Document Open Access Logo

Characterising Memory in Infinite Games

Authors Antonio Casares , Pierre Ohlmann

Thumbnail PDF


  • Filesize: 1.13 MB
  • 18 pages

Document Identifiers

Author Details

Antonio Casares
  • LaBRI, Université de Bordeaux, France
Pierre Ohlmann
  • University of Warsaw, Poland


We thank Nathanaël Fijalkow, Rémi Morvan and Pierre Vandenhove for stimulating discussions around the topic.

Cite AsGet BibTex

Antonio Casares and Pierre Ohlmann. Characterising Memory in Infinite Games. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 122:1-122:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


This paper is concerned with games of infinite duration played over potentially infinite graphs. Recently, Ohlmann (TheoretiCS 2023) presented a characterisation of objectives admitting optimal positional strategies, by means of universal graphs: an objective is positional if and only if it admits well-ordered monotone universal graphs. We extend Ohlmann’s characterisation to encompass (finite or infinite) memory upper bounds. We prove that objectives admitting optimal strategies with ε-memory less than m (a memory that cannot be updated when reading an ε-edge) are exactly those which admit well-founded monotone universal graphs whose antichains have size bounded by m. We also give a characterisation of chromatic memory by means of appropriate universal structures. Our results apply to finite as well as infinite memory bounds (for instance, to objectives with finite but unbounded memory, or with countable memory strategies). We illustrate the applicability of our framework by carrying out a few case studies, we provide examples witnessing limitations of our approach, and we discuss general closure properties which follow from our results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Verification by model checking
  • Infinite duration games
  • Memory
  • Universal graphs


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


  1. Patricia Bouyer, Antonio Casares, Mickael Randour, and Pierre Vandenhove. Half-positional objectives recognized by deterministic Büchi automata. In CONCUR, volume 243, pages 20:1-20:18, 2022. URL:
  2. Patricia Bouyer, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Arena-independent finite-memory determinacy in stochastic games. In CONCUR, volume 203, pages 26:1-26:18, 2021. URL:
  3. Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. Characterizing omega-regularity through finite-memory determinacy of games on infinite graphs. In STACS, volume 219, pages 16:1-16:16, 2022. URL:
  4. Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Games where you can play optimally with arena-independent finite memory. Log. Methods Comput. Sci., 18(1), 2022. URL:
  5. Antonio Casares. On the minimisation of transition-based Rabin automata and the chromatic memory requirements of Muller conditions. In CSL, volume 216, pages 12:1-12:17, 2022. URL:
  6. Antonio Casares, Thomas Colcombet, and Karoliina Lehtinen. On the size of good-for-games Rabin automata and its link with the memory in Muller games. In ICALP, volume 229, pages 117:1-117:20, 2022. URL:
  7. Antonio Casares and Pierre Ohlmann. Characterising memory in infinite games. CoRR, abs/2209.12044, 2022. URL:
  8. Thomas Colcombet, Nathanaël Fijalkow, and Florian Horn. Playing safe. In FSTTCS, volume 29, pages 379-390, 2014. URL:
  9. Thomas Colcombet and Damian Niwiński. On the positional determinacy of edge-labeled games. Theor. Comput. Sci., 352(1-3):190-196, 2006. URL:
  10. Stéphane Demri, Alain Finkel, Jean Goubault-Larrecq, Sylvain Schmitz, and Philippe Schnoebelen. Well-quasi-orders for algorithms. Lecture notes, Master MPRI, 2017. URL:
  11. Robert P. Dilworth. A decomposition theorem for partially ordered sets. Annals of Mathematics, 51(1):161-166, 1950. URL:
  12. Stefan Dziembowski, Marcin Jurdzinski, and Igor Walukiewicz. How much memory is needed to win infinite games? In LICS, pages 99-110. IEEE Computer Society, 1997. URL:
  13. Hugo Gimbert and Wieslaw Zielonka. Games where you can play optimally without any memory. In CONCUR, volume 3653 of Lecture Notes in Computer Science, pages 428-442. Springer, 2005. URL:
  14. Yuri Gurevich and Leo Harrington. Trees, automata, and games. In STOC, pages 60-65, 1982. URL:
  15. Eryk Kopczyński. Half-positional Determinacy of Infinite Games. PhD thesis, Warsaw University, 2008. Google Scholar
  16. Alexander Kozachinskiy. Energy games over totally ordered groups. CoRR, abs/2205.04508, 2022. URL:
  17. Alexander Kozachinskiy. Infinite separation between general and chromatic memory. CoRR, abs/2208.02691, 2022. URL:
  18. Alexander Kozachinskiy. State complexity of chromatic memory in infinite-duration games. CoRR, abs/2201.09297, 2022. URL:
  19. Pierre Ohlmann. Characterizing positionality in games of infinite duration over infinite graphs. In LICS, pages 22:1-22:12, 2022. URL:
  20. Pierre Ohlmann. Characterizing Positionality in Games of Infinite Duration over Infinite Graphs. TheoretiCS, Volume 2, January 2023. URL:
  21. Dominique Perrin and Jean-Éric Pin. Infinite words - automata, semigroups, logic and games, volume 141 of Pure and applied mathematics series. Elsevier Morgan Kaufmann, 2004. Google Scholar
  22. Michał Skrzypczak. Topological extension of parity automata. Information and Computation, 228-229:16-27, 2013. URL:
  23. Wiesław Zielonka. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoretical Computer Science, 200(1-2):135-183, 1998. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail