The Variable Hierarchy for the Games mu-Calculus

Abstract : 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.
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Submitted on : Thursday, March 13, 2008 - 1:49:26 PM
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Walid Belkhir, Luigi Santocanale. The Variable Hierarchy for the Games mu-Calculus. Annals of Pure and Applied Logic, Elsevier Masson, 2010, 161 (5), pp.690-707. ⟨10.1016/j.apal.2009.07.015⟩. ⟨hal-00178806v2⟩

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