The Variable Hierarchy for the Games mu-Calculus

Walid Belkhir (1) , Luigi Santocanale (1)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille - UMR 6166

Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables.

Mots clés: variable hierarchy; parity games; mu-lattices

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]; Informatique et théorie des jeux [cs.GT]; Calcul parallèle, distribué et partagé [cs.DC]

Walid Belkhir, Luigi Santocanale. The Variable Hierarchy for the Games mu-Calculus. 2007. ⟨hal-00178806v1⟩