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

TL;DR

This paper explores the relationship between the semantic complexity of modal mu calculus formulas and the descriptive complexity of their model-checking parity games, revealing a precise correspondence up to co-Büchi complexity.

Contribution

It demonstrates that the descriptive complexity of model-checking games matches the semantic complexity for formulas up to co-Büchi, and provides bounds beyond that.

Findings

Descriptive complexity matches semantic complexity up to co-Büchi.

Beyond co-Büchi, descriptive complexity bounds semantic complexity from above.

Open question on whether bounds are tight beyond co-Büchi.

Abstract

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.