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Reverse Mathematics Of Combinatorial Principles
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Book Synopsis Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles by : Denis R Hirschfeldt
Download or read book Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles written by Denis R Hirschfeldt and published by World Scientific. This book was released on 2014-07-18 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions.
Book Synopsis Reverse Mathematics by : Damir D. Dzhafarov
Download or read book Reverse Mathematics written by Damir D. Dzhafarov and published by Springer Nature. This book was released on 2022-07-25 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features: Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments Includes a large number of exercises of varying levels of difficulty, supplementing each chapter The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas. Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.
Book Synopsis Reverse Mathematics of Combinatorial Principles by : Damir Dzhalil Dzhafarov
Download or read book Reverse Mathematics of Combinatorial Principles written by Damir Dzhalil Dzhafarov and published by . This book was released on 2011 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the logical strength of various weak combinatorial principles, using the tools of reverse mathematics, computability theory, and effective measure theory. Our focus is on Ramsey's theorem, various equivalents of the axiom of choice, and theorems arising from problems in cognitive science. We obtain new results concerning the effective content of previously studied principles, and show how these relate to several new principles we introduce.
Book Synopsis Slicing the Truth by : Denis R. Hirschfeldt
Download or read book Slicing the Truth written by Denis R. Hirschfeldt and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Calibrating the Complexity of Combinatorics by : Giovanni Solda
Download or read book Calibrating the Complexity of Combinatorics written by Giovanni Solda and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem by : Denis R. Hirschfeldt
Download or read book Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem written by Denis R. Hirschfeldt and published by American Mathematical Soc.. This book was released on 2017-09-25 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.
Book Synopsis Subsystems of Second Order Arithmetic by : Stephen George Simpson
Download or read book Subsystems of Second Order Arithmetic written by Stephen George Simpson and published by Cambridge University Press. This book was released on 2009-05-29 with total page 461 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Book Synopsis Reverse Mathematics by : John Stillwell
Download or read book Reverse Mathematics written by John Stillwell and published by Princeton University Press. This book was released on 2019-09-24 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents reverse mathematics to a general mathematical audience for the first time. Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. to logic.
Book Synopsis Principles And Techniques In Combinatorics by : Chuan Chong Chen
Download or read book Principles And Techniques In Combinatorics written by Chuan Chong Chen and published by World Scientific. This book was released on 1992-07-22 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: A textbook suitable for undergraduate courses. The materials are presented very explicitly so that students will find it very easy to read. A wide range of examples, about 500 combinatorial problems taken from various mathematical competitions and exercises are also included.
Book Synopsis Well-Quasi Orders in Computation, Logic, Language and Reasoning by : Peter M. Schuster
Download or read book Well-Quasi Orders in Computation, Logic, Language and Reasoning written by Peter M. Schuster and published by Springer Nature. This book was released on 2020-01-01 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven to be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students.
Book Synopsis Combinatorics: The Art of Counting by : Bruce E. Sagan
Download or read book Combinatorics: The Art of Counting written by Bruce E. Sagan and published by American Mathematical Soc.. This book was released on 2020-10-16 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a gentle introduction to the enumerative part of combinatorics suitable for study at the advanced undergraduate or beginning graduate level. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature which has never before appeared in print, such as the use of quotient posets to study the Möbius function and characteristic polynomial of a partially ordered set, or the connection between quasisymmetric functions and pattern avoidance. The book assumes minimal background, and a first course in abstract algebra should suffice. The exposition is very reader friendly: keeping a moderate pace, using lots of examples, emphasizing recurring themes, and frankly expressing the delight the author takes in mathematics in general and combinatorics in particular.
Book Synopsis Bounded Reverse Mathematics by : Phuong Nguyen
Download or read book Bounded Reverse Mathematics written by Phuong Nguyen and published by . This book was released on 2008 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: First we provide a unified framework for developing theories of Bounded Arithmetic that are associated with uniform classes inside polytime (P) in the same way that Buss's theory S12 is associated with P. We obtain finitely axiomatized theories many of which turn out to be equivalent to a number of existing systems. By formalizing the proof of Barrington's Theorem (that the functions computable by polynomial-size bounded-width branching programs are precisely functions computable in ALogTime, or equivalently NC 1) we prove one such equivalence between the theories associated with ALogTime, solving a problem that remains open in [Ara00, Pit00]. Our theories demonstrate an advantage of the simplicity of Zambella's two-sorted setting for small theories of Bounded Arithmetic. Then we give the first definitions for the relativizations of small classes such as NC1, L, NL that preserve their inclusion order. Separating these relativized classes is shown to be as hard as separating the corresponding non-relativized classes. Our framework also allows us to obtain relativized theories that characterize the newly defined relativized classes. Finally we formalize and prove a number of mathematical theorems in our theories. In particular, we prove the discrete versions of the Jordan Curve Theorem in the theories V0 and V0 (2), and establish some facts about the distribution of prime numbers in the theory VTC0. Our V 0- and V0(2)-proofs improve a number of existing upper bounds for the propositional complexity of combinatorial principles related to grid graphs. Overall, this thesis is a contribution to Bounded Reverse Mathematics, a theme whose purpose is to formalize and prove (discrete versions of) mathematical theorems in the weakest possible theories of bounded arithmetic.
