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

This paper revisits the well-established relationship between the modal μ calculus Lμ and parity games to show that it is evenmore robust than previously known. It addresses the question of whether the descriptive complexity of Lμ 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 complexity Σ μ 2 , the descriptive complexity of their model-checking games coincides exactly with their semantic complexity. Beyond Σ μ 2 , the descriptive complexity of the model-checking parity games of a formulaΨ is shown to be an upper bound on the semantic complexity of Ψ; whether it is also a lower bound remains an open question.