The Variable Hierarchy for the Games mu-Calculus

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

Ambiguous Classes in the Games µ-Calculus Hierarchy

Every parity game is a combinatorial representation of a closed Boolean µ-term. When interpreted in a distributive lattice every Boolean µ-term is equivalent to a fixed-point free term. The alternationdepth hierarchy is therefore trivial in this case. This is not the case for non distributive lattices, as the second author has shown that the alternation-depth hierarchy is infinite. In this pape