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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.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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Complete Volume

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

LIPIcs, Volume 254, STACS 2023, Complete Volume

Petra Berenbrink, Mamadou Moustapha Kanté, Patricia Bouyer, and Anuj Dawar. LIPIcs, Volume 254, STACS 2023, Complete Volume. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 1-1026, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@Proceedings{berenbrink_et_al:LIPIcs.STACS.2023, title = {{LIPIcs, Volume 254, STACS 2023, Complete Volume}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {1--1026}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023}, URN = {urn:nbn:de:0030-drops-176515}, doi = {10.4230/LIPIcs.STACS.2023}, annote = {Keywords: LIPIcs, Volume 254, STACS 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Front Matter, Table of Contents, Preface, Conference Organization

Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté. Front Matter, Table of Contents, Preface, Conference Organization. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 0:i-0:xxii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{berenbrink_et_al:LIPIcs.STACS.2023.0, author = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {0:i--0:xxii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.0}, URN = {urn:nbn:de:0030-drops-176525}, doi = {10.4230/LIPIcs.STACS.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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

We study two-player concurrent stochastic games on finite graphs, with Büchi and co-Büchi objectives. The goal of the first player is to maximize the probability of satisfying the given objective. Following Martin’s determinacy theorem for Blackwell games, we know that such games have a value. Natural questions are then: does there exist an optimal strategy, that is, a strategy achieving the value of the game? what is the memory required for playing (almost-)optimally?
The situation is rather simple to describe for turn-based games, where positional pure strategies suffice to play optimally in games with parity objectives. Concurrency makes the situation intricate and heterogeneous. For most ω-regular objectives, there do indeed not exist optimal strategies in general. For some objectives (that we will mention), infinite memory might also be required for playing optimally or almost-optimally.
We also provide characterizations of local interactions of the players to ensure positionality of (almost-)optimal strategies for Büchi and co-Büchi objectives. This characterization relies on properties of game forms underpinning the formalism for defining local interactions of the two players. These well-behaved game forms are like elementary bricks which, when they behave well in isolation, can be assembled in graph games and ensure the good property for the whole game.

Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. Playing (Almost-)Optimally in Concurrent Büchi and Co-Büchi Games. 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. 33:1-33:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bordais_et_al:LIPIcs.FSTTCS.2022.33, author = {Bordais, Benjamin and Bouyer, Patricia and Le Roux, St\'{e}phane}, title = {{Playing (Almost-)Optimally in Concurrent B\"{u}chi and Co-B\"{u}chi Games}}, booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)}, pages = {33:1--33:18}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.33}, URN = {urn:nbn:de:0030-drops-174258}, doi = {10.4230/LIPIcs.FSTTCS.2022.33}, annote = {Keywords: Concurrent Games, Optimal Strategies, B\"{u}chi Objective, co-B\"{u}chi Objective} }

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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.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 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.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 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

We study two-player reachability games on finite graphs. At each state the interaction between the players is concurrent and there is a stochastic Nature. Players also play stochastically. The literature tells us that 1) Player 𝖡, who wants to avoid the target state, has a positional strategy that maximizes the probability to win (uniformly from every state) and 2) from every state, for every ε > 0, Player 𝖠 has a strategy that maximizes up to ε the probability to win. Our work is two-fold.
First, we present a double-fixed-point procedure that says from which state Player 𝖠 has a strategy that maximizes (exactly) the probability to win. This is computable if Nature’s probability distributions are rational. We call these states maximizable. Moreover, we show that for every ε > 0, Player 𝖠 has a positional strategy that maximizes the probability to win, exactly from maximizable states and up to ε from sub-maximizable states.
Second, we consider three-state games with one main state, one target, and one bin. We characterize the local interactions at the main state that guarantee the existence of an optimal Player 𝖠 strategy. In this case there is a positional one. It turns out that in many-state games, these local interactions also guarantee the existence of a uniform optimal Player 𝖠 strategy. In a way, these games are well-behaved by design of their elementary bricks, the local interactions. It is decidable whether a local interaction has this desirable property.

Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. Optimal Strategies in Concurrent Reachability Games. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 7:1-7:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bordais_et_al:LIPIcs.CSL.2022.7, author = {Bordais, Benjamin and Bouyer, Patricia and Le Roux, St\'{e}phane}, title = {{Optimal Strategies in Concurrent Reachability Games}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.7}, URN = {urn:nbn:de:0030-drops-157278}, doi = {10.4230/LIPIcs.CSL.2022.7}, annote = {Keywords: Concurrent reachability games, Game forms, Optimal strategies} }

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**Published in:** LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

We study infinite two-player win/lose games (A,B,W) where A,B are finite and W ⊆ (A×B)^ω. At each round Player 1 and Player 2 concurrently choose one action in A and B, respectively. Player 1 wins iff the generated sequence is in W. Each history h ∈ (A×B)^* induces a game (A,B,W_h) with W_h : = {ρ ∈ (A×B)^ω ∣ h ρ ∈ W}. We show the following: if W is in Δ⁰₂ (for the usual topology), if the inclusion relation induces a well partial order on the W_h’s, and if Player 1 has a winning strategy, then she has a finite-memory winning strategy. Our proof relies on inductive descriptions of set complexity, such as the Hausdorff difference hierarchy of the open sets.
Examples in Σ⁰₂ and Π⁰₂ show some tightness of our result. Our result can be translated to games on finite graphs: e.g. finite-memory determinacy of multi-energy games is a direct corollary, whereas it does not follow from recent general results on finite memory strategies.

Patricia Bouyer, Stéphane Le Roux, and Nathan Thomasset. Finite-Memory Strategies in Two-Player Infinite Games. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 8:1-8:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bouyer_et_al:LIPIcs.CSL.2022.8, author = {Bouyer, Patricia and Le Roux, St\'{e}phane and Thomasset, Nathan}, title = {{Finite-Memory Strategies in Two-Player Infinite Games}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {8:1--8:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.8}, URN = {urn:nbn:de:0030-drops-157285}, doi = {10.4230/LIPIcs.CSL.2022.8}, annote = {Keywords: Two-player win/lose games, Infinite trees, Finite-memory winning strategies, Well partial orders, Hausdorff difference hierarchy} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

In general, finite concurrent two-player reachability games are only determined in a weak sense: the supremum probability to win can be approached via stochastic strategies, but cannot be realized.
We introduce a class of concurrent games that are determined in a much stronger sense, and in a way, it is the largest class with this property. To this end, we introduce the notion of local interaction at a state of a graph game: it is a game form whose outcomes (i.e. a table whose entries) are the next states, which depend on the concurrent actions of the players. By definition, a game form is determined iff it always yields games that are determined via deterministic strategies when used as a local interaction in a Nature-free, one-shot reachability game. We show that if all the local interactions of a graph game with Borel objective are determined game forms, the game itself is determined: if Nature does not play, one player has a winning strategy; if Nature plays, both players have deterministic strategies that maximize the probability to win. This constitutes a clear-cut separation: either a game form behaves poorly already when used alone with basic objectives, or it behaves well even when used together with other well-behaved game forms and complex objectives.
Existing results for positional and finite-memory determinacy in turn-based games are extended this way to concurrent games with determined local interactions (CG-DLI).

Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. From Local to Global Determinacy in Concurrent Graph Games. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 41:1-41:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bordais_et_al:LIPIcs.FSTTCS.2021.41, author = {Bordais, Benjamin and Bouyer, Patricia and Le Roux, St\'{e}phane}, title = {{From Local to Global Determinacy in Concurrent Graph Games}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.41}, URN = {urn:nbn:de:0030-drops-155522}, doi = {10.4230/LIPIcs.FSTTCS.2021.41}, annote = {Keywords: Concurrent games, Game forms, Local interaction} }

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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.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 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Concurrent games with a fixed number of agents have been thoroughly studied, with various solution concepts and objectives for the agents. In this paper, we consider concurrent games with an arbitrary number of agents, and study the problem of synthesizing a coalition strategy to achieve a global safety objective. The problem is non-trivial since the agents do not know a priori how many they are when they start the game. We prove that the existence of a safe arbitrary-large coalition strategy for safety objectives is a PSPACE-hard problem that can be decided in exponential space.

Nathalie Bertrand, Patricia Bouyer, and Anirban Majumdar. Synthesizing Safe Coalition Strategies. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 39:1-39:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bertrand_et_al:LIPIcs.FSTTCS.2020.39, author = {Bertrand, Nathalie and Bouyer, Patricia and Majumdar, Anirban}, title = {{Synthesizing Safe Coalition Strategies}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {39:1--39:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.39}, URN = {urn:nbn:de:0030-drops-132807}, doi = {10.4230/LIPIcs.FSTTCS.2020.39}, annote = {Keywords: concurrent games, parameterized verification, strategy synthesis} }

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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.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 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Traditional concurrent games on graphs involve a fixed number of players, who take decisions simultaneously, determining the next state of the game. In this paper, we introduce a parameterized variant of concurrent games on graphs, where the parameter is precisely the number of players. Parameterized concurrent games are described by finite graphs, in which the transitions bear regular languages to describe the possible move combinations that lead from one vertex to another.
We consider the problem of determining whether the first player, say Eve, has a strategy to ensure a reachability objective against any strategy profile of her opponents as a coalition. In particular Eve’s strategy should be independent of the number of opponents she actually has. Technically, this paper focuses on an a priori simpler setting where the languages labeling transitions only constrain the number of opponents (but not their precise action choices). These constraints are described as semilinear sets, finite unions of intervals, or intervals.
We establish the precise complexities of the parameterized reachability game problem, ranging from PTIME-complete to PSPACE-complete, in a variety of situations depending on the contraints (semilinear predicates, unions of intervals, or intervals) and on the presence or not of non-determinism.

Nathalie Bertrand, Patricia Bouyer, and Anirban Majumdar. Concurrent Parameterized Games. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 31:1-31:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bertrand_et_al:LIPIcs.FSTTCS.2019.31, author = {Bertrand, Nathalie and Bouyer, Patricia and Majumdar, Anirban}, title = {{Concurrent Parameterized Games}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {31:1--31:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.31}, URN = {urn:nbn:de:0030-drops-115931}, doi = {10.4230/LIPIcs.FSTTCS.2019.31}, annote = {Keywords: concurrent games, parameterized verification} }

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Invited Talk

**Published in:** LIPIcs, Volume 147, 26th International Symposium on Temporal Representation and Reasoning (TIME 2019)

In this talk, I will show how one can characterize and compute Nash equilibria in multiplayer games played on graphs. I will present in particular a construction, called the suspect game construction, which allows to reduce the computation of Nash equilibria to the computation of winning strategies in a two-player zero-sum game.

Patricia Bouyer. On the Computation of Nash Equilibria in Games on Graphs (Invited Talk). In 26th International Symposium on Temporal Representation and Reasoning (TIME 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 147, pp. 3:1-3:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bouyer:LIPIcs.TIME.2019.3, author = {Bouyer, Patricia}, title = {{On the Computation of Nash Equilibria in Games on Graphs}}, booktitle = {26th International Symposium on Temporal Representation and Reasoning (TIME 2019)}, pages = {3:1--3:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-127-6}, ISSN = {1868-8969}, year = {2019}, volume = {147}, editor = {Gamper, Johann and Pinchinat, Sophie and Sciavicco, Guido}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2019.3}, URN = {urn:nbn:de:0030-drops-113616}, doi = {10.4230/LIPIcs.TIME.2019.3}, annote = {Keywords: Multiplayer games, Nash equilibria} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We study pure Nash equilibria in infinite-duration games on graphs, with partial visibility of actions but communication (based on a graph) among the players. We show that a simple communication mechanism consisting in reporting the deviator when seeing it and propagating this information is sufficient for characterizing Nash equilibria. We propose an epistemic game construction, which conveniently records important information about the knowledge of the players. With this abstraction, we are able to characterize Nash equilibria which follow the simple communication pattern via winning strategies. We finally discuss the size of the construction, which would allow efficient algorithmic solutions to compute Nash equilibria in the original game.

Patricia Bouyer and Nathan Thomasset. Nash Equilibria in Games over Graphs Equipped with a Communication Mechanism. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 9:1-9:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bouyer_et_al:LIPIcs.MFCS.2019.9, author = {Bouyer, Patricia and Thomasset, Nathan}, title = {{Nash Equilibria in Games over Graphs Equipped with a Communication Mechanism}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.9}, URN = {urn:nbn:de:0030-drops-109532}, doi = {10.4230/LIPIcs.MFCS.2019.9}, annote = {Keywords: Multiplayer games, Nash equilibria, partial information} }

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

Broadcast networks allow one to model networks of identical nodes communicating through message broadcasts. Their parameterized verification aims at proving a property holds for any number of nodes, under any communication topology, and on all possible executions. We focus on the coverability problem which dually asks whether there exists an execution that visits a configuration exhibiting some given state of the broadcast protocol. Coverability is known to be undecidable for static networks, i.e. when the number of nodes and communication topology is fixed along executions. In contrast, it is decidable in PTIME when the communication topology may change arbitrarily along executions, that is for reconfigurable networks. Surprisingly, no lower nor upper bounds on the minimal number of nodes, or the minimal length of covering execution in reconfigurable networks, appear in the literature.
In this paper we show tight bounds for cutoff and length, which happen to be linear and quadratic, respectively, in the number of states of the protocol. We also introduce an intermediary model with static communication topology and non-deterministic message losses upon sending. We show that the same tight bounds apply to lossy networks, although, reconfigurable executions may be linearly more succinct than lossy executions. Finally, we show NP-completeness for the natural optimisation problem associated with the cutoff.

Nathalie Bertrand, Patricia Bouyer, and Anirban Majumdar. Reconfiguration and Message Losses in Parameterized Broadcast Networks. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 32:1-32:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bertrand_et_al:LIPIcs.CONCUR.2019.32, author = {Bertrand, Nathalie and Bouyer, Patricia and Majumdar, Anirban}, title = {{Reconfiguration and Message Losses in Parameterized Broadcast Networks}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {32:1--32:15}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.32}, URN = {urn:nbn:de:0030-drops-109345}, doi = {10.4230/LIPIcs.CONCUR.2019.32}, annote = {Keywords: model checking, parameterized verification, broadcast networks} }

Document

**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Having a finite bisimulation is a good feature for a dynamical system, since it can lead to the decidability of the verification of reachability properties. We investigate a new class of o-minimal dynamical systems with very general flows, where the classical restrictions on trajectory intersections are partly lifted. We identify conditions, that we call Finite and Uniform Crossing: When Finite Crossing holds, the time-abstract bisimulation is computable and, under the stronger Uniform Crossing assumption, this bisimulation is finite and definable.

Béatrice Bérard, Patricia Bouyer, and Vincent Jugé. Finite Bisimulations for Dynamical Systems with Overlapping Trajectories. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 26:1-26:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{berard_et_al:LIPIcs.CSL.2018.26, author = {B\'{e}rard, B\'{e}atrice and Bouyer, Patricia and Jug\'{e}, Vincent}, title = {{Finite Bisimulations for Dynamical Systems with Overlapping Trajectories}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {26:1--26:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.26}, URN = {urn:nbn:de:0030-drops-96932}, doi = {10.4230/LIPIcs.CSL.2018.26}, annote = {Keywords: Reachability properties, dynamical systems, o-minimal structures, intersecting trajectories, finite bisimulations} }

Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Strategy Logic (SL) is a very expressive temporal logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express LTL properties of the resulting plays. Such a powerful framework has two drawbacks: First, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula, revisiting the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014]. We explain why elementary dependences, as defined by Mogavero et al., do not exactly capture the intended concept of behavioral strategies. We address this discrepancy by introducing timeline dependences, and exhibit a large fragment of SL for which model checking can be performed in 2-EXPTIME under this new semantics.

Patrick Gardy, Patricia Bouyer, and Nicolas Markey. Dependences in Strategy Logic. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 34:1-34:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gardy_et_al:LIPIcs.STACS.2018.34, author = {Gardy, Patrick and Bouyer, Patricia and Markey, Nicolas}, title = {{Dependences in Strategy Logic}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.34}, URN = {urn:nbn:de:0030-drops-85320}, doi = {10.4230/LIPIcs.STACS.2018.34}, annote = {Keywords: strategic reasoning, strategy logic, dependences, behavioural strategies.} }

Document

**Published in:** LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

Computing steady-state distributions in infinite-state stochastic systems is in general a very difficult task. Product-form Petri nets are those Petri nets for which the steady-state distribution can be described as a natural product corresponding, up to a normalising constant, to an exponentiation of the markings. However, even though some classes of nets are known to have a product-form distribution, computing the normalising constant can be hard. The class of (closed) \Pi^3-nets has been proposed in an earlier work, for which it is shown that one can compute the steady-state distribution efficiently. However these nets are bounded. In this paper, we generalise queuing Markovian networks and closed \Pi^3-nets to obtain the class of open \Pi^3-nets, that generate infinite-state systems. We show interesting properties of these nets: (1) we prove that liveness can be decided in polynomial time, and that reachability in live \Pi^3-nets can be decided in polynomial time; (2) we show that we can decide ergodicity of such nets in polynomial time as well; (3) we provide a pseudo-polynomial time algorithm to compute the normalising constant.

Patricia Bouyer, Serge Haddad, and Vincent Jugé. Unbounded Product-Form Petri Nets. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 31:1-31:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bouyer_et_al:LIPIcs.CONCUR.2017.31, author = {Bouyer, Patricia and Haddad, Serge and Jug\'{e}, Vincent}, title = {{Unbounded Product-Form Petri Nets}}, booktitle = {28th International Conference on Concurrency Theory (CONCUR 2017)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-048-4}, ISSN = {1868-8969}, year = {2017}, volume = {85}, editor = {Meyer, Roland and Nestmann, Uwe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.31}, URN = {urn:nbn:de:0030-drops-77957}, doi = {10.4230/LIPIcs.CONCUR.2017.31}, annote = {Keywords: Performance evaluation, infinite-state systems, Petri nets, steady-state distribution} }

Document

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

In 2007, Abdulla et al. introduced the elegant concept of decisive Markov chain. Intuitively, decisiveness allows one to lift the good properties of finite Markov chains to infinite Markov chains. For instance, the approximate quantitative reachability problem can be solved for decisive Markov chains (enjoying reasonable effectiveness assumptions) including probabilistic lossy channel systems and probabilistic vector addition systems with states. In this paper, we extend the concept of decisiveness to more general stochastic processes. This extension is non trivial as we consider stochastic processes with a potentially continuous set of states and uncountable branching (common features of real-time stochastic processes). This allows us to obtain decidability results for both qualitative and quantitative verification problems on some classes of real-time stochastic processes, including generalized semi-Markov processes and stochastic timed
automata

Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, and Pierre Carlier. Analysing Decisive Stochastic Processes. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 101:1-101:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bertrand_et_al:LIPIcs.ICALP.2016.101, author = {Bertrand, Nathalie and Bouyer, Patricia and Brihaye, Thomas and Carlier, Pierre}, title = {{Analysing Decisive Stochastic Processes}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {101:1--101:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.101}, URN = {urn:nbn:de:0030-drops-62362}, doi = {10.4230/LIPIcs.ICALP.2016.101}, annote = {Keywords: Real-time stochastic processes, Decisiveness, Approximation Scheme} }

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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.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 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Stochastic timed games (STGs), introduced by Bouyer and Forejt, naturally generalize both continuous-time Markov chains and timed automata by providing a partition of the locations between those controlled by two players (Player Box and Player Diamond) with competing objectives and those governed by stochastic laws. Depending on the number of players - 2, 1, or 0 - subclasses of stochastic timed games are often classified as 2 1/2-player, 1 1/2-player, and 1/2-player games where the 1/2 symbolizes the presence of the stochastic "nature" player. For STGs with reachability objectives it is known that 1 1/2-player one-clock STGs are decidable for qualitative objectives, and that 2 1/2-player three-clock STGs are undecidable for quantitative reachability objectives. This paper further refines the gap in this decidability spectrum. We show that quantitative reachability objectives are already undecidable for 1 1/2 player four-clock STGs, and even under the time-bounded restriction for 2 1/2-player five-clock STGs. We also obtain a class of 1 1/2, 2 1/2 player STGs for which the quantitative reachability problem is decidable.

S. Akshay, Patricia Bouyer, Shankara Narayanan Krishna, Lakshmi Manasa, and Ashutosh Trivedi. Stochastic Timed Games Revisited. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{akshay_et_al:LIPIcs.MFCS.2016.8, author = {Akshay, S. and Bouyer, Patricia and Krishna, Shankara Narayanan and Manasa, Lakshmi and Trivedi, Ashutosh}, title = {{Stochastic Timed Games Revisited}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.8}, URN = {urn:nbn:de:0030-drops-64985}, doi = {10.4230/LIPIcs.MFCS.2016.8}, annote = {Keywords: timed automata, stochastic games, two-counter machines} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Strategy Logic is a powerful specification language for expressing non-zero-sum properties of multi-player games. SL conveniently extends the logic ATL with explicit quantification and assignment of strategies. In this paper, we consider games over one-counter automata, and a quantitative extension 1cSL of SL with assertions over the value of the counter. We prove two results: we first show that, if decidable, model checking the so-called Boolean-goal fragment of 1cSL has non-elementary complexity; we actually prove the result for the Boolean-goal fragment of SL over finite-state games, which was an open question in [Mogavero et al. Reasoning about strategies: On the model-checking problem. ACM ToCL 15(4),2014]. As a first step towards proving decidability, we then show that the Boolean-goal fragment of 1cSL over one-counter games enjoys a nice periodicity property.

Patricia Bouyer, Patrick Gardy, and Nicolas Markey. Weighted Strategy Logic with Boolean Goals Over One-Counter Games. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 69-83, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bouyer_et_al:LIPIcs.FSTTCS.2015.69, author = {Bouyer, Patricia and Gardy, Patrick and Markey, Nicolas}, title = {{Weighted Strategy Logic with Boolean Goals Over One-Counter Games}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {69--83}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.69}, URN = {urn:nbn:de:0030-drops-56537}, doi = {10.4230/LIPIcs.FSTTCS.2015.69}, annote = {Keywords: Temporal logics, multi-player games, strategy logic, quantitative games} }

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**Published in:** LIPIcs, Volume 42, 26th International Conference on Concurrency Theory (CONCUR 2015)

A weighted timed game is a timed game with extra quantitative information representing e.g. energy consumption. Optimizing the weight for reaching a target is a natural question, which has already been investigated for ten years. Existence of optimal strategies is known to be undecidable in general, and only very restricted classes of games have been identified for which optimal weight and almost-optimal strategies can be computed. In this paper, we show that the value problem is undecidable in weighted timed games. We then introduce a large subclass of weighted timed games (for which the undecidability proof above applies), and provide an algorithm to compute arbitrary approximations of the value in such games. To the best of our knowledge, this is the first approximation scheme for an undecidable class of weighted timed games.

Patricia Bouyer, Samy Jaziri, and Nicolas Markey. On the Value Problem in Weighted Timed Games. In 26th International Conference on Concurrency Theory (CONCUR 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 42, pp. 311-324, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bouyer_et_al:LIPIcs.CONCUR.2015.311, author = {Bouyer, Patricia and Jaziri, Samy and Markey, Nicolas}, title = {{On the Value Problem in Weighted Timed Games}}, booktitle = {26th International Conference on Concurrency Theory (CONCUR 2015)}, pages = {311--324}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-91-0}, ISSN = {1868-8969}, year = {2015}, volume = {42}, editor = {Aceto, Luca and de Frutos Escrig, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2015.311}, URN = {urn:nbn:de:0030-drops-53863}, doi = {10.4230/LIPIcs.CONCUR.2015.311}, annote = {Keywords: Timed games, undecidability, approximation} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We study mixed-strategy Nash equilibria in multiplayer deterministic concurrent games played on graphs, with terminal-reward payoffs (that is, absorbing states with a value for each player). We show undecidability of the existence of a constrained Nash equilibrium (the constraint requiring that one player should have maximal payoff), with only three players and 0/1-rewards (i.e., reachability objectives). This has to be compared with the undecidability result by Ummels and Wojtczak for turn-based games which requires 14 players and general rewards. Our proof has various interesting consequences: (i) the undecidability of the existence of a Nash equilibrium with a constraint on the social welfare; (ii) the undecidability of the existence of an (unconstrained) Nash equilibrium in concurrent games with terminal-reward payoffs.

