License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2022.8
URN: urn:nbn:de:0030-drops-157285
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15728/
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Bouyer, Patricia ; Le Roux, Stéphane ; Thomasset, Nathan

Finite-Memory Strategies in Two-Player Infinite Games

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LIPIcs-CSL-2022-8.pdf (0.7 MB)

Abstract

We study infinite two-player win/lose games (A,B,W) where A,B are finite and W ⊆ (A×B)^ω. At each round Player 1 and Player 2 concurrently choose one action in A and B, respectively. Player 1 wins iff the generated sequence is in W. Each history h ∈ (A×B)^* induces a game (A,B,W_h) with W_h : = {ρ ∈ (A×B)^ω ∣ h ρ ∈ W}. We show the following: if W is in Δ⁰₂ (for the usual topology), if the inclusion relation induces a well partial order on the W_h’s, and if Player 1 has a winning strategy, then she has a finite-memory winning strategy. Our proof relies on inductive descriptions of set complexity, such as the Hausdorff difference hierarchy of the open sets.
Examples in Σ⁰₂ and Π⁰₂ show some tightness of our result. Our result can be translated to games on finite graphs: e.g. finite-memory determinacy of multi-energy games is a direct corollary, whereas it does not follow from recent general results on finite memory strategies.

BibTeX - Entry

@InProceedings{bouyer_et_al:LIPIcs.CSL.2022.8,
  author =	{Bouyer, Patricia and Le Roux, St\'{e}phane and Thomasset, Nathan},
  title =	{{Finite-Memory Strategies in Two-Player Infinite Games}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15728},
  URN =		{urn:nbn:de:0030-drops-157285},
  doi =		{10.4230/LIPIcs.CSL.2022.8},
  annote =	{Keywords: Two-player win/lose games, Infinite trees, Finite-memory winning strategies, Well partial orders, Hausdorff difference hierarchy}
}

Keywords: Two-player win/lose games, Infinite trees, Finite-memory winning strategies, Well partial orders, Hausdorff difference hierarchy
Collection: 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)
Issue Date: 2022
Date of publication: 27.01.2022

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