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

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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.

BibTeX - Entry

@InProceedings{bouyer_et_al:LIPIcs.STACS.2022.16,
author =	{Bouyer, Patricia and Randour, Mickael and Vandenhove, Pierre},
title =	{{Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs}},
booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages =	{16:1--16:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-222-8},
ISSN =	{1868-8969},
year =	{2022},
volume =	{219},
editor =	{Berenbrink, Petra and Monmege, Benjamin},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
}