The \muCalculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity
Abstract
It is known that the alternation hierarchy of least and greatest fixpoint operators in the mucalculus 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 alternationfree fragment is already as expressive as the full mucalculus. Our current understanding of when and why the mucalculus alternation hierarchy is not strict is limited. This paper makes progress in answering these questions by showing that the alternation hierarchy of the mucalculus collapses to the alternationfree 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 automatatheoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mucalculus over restricted classes of structures.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.2455
 Bibcode:
 2012arXiv1210.2455G
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Computational Complexity;
 Computer Science  Formal Languages and Automata Theory;
 F.1.1;
 F.4.1
 EPrint:
 In Proceedings GandALF 2012, arXiv:1210.2028