flat coalgebraic fixpoint logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The μ-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability and in fact often live within the single variable fragment of the μ-calculus. The family of such flat fixpoint logics includes, e.g., Linear Temporal Logic (LTL), Computation Tree Logic (CTL), and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard μ-calculus including, e.g., flat fragments of the graded μ-calculus and the alternating-time μ-calculus (such as alternating-time temporal logic), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.