Decomposition theorems and model-checking for the modal μ-calculus

We prove a general decomposition theorem for the modal μ-calculus Lμ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two substructures M1 and M2 plus edges from M1 to M2, then the formulas true at a node in M only depend on the formulas true in the respective substructures in a sense made precise below. As a consequence we show that the model-checking problem for Lμ is fixed-parameter tractable (fpt) on classes of structures of bounded Kelly-width or bounded DAG-width. As far as we are aware, these are the first fpt results for Lμ which do not follow from embedding into monadic second-order logic.