Optimal Transformations of Games and Automata Using Muller Conditions

Authors Antonio Casares , Thomas Colcombet , Nathanaël Fijalkow

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Antonio Casares
  • LaBRI, Université de Bordeaux, France
Thomas Colcombet
  • CNRS, IRIF, Université de Paris, France
Nathanaël Fijalkow
  • CNRS, LaBRI, Université de Bordeaux, France
  • The Alan Turing Institute of Data Science, London, UK


We want to thank Philipp Meyer and Salomon Sickert for their comments and for spotting a mistake on a previous version of this paper.

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Antonio Casares, Thomas Colcombet, and Nathanaël Fijalkow. Optimal Transformations of Games and Automata Using Muller Conditions. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 123:1-123:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We consider the following question: given an automaton or a game with a Muller condition, how can we efficiently construct an equivalent one with a parity condition? There are several examples of such transformations in the literature, including in the determinisation of Büchi automata. We define a new transformation called the alternating cycle decomposition, inspired and extending Zielonka’s construction. Our transformation operates on transition systems, encompassing both automata and games, and preserves semantic properties through the existence of a locally bijective morphism. We show a strong optimality result: the obtained parity transition system is minimal both in number of states and number of priorities with respect to locally bijective morphisms. We give two applications: the first is related to the determinisation of Büchi automata, and the second is to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Automata over infinite words
  • Omega regular languages
  • Determinisation of automata


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