Characterizing Positionality in Games of Infinite Duration over Infinite Graphs
Article No.: 22, Pages 1 - 12
Abstract
We study turn-based quantitative games of infinite duration opposing two antagonistic players and played over graphs. This model is widely accepted as providing the adequate framework for formalizing the synthesis question for reactive systems. This important application motivates the question of strategy complexity: which valuations (or payoff functions) admit optimal positional strategies (without memory)? Valuations for which both players have optimal positional strategies have been characterized by Gimbert and Zielonka [16] for finite graphs and by Colcombet and Niwiński [12] for infinite graphs.
However, for reactive synthesis, existence of optimal positional strategies for the opponent (which models an antagonistic environment) is irrelevant. Despite this fact, not much is known about valuations for which the protagonist admits optimal positional strategies, regardless of the opponent. In this work, we characterize valuations which admit such strategies over infinite graphs. Our characterization uses the vocabulary of universal graphs, which has also proved useful in understanding recent breakthrough results regarding the complexity of parity games.
More precisely, we show that a valuation admitting universal graphs which are monotonic and well-ordered is positional over all game graphs, and – more surprisingly – that the converse is also true for valuations admitting neutral colors. We prove the applicability and elegance of the framework by unifying a number of known positionality results, proving a few stronger ones, and establishing closure under lexicographical products.
References
[1]
Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. 2021. From local to global determinacy in concurrent graph games. CoRR abs/2107.04081(2021). arxiv:2107.04081https://arxiv.org/abs/2107.04081
[2]
Patricia Bouyer, Ulrich Fahrenberg, Kim Guldstrand Larsen, Nicolas Markey, and Jirí Srba. 2008. Infinite Runs in Weighted Timed Automata with Energy Constraints. In FORMATS(Lecture Notes in Computer Science, Vol. 5215), Franck Cassez and Claude Jard (Eds.). Springer, 33–47. https://doi.org/10.1007/978-3-540-85778-5_4
[3]
Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. 2020. Games Where You Can Play Optimally with Arena-Independent Finite Memory. In CONCUR(LIPIcs, Vol. 171). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 24:1–24:22. https://doi.org/10.4230/LIPIcs.CONCUR.2020.24
[4]
Patricia Bouyer, Stéphane Le Roux, and Nathan Thomasset. 2021. Finite-memory strategies in two-player infinite games. CoRR abs/2107.09945(2021). arxiv:2107.09945https://arxiv.org/abs/2107.09945
[5]
Patricia Bouyer, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. 2021. Arena-Independent Finite-Memory Determinacy in Stochastic Games. In CONCUR(LIPIcs, Vol. 203). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 26:1–26:18. https://doi.org/10.4230/LIPIcs.CONCUR.2021.26
[6]
Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. 2021. Characterizing Omega-Regularity through Finite-Memory Determinacy of Games on Infinite Graphs. CoRR abs/2110.01276(2021). https://arxiv.org/abs/2110.01276
[7]
J. Richard Büchi. 1977. Using Determinancy of Games to Eliminate Quantifiers. In FCT(Lecture Notes in Computer Science, Vol. 56). Springer, 367–378. https://doi.org/10.1007/3-540-08442-8_104
[8]
J. R. Büchi and L. Landweber. 1969. Solving sequential conditions by finite-state strategies. Trans. Amer. Math. Soc. 138 (1969), 295–311.
[9]
Antonio Casares. 2021. On the Minimisation of Transition-Based Rabin Automata and the Chromatic Memory Requirements of Muller Conditions. CoRR abs/2105.12009(2021). arxiv:2105.12009https://arxiv.org/abs/2105.12009
[10]
Thomas Colcombet, Nathanaël Fijalkow, Pawel Gawrychowski, and Pierre Ohlmann. 2021. The Theory of Universal Graphs for Infinite Duration Games. CoRR abs/2104.05262(2021). arxiv:2104.05262https://arxiv.org/abs/2104.05262
[11]
Thomas Colcombet, Nathanaël Fijalkow, and Florian Horn. 2014. Playing Safe. In FSTTCS(LIPIcs, Vol. 29). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 379–390. https://doi.org/10.4230/LIPIcs.FSTTCS.2014.379
[12]
Thomas Colcombet and Damian Niwiński. 2006. On the positional determinacy of edge-labeled games. Theor. Comput. Sci. 352, 1-3 (2006), 190–196. https://doi.org/10.1016/j.tcs.2005.10.046
[13]
Stefan Dziembowski, Marcin Jurdzinski, and Igor Walukiewicz. 1997. How Much Memory is Needed to Win Infinite Games?. In LICS. IEEE Computer Society, 99–110. https://doi.org/10.1109/LICS.1997.614939
[14]
A. Ehrenfeucht and J. Mycielski. 1973. Positional games over a graph. Notices of the American Mathematical Society 20 (1973), A–334.
[15]
E. Allen Emerson and Charanjit S. Jutla. 1991. Tree Automata, μ-Calculus and Determinacy. In FOCS. IEEE Computer Society, 368–377.
[16]
Hugo Gimbert and Wieslaw Zielonka. 2005. Games Where You Can Play Optimally Without Any Memory. In CONCUR(Lecture Notes in Computer Science, Vol. 3653). Springer, 428–442. https://doi.org/10.1007/11539452_33
[17]
Erich Grädel and Igor Walukiewicz. 2006. Positional Determinacy of Games with Infinitely Many Priorities. Log. Methods Comput. Sci. 2, 4 (2006). https://doi.org/10.2168/LMCS-2(4:6)2006
[18]
Yuri Gurevich and Leo Harrington. 1982. Trees, Automata, and Games(STOC). Association for Computing Machinery, 60–65. https://doi.org/10.1145/800070.802177
[19]
Nils Klarlund. 1991. Progress Measures for Complementation of omega-Automata with Applications to Temporal Logic. In FOCS. IEEE Computer Society, 358–367. https://doi.org/10.1109/SFCS.1991.185391
[20]
Nils Klarlund. 1992. Progress Measures, Immediate Determinacy, and a Subset Construction for Tree Automata. In LICS. IEEE Computer Society, 382–393. https://doi.org/10.1109/LICS.1992.185550
[21]
Eryk Kopczyński. 2006. Half-Positional Determinacy of Infinite Games. In ICALP. 336–347.
[22]
Robert McNaughton. 1993. Infinite Games Played on Finite Graphs. Annals of Pure and Applied Logic 65, 2 (1993), 149–184.
[23]
Pierre Ohlmann. 2022. Characterizing Positionality in Games of Infinite Duration over Infinite Graphs. CoRR abs/2205.04309(2022). https://arxiv.org/abs/2205.04309
[24]
Stéphane Le Roux, Arno Pauly, and Mickael Randour. 2018. Extending Finite-Memory Determinacy by Boolean Combination of Winning Conditions. In FSTTCS(LIPIcs, Vol. 122). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 38:1–38:20. https://doi.org/10.4230/LIPIcs.FSTTCS.2018.38
[25]
L. S. Shapley. 1953. Stochastic Games. 39, 10 (1953), 1095–1100. https://doi.org/10.1073/pnas.39.10.1095
[26]
Igor Walukiewicz. 1996. Pushdown Processes: Games and Model Checking. In CAV(Lecture Notes in Computer Science, Vol. 1102). Springer, 62–74. https://doi.org/10.1007/3-540-61474-5_58
[27]
Wiesław Zielonka. 1998. Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees. Theoretical Computer Science 200, 1-2 (1998), 135–183. https://doi.org/10.1016/S0304-3975(98)00009-7
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Published: 04 August 2022
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LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science
August 2 - 5, 2022
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