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Eve-Positional Languages: Putting Order into Büchi Automata

Author Olivier Idir

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ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Büchi automata
  • parity automata
  • Eve-positional language
  • ε-complete automata
  • positional strategy
  • ordered Büchi automata

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Abstract

An ω-regular language is Eve-positional if, in all games with this language as objective, the existential player can play optimally without keeping any information from the previous moves. This notion plays a crucial role in verification, automata theory and synthesis.
Casares and Ohlmann recently gave several characterisations of Eve-positionality of ω-regular languages. For this, they introduce the notion of ε-complete parity automaton and show (among other results) that an ω-regular language is Eve-positional if and only if it can be recognised by some ε-completion of a deterministic parity automaton. Colcombet and Idir built on their work, and obtained a more direct algebraic characterisation of Eve-positionality.
We introduce a new formalism that characterises the Eve-positional languages, consisting of a restriction of non-deterministic Büchi automata. This allows us to complete a missing implication in Casares and Ohlmann’s work. We then use this formalism to describe a determinization procedure for non-deterministic Büchi automata recognising such languages, with size blow-up at most factorial. We also show that this construction is state-wise optimal for languages over sufficiently complete alphabets.

Cite As Get BibTex

Olivier Idir. Eve-Positional Languages: Putting Order into Büchi Automata. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.33

Author Details

Olivier Idir
  • Université Paris Cité, CNRS, IRIF, France

Acknowledgements

I thank Thomas Colcombet for his many insights during the elaboration of this paper. I also thank Antonio Casares and Pierre Ohlmann for their enthusiastic answers concerning their article.

References

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