When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.24
URN: urn:nbn:de:0030-drops-144643
URL: https://drops.dagstuhl.de/opus/volltexte/2021/14464/
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### Finite Convergence of μ-Calculus Fixpoints on Genuinely Infinite Structures

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### Abstract

The modal μ-calculus can only express bisimulation-invariant properties. It is a simple consequence of Kleene’s Fixpoint Theorem that on structures with finite bisimulation quotients, the fixpoint iteration of any formula converges after finitely many steps. We show that the converse does not hold: we construct a word with an infinite bisimulation quotient that is locally regular so that the iteration for any fixpoint formula of the modal μ-calculus on it converges after finitely many steps. This entails decidability of μ-calculus model-checking over this word. We also show that the reason for the discrepancy between infinite bisimulation quotients and trans-finite fixpoint convergence lies in the fact that the μ-calculus can only express regular properties.

### BibTeX - Entry

@InProceedings{bruse_et_al:LIPIcs.MFCS.2021.24,
author =	{Bruse, Florian and S\"{a}lzer, Marco and Lange, Martin},
title =	{{Finite Convergence of \mu-Calculus Fixpoints on Genuinely Infinite Structures}},
booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
pages =	{24:1--24:19},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-201-3},
ISSN =	{1868-8969},
year =	{2021},
volume =	{202},
editor =	{Bonchi, Filippo and Puglisi, Simon J.},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
}