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Games Where You Can Play Optimally with Arena-Independent Finite Memory

Authors Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, Pierre Vandenhove



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Author Details

Patricia Bouyer
  • LSV – CNRS & ENS Paris-Saclay, Université Paris-Saclay, Saint-Aubin, France
Stéphane Le Roux
  • LSV – CNRS & ENS Paris-Saclay, Université Paris-Saclay, Saint-Aubin, France
Youssouf Oualhadj
  • LACL, UPEC, Créteil, France
Mickael Randour
  • F.R.S.-FNRS & UMONS – Université de Mons, Mons, Belgium
Pierre Vandenhove
  • F.R.S.-FNRS & UMONS – Université de Mons, Mons, Belgium
  • LSV – CNRS & ENS Paris-Saclay, Université Paris-Saclay, Saint-Aubin, France

Acknowledgements

We extend our warmest thanks to Mathieu Sassolas, for inspiring discussions that were essential in starting this work.

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 24:1-24:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CONCUR.2020.24

Abstract

For decades, two-player (antagonistic) games on graphs have been a framework of choice for many important problems in theoretical computer science. A notorious one is controller synthesis, which can be rephrased through the game-theoretic metaphor as the quest for a winning strategy of the system in a game against its antagonistic environment. Depending on the specification, optimal strategies might be simple or quite complex, for example having to use (possibly infinite) memory. Hence, research strives to understand which settings allow for simple strategies. In 2005, Gimbert and Zielonka [Hugo Gimbert and Wieslaw Zielonka, 2005] provided a complete characterization of preference relations (a formal framework to model specifications and game objectives) that admit memoryless optimal strategies for both players. In the last fifteen years however, practical applications have driven the community toward games with complex or multiple objectives, where memory - finite or infinite - is almost always required. Despite much effort, the exact frontiers of the class of preference relations that admit finite-memory optimal strategies still elude us. In this work, we establish a complete characterization of preference relations that admit optimal strategies using arena-independent finite memory, generalizing the work of Gimbert and Zielonka to the finite-memory case. We also prove an equivalent to their celebrated corollary of great practical interest: if both players have optimal (arena-independent-)finite-memory strategies in all one-player games, then it is also the case in all two-player games. Finally, we pinpoint the boundaries of our results with regard to the literature: our work completely covers the case of arena-independent memory (e.g., multiple parity objectives, lower- and upper-bounded energy objectives), and paves the way to the arena-dependent case (e.g., multiple lower-bounded energy objectives).

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • two-player games on graphs
  • finite-memory determinacy
  • optimal strategies

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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. 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.
  3. Patricia Bouyer, Ulrich Fahrenberg, Kim Guldstrand Larsen, Nicolas Markey, and Jirí Srba. Infinite runs in weighted timed automata with energy constraints. In Franck Cassez and Claude Jard, editors, Formal Modeling and Analysis of Timed Systems, 6th International Conference, FORMATS 2008, Saint Malo, France, September 15-17, 2008. Proceedings, volume 5215 of Lecture Notes in Computer Science, pages 33-47. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-85778-5_4.
  4. Patricia Bouyer, Piotr Hofman, Nicolas Markey, Mickael Randour, and Martin Zimmermann. Bounding average-energy games. In Javier Esparza and Andrzej S. Murawski, editors, Foundations of Software Science and Computation Structures - 20th International Conference, FOSSACS 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2017, Uppsala, Sweden, April 22-29, 2017, Proceedings, volume 10203 of Lecture Notes in Computer Science, pages 179-195, 2017. URL: https://doi.org/10.1007/978-3-662-54458-7_11.
  5. Patricia Bouyer, Nicolas Markey, Mickael Randour, Kim G. Larsen, and Simon Laursen. Average-energy games. Acta Inf., 55(2):91-127, 2018. URL: https://doi.org/10.1007/s00236-016-0274-1.
  6. Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Games where you can play optimally with arena-independent finite memory. CoRR, abs/2001.03894, 2020. URL: http://arxiv.org/abs/2001.03894.
  7. Romain Brenguier, Lorenzo Clemente, Paul Hunter, Guillermo A. Pérez, Mickael Randour, Jean-François Raskin, Ocan Sankur, and Mathieu Sassolas. Non-zero sum games for reactive synthesis. In Adrian-Horia Dediu, Jan Janousek, Carlos Martín-Vide, and Bianca Truthe, editors, Language and Automata Theory and Applications - 10th International Conference, LATA 2016, Prague, Czech Republic, March 14-18, 2016, Proceedings, volume 9618 of Lecture Notes in Computer Science, pages 3-23. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-30000-9_1.
  8. Thomas Brihaye, Florent Delgrange, Youssouf Oualhadj, and Mickael Randour. Life is random, time is not: Markov decision processes with window objectives. In Wan Fokkink and Rob van Glabbeek, editors, 30th International Conference on Concurrency Theory, CONCUR 2019, August 27-30, 2019, Amsterdam, the Netherlands, volume 140 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2019.8.
  9. Véronique Bruyère, Emmanuel Filiot, Mickael Randour, and Jean-François Raskin. Meet your expectations with guarantees: Beyond worst-case synthesis in quantitative games. Inf. Comput., 254:259-295, 2017. URL: https://doi.org/10.1016/j.ic.2016.10.011.
  10. Véronique Bruyère, Quentin Hautem, and Mickael Randour. Window parity games: an alternative approach toward parity games with time bounds. In Domenico Cantone and Giorgio Delzanno, editors, Proceedings of the Seventh International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2016, Catania, Italy, 14-16 September 2016, volume 226 of EPTCS, pages 135-148, 2016. URL: https://doi.org/10.4204/EPTCS.226.10.
  11. Véronique Bruyère, Quentin Hautem, Mickael Randour, and Jean-François Raskin. Energy mean-payoff games. In Wan Fokkink and Rob van Glabbeek, editors, 30th International Conference on Concurrency Theory, CONCUR 2019, August 27-30, 2019, Amsterdam, the Netherlands, volume 140 of LIPIcs, pages 21:1-21:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2019.21.
  12. Arindam Chakrabarti, Luca de Alfaro, Thomas A. Henzinger, and Mariëlle Stoelinga. Resource interfaces. In Rajeev Alur and Insup Lee, editors, EMSOFT, 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.
  13. Krishnendu Chatterjee and Laurent Doyen. Energy parity games. Theor. Comput. Sci., 458:49-60, 2012. URL: https://doi.org/10.1016/j.tcs.2012.07.038.
  14. Krishnendu Chatterjee, Laurent Doyen, Mickael Randour, and Jean-François Raskin. Looking at mean-payoff and total-payoff through windows. Inf. Comput., 242:25-52, 2015. URL: https://doi.org/10.1016/j.ic.2015.03.010.
  15. Krishnendu Chatterjee, Thomas A. Henzinger, and Marcin Jurdzinski. Mean-payoff parity games. In 20th IEEE Symposium on Logic in Computer Science (LICS 2005), 26-29 June 2005, Chicago, IL, USA, Proceedings, pages 178-187. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/LICS.2005.26.
  16. Krishnendu Chatterjee, Thomas A. Henzinger, and Nir Piterman. Generalized parity games. In Helmut Seidl, editor, Foundations of Software Science and Computational Structures, 10th International Conference, FOSSACS 2007, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2007, Braga, Portugal, March 24-April 1, 2007, Proceedings, volume 4423 of Lecture Notes in Computer Science, pages 153-167. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-71389-0_12.
  17. Krishnendu Chatterjee, Mickael Randour, and Jean-François Raskin. Strategy synthesis for multi-dimensional quantitative objectives. Acta Inf., 51(3-4):129-163, 2014. URL: https://doi.org/10.1007/s00236-013-0182-6.
