Abstract
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 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 πn+1 is also equivalent to a game obtained by composing games in σn and πn.
The second author acknowledges .nancial support from the European Commission through an individual Marie Curie fellowship.
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Arnold, A., Santocanale, L. (2003). Ambiguous Classes in the Games μ-Calculus Hierarchy. In: Gordon, A.D. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2003. Lecture Notes in Computer Science, vol 2620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36576-1_5
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