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Author : Rob Nederpelt,Herman Geuvers
Publisher : Cambridge University Press
Page : 465 pages
File Size : 46,7 Mb
Release : 2014-11-06
Category : Computers
ISBN : 9781107036505
A gentle introduction for graduate students and researchers in the art of formalizing mathematics on the basis of type theory.
Author : Rob Nederpelt,Herman Geuvers
Publisher : Cambridge University Press
Page : 465 pages
File Size : 49,8 Mb
Release : 2014-11-06
Category : Computers
ISBN : 9781316061084
Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.
Author : Peter B. Andrews
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 52,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401599344
In case you are considering to adopt this book for courses with over 50 students, please contact [email protected] for more information. This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.
Author : Anonim
Publisher : Univalent Foundations
Page : 484 pages
File Size : 53,5 Mb
Release : 2024-06-17
Category : Electronic
ISBN : 8210379456XXX
Author : Per Martin-Löf,Giovanni Sambin
Publisher : Unknown
Page : 116 pages
File Size : 43,7 Mb
Release : 1984
Category : Mathematics
ISBN : STANFORD:36105021234930
Author : Alfred North Whitehead,Bertrand Russell
Publisher : Cambridge University Press
Page : 524 pages
File Size : 43,7 Mb
Release : 1927
Category : Mathematics
ISBN : 052106791X
The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century.
Author : Bengt Nordström,Kent Petersson,Jan M. Smith
Publisher : Oxford University Press, USA
Page : 240 pages
File Size : 52,5 Mb
Release : 1990
Category : Computers
ISBN : UOM:39015018505134
In recent years, several formalisms for program construction have appeared. One such formalism is the type theory developed by Per Martin-Löf. Well suited as a theory for program construction, it makes possible the expression of both specifications and programs within the same formalism. Furthermore, the proof rules can be used to derive a correct program from a specification as well as to verify that a given program has a certain property. This book contains a thorough introduction to type theory, with information on polymorphic sets, subsets, monomorphic sets, and a full set of helpful examples.
Author : Simon Thompson
Publisher : Addison Wesley Publishing Company
Page : 396 pages
File Size : 50,6 Mb
Release : 1991
Category : Computers
ISBN : UOM:39015021884732
This book explores the role of Martin-Lof s constructive type theory in computer programming. The main focus of the book is how the theory can be successfully applied in practice. Introductory sections provide the necessary background in logic, lambda calculus and constructive mathematics, and exercises and chapter summaries are included to reinforce understanding.
Author : Martin Hofmann
Publisher : Springer Science & Business Media
Page : 221 pages
File Size : 44,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447109631
Extensional Constructs in Intensional Type Theory presents a novel approach to the treatment of equality in Martin-Loef type theory (a basis for important work in mechanised mathematics and program verification). Martin Hofmann attempts to reconcile the two different ways that type theories deal with identity types. The book will be of interest particularly to researchers with mainly theoretical interests and implementors of type theory based proof assistants, and also fourth year undergraduates who will find it useful as part of an advanced course on type theory.
Author : Benjamin C. Pierce
Publisher : MIT Press
Page : 117 pages
File Size : 44,7 Mb
Release : 1991-08-07
Category : Computers
ISBN : 9780262326452
Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading
Author : Stergios Chatzikyriakidis,Zhaohui Luo
Publisher : John Wiley & Sons
Page : 256 pages
File Size : 51,9 Mb
Release : 2021-02-17
Category : Language Arts & Disciplines
ISBN : 9781786301284
This book studies formal semantics in modern type theories (MTTsemantics). Compared with simple type theory, MTTs have much richer type structures and provide powerful means for adequate semantic constructions. This offers a serious alternative to the traditional settheoretical foundation for linguistic semantics and opens up a new avenue for developing formal semantics that is both model-theoretic and proof-theoretic, which was not available before the development of MTTsemantics. This book provides a reader-friendly and precise description of MTTs and offers a comprehensive introduction to MTT-semantics. It develops several case studies, such as adjectival modification and copredication, to exemplify the attractiveness of using MTTs for the study of linguistic meaning. It also examines existing proof assistant technology based on MTT-semantics for the verification of semantic constructions and reasoning in natural language. Several advanced topics are also briefly studied, including dependent event types, an application of dependent typing to event semantics.
Author : Daniel J. Velleman
Publisher : Cambridge University Press
Page : 401 pages
File Size : 40,6 Mb
Release : 2006-01-16
Category : Mathematics
ISBN : 9780521861243
Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.
Author : B. Jacobs
Publisher : Gulf Professional Publishing
Page : 784 pages
File Size : 47,6 Mb
Release : 2001-05-10
Category : Computers
ISBN : 0444508538
This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.
Author : J. Roger Hindley
Publisher : Cambridge University Press
Page : 200 pages
File Size : 53,5 Mb
Release : 1997
Category : Computers
ISBN : 9780521465182
Type theory is one of the most important tools in the design of higher-level programming languages, such as ML. This book introduces and teaches its techniques by focusing on one particularly neat system and studying it in detail. By concentrating on the principles that make the theory work in practice, the author covers all the key ideas without getting involved in the complications of more advanced systems. This book takes a type-assignment approach to type theory, and the system considered is the simplest polymorphic one. The author covers all the basic ideas, including the system's relation to propositional logic, and gives a careful treatment of the type-checking algorithm that lies at the heart of every such system. Also featured are two other interesting algorithms that until now have been buried in inaccessible technical literature. The mathematical presentation is rigorous but clear, making it the first book at this level that can be used as an introduction to type theory for computer scientists.
Author : S.R. Buss
Publisher : Elsevier
Page : 810 pages
File Size : 43,6 Mb
Release : 1998-07-09
Category : Mathematics
ISBN : 0080533183
This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.
