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Invited Talk

**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Reachability objectives are arguably the most basic ones in the theory of games on graphs (and beyond). But far from being bland, they constitute the cornerstone of this field. Reachability is everywhere, as are the tools we use to reason about it. In this invited contribution, we take the reader on a journey through a zoo of models that have reachability objectives at their core. Our goal is to illustrate how model complexity impacts the complexity of strategies needed to play optimally in the corresponding games and computational complexity.

Thomas Brihaye, Aline Goeminne, James C. A. Main, and Mickael Randour. Reachability Games and Friends: A Journey Through the Lens of Memory and Complexity (Invited Talk). In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 1:1-1:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{brihaye_et_al:LIPIcs.FSTTCS.2023.1, author = {Brihaye, Thomas and Goeminne, Aline and Main, James C. A. and Randour, Mickael}, title = {{Reachability Games and Friends: A Journey Through the Lens of Memory and Complexity}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {1:1--1:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.1}, URN = {urn:nbn:de:0030-drops-193747}, doi = {10.4230/LIPIcs.FSTTCS.2023.1}, annote = {Keywords: Games on graphs, reachability, finite-memory strategies, complexity} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

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.

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)

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@InProceedings{bouyer_et_al:LIPIcs.ICALP.2023.118, author = {Bouyer, Patricia and Fijalkow, Nathana\"{e}l and Randour, Mickael and Vandenhove, Pierre}, title = {{How to Play Optimally for Regular Objectives?}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {118:1--118:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.118}, URN = {urn:nbn:de:0030-drops-181700}, doi = {10.4230/LIPIcs.ICALP.2023.118}, annote = {Keywords: two-player games on graphs, strategy complexity, regular languages, finite-memory strategies, NP-completeness} }

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Invited Talk

**Published in:** LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Two-player turn-based zero-sum games on (finite or infinite) graphs are a central framework in theoretical computer science - notably as a tool for controller synthesis, but also due to their connection with logic and automata theory. A crucial challenge in the field is to understand how complex strategies need to be to play optimally, given a type of game and a winning objective. In this invited contribution, we give a tour of recent advances aiming to characterize games where finite-memory strategies suffice (i.e., using a limited amount of information about the past). We mostly focus on so-called chromatic memory, which is limited to using colors - the basic building blocks of objectives - seen along a play to update itself. Chromatic memory has the advantage of being usable in different game graphs, and the corresponding class of strategies turns out to be of great interest to both the practical and the theoretical sides.

Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. The True Colors of Memory: A Tour of Chromatic-Memory Strategies in Zero-Sum Games on Graphs (Invited Talk). In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bouyer_et_al:LIPIcs.FSTTCS.2022.3, author = {Bouyer, Patricia and Randour, Mickael and Vandenhove, Pierre}, title = {{The True Colors of Memory: A Tour of Chromatic-Memory Strategies in Zero-Sum Games on Graphs}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {3:1--3:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-261-7}, ISSN = {1868-8969}, year = {2022}, volume = {250}, editor = {Dawar, Anuj and Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.3}, URN = {urn:nbn:de:0030-drops-173957}, doi = {10.4230/LIPIcs.FSTTCS.2022.3}, annote = {Keywords: two-player games on graphs, finite-memory strategies, chromatic memory, parity automata, \omega-regularity} }

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Invited Paper

**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

This short article recaps the purpose of the CONCUR Test-of-Time Award and presents the four papers that received the Award in 2022.

Ilaria Castellani, Paul Gastin, Orna Kupferman, Mickael Randour, and Davide Sangiorgi. CONCUR Test-Of-Time Award 2022 (Invited Paper). In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 1:1-1:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{castellani_et_al:LIPIcs.CONCUR.2022.1, author = {Castellani, Ilaria and Gastin, Paul and Kupferman, Orna and Randour, Mickael and Sangiorgi, Davide}, title = {{CONCUR Test-Of-Time Award 2022}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {1:1--1:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.1}, URN = {urn:nbn:de:0030-drops-170644}, doi = {10.4230/LIPIcs.CONCUR.2022.1}, annote = {Keywords: CONCUR Test-of-Time Award} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

A central question in the theory of two-player games over graphs is to understand which objectives are half-positional, that is, which are the objectives for which the protagonist does not need memory to implement winning strategies. Objectives for which both players do not need memory have already been characterized (both in finite and infinite graphs); however, less is known about half-positional objectives. In particular, no characterization of half-positionality is known for the central class of ω-regular objectives.
In this paper, we characterize objectives recognizable by deterministic Büchi automata (a class of ω-regular objectives) that are half-positional, in both finite and infinite graphs. Our characterization consists of three natural conditions linked to the language-theoretic notion of right congruence. Furthermore, this characterization yields a polynomial-time algorithm to decide half-positionality of an objective recognized by a given deterministic Büchi automaton.

Patricia Bouyer, Antonio Casares, Mickael Randour, and Pierre Vandenhove. Half-Positional Objectives Recognized by Deterministic Büchi Automata. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bouyer_et_al:LIPIcs.CONCUR.2022.20, author = {Bouyer, Patricia and Casares, Antonio and Randour, Mickael and Vandenhove, Pierre}, title = {{Half-Positional Objectives Recognized by Deterministic B\"{u}chi Automata}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.20}, URN = {urn:nbn:de:0030-drops-170833}, doi = {10.4230/LIPIcs.CONCUR.2022.20}, annote = {Keywords: two-player games on graphs, half-positionality, memoryless optimal strategies, B\"{u}chi automata, \omega-regularity} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

Two-player (antagonistic) games on (possibly stochastic) graphs are a prevalent model in theoretical computer science, notably as a framework for reactive synthesis.
Optimal strategies may require randomisation when dealing with inherently probabilistic goals, balancing multiple objectives, or in contexts of partial information. There is no unique way to define randomised strategies. For instance, one can use so-called mixed strategies or behavioural ones. In the most general settings, these two classes do not share the same expressiveness. A seminal result in game theory - Kuhn’s theorem - asserts their equivalence in games of perfect recall.
This result crucially relies on the possibility for strategies to use infinite memory, i.e., unlimited knowledge of all past observations. However, computer systems are finite in practice. Hence it is pertinent to restrict our attention to finite-memory strategies, defined as automata with outputs. Randomisation can be implemented in these in different ways: the initialisation, outputs or transitions can be randomised or deterministic respectively. Depending on which aspects are randomised, the expressiveness of the corresponding class of finite-memory strategies differs.
In this work, we study two-player turn-based stochastic games and provide a complete taxonomy of the classes of finite-memory strategies obtained by varying which of the three aforementioned components are randomised. Our taxonomy holds both in settings of perfect and imperfect information, and in games with more than two players.

James C. A. Main and Mickael Randour. Different Strokes in Randomised Strategies: Revisiting Kuhn’s Theorem Under Finite-Memory Assumptions. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{main_et_al:LIPIcs.CONCUR.2022.22, author = {Main, James C. A. and Randour, Mickael}, title = {{Different Strokes in Randomised Strategies: Revisiting Kuhn’s Theorem Under Finite-Memory Assumptions}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.22}, URN = {urn:nbn:de:0030-drops-170854}, doi = {10.4230/LIPIcs.CONCUR.2022.22}, annote = {Keywords: two-player games on graphs, stochastic games, Markov decision processes, finite-memory strategies, randomised strategies} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of ω-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is ω-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be ω-regular. This provides a game-theoretic characterization of ω-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which ω-regularity depends on the value of some parameters.

Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bouyer_et_al:LIPIcs.STACS.2022.16, author = {Bouyer, Patricia and Randour, Mickael and Vandenhove, Pierre}, title = {{Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.16}, URN = {urn:nbn:de:0030-drops-158262}, doi = {10.4230/LIPIcs.STACS.2022.16}, annote = {Keywords: two-player games on graphs, infinite arenas, finite-memory determinacy, optimal strategies, \omega-regular languages} }

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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

The window mechanism was introduced by Chatterjee et al. to reinforce mean-payoff and total-payoff objectives with time bounds in two-player turn-based games on graphs [Krishnendu Chatterjee et al., 2015]. It has since proved useful in a variety of settings, including parity objectives in games [Véronique Bruyère et al., 2016] and both mean-payoff and parity objectives in Markov decision processes [Thomas Brihaye et al., 2020].
We study window parity objectives in timed automata and timed games: given a bound on the window size, a path satisfies such an objective if, in all states along the path, we see a sufficiently small window in which the smallest priority is even. We show that checking that all time-divergent paths of a timed automaton satisfy such a window parity objective can be done in polynomial space, and that the corresponding timed games can be solved in exponential time. This matches the complexity class of timed parity games, while adding the ability to reason about time bounds. We also consider multi-dimensional objectives and show that the complexity class does not increase. To the best of our knowledge, this is the first study of the window mechanism in a real-time setting.

James C. A. Main, Mickael Randour, and Jeremy Sproston. Time Flies When Looking out of the Window: Timed Games with Window Parity Objectives. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{main_et_al:LIPIcs.CONCUR.2021.25, author = {Main, James C. A. and Randour, Mickael and Sproston, Jeremy}, title = {{Time Flies When Looking out of the Window: Timed Games with Window Parity Objectives}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {25:1--25:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.25}, URN = {urn:nbn:de:0030-drops-144021}, doi = {10.4230/LIPIcs.CONCUR.2021.25}, annote = {Keywords: Window objectives, timed automata, timed games, parity games} }

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**Published in:** LIPIcs, Volume 203, 32nd International Conference on Concurrency Theory (CONCUR 2021)

We study stochastic zero-sum games on graphs, which are prevalent tools to model decision-making in presence of an antagonistic opponent in a random environment. In this setting, an important question is the one of strategy complexity: what kinds of strategies are sufficient or required to play optimally (e.g., randomization or memory requirements)? Our contributions further the understanding of arena-independent finite-memory (AIFM) determinacy, i.e., the study of objectives for which memory is needed, but in a way that only depends on limited parameters of the game graphs. First, we show that objectives for which pure AIFM strategies suffice to play optimally also admit pure AIFM subgame perfect strategies. Second, we show that we can reduce the study of objectives for which pure AIFM strategies suffice in two-player stochastic games to the easier study of one-player stochastic games (i.e., Markov decision processes). Third, we characterize the sufficiency of AIFM strategies through two intuitive properties of objectives. This work extends a line of research started on deterministic games in [Bouyer et al., 2020] to stochastic ones.

Patricia Bouyer, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Arena-Independent Finite-Memory Determinacy in Stochastic Games. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bouyer_et_al:LIPIcs.CONCUR.2021.26, author = {Bouyer, Patricia and Oualhadj, Youssouf and Randour, Mickael and Vandenhove, Pierre}, title = {{Arena-Independent Finite-Memory Determinacy in Stochastic Games}}, booktitle = {32nd International Conference on Concurrency Theory (CONCUR 2021)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-203-7}, ISSN = {1868-8969}, year = {2021}, volume = {203}, editor = {Haddad, Serge and Varacca, Daniele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2021.26}, URN = {urn:nbn:de:0030-drops-144037}, doi = {10.4230/LIPIcs.CONCUR.2021.26}, annote = {Keywords: two-player games on graphs, stochastic games, Markov decision processes, finite-memory determinacy, optimal strategies} }

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**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

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).

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)

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@InProceedings{bouyer_et_al:LIPIcs.CONCUR.2020.24, author = {Bouyer, Patricia and Le Roux, St\'{e}phane and Oualhadj, Youssouf and Randour, Mickael and Vandenhove, Pierre}, title = {{Games Where You Can Play Optimally with Arena-Independent Finite Memory}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {24:1--24:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.24}, URN = {urn:nbn:de:0030-drops-128360}, doi = {10.4230/LIPIcs.CONCUR.2020.24}, annote = {Keywords: two-player games on graphs, finite-memory determinacy, optimal strategies} }

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**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

The window mechanism was introduced by Chatterjee et al. [Krishnendu Chatterjee et al., 2015] to strengthen classical game objectives with time bounds. It permits to synthesize system controllers that exhibit acceptable behaviors within a configurable time frame, all along their infinite execution, in contrast to the traditional objectives that only require correctness of behaviors in the limit. The window concept has proved its interest in a variety of two-player zero-sum games, thanks to the ability to reason about such time bounds in system specifications, but also the increased tractability that it usually yields. In this work, we extend the window framework to stochastic environments by considering the fundamental threshold probability problem in Markov decision processes for window objectives. That is, given such an objective, we want to synthesize strategies that guarantee satisfying runs with a given probability. We solve this problem for the usual variants of window objectives, where either the time frame is set as a parameter, or we ask if such a time frame exists. We develop a generic approach for window-based objectives and instantiate it for the classical mean-payoff and parity objectives, already considered in games. Our work paves the way to a wide use of the window mechanism in stochastic models.

Thomas Brihaye, Florent Delgrange, Youssouf Oualhadj, and Mickael Randour. Life Is Random, Time Is Not: Markov Decision Processes with Window Objectives. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brihaye_et_al:LIPIcs.CONCUR.2019.8, author = {Brihaye, Thomas and Delgrange, Florent and Oualhadj, Youssouf and Randour, Mickael}, title = {{Life Is Random, Time Is Not: Markov Decision Processes with Window Objectives}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.8}, URN = {urn:nbn:de:0030-drops-109103}, doi = {10.4230/LIPIcs.CONCUR.2019.8}, annote = {Keywords: Markov decision processes, window mean-payoff, window parity} }

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**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

In this paper, we study one-player and two-player energy mean-payoff games. Energy mean-payoff games are games of infinite duration played on a finite graph with edges labeled by 2-dimensional weight vectors. The objective of the first player (the protagonist) is to satisfy an energy objective on the first dimension and a mean-payoff objective on the second dimension. We show that optimal strategies for the first player may require infinite memory while optimal strategies for the second player (the antagonist) do not require memory. In the one-player case (where only the first player has choices), the problem of deciding who is the winner can be solved in polynomial time while for the two-player case we show co-NP membership and we give effective constructions for the infinite-memory optimal strategies of the protagonist.

Véronique Bruyère, Quentin Hautem, Mickael Randour, and Jean-François Raskin. Energy Mean-Payoff Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bruyere_et_al:LIPIcs.CONCUR.2019.21, author = {Bruy\`{e}re, V\'{e}ronique and Hautem, Quentin and Randour, Mickael and Raskin, Jean-Fran\c{c}ois}, title = {{Energy Mean-Payoff Games}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {21:1--21:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.21}, URN = {urn:nbn:de:0030-drops-109239}, doi = {10.4230/LIPIcs.CONCUR.2019.21}, annote = {Keywords: two-player zero-sum games played on graphs, energy and mean-payoff objectives, complexity study and construction of optimal strategies} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

We study finite-memory (FM) determinacy in games on finite graphs, a central question for applications in controller synthesis, as FM strategies correspond to implementable controllers. We establish general conditions under which FM strategies suffice to play optimally, even in a broad multi-objective setting. We show that our framework encompasses important classes of games from the literature, and permits to go further, using a unified approach. While such an approach cannot match ad-hoc proofs with regard to tightness of memory bounds, it has two advantages: first, it gives a widely-applicable criterion for FM determinacy; second, it helps to understand the cornerstones of FM determinacy, which are often hidden but common in proofs for specific (combinations of) winning conditions.

Stéphane Le Roux, Arno Pauly, and Mickael Randour. Extending Finite-Memory Determinacy by Boolean Combination of Winning Conditions. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{leroux_et_al:LIPIcs.FSTTCS.2018.38, author = {Le Roux, St\'{e}phane and Pauly, Arno and Randour, Mickael}, title = {{Extending Finite-Memory Determinacy by Boolean Combination of Winning Conditions}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {38:1--38:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.38}, URN = {urn:nbn:de:0030-drops-99373}, doi = {10.4230/LIPIcs.FSTTCS.2018.38}, annote = {Keywords: games on graphs, finite-memory determinacy, multiple objectives} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The beyond worst-case synthesis problem was introduced recently by Bruyère et al. [BFRR14]: it aims at building system controllers that provide strict worst-case performance guarantees against an antagonistic environment while ensuring higher expected performance against a stochastic model of the environment. Our work extends the framework of [Bruyère/Filiot/Randour/Raskin, STACS 2014] and follow-up papers, which focused on quantitative objectives, by addressing the case of omega-regular conditions encoded as parity objectives, a natural way to represent functional requirements of systems.
We build strategies that satisfy a main parity objective on all plays, while ensuring a secondary one with sufficient probability. This setting raises new challenges in comparison to quantitative objectives, as one cannot easily mix different strategies without endangering the functional properties of the system. We establish that, for all variants of this problem, deciding the existence of a strategy lies in NP and in coNP, the same complexity class as classical parity games. Hence, our framework provides additional modeling power while staying in the same complexity class.

Raphaël Berthon, Mickael Randour, and Jean-François Raskin. Threshold Constraints with Guarantees for Parity Objectives in Markov Decision Processes. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 121:1-121:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{berthon_et_al:LIPIcs.ICALP.2017.121, author = {Berthon, Rapha\"{e}l and Randour, Mickael and Raskin, Jean-Fran\c{c}ois}, title = {{Threshold Constraints with Guarantees for Parity Objectives in Markov Decision Processes}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {121:1--121:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.121}, URN = {urn:nbn:de:0030-drops-74360}, doi = {10.4230/LIPIcs.ICALP.2017.121}, annote = {Keywords: Markov decision processes, parity objectives, beyond worst-case synthesis} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We study the almost-sure reachability problem in a distributed system obtained as the asynchronous composition of N copies (called processes) of the same automaton (called protocol), that can communicate via a shared register with finite domain. The automaton has two types of transitions: write-transitions update the value of the register, while read-transitions move to a new state depending on the content of the register. Non-determinism is resolved by a stochastic scheduler. Given a protocol, we focus on almost-sure reachability of a target state by one of the processes. The answer to this problem naturally depends on the number N of processes. However, we prove that our setting has a cut-off property: the answer to the almost-sure reachability problem is constant when N is large enough; we then develop an EXPSPACE algorithm deciding whether this constant answer is positive or negative.

Patricia Bouyer, Nicolas Markey, Mickael Randour, Arnaud Sangnier, and Daniel Stan. Reachability in Networks of Register Protocols under Stochastic Schedulers. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 106:1-106:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bouyer_et_al:LIPIcs.ICALP.2016.106, author = {Bouyer, Patricia and Markey, Nicolas and Randour, Mickael and Sangnier, Arnaud and Stan, Daniel}, title = {{Reachability in Networks of Register Protocols under Stochastic Schedulers}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {106:1--106:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.106}, URN = {urn:nbn:de:0030-drops-62416}, doi = {10.4230/LIPIcs.ICALP.2016.106}, annote = {Keywords: Networks of Processes, Parametrized Systems, Stochastic Scheduler, Almost-sure Reachability, Cut-Off Property} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Classical analysis of two-player quantitative games involves an adversary (modeling the environment of the system) which is purely antagonistic and asks for strict guarantees while Markov decision processes model systems facing a purely randomized environment: the aim is then to optimize the expected payoff, with no guarantee on individual outcomes. We introduce the beyond worst-case synthesis problem, which is to construct strategies that guarantee some quantitative requirement in the worst-case while providing an higher expected value against a particular stochastic model of the environment given as input. We consider both the mean-payoff value problem and the shortest path problem. In both cases, we show how to decide the existence of finite-memory strategies satisfying the problem and how to synthesize one if one exists. We establish algorithms and we study complexity bounds and memory requirements.

Véronique Bruyère, Emmanuel Filiot, Mickael Randour, and Jean-François Raskin. Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 199-213, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bruyere_et_al:LIPIcs.STACS.2014.199, author = {Bruy\`{e}re, V\'{e}ronique and Filiot, Emmanuel and Randour, Mickael and Raskin, Jean-Fran\c{c}ois}, title = {{Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {199--213}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.199}, URN = {urn:nbn:de:0030-drops-44589}, doi = {10.4230/LIPIcs.STACS.2014.199}, annote = {Keywords: two-player games on graphs, Markov decision processes, quantitative objectives, synthesis, worst-case and expected value, mean-payoff, shortest path} }

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