LIPIcs.CSL.2022.8.pdf
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Finite-Memory Strategies in Two-Player Infinite Games
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30th EACSL Annual Conference on Computer Science Logic (CSL 2022)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computer Science Logic (CSL) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2022-01-27
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Patricia Bouyer, Stéphane Le Roux, and Nathan Thomasset. Finite-Memory Strategies in Two-Player Infinite Games. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CSL.2022.8
https://doi.org/10.4230/LIPIcs.CSL.2022.8
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.
Subject Classification
ACM Subject Classification
- Theory of computation → Verification by model checking
Keywords
- Two-player win/lose games
- Infinite trees
- Finite-memory winning strategies
- Well partial orders
- Hausdorff difference hierarchy
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References
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