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Cut Elimination In Categories
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Book Synopsis Cut Elimination in Categories by : K. Dosen
Download or read book Cut Elimination in Categories written by K. Dosen and published by Springer Science & Business Media. This book was released on 2013-04-18 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proof theory and category theory were first drawn together by Lambek some 30 years ago but, until now, the most fundamental notions of category theory (as opposed to their embodiments in logic) have not been explained systematically in terms of proof theory. Here it is shown that these notions, in particular the notion of adjunction, can be formulated in such as way as to be characterised by composition elimination. Among the benefits of these composition-free formulations are syntactical and simple model-theoretical, geometrical decision procedures for the commuting of diagrams of arrows. Composition elimination, in the form of Gentzen's cut elimination, takes in categories, and techniques inspired by Gentzen are shown to work even better in a purely categorical context than in logic. An acquaintance with the basic ideas of general proof theory is relied on only for the sake of motivation, however, and the treatment of matters related to categories is also in general self contained. Besides familiar topics, presented in a novel, simple way, the monograph also contains new results. It can be used as an introductory text in categorical proof theory.
Book Synopsis An Introduction to Proof Theory by : Paolo Mancosu
Download or read book An Introduction to Proof Theory written by Paolo Mancosu and published by Oxford University Press. This book was released on 2021-08-12 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.
Book Synopsis The Blind Spot by : Jean-Yves Girard
Download or read book The Blind Spot written by Jean-Yves Girard and published by European Mathematical Society. This book was released on 2011 with total page 554 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lectures on logic, more specifically proof theory, are basically intended for postgraduate students and researchers in logic. The question at stake is the nature of mathematical knowledge and the difference between a question and an answer, i.e., the implicit and the explicit. The problem is delicate mathematically and philosophically as well: the relation between a question and its answer is a sort of equality where one side is ``more equal than the other'': one thus discovers essentialist blind spots. Starting with Godel's paradox (1931)--so to speak, the incompleteness of answers with respect to questions--the book proceeds with paradigms inherited from Gentzen's cut-elimination (1935). Various settings are studied: sequent calculus, natural deduction, lambda calculi, category-theoretic composition, up to geometry of interaction (GoI), all devoted to explicitation, which eventually amounts to inverting an operator in a von Neumann algebra. Mathematical language is usually described as referring to a preexisting reality. Logical operations can be given an alternative procedural meaning: typically, the operators involved in GoI are invertible, not because they are constructed according to the book, but because logical rules are those ensuring invertibility. Similarly, the durability of truth should not be taken for granted: one should distinguish between imperfect (perennial) and perfect modes. The procedural explanation of the infinite thus identifies it with the unfinished, i.e., the perennial. But is perenniality perennial? This questioning yields a possible logical explanation for algorithmic complexity. This highly original course on logic by one of the world's leading proof theorists challenges mathematicians, computer scientists, physicists, and philosophers to rethink their views and concepts on the nature of mathematical knowledge in an exceptionally profound way.
Book Synopsis Applications of Categories in Computer Science by : M. P. Fourman
Download or read book Applications of Categories in Computer Science written by M. P. Fourman and published by Cambridge University Press. This book was released on 1992-06-26 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the subject which was held at the University of Durham. Participants at the conference were leading computer scientists and mathematicians working in the area and this volume reflects the excitement and importance of the meeting. All the papers have been refereed and represent some of the most important and current ideas. Hence this book will be essential to mathematicians and computer scientists working in the applications of category theory.
Book Synopsis Towards Higher Categories by : John C. Baez
Download or read book Towards Higher Categories written by John C. Baez and published by Springer Science & Business Media. This book was released on 2009-09-24 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to give background for those who would like to delve into some higher category theory. It is not a primer on higher category theory itself. It begins with a paper by John Baez and Michael Shulman which explores informally, by analogy and direct connection, how cohomology and other tools of algebraic topology are seen through the eyes of n-category theory. The idea is to give some of the motivations behind this subject. There are then two survey articles, by Julie Bergner and Simona Paoli, about (infinity,1) categories and about the algebraic modelling of homotopy n-types. These are areas that are particularly well understood, and where a fully integrated theory exists. The main focus of the book is on the richness to be found in the theory of bicategories, which gives the essential starting point towards the understanding of higher categorical structures. An article by Stephen Lack gives a thorough, but informal, guide to this theory. A paper by Larry Breen on the theory of gerbes shows how such categorical structures appear in differential geometry. This book is dedicated to Max Kelly, the founder of the Australian school of category theory, and an historical paper by Ross Street describes its development.
Book Synopsis Proof-net Categories by : Kosta Dosen
Download or read book Proof-net Categories written by Kosta Dosen and published by Polimetrica s.a.s.. This book was released on 2007 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Methods of Cut-Elimination by : Matthias Baaz
Download or read book Methods of Cut-Elimination written by Matthias Baaz and published by Springer Science & Business Media. This book was released on 2011-01-07 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book on cut-elimination in first-order predicate logic from an algorithmic point of view. Instead of just proving the existence of cut-free proofs, it focuses on the algorithmic methods transforming proofs with arbitrary cuts to proofs with only atomic cuts (atomic cut normal forms, so-called ACNFs). The first part investigates traditional reductive methods from the point of view of proof rewriting. Within this general framework, generalizations of Gentzen's and Sch\”utte-Tait's cut-elimination methods are defined and shown terminating with ACNFs of the original proof. Moreover, a complexity theoretic comparison of Gentzen's and Tait's methods is given. The core of the book centers around the cut-elimination method CERES (cut elimination by resolution) developed by the authors. CERES is based on the resolution calculus and radically differs from the reductive cut-elimination methods. The book shows that CERES asymptotically outperforms all reductive methods based on Gentzen's cut-reduction rules. It obtains this result by heavy use of subsumption theorems in clause logic. Moreover, several applications of CERES are given (to interpolation, complexity analysis of cut-elimination, generalization of proofs, and to the analysis of real mathematical proofs). Lastly, the book demonstrates that CERES can be extended to nonclassical logics, in particular to finitely-valued logics and to G\"odel logic.
Book Synopsis Rewriting Techniques and Applications by : Ralf Treinen
Download or read book Rewriting Techniques and Applications written by Ralf Treinen and published by Springer. This book was released on 2009-06-19 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 20th International Conference on Rewriting Techniques and Applications, RTA 2009, held in Brasília, Brazil, during June 29 - July 1, 2009. The 22 revised full papers and four system descriptions presented were carefully reviewed and selected from 59 initial submissions. The papers cover current research on all aspects of rewriting including typical areas of interest such as applications, foundational issues, frameworks, implementations, and semantics.
Book Synopsis Categories in Computer Science and Logic by : John Walker Gray
Download or read book Categories in Computer Science and Logic written by John Walker Gray and published by American Mathematical Soc.. This book was released on 1989 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the proceedings of AMS-IMS-SIAM Summer Research Conference on Categories in Computer Science and Logic that was held at the University of Colorado in Boulder. This book discusses the use of category theory in formalizing aspects of computer programming and program design.
Book Synopsis Categories and Types in Logic, Language, and Physics by : Claudia Casadio
Download or read book Categories and Types in Logic, Language, and Physics written by Claudia Casadio and published by Springer. This book was released on 2014-04-03 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory. This Festschrift was put together on the occasion of his 90th birthday. The papers in it give a good picture of the multiple research areas where the impact of Jim Lambek's work can be felt. The volume includes contributions by prominent researchers and by their students, showing how Jim Lambek's ideas keep inspiring upcoming generations of scholars.
