Countdown μ-Calculus

Authors Jędrzej Kołodziejski , Bartek Klin

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Author Details

Jędrzej Kołodziejski
  • University of Warsaw, Poland
Bartek Klin
  • University of Oxford, UK


We are grateful to Mikołaj Bojańczyk for numerous helpful suggestions.

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Jędrzej Kołodziejski and Bartek Klin. Countdown μ-Calculus. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We introduce the countdown μ-calculus, an extension of the modal μ-calculus with ordinal approximations of fixpoint operators. In addition to properties definable in the classical calculus, it can express (un)boundedness properties such as the existence of arbitrarily long sequences of specific actions. The standard correspondence with parity games and automata extends to suitably defined countdown games and automata. However, unlike in the classical setting, the scalar fragment is provably weaker than the full vectorial calculus and corresponds to automata satisfying a simple syntactic condition. We establish some facts, in particular decidability of the model checking problem and strictness of the hierarchy induced by the maximal allowed nesting of our new operators.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Automata over infinite objects
  • countdown μ-calculus
  • games
  • automata


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