The Descriptive Complexity of Modal $\mu$ Model-checking Games

This paper revisits the well-established relationship between the modal mu calculus and parity games to show that it is even more robust than previously known. It addresses the question of whether the descriptive complexity of modal mu model-checking games, previously known to depend on the syntactic complexity of a formula, depends in fact on its semantic complexity. It shows that up to formulas of semantic co-B\"uchi complexity, the descriptive complexity of their model-checking games coincides exactly with their semantic complexity. Beyond co-B\"uchi, the descriptive complexity of the model-checking parity games of a formula is shown to be an upper bound on its semantic complexity; whether it is also a lower bound remains an open question.