Patricia Bouyer, Nicolas Markey, and Daniel Stan. Mixed Nash Equilibria in Concurrent Terminal-Reward Games. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 351-363, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bouyer_et_al:LIPIcs.FSTTCS.2014.351, author = {Bouyer, Patricia and Markey, Nicolas and Stan, Daniel}, title = {{Mixed Nash Equilibria in Concurrent Terminal-Reward Games}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {351--363}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.351}, URN = {urn:nbn:de:0030-drops-48550}, doi = {10.4230/LIPIcs.FSTTCS.2014.351}, annote = {Keywords: concurrent games, randomized strategy, Nash equilibria, undecidability} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We define and study a new approach to the implementability of timed automata, where the semantics is perturbed by imprecisions and finite frequency of the hardware. In order to circumvent these effects, we introduce parametric shrinking of clock constraints, which corresponds to tightening these. We propose symbolic procedures to decide the existence of (and then compute) parameters under which the shrunk version of a given timed automaton is non-blocking and can time-abstract simulate the exact semantics. We then define an implementation semantics for timed automata with a digital clock and positive reaction times, and show that for shrinkable timed automata, non-blockingness and time-abstract simulation are preserved in implementation.

Ocan Sankur, Patricia Bouyer, and Nicolas Markey. Shrinking Timed Automata. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 90-102, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{sankur_et_al:LIPIcs.FSTTCS.2011.90, author = {Sankur, Ocan and Bouyer, Patricia and Markey, Nicolas}, title = {{Shrinking Timed Automata}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {90--102}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.90}, URN = {urn:nbn:de:0030-drops-33627}, doi = {10.4230/LIPIcs.FSTTCS.2011.90}, annote = {Keywords: Timed automata, implementability, robustness} }

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**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We study the problem of computing pure-strategy Nash equilibria in multiplayer concurrent games with Büchi-definable objectives. First, when the objectives are Büchi conditions on the game, we prove that the existence problem can be solved in polynomial time. In a second part, we extend our technique to objectives defined by deterministic Büchi automata, and prove that the problem then becomes EXPTIME-complete. We prove PSPACE-completeness for the case where the Büchi automata are 1-weak.

Patricia Bouyer, Romain Brenguier, Nicolas Markey, and Michael Ummels. Nash Equilibria in Concurrent Games with Büchi Objectives. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 375-386, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{bouyer_et_al:LIPIcs.FSTTCS.2011.375, author = {Bouyer, Patricia and Brenguier, Romain and Markey, Nicolas and Ummels, Michael}, title = {{Nash Equilibria in Concurrent Games with B\"{u}chi Objectives}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {375--386}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.375}, URN = {urn:nbn:de:0030-drops-33340}, doi = {10.4230/LIPIcs.FSTTCS.2011.375}, annote = {Keywords: Concurrent games, Nash equilibria, B\"{u}chi Objectives} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether $\sum_{i=1}^m C_i A_i^{X_i}$ is zero for given rational numbers $A_i$, $C_i$, $X_i$. It has been known for almost twenty years that this can be decided in polynomial time. In this paper we improve this result by showing membership in uniform TC0. This requires several significant departures from Blömer's polynomial-time algorithm as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC0.

Paul Hunter, Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. Computing Rational Radical Sums in Uniform TC^0. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 308-316, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{hunter_et_al:LIPIcs.FSTTCS.2010.308, author = {Hunter, Paul and Bouyer, Patricia and Markey, Nicolas and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Computing Rational Radical Sums in Uniform TC^0}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {308--316}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.308}, URN = {urn:nbn:de:0030-drops-28739}, doi = {10.4230/LIPIcs.FSTTCS.2010.308}, annote = {Keywords: Sum of square roots, Threshold circuits, Complexity} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

A channel machine consists of a finite controller together with
several fifo channels; the controller can read messages from the
head of a channel and write messages to the tail of a channel. In
this paper, we focus on channel machines with insertion errors,
i.e., machines in whose channels messages can spontaneously appear.
Such devices have been previously introduced in the study of Metric
Temporal Logic. We consider the termination problem: are all the
computations of a given insertion channel machine finite? We show
that this problem has non-elementary, yet primitive recursive
complexity.

Patricia Bouyer, Nicolas Markey, Joël Ouaknine, Philippe Schnoebelen, and James Worrell. On Termination for Faulty Channel Machines. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 121-132, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{bouyer_et_al:LIPIcs.STACS.2008.1339, author = {Bouyer, Patricia and Markey, Nicolas and Ouaknine, Jo\"{e}l and Schnoebelen, Philippe and Worrell, James}, title = {{On Termination for Faulty Channel Machines}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {121--132}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1339}, URN = {urn:nbn:de:0030-drops-13390}, doi = {10.4230/LIPIcs.STACS.2008.1339}, annote = {Keywords: Automated Verification, Computational Complexity} }

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