Document Open Access Logo

Finite-Memory Strategies in Two-Player Infinite Games

Authors Patricia Bouyer, Stéphane Le Roux, Nathan Thomasset

PDF

File

Thumbnail PDF
PDF
LIPIcs.CSL.2022.8.pdf
  • Filesize: 0.67 MB
  • 16 pages

Document Identifiers

Related Versions

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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

Author Details

Patricia Bouyer
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France
Stéphane Le Roux
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France
Nathan Thomasset
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France

References

  1. Benjamin Aminof and Sasha Rubin. First-cycle games. Inf. Comput., 254:195-216, 2017. URL: https://doi.org/10.1016/j.ic.2016.10.008.
  2. Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Games where you can play optimally with arena-independent finite memory. In 31st International Conference on Concurrency Theory, CONCUR 2020, September 1-4, 2020, Vienna, Austria (Virtual Conference), pages 24:1-24:22, 2020. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2020.24.
  3. Patricia Bouyer, Stéphane Le Roux, and Nathan Thomasset. Finite-memory strategies in two-player infinite games. CoRR, abs/2107.09945, 2021. URL: http://arxiv.org/abs/2107.09945.
  4. John Fearnley and Martin Zimmermann. Playing Muller games in a hurry. Int. J. Found. Comput. Sci., 23(3):649-668, 2012. URL: https://doi.org/10.1142/S0129054112400321.
  5. Alexander Kechris. Classical Descriptive Set Theory. Springer, New York, NY, 1995. URL: https://doi.org/10.1007/978-1-4612-4190-4.
  6. Dénes Kőnig. Über eine Schlussweise aus dem Endlichen ins Unendliche (in german). Acta Sci. Math. (Szeged), 1927. Google Scholar
  7. Stéphane Le Roux, Arno Pauly, and Mickael Randour. Extending finite-memory determinacy by boolean combination of winning conditions. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2018, December 11-13, 2018, Ahmedabad, India, pages 38:1-38:20, 2018. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2018.38.
  8. Robert McNaughton. Playing infinite games in finite time. In A Half-Century of Automata Theory: Celebration and Inspiration, pages 73-91. World Scientific, 2000. Google Scholar
  9. Stéphane Le Roux. Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy. In Mohammad Taghi Hajiaghayi and Mohammad Reza Mousavi, editors, Topics in Theoretical Computer Science - The First IFIP WG 1.8 International Conference, TTCS 2015, Tehran, Iran, August 26-28, 2015, Revised Selected Papers, volume 9541 of Lecture Notes in Computer Science, pages 147-163. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-28678-5_11.
  10. Yaron Velner, Krishnendu Chatterjee, Laurent Doyen, Thomas A. Henzinger, Alexander Moshe Rabinovich, and Jean-François Raskin. The complexity of multi-mean-payoff and multi-energy games. Inf. Comput., 241:177-196, 2015. URL: https://doi.org/10.1016/j.ic.2015.03.001.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact