The Variable Hierarchy for the Games mu-Calculus

TL;DR

This paper investigates the hierarchy of parity games in the mu-calculus, demonstrating that the hierarchy does not collapse by constructing specific games that require a certain number of fixed-point variables.

Contribution

It provides a negative answer to the hierarchy collapse question by constructing parity games that require exactly n fixed-point variables, showing the hierarchy's strictness.

Findings

Hierarchy of parity games does not collapse.

Constructed games require exactly n fixed-point variables.

Hierarchy remains strict for all n >= 1.

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.