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 alternation-depth 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 paper we show that the alternation-depth hierarchy of the games µ-calculus, with its interpretation in the class of all complete lattices, has a nice characterization of ambiguous classes: every parity game which is equivalent both to a game in Σn+1 and to a game in Πn+1 is also equivalent to a game obtained by composing games in Σn and Πn.

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