Patey Ludovic (23 results)
Milliken's Tree Theorem and Its Applications: a Computability-Theoretic Perspective
Paul-Elliot Angles D'Auriac, Peter A. Cholak, Damir D. Dzhafarov, Benoit Monin, Ludovic Patey
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Paperback. Condition: New. Milliken's tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey's theorem and its many variants and consequences. In this sense, Milliken's tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the… common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive.Motivated by a question of Dobrinen, we initiate the study of Milliken's tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey's theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken's tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-La¨uchli theorem, which is a key ingredient.Our advance here stems from a careful analysis of the Halpern-Läuchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken's tree theorem that permits us to gauge its effectivity in turn. The key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken's tree theorem. This enables us to build solutions to the full Milliken's tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken's tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove Milliken's tree theorem.We apply our analysis also to several well-known applications of Milliken's tree theorem, namely Devlin's theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey's theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken's tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. In particular, we establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker's notion of big Ramsey structure.
Twenty Years of Theoretical and Practical Synergies : 20th Conference on Computability in Europe, Cie 2024, Amsterdam, the Netherlands, July 8?12, 2024, Proceedings
Levy Patey, Ludovic (EDT); Pimentel, Elaine (EDT); Galeotti, Lorenzo (EDT); Manea, Florin (EDT)
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Milliken's Tree Theorem and Its Applications: A Computability-Theoretic Perspective: Volume: 293 Number: 1457 (Memoirs of the American Mathematical Society)
D'Auriac, Paul-Elliot Angles/ Cholak, Peter A./ Dzhafarov, Damir D./ Monin, Benoit/ Patey, Ludovic
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Twenty Years of Theoretical and Practical Synergies : 20th Conference on Computability in Europe, Cie 2024, Amsterdam, the Netherlands, July 8?12, 2024, Proceedings
Levy Patey, Ludovic (EDT); Pimentel, Elaine (EDT); Galeotti, Lorenzo (EDT); Manea, Florin (EDT)
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Twenty Years of Theoretical and Practical Synergies : 20th Conference on Computability in Europe, Cie 2024, Amsterdam, the Netherlands, July 8?12, 2024, Proceedings
Levy Patey, Ludovic (EDT); Pimentel, Elaine (EDT); Galeotti, Lorenzo (EDT); Manea, Florin (EDT)
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Twenty Years of Theoretical and Practical Synergies : 20th Conference on Computability in Europe, Cie 2024, Amsterdam, the Netherlands, July 8?12, 2024, Proceedings
Levy Patey, Ludovic (EDT); Pimentel, Elaine (EDT); Galeotti, Lorenzo (EDT); Manea, Florin (EDT)
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Taschenbuch. Condition: Neu. Twenty Years of Theoretical and Practical Synergies | 20th Conference on Computability in Europe, CiE 2024, Amsterdam, The Netherlands, July 8-12, 2024, Proceedings | Ludovic Levy Patey (u. a.) | Taschenbuch | Lecture Notes in Computer Science | xxxix | Englisch | 2024 | Springer | EAN 9783031643088…| Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu.
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Taschenbuch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - This book constitutes the refereed proceedings of the 20th Conference on Computabilitty in Europe, CiE 2024, held in Amsterdam, The Netherlands, during July 8-12, 2024The 26 full papers and 2 short papers included in this book were carefully review…ed and selected from 43submissions. This CIE 2024 focus ondevelopment of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences.
Milliken's Tree Theorem and Its Applications: a Computability-Theoretic Perspective
Paul-Elliot Angles D'Auriac, Peter A. Cholak, Damir D. Dzhafarov, Benoit Monin, Ludovic Patey
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Paperback. Condition: New. Milliken's tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey's theorem and its many variants and consequences. In this sense, Milliken's tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the… common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive.Motivated by a question of Dobrinen, we initiate the study of Milliken's tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey's theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken's tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-La¨uchli theorem, which is a key ingredient.Our advance here stems from a careful analysis of the Halpern-Läuchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken's tree theorem that permits us to gauge its effectivity in turn. The key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken's tree theorem. This enables us to build solutions to the full Milliken's tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken's tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove Milliken's tree theorem.We apply our analysis also to several well-known applications of Milliken's tree theorem, namely Devlin's theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey's theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken's tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. In particular, we establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker's notion of big Ramsey structure.
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Paperback. Condition: new. Paperback. This book constitutes the refereed proceedings of the 20th Conference on Computabilitty in Europe, CiE 2024, held in Amsterdam, The Netherlands, during July 812, 2024The 26 full papers and 2 short papers included in this book were carefully reviewed and selected from 43 submissions. This CIE… 2024 focus on development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences. This book constitutes the refereed proceedings of the 20th Conference on Computabilitty in Europe, CiE 2024, held in Amsterdam, The Netherlands, during July 812, 2024The 26 full papers and 2 short papers included in this book were carefully reviewed and selected from 43 submissions. Shipping may be from multiple locations in the US or from the UK, depending on stock availability.
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Paperback. Condition: new. Paperback. This book constitutes the refereed proceedings of the 20th Conference on Computabilitty in Europe, CiE 2024, held in Amsterdam, The Netherlands, during July 812, 2024The 26 full papers and 2 short papers included in this book were carefully reviewed and selected from 43 submissions. This CIE… 2024 focus on development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences. This book constitutes the refereed proceedings of the 20th Conference on Computabilitty in Europe, CiE 2024, held in Amsterdam, The Netherlands, during July 812, 2024The 26 full papers and 2 short papers included in this book were carefully reviewed and selected from 43 submissions. Shipping may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability.
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Taschenbuch. Condition: Neu. This item is printed on demand - Print on Demand Titel. Neuware -This book constitutes the refereed proceedings of the 20th Conference on Computabilitty in Europe, CiE 2024, held in Amsterdam, The Netherlands, during July 8-12, 2024The 26 full papers and 2 short papers included in this book were care…fully reviewed and selected from 43 submissions. This CIE 2024 focus on development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences.Springer-Verlag KG, Sachsenplatz 4-6, 1201 Wien 504 pp. Englisch.
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