The Topological Complexity of Models of the Modal μ-Calculus: On The Alternation Free Fragment and Beyond

Jacques Duparc (1) , Alessandro Facchini (2)

1 ISI - Institut des systèmes d'information
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

This is the extended journal version of [C1]. In this paper we define set theoretical operations in terms of µ-formulae. In particular, we introduce the operation given by an action of a µ-definable topological property over a class of models. When considering definability by alternation free formulae, we obtain the $\mu$-calculus counterpart of the Wadge hierarchy for weakly alternating tree automata. It was conjectured that the height of this hierarchy is exactly $\varepsilon_{0}$. We prove that the degree of a tree language definable by an alternation free formula is either below $ \omega^\omega$ or above $\omega_1$. However, very little is known about the Wadge hierarchy for the full µ-calculus, the problem being that most of the sets definable by a µ-formula are even not Borel. We make a first step in this direction by introducing the Wadge hierarchy extending the one for the alternating free fragment with an action given by a difference of two $\Pi^1_1$ complete sets.

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Jacques Duparc, Alessandro Facchini. The Topological Complexity of Models of the Modal μ-Calculus: On The Alternation Free Fragment and Beyond. 2008. ⟨hal-00396433⟩