Positionality in Σ⁰₂ and a Completeness Result

Authors Pierre Ohlmann , Michał Skrzypczak

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Pierre Ohlmann
  • Institute of Informatics, University of Warsaw, Poland
Michał Skrzypczak
  • Institute of Informatics, University of Warsaw, Poland


We thank Antonio Casares and Lorenzo Clemente for discussions on and around the topic.

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Pierre Ohlmann and Michał Skrzypczak. Positionality in Σ⁰₂ and a Completeness Result. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 54:1-54:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study the existence of positional strategies for the protagonist in infinite duration games over arbitrary game graphs. We prove that prefix-independent objectives in Σ⁰₂ which are positional and admit a (strongly) neutral letter are exactly those that are recognised by history-deterministic monotone co-Büchi automata over countable ordinals. This generalises a criterion proposed by [Kopczyński, ICALP 2006] and gives an alternative proof of closure under union for these objectives, which was known from [Ohlmann, TheoretiCS 2023]. We then give two applications of our result. First, we prove that the mean-payoff objective is positional over arbitrary game graphs. Second, we establish the following completeness result: for any objective W which is prefix-independent, admits a (weakly) neutral letter, and is positional over finite game graphs, there is an objective W' which is equivalent to W over finite game graphs and positional over arbitrary game graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • infinite duration games
  • positionality
  • Borel class Σ⁰₂
  • history determinism


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