2007/10/12 by Walid Belkhir, Belkhir, Walid, Luigi Santocanale +1
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #Computer Science and Game Theory (cs.GT) #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems #cs.GT #cs.LO #math.LO
paper · pdf · doi:10.48550/arxiv.0710.2419
openalex publication_date 2007/10/12 · arxiv created 2008/03/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.