Ambiguous classes in the games µ-calculus hierarchy

Authors:
André Arnold;Luigi Santocanale
Affiliations:
LaBRI - Université Bordeaux 1;LaBRI - Université Bordeaux 1
Venue:
FOSSACS'03/ETAPS'03 Proceedings of the 6th International conference on Foundations of Software Science and Computation Structures and joint European conference on Theory and practice of software
Year:
2003

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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 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.