Book Synopsis Categories for the Working Philosopher by : Elaine Landry
Download or read book Categories for the Working Philosopher written by Elaine Landry and published by Oxford University Press. This book was released on 2017-11-17 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: Often people have wondered why there is no introductory text on category theory aimed at philosophers working in related areas. The answer is simple: what makes categories interesting and significant is their specific use for specific purposes. These uses and purposes, however, vary over many areas, both "pure", e.g., mathematical, foundational and logical, and "applied", e.g., applied to physics, biology and the nature and structure of mathematical models. Borrowing from the title of Saunders Mac Lane's seminal work "Categories for the Working Mathematician", this book aims to bring the concepts of category theory to philosophers working in areas ranging from mathematics to proof theory to computer science to ontology, from to physics to biology to cognition, from mathematical modeling to the structure of scientific theories to the structure of the world. Moreover, it aims to do this in a way that is accessible to non-specialists. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, and in a way that builds on the concepts that are already familiar to philosophers working in these areas.
Book Synopsis Category Neutrality by : Neil Whitman
Download or read book Category Neutrality written by Neil Whitman and published by Routledge. This book was released on 2004-08 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: It has been claimed that 'category neutrality' where a word or a phrase is used simultaneously with more than one syntactic category, does not exist. This work shows that it does exist in English. This work not only challenges the current thinking, but also raises foundational questions about the nature of ambiguity.
Book Synopsis Basic Proof Theory by : A. S. Troelstra
Download or read book Basic Proof Theory written by A. S. Troelstra and published by Cambridge University Press. This book was released on 2000-07-27 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.
Book Synopsis Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics by : Claudia Casadio
Download or read book Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics written by Claudia Casadio and published by Springer Nature. This book was released on 2021-04-21 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is dedicated to the life and work of the mathematician Joachim Lambek (1922–2014). The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.
Author :Yiannis N. Moschovakis Publisher :Springer Science & Business Media ISBN 13 :1461228220 Total Pages :607 pages Book Rating :4.4/5 (612 download)
Book Synopsis Logic from Computer Science by : Yiannis N. Moschovakis
Download or read book Logic from Computer Science written by Yiannis N. Moschovakis and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 607 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume is the outgrowth of a workshop with the same title held at MSRI in the week of November 13-17, 1989, and for those who did not get it, Logic from Computer Science is the converse of Logic in Computer Science, the full name of the highly successful annual LICS conferences. We meant to have a conference which would bring together the LICS commu nity with some of the more traditional "mathematical logicians" and where the emphasis would be on the flow of ideas from computer science to logic rather than the other way around. In a LICS talk, sometimes, the speaker presents a perfectly good theorem about (say) the A-calculus or finite model theory in terms of its potential applications rather than its (often more ob vious) intrinsic, foundational interest and intricate proof. This is not meant to be a criticism; the LICS meetings are, after all, organized by the IEEE Computer Society. We thought, for once, it would be fun to see what we would get if we asked the speakers to emphasize the relevance of their work for logic rather than computer science and to point out what is involved in the proofs. I think, mostly, it worked. In any case, the group of people represented as broad a selection of logicians as I have seen in recent years, and the quality of the talks was (in my view) exceptionally, unusually high. I learned a lot and (I think) others did too.
Book Synopsis Coherence in Categories by : Saunders Mac Lane
Download or read book Coherence in Categories written by Saunders Mac Lane and published by Springer. This book was released on 2006-11-15 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Category Theory and Computer Science by : David H. Pitt
Download or read book Category Theory and Computer Science written by David H. Pitt and published by Springer Science & Business Media. This book was released on 1991-08-21 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers in this volume were presented at the fourth biennial Summer Conference on Category Theory and Computer Science, held in Paris, September3-6, 1991. Category theory continues to be an important tool in foundationalstudies in computer science. It has been widely applied by logicians to get concise interpretations of many logical concepts. Links between logic and computer science have been developed now for over twenty years, notably via the Curry-Howard isomorphism which identifies programs with proofs and types with propositions. The triangle category theory - logic - programming presents a rich world of interconnections. Topics covered in this volume include the following. Type theory: stratification of types and propositions can be discussed in a categorical setting. Domain theory: synthetic domain theory develops domain theory internally in the constructive universe of the effective topos. Linear logic: the reconstruction of logic based on propositions as resources leads to alternatives to traditional syntaxes. The proceedings of the previous three category theory conferences appear as Lecture Notes in Computer Science Volumes 240, 283 and 389.
