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The Power of Counting Steps in Quantitative Games

Authors Sougata Bose , Rasmus Ibsen-Jensen , David Purser , Patrick Totzke , Pierre Vandenhove

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Author Details

Sougata Bose
  • University of Liverpool, UK
Rasmus Ibsen-Jensen
  • University of Liverpool, UK
David Purser
  • University of Liverpool, UK
Patrick Totzke
  • University of Liverpool, UK
Pierre Vandenhove
  • LaBRI, Université de Bordeaux, France

Funding

P. Vandenhove is funded by ANR project G4S (ANR-21-CE48-0010-01). S. Bose and P. Totzke acknowledge support from the EPSRC (EP/V025848/1, EP/X042596/1). Collaboration initiated via Royal Society International Exchanges grant (IES{backslash}R1{backslash}201211).

Cite AsGet BibTex

Sougata Bose, Rasmus Ibsen-Jensen, David Purser, Patrick Totzke, and Pierre Vandenhove. The Power of Counting Steps in Quantitative Games. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CONCUR.2024.13

Abstract

We study deterministic games of infinite duration played on graphs and focus on the strategy complexity of quantitative objectives. Such games are known to admit optimal memoryless strategies over finite graphs, but require infinite-memory strategies in general over infinite graphs. We provide new lower and upper bounds for the strategy complexity of mean-payoff and total-payoff objectives over infinite graphs, focusing on whether step-counter strategies (sometimes called Markov strategies) suffice to implement winning strategies. In particular, we show that over finitely branching arenas, three variants of limsup mean-payoff and total-payoff objectives admit winning strategies that are based either on a step counter or on a step counter and an additional bit of memory. Conversely, we show that for certain liminf total-payoff objectives, strategies resorting to a step counter and finite memory are not sufficient. For step-counter strategies, this settles the case of all classical quantitative objectives up to the second level of the Borel hierarchy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Games on graphs
  • Markov strategies
  • quantitative objectives
  • infinite-state systems