  18. Florent Delgrange, Joost-Pieter Katoen, Tim Quatmann, and Mickael Randour. Simple strategies in multi-objective MDPs. In Armin Biere and David Parker, editors, Tools and Algorithms for the Construction and Analysis of Systems - 26th International Conference, TACAS 2020, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020, Dublin, Ireland, April 25-30, 2020, Proceedings, Part I, volume 12078 of Lecture Notes in Computer Science, pages 346-364. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-45190-5_19.
  19. Andrzej Ehrenfeucht and Jan Mycielski. Positional strategies for mean payoff games. Int. Journal of Game Theory, 8(2):109-113, 1979. Google Scholar
  20. E. Allen Emerson and Charanjit S. Jutla. The complexity of tree automata and logics of programs. In FOCS, pages 328-337. IEEE Computer Society, 1988. URL: https://doi.org/10.1109/SFCS.1988.21949.
  21. Nathanaël Fijalkow and Florian Horn. The surprizing complexity of reachability games. CoRR, abs/1010.2420, 2010. URL: http://arxiv.org/abs/1010.2420.
  22. Nathanaël Fijalkow, Florian Horn, Denis Kuperberg, and Michal Skrzypczak. Trading bounds for memory in games with counters. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 197-208. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47666-6_16.
  23. Hugo Gimbert. Pure stationary optimal strategies in Markov decision processes. In Wolfgang Thomas and Pascal Weil, editors, STACS 2007, 24th Annual Symposium on Theoretical Aspects of Computer Science, Aachen, Germany, February 22-24, 2007, Proceedings, volume 4393 of Lecture Notes in Computer Science, pages 200-211. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-70918-3_18.
  24. Hugo Gimbert and Edon Kelmendi. Two-player perfect-information shift-invariant submixing stochastic games are half-positional. Unpublished, 2014. Google Scholar
  25. Hugo Gimbert and Wieslaw Zielonka. When can you play positionally? In Jirí Fiala, Václav Koubek, and Jan Kratochvíl, editors, Mathematical Foundations of Computer Science 2004, 29th International Symposium, MFCS 2004, Prague, Czech Republic, August 22-27, 2004, Proceedings, 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.
  26. Hugo Gimbert and Wieslaw Zielonka. Games where you can play optimally without any memory. In Martín Abadi and Luca de Alfaro, editors, CONCUR 2005 - Concurrency Theory, 16th International Conference, CONCUR 2005, San Francisco, CA, USA, August 23-26, 2005, Proceedings, volume 3653 of Lecture Notes in Computer Science, pages 428-442. Springer, 2005. URL: https://doi.org/10.1007/11539452_33.
  27. Hugo Gimbert and Wieslaw Zielonka. Pure and Stationary Optimal Strategies in Perfect-Information Stochastic Games with Global Preferences. Unpublished, December 2009. URL: https://hal.archives-ouvertes.fr/hal-00438359.
  28. 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.
  29. Marcin Jurdzinski. Deciding the winner in parity games is in UP ∩ co-UP. Inf. Process. Lett., 68(3):119-124, 1998. URL: https://doi.org/10.1016/S0020-0190(98)00150-1.
  30. Marcin Jurdzinski, Ranko Lazic, and Sylvain Schmitz. Fixed-dimensional energy games are in pseudo-polynomial time. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 260-272. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47666-6_21.
  31. Eryk Kopczyński. Half-positional determinacy of infinite games. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part II, volume 4052 of Lecture Notes in Computer Science, pages 336-347. Springer, 2006. URL: https://doi.org/10.1007/11787006_29.
  32. Eryk Kopczyński. Half-positional Determinacy of Infinite Games. PhD thesis, Warsaw University, 2008. Google Scholar
  33. Donald A. Martin. Borel determinacy. Annals of Mathematics, 102(2):363-371, 1975. Google Scholar
  34. Martin J. Osborne and Ariel Rubinstein. A course in game theory. The MIT Press, Cambridge, USA, 1994. Google Scholar
  35. Mickael Randour. Automated synthesis of reliable and efficient systems through game theory: A case study. In Proc. of ECCS 2012, Springer Proceedings in Complexity XVII, pages 731-738. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-00395-5_90.
  36. Stéphane Le Roux. Infinite sequential Nash equilibrium. Logical Methods in Computer Science, 9(2), 2013. URL: https://doi.org/10.2168/LMCS-9(2:3)2013.
  37. Stéphane Le Roux. Concurrent games and semi-random determinacy. In Igor Potapov, Paul G. Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, volume 117 of LIPIcs, pages 40:1-40:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.40.
  38. Stéphane Le Roux and Arno Pauly. Extending finite-memory determinacy to multi-player games. Inf. Comput., 261(Part):676-694, 2018. URL: https://doi.org/10.1016/j.ic.2018.02.024.
  39. 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, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2018, December 11-13, 2018, Ahmedabad, India, volume 122 of LIPIcs, pages 38:1-38:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2018.38.
  40. 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.
  41. Wieslaw Zielonka. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci., 200(1-2):135-183, 1998. URL: https://doi.org/10.1016/S0304-3975(98)00009-7.
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