The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

TL;DR

This paper demonstrates that the mu-calculus alternation hierarchy collapses to the alternation-free fragment over certain classes of structures with restricted connectivity, such as infinite nested words and finite graphs with bounded feedback vertex sets.

Contribution

It shows the collapse of the mu-calculus alternation hierarchy over classes with restricted connectivity, extending and strengthening previous results.

Findings

Hierarchy collapses over infinite nested words

Hierarchy collapses over finite graphs with bounded feedback vertex sets

Results are obtained via automata-theoretic methods

Abstract

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of structures. A prominent instance is the class of infinite words over which the alternation-free fragment is already as expressive as the full mu-calculus. Our current understanding of when and why the mu-calculus alternation hierarchy is not strict is limited. This paper 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 structure's graph decomposes it into graphs in which all paths are of finite length. Our collapse results are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.