Book Synopsis Computability Theory And Foundations Of Mathematics - Proceedings Of The 9th International Conference On Computability Theory And Foundations Of Mathematics by : Ningning Peng
Download or read book Computability Theory And Foundations Of Mathematics - Proceedings Of The 9th International Conference On Computability Theory And Foundations Of Mathematics written by Ningning Peng and published by World Scientific. This book was released on 2022-05-18 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume features the latest scientific developments in the fields of computability theory and logical foundations of mathematics as well as applications. The scope involves the topics of Computability Theory, Reverse Mathematics, Nonstandard Analysis, Proof Theory, Set Theory, Philosophy of Mathematics, Constructive Mathematics, Theory of Randomness and Computational Complexity Theory.
Book Synopsis Reverse Mathematics 2001 by : Stephen G. Ross
Download or read book Reverse Mathematics 2001 written by Stephen G. Ross and published by CRC Press. This book was released on 2005-09-01 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reverse Mathematics is a program of research in the foundations of mathematics, motivated by the foundational questions of what are appropriate axioms for mathematics, and what are the logical strengths of particular axioms and particular theorems. The book contains 24 original papers by leading researchers. These articles exhibit the exciting rece
Book Synopsis Bounded Reverse Mathematics by : Nicholas Byrne
Download or read book Bounded Reverse Mathematics written by Nicholas Byrne and published by . This book was released on 2019-02-27 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: First we provide a unified framework for developing theories of Bounded Arithmetic that are associated with uniform classes inside polytime (P) in the same way that Buss's theory S 12 is associated with P. We obtain finitely axiomatized theories many of which turn out to be equivalent to a number of existing systems. By formalizing the proof of Barrington's Theorem (that the functions computable by polynomial-size bounded-width branching programs are precisely functions computable in ALogTime, or equivalently NC 1 ) we prove one such equivalence between the theories associated with ALogTime, solving a problem that remains open in [Ara00, Pit00]. Our theories demonstrate an advantage of the simplicity of Zambella's two-sorted setting for small theories of Bounded Arithmetic. Then we give the first definitions for the relativizations of small classes such as NC 1 , L, NL that preserve their inclusion order. Separating these relativized classes is shown to be as hard as separating the corresponding non-relativized classes. Our framework also allows us to obtain relativized theories that characterize the newly defined relativized classes. Finally we formalize and prove a number of mathematical theorems in our theories. In particular, we prove the discrete versions of the Jordan Curve Theorem in the theories V 0 and V 0 (2), and establish some facts about the distribution of prime numbers in the theory VTC 0 . Our V 0 - and V 0 (2)-proofs improve a number of existing upper bounds for the propositional complexity of combinatorial principles related to grid graphs. Overall, this thesis is a contribution to Bounded Reverse Mathematics, a theme iiwhose purpose is to formalize and prove (discrete versions of) mathematical theorems in the weakest possible theories of bounded arithmetic.
Book Synopsis Connecting with Computability by : Liesbeth De Mol
Download or read book Connecting with Computability written by Liesbeth De Mol and published by Springer Nature. This book was released on 2021-07-01 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 17th Conference on Computability in Europe, CiE 2021, organized by the University of Ghent in July 2021. Due to COVID-19 pandemic the conference was held virtually. The 48 full papers presented in this volume were carefully reviewed and selected from 50 submissions. CiE promotes the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences, such as physics and biology, as well as related fields, such as philosophy and history of computing. CiE 2021 had as its motto Connecting with Computability, a clear acknowledgement of the connecting and interdisciplinary nature of the conference series which is all the more important in a time where people are more than ever disconnected from one another due to the COVID-19 pandemic.
Book Synopsis Combinatorial And Toric Homotopy: Introductory Lectures by : Alastair Darby
Download or read book Combinatorial And Toric Homotopy: Introductory Lectures written by Alastair Darby and published by World Scientific. This book was released on 2017-10-20 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume consists of introductory lectures on the topics in the new and rapidly developing area of toric homotopy theory, and its applications to the current research in configuration spaces and braids, as well as to more applicable mathematics such as fr-codes and robot motion planning.The book starts intertwining homotopy theoretical and combinatorial ideas within the remits of toric topology and illustrates an attempt to classify in a combinatorial way polytopes known as fullerenes, which are important objects in quantum physics, quantum chemistry and nanotechnology. Toric homotopy theory is then introduced as a further development of toric topology, which describes properties of Davis-Januszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. The book also displays the current research on configuration spaces, braids, the theory of limits over the category of presentations and the theory of fr-codes. As an application to robotics, the book surveys topological problems relevant to the motion planning problem of robotics and includes new results and constructions, which enrich the emerging area of topological robotics.The book is at research entry level addressing the core components in homotopy theory and their important applications in the sciences and thus suitable for advanced undergraduate and graduate students.