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References

  1. Roderick Bloem, Krishnendu Chatterjee, and Barbara Jobstmann. Graph games and reactive synthesis. In Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem, editors, Handbook of Model Checking, pages 921-962. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-10575-8_27.
  2. Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. Optimal strategies in concurrent reachability games. In Florin Manea and Alex Simpson, editors, 30th EACSL Annual Conference on Computer Science Logic, CSL 2022, February 14-19, 2022, Göttingen, Germany (Virtual Conference), volume 216 of LIPIcs, pages 7:1-7:17, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CSL.2022.7.
  3. Sougata Bose, Rasmus Ibsen-Jensen, David Purser, Patrick Totzke, and Pierre Vandenhove. The power of counting steps in quantitative games. CoRR, abs/2406.17482, 2024. URL: https://doi.org/10.48550/arXiv.2406.17482.
  4. Sougata Bose, Rasmus Ibsen-Jensen, and Patrick Totzke. Bounded-memory strategies in partial-information games. In ACM/IEEE Symposium on Logic in Computer Science, LICS '24, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3661814.3662096.
  5. Patricia Bouyer, Nathanaël Fijalkow, Mickael Randour, and Pierre Vandenhove. How to play optimally for regular objectives? In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 118:1-118:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.118.
  6. J. Richard Büchi and Lawrence H. Landweber. Definability in the monadic second-order theory of successor. Journal of Symbolic Logic, 34(2):166-170, 1969. URL: https://doi.org/10.2307/2271090.
  7. Antonio Casares. On the minimisation of transition-based Rabin automata and the chromatic memory requirements of Muller conditions. In Florin Manea and Alex Simpson, editors, 30th EACSL Annual Conference on Computer Science Logic, CSL 2022, February 14-19, 2022, Göttingen, Germany (Virtual Conference), volume 216 of LIPIcs, pages 12:1-12:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.CSL.2022.12.
  8. Arindam Chakrabarti, Luca de Alfaro, Thomas A. Henzinger, and Mariëlle Stoelinga. Resource interfaces. In Rajeev Alur and Insup Lee, editors, Proceedings of the 3rd International Conference on Embedded Software, EMSOFT 2003, Philadelphia, PA, USA, October 13-15, 2003, volume 2855 of Lecture Notes in Computer Science, pages 117-133. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-45212-6_9.
  9. Thomas Colcombet, Nathanaël Fijalkow, and Florian Horn. Playing safe, ten years later. Logical Methods in Computer Science, 20(1), 2024. URL: https://doi.org/10.46298/LMCS-20(1:10)2024.
  10. Stefan Dziembowski, Marcin Jurdziński, and Igor Walukiewicz. How much memory is needed to win infinite games? In Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science, LICS 1997, Warsaw, Poland, June 29 - July 2, 1997, pages 99-110, 1997. URL: https://doi.org/10.1109/.1997.614939.
  11. Andrzej Ehrenfeucht and Jan Mycielski. Positional strategies for mean payoff games. International Journal of Game Theory, 8(2):109-113, 1979. URL: https://doi.org/10.1007/BF01768705.
  12. E. Allen Emerson and Charanjit S. Jutla. Tree automata, mu-calculus and determinacy (extended abstract). In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, FOCS 1991, San Juan, Puerto Rico, October, 1991, pages 368-377. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185392.
  13. Nathanaël Fijalkow, Nathalie Bertrand, Patricia Bouyer-Decitre, Romain Brenguier, Arnaud Carayol, John Fearnley, Hugo Gimbert, Florian Horn, Rasmus Ibsen-Jensen, Nicolas Markey, Benjamin Monmege, Petr Novotný, Mickael Randour, Ocan Sankur, Sylvain Schmitz, Olivier Serre, and Mateusz Skomra. Games on graphs, 2023. URL: https://arxiv.org/abs/2305.10546.
  14. Jerzy Filar and Koos Vrieze. Competitive Markov Decision Processes. Springer New York, 1996. URL: https://books.google.fr/books?id=21lcbnzDNwsC.
  15. Dean Gillette. Stochastic Games with Zero Stop Probabilities, pages 179-188. Princeton University Press, Princeton, 1957. URL: https://doi.org/10.1515/9781400882151-011.
  16. Hugo Gimbert and Wiesław Zielonka. When can you play positionally? In Jiří Fiala, Václav Koubek, and Jan Kratochvíl, editors, Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, MFCS 2004, Prague, Czech Republic, August 22-27, 2004, volume 3153 of Lecture Notes in Computer Science, pages 686-697. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-28629-5_53.
  17. Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors. Automata, Logics, and Infinite Games: A Guide to Current Research [outcome of a Dagstuhl seminar, February 2001], volume 2500 of Lecture Notes in Computer Science. Springer, 2002. URL: https://doi.org/10.1007/3-540-36387-4.
  18. Yuri Gurevich and Leo Harrington. Trees, automata, and games. In Harry R. Lewis, Barbara B. Simons, Walter A. Burkhard, and Lawrence H. Landweber, editors, Proceedings of the 14th Annual ACM Symposium on Theory of Computing, STOC 1982, San Francisco, CA, USA, May 5-7, 1982, pages 60-65. ACM, 1982. URL: https://doi.org/10.1145/800070.802177.
  19. Kristoffer Arnsfelt Hansen, Rasmus Ibsen-Jensen, and Abraham Neyman. The big match with a clock and a bit of memory. Mathematics of Operations Research, 48(1):419-432, 2023. URL: https://doi.org/10.1287/moor.2022.1267.
  20. Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, and Patrick Totzke. Strategy complexity of parity objectives in countable MDPs. In Igor Konnov and Laura Kovács, editors, Proceedings of the 31st International Conference on Concurrency Theory, CONCUR 2020, September 1-4, 2020, Vienna, Austria (Virtual Conference), volume 171 of LIPIcs, pages 39:1-39:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2020.39.
  21. Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, and Patrick Totzke. Memoryless strategies in stochastic reachability games, 2024. To appear in Lecture Notes in Computer Science. Google Scholar
  22. Stefan Kiefer, Richard Mayr, Mahsa Shirmohammadi, and Patrick Totzke. Strategy complexity of Büchi objectives in concurrent stochastic games, 2024. URL: https://arxiv.org/abs/2404.15483.
  23. Ashok P. Maitra and William D. Sudderth. Discrete Gambling and Stochastic Games. Springer-Verlag, 1996. Google Scholar
  24. Donald A. Martin. Borel determinacy. Annals of Mathematics, 102(2):363-371, 1975. URL: http://www.jstor.org/stable/1971035.
  25. Richard Mayr and Eric Munday. Strategy complexity of point payoff, mean payoff and total payoff objectives in countable MDPs. Logical Methods in Computer Science, 19(1), 2023. URL: https://doi.org/10.46298/LMCS-19(1:16)2023.
  26. Pierre Ohlmann. Monotonic graphs for parity and mean-payoff games. PhD thesis, IRIF - Research Institute on the Foundations of Computer Science, 2021. Google Scholar
  27. Pierre Ohlmann and Michał Skrzypczak. Positionality in Σ₂⁰ and a completeness result. In Olaf Beyersdorff, Mamadou Moustapha Kanté, Orna Kupferman, and Daniel Lokshtanov, editors, Proceedings of the 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024, March 12-14, 2024, Clermont-Ferrand, France, volume 289 of LIPIcs, pages 54:1-54:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.STACS.2024.54.
  28. Anuj Puri. Theory of Hybrid Systems and Discrete Event Systems. PhD thesis, EECS Department, University of California, Berkeley, December 1995. URL: http://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/2950.html.
  29. Martin L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series in Probability and Statistics. Wiley, 1994. URL: https://doi.org/10.1002/9780470316887.
  30. Michael O. Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141:1-35, 1969. URL: https://doi.org/10.2307/1995086.
  31. Lloyd S. Shapley. Stochastic games. Proceedings of the National Academy of Sciences, 39(10):1095-1100, 1953. URL: https://doi.org/10.1073/pnas.39.10.1095.
  32. Frank Thuijsman. Optimality and Equilibria in Stochastic Games. Number no. 82 in CWI Tract - Centrum voor Wiskunde en Informatica. Centrum voor Wiskunde en Informatica, 1992. URL: https://books.google.co.uk/books?id=sfzuAAAAMAAJ.
  33. Wiesław Zielonka. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoretical Computer Science, 200(1-2):135-183, 1998. URL: https://doi.org/10.1016/S0304-3975(98)00009-7.
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