Document Open Access Logo

How to Play Optimally for Regular Objectives?

Authors Patricia Bouyer , Nathanaël Fijalkow , Mickael Randour , Pierre Vandenhove

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2023.118.pdf
  • Filesize: 0.83 MB
  • 18 pages

Document Identifiers

Author Details

Patricia Bouyer
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France
Nathanaël Fijalkow
  • CNRS, LaBRI and Université de Bordeaux, France
  • University of Warsaw, Poland
Mickael Randour
  • F.R.S.-FNRS & UMONS - Université de Mons, Belgium
Pierre Vandenhove
  • F.R.S.-FNRS & UMONS - Université de Mons, Belgium
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

Funding

This work has been partially supported by the ANR Project MAVeriQ (ANR-20-CE25-0012) and by the Fonds de la Recherche Scientifique - FNRS under Grant n^∘ T.0188.23 (PDR ControlleRS). Mickael Randour is an F.R.S.-FNRS Research Associate and a member of the TRAIL Institute. Pierre Vandenhove is an F.R.S.-FNRS Research Fellow.

Acknowledgements

We thank Antonio Casares and Igor Walukiewicz for valuable discussions about this article.

Cite As Get BibTex

Patricia Bouyer, Nathanaël Fijalkow, Mickael Randour, and Pierre Vandenhove. How to Play Optimally for Regular Objectives?. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 118:1-118:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.118

Abstract

This paper studies two-player zero-sum games played on graphs and makes contributions toward the following question: given an objective, how much memory is required to play optimally for that objective? We study regular objectives, where the goal of one of the two players is that eventually the sequence of colors along the play belongs to some regular language of finite words. We obtain different characterizations of the chromatic memory requirements for such objectives for both players, from which we derive complexity-theoretic statements: deciding whether there exist small memory structures sufficient to play optimally is NP-complete for both players. Some of our characterization results apply to a more general class of objectives: topologically closed and topologically open sets.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • two-player games on graphs
  • strategy complexity
  • regular languages
  • finite-memory strategies
  • NP-completeness

Metrics

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

Related Versions

References

  1. Alessandro Bianco, Marco Faella, Fabio Mogavero, and Aniello Murano. Exploring the boundary of half-positionality. Annals of Mathematics and Artificial Intelligence, 62(1-2):55-77, 2011. URL: https://doi.org/10.1007/s10472-011-9250-1.
  2. Kellogg S. Booth. Isomorphism testing for graphs, semigroups, and finite automata are polynomially equivalent problems. SIAM Journal on Computing, 7(3):273-279, 1978. URL: https://doi.org/10.1137/0207023.
  3. Patricia Bouyer, Antonio Casares, Mickael Randour, and Pierre Vandenhove. Half-positional objectives recognized by deterministic Büchi automata. In Bartek Klin, Sławomir Lasota, and Anca Muscholl, editors, Proceedings of the 33rd International Conference on Concurrency Theory, CONCUR 2022, Warsaw, Poland, September 12-16, 2022, volume 243 of LIPIcs, pages 20:1-20:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2022.20.
  4. Patricia Bouyer, Nathanaël Fijalkow, Mickael Randour, and Pierre Vandenhove. How to play optimally for regular objectives? CoRR, abs/2210.09703, 2022. URL: https://doi.org/10.48550/arXiv.2210.09703.
  5. Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Games where you can play optimally with arena-independent finite memory. Logical Methods in Computer Science, 18(1), 2022. URL: https://doi.org/10.46298/lmcs-18(1:11)2022.
  6. Patricia Bouyer, Stéphane Le Roux, and Nathan Thomasset. Finite-memory strategies in two-player infinite games. In Florin Manea and Alex Simpson, editors, Proceedings of the 30th EACSL Annual Conference on Computer Science Logic, CSL 2022, Göttingen, Germany, February 14-19, 2022, volume 216 of LIPIcs, pages 8:1-8:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CSL.2022.8.
  7. Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. Characterizing omega-regularity through finite-memory determinacy of games on infinite graphs. TheoretiCS, 2:1-48, 2023. URL: https://doi.org/10.46298/theoretics.23.1.
  8. 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, Proceedings of the 30th EACSL Annual Conference on Computer Science Logic, CSL 2022, Göttingen, Germany, February 14-19, 2022, 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.
  9. 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 Mikołaj Bojańczyk, Emanuela Merelli, and David P. Woodruff, editors, Proceedings of the 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, Paris, France, July 4-8, 2022, volume 229 of LIPIcs, pages 117:1-117:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.117.
  10. Thomas Colcombet, Nathanaël Fijalkow, and Florian Horn. Playing safe. In Venkatesh Raman and S. P. Suresh, editors, Proceedings of the 34th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2014, New Delhi, India, December 15-17, 2014, volume 29 of LIPIcs, pages 379-390. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2014. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2014.379.
  11. Thomas Colcombet, Nathanaël Fijalkow, and Florian Horn. Playing safe, ten years later. CoRR, abs/2212.12024, 2022. URL: https://doi.org/10.48550/arXiv.2212.12024.
  12. Thomas Colcombet and Damian Niwiński. On the positional determinacy of edge-labeled games. Theoretical Computer Science, 352(1-3):190-196, 2006. URL: https://doi.org/10.1016/j.tcs.2005.10.046.
  13. 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. IEEE Computer Society, 1997. URL: https://doi.org/10.1109/LICS.1997.614939.
  14. Hugo Gimbert and Wieslaw Zielonka. Games where you can play optimally without any memory. In Martín Abadi and Luca de Alfaro, editors, Proceedings of the 16th International Conference on Concurrency Theory, CONCUR 2005, San Francisco, CA, USA, August 23-26, 2005, volume 3653 of Lecture Notes in Computer Science, pages 428-442. Springer, 2005. URL: https://doi.org/10.1007/11539452_33.
  15. Abraham Ginzburg and Michael Yoeli. Products of automata and the problem of covering. Transactions of the American Mathematical Society, 116:253-266, 1965. URL: http://www.jstor.org/stable/1994117.
  16. 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.
  17. Alexey Ignatiev, António Morgado, and João Marques-Silva. PySAT: A Python toolkit for prototyping with SAT oracles. In Olaf Beyersdorff and Christoph M. Wintersteiger, editors, Proceedings of the 21st International Conference on the Theory and Applications of Satisfiability Testing, SAT 2018, Held as Part of FloC 2018, Oxford, UK, July 9-12, 2018, volume 10929 of Lecture Notes in Computer Science, pages 428-437. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-94144-8_26.
  18. Eryk Kopczyński. Half-positional Determinacy of Infinite Games. PhD thesis, Warsaw University, 2008. Google Scholar
  19. Stéphane Le Roux, Arno Pauly, and Mickael Randour. Extending finite-memory determinacy by Boolean combination of winning conditions. In Sumit Ganguly and Paritosh K. Pandya, editors, Proceedings of the 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2018, Ahmedabad, India, December 11-13, 2018, volume 122 of LIPIcs, pages 38:1-38:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2018.38.
  20. Anil Nerode. Linear automaton transformations. Proceedings of the American Mathematical Society, 9(4):541-544, 1958. URL: https://doi.org/10.2307/2033204.
  21. Pierre Ohlmann. Characterizing positionality in games of infinite duration over infinite graphs. TheoretiCS, 2, 2023. URL: https://doi.org/10.46298/theoretics.23.3.
  22. 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.
  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. URL: https://doi.org/10.1016/S0304-3975(98)00009-7.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact