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

Julian Gutierrez
(University of Cambridge, United Kingdom)
Felix Klaedtke
(ETH Zurich, Switzerland)
Martin Lange
(University of Kassel, Germany)

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.

In Marco Faella and Aniello Murano: Proceedings Third International Symposium on Games, Automata, Logics and Formal Verification (GandALF 2012), Napoli, Italy, September 6-8, 2012, Electronic Proceedings in Theoretical Computer Science 96, pp. 113–126.
Published: 7th October 2012.

ArXived at: http://dx.doi.org/10.4204/EPTCS.96.9 bibtex PDF
References in reconstructed bibtex, XML and HTML format (approximated).
Comments and questions to: eptcs@eptcs.org
For website issues: webmaster@eptcs.org