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Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs

Authors Patricia Bouyer , 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
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-ANR-20-CE25-0012).
  • Randour, Mickael: Mickael Randour is an F.R.S.-FNRS Research Associate.
  • Vandenhove, Pierre: Pierre Vandenhove is an F.R.S.-FNRS Research Fellow.

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Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.16

Abstract

We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of ω-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is ω-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be ω-regular. This provides a game-theoretic characterization of ω-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which ω-regularity depends on the value of some parameters.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • two-player games on graphs
  • infinite arenas
  • finite-memory determinacy
  • optimal strategies
  • ω-regular languages

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