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How to Play Optimally for Regular Objectives?

Authors Patricia Bouyer , Nathanaël Fijalkow , Mickael Randour , Pierre Vandenhove

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Author Details

Patricia Bouyer
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France
Nathanaël Fijalkow
  • CNRS, LaBRI and Université de Bordeaux, France
  • University of Warsaw, Poland
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

Funding

This work has been partially supported by the ANR Project MAVeriQ (ANR-20-CE25-0012) and by the Fonds de la Recherche Scientifique - FNRS under Grant n^∘ T.0188.23 (PDR ControlleRS). Mickael Randour is an F.R.S.-FNRS Research Associate and a member of the TRAIL Institute. Pierre Vandenhove is an F.R.S.-FNRS Research Fellow.

Acknowledgements

We thank Antonio Casares and Igor Walukiewicz for valuable discussions about this article.

Cite AsGet BibTex

Patricia Bouyer, Nathanaël Fijalkow, Mickael Randour, and Pierre Vandenhove. How to Play Optimally for Regular Objectives?. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 118:1-118:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.118

Abstract

This paper studies two-player zero-sum games played on graphs and makes contributions toward the following question: given an objective, how much memory is required to play optimally for that objective? We study regular objectives, where the goal of one of the two players is that eventually the sequence of colors along the play belongs to some regular language of finite words. We obtain different characterizations of the chromatic memory requirements for such objectives for both players, from which we derive complexity-theoretic statements: deciding whether there exist small memory structures sufficient to play optimally is NP-complete for both players. Some of our characterization results apply to a more general class of objectives: topologically closed and topologically open sets.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • two-player games on graphs
  • strategy complexity
  • regular languages
  • finite-memory strategies
  • NP-completeness

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