Half-Positional Objectives Recognized by Deterministic Büchi Automata

Authors Patricia Bouyer , Antonio Casares , Mickael Randour , Pierre Vandenhove

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Patricia Bouyer
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France
Antonio Casares
  • LaBRI, Université de Bordeaux, France
Mickael Randour
  • F.R.S.-FNRS & UMONS - Université de Mons, Belgium
Pierre Vandenhove
  • F.R.S.-FNRS & UMONS - Université de Mons, Belgium
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France


We would like to thank Igor Walukiewicz for suggesting a simplification of the proof of Lemma 17 and Pierre Ohlmann for interesting discussions on the subject.

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Patricia Bouyer, Antonio Casares, Mickael Randour, and Pierre Vandenhove. Half-Positional Objectives Recognized by Deterministic Büchi Automata. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A central question in the theory of two-player games over graphs is to understand which objectives are half-positional, that is, which are the objectives for which the protagonist does not need memory to implement winning strategies. Objectives for which both players do not need memory have already been characterized (both in finite and infinite graphs); however, less is known about half-positional objectives. In particular, no characterization of half-positionality is known for the central class of ω-regular objectives. In this paper, we characterize objectives recognizable by deterministic Büchi automata (a class of ω-regular objectives) that are half-positional, in both finite and infinite graphs. Our characterization consists of three natural conditions linked to the language-theoretic notion of right congruence. Furthermore, this characterization yields a polynomial-time algorithm to decide half-positionality of an objective recognized by a given deterministic Büchi automaton.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • two-player games on graphs
  • half-positionality
  • memoryless optimal strategies
  • Büchi automata
  • ω-regularity


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