The Mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

Abstract
The alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the hierarchy does not necessarily carry over when considering restricted classes of structures. For instance, over the class of infinite words the alternation-free fragment of the mu-calculus is already as expressive as the full logic. Our current understanding of when and why the mu-calculus alternation hierarchy is (and is not) strict is limited. This article makes progress in answering these questions by showing that the alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the graph of the structure decomposes it into graphs in which all paths are of finite length. The collapse results herein are obtained in an automatatheoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.