Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)
Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. From Local to Global Optimality in Concurrent Parity Games. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{bordais_et_al:LIPIcs.CSL.2024.18, author = {Bordais, Benjamin and Bouyer, Patricia and Le Roux, St\'{e}phane}, title = {{From Local to Global Optimality in Concurrent Parity Games}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {18:1--18:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.18}, URN = {urn:nbn:de:0030-drops-196612}, doi = {10.4230/LIPIcs.CSL.2024.18}, annote = {Keywords: Game forms, stochastic games, parity games, Blackwell/Martin values} }
Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)
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)
@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} }
Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)
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)
@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} }
Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)
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)
@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} }
Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)
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)
@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} }
Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)
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)
@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} }
Published in: LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)
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)
@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} }
Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)
Guillaume Bertholon, Érik Martin-Dorel, and Pierre Roux. Primitive Floats in Coq. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
@InProceedings{bertholon_et_al:LIPIcs.ITP.2019.7, author = {Bertholon, Guillaume and Martin-Dorel, \'{E}rik and Roux, Pierre}, title = {{Primitive Floats in Coq}}, booktitle = {10th International Conference on Interactive Theorem Proving (ITP 2019)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-122-1}, ISSN = {1868-8969}, year = {2019}, volume = {141}, editor = {Harrison, John and O'Leary, John and Tolmach, Andrew}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.7}, URN = {urn:nbn:de:0030-drops-110629}, doi = {10.4230/LIPIcs.ITP.2019.7}, annote = {Keywords: Coq formal proofs, floating-point arithmetic, reflexive tactics, Cholesky decomposition} }
Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)
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)
@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.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} }
Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)
Thomas Brihaye, Véronique Bruyère, Aline Goeminne, Jean-François Raskin, and Marie van den Bogaard. The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
@InProceedings{brihaye_et_al:LIPIcs.CONCUR.2019.13, author = {Brihaye, Thomas and Bruy\`{e}re, V\'{e}ronique and Goeminne, Aline and Raskin, Jean-Fran\c{c}ois and van den Bogaard, Marie}, title = {{The Complexity of Subgame Perfect Equilibria in Quantitative Reachability Games}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {13:1--13:16}, 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.2019.13}, URN = {urn:nbn:de:0030-drops-109153}, doi = {10.4230/LIPIcs.CONCUR.2019.13}, annote = {Keywords: multiplayer non-zero-sum games played on graphs, quantitative reachability objectives, subgame perfect equilibria, constrained existence problem} }
Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)
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)
@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.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} }
Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Stéphane Le Roux. Concurrent Games and Semi-Random Determinacy. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 40:1-40:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
@InProceedings{leroux:LIPIcs.MFCS.2018.40, author = {Le Roux, St\'{e}phane}, title = {{Concurrent Games and Semi-Random Determinacy}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.40}, URN = {urn:nbn:de:0030-drops-96220}, doi = {10.4230/LIPIcs.MFCS.2018.40}, annote = {Keywords: Two-player win/lose, graph, infinite duration, abstract winning condition} }
Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)
Stéphane Le Roux, Arno Pauly, and Jean-François Raskin. Minkowski Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 50:1-50:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
@InProceedings{leroux_et_al:LIPIcs.STACS.2017.50, author = {Le Roux, St\'{e}phane and Pauly, Arno and Raskin, Jean-Fran\c{c}ois}, title = {{Minkowski Games}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert 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.2017.50}, URN = {urn:nbn:de:0030-drops-69849}, doi = {10.4230/LIPIcs.STACS.2017.50}, annote = {Keywords: Control in R^d, determinacy, polytopic/arbitrary, coNP-complete, undecidable} }
Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)
Volker Betz and Stéphane Le Roux. Stable States of Perturbed Markov Chains. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 18:1-18:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
@InProceedings{betz_et_al:LIPIcs.MFCS.2016.18, author = {Betz, Volker and Le Roux, St\'{e}phane}, title = {{Stable States of Perturbed Markov Chains}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {18:1--18:14}, 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.2016.18}, URN = {urn:nbn:de:0030-drops-64335}, doi = {10.4230/LIPIcs.MFCS.2016.18}, annote = {Keywords: evolution, metastability, tropical, shortest path, SCC, cubic time} }
Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)
Thomas Brihaye, Véronique Bruyère, Noémie Meunier, and Jean-Francois Raskin. Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 504-518, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
@InProceedings{brihaye_et_al:LIPIcs.CSL.2015.504, author = {Brihaye, Thomas and Bruy\`{e}re, V\'{e}ronique and Meunier, No\'{e}mie and Raskin, Jean-Francois}, title = {{Weak Subgame Perfect Equilibria and their Application to Quantitative Reachability}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {504--518}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.504}, URN = {urn:nbn:de:0030-drops-54345}, doi = {10.4230/LIPIcs.CSL.2015.504}, annote = {Keywords: multi-player games on graphs, quantitative objectives, Nash equilibrium, subgame perfect equilibrium, quantitative reachability} }
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