On One-Variable Fragments of Modal μ-Calculus

In this paper, we study one-variable fragments of modal mu-calculus and their relations to parity games. We first introduce the weak modal mu-calculus as an extension of the one-variable modal mu-calculus. We apply weak parity games to show the strictness of the one-variable hierarchy as well as its extension. We also consider games with infinitely many priorities and show that their winning positions can be expressed by both Sigma(mu)(2) and Pi(mu)(2) formulas with two variables, but requires a transfinite extension of the L-mu-formulas to be expressed with only one variable. At last, we define the mu-arithmetic and show that a set of natural numbers is definable by both a Sigma(mu)(2) and a Pi(mu)(2) formula of mu-arithmetic if and only if it is definable by a formula of the one-variable transfinite mu-arithmetic.