Global Caching for the Flat Coalgebraic µ-Calculus

Branching-time temporal logics generalizing relational temporal logics such as CTL have been proposed for various system types beyond the purely relational world. This includes, e.g., alternating-time logics, which talk about winning strategies over concurrent game structures, and Parikh´s game logic, which is interpreted over monotone neighbourhood frames, as well as probabilistic fixpoint logics. Coalgebraic logic has emerged as a unifying semantic and algorithmic framework for logics featuring generalized modalities of this type. Here, we present a generic global caching algorithm for satisfiability checking in the flat coalgebraic mu-calculus, which realizes known tight exponential-time upper complexity bounds but offers potential for heuristic optimization. It is based on a tableau system that makes do without additional labelling of nodes beyond formulas from the standard Fischer-Ladner closure, such as foci or termination counters for eventualities. Moreover, the tableau system is single-pass, i.e. avoids building an exponential-sized structure in a first pass, to our best knowledge, optimal single-pass systems without numeric time-outs were not previously available even for CTL.