Enriched µ-Calculi Module Checking

The model checking problem for open systems has been widely studied in the literature, for both finite–state (module checking) and infinite–state (pushdown module checking) systems, with respect to CTL and CTL. In this paper, we further investigate this problem with respect to the μ-calculus enriched with nominals and graded modalities (hybrid graded μ-calculus), in both the finite–state and infinite-state settings. Using an automata-theoretic approach, we show that hybrid graded μ-calculus module checking is solvable in exponential time, while hybrid graded μ-calculus pushdown module checking is solvable in double-exponential time. These results are also tight since they match the known lower bounds for CTL. We also investigate the module checking problem with respect to the hybrid graded μ-calculus enriched with inverse programs (Fully enriched μ-calculus): by showing a reduction from the tiling problem, we show its undecidability. We conclude with a short overview of the model checking problem for the Fully enriched μ-calculus and the fragments obtained by dropping at least one of the additional constructs.