Finite Convergence of μ-Calculus Fixpoints on Genuinely Infinite Structures

Authors Florian Bruse, Marco Sälzer, Martin Lange

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Florian Bruse
  • School of Electrical Engineering and Computer Science, University of Kassel, Germany
Marco Sälzer
  • School of Electrical Engineering and Computer Science, University of Kassel, Germany
Martin Lange
  • School of Electrical Engineering and Computer Science, University of Kassel, Germany

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Florian Bruse, Marco Sälzer, and Martin Lange. Finite Convergence of μ-Calculus Fixpoints on Genuinely Infinite Structures. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Higher order logic
  • Theory of computation → Logic and verification
  • Theory of computation → Automata over infinite objects
  • temporal logic
  • fixpoint iteration
  • bisimulation


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