introduction to higher order categorical logic cambridge studies in advanced mathematics

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Introduction To Higher Order Categorical Logic

Author : J. Lambek
ISBN : 0521356539
Genre : Mathematics
File Size : 26. 40 MB
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Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.

Categorical Logic And Type Theory

Author : Bart Jacobs
ISBN : 0444508538
Genre : Mathematics
File Size : 64. 73 MB
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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.

Generic Programming

Author : Roland Backhouse
ISBN : 9783540201946
Genre : Computers
File Size : 55. 75 MB
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Generic programming attempts to make programming more efficient by making it more general. This book is devoted to a novel form of genericity in programs, based on parameterizing programs by the structure of the data they manipulate. The book presents the following four revised and extended chapters first given as lectures at the Generic Programming Summer School held at the University of Oxford, UK in August 2002: - Generic Haskell: Practice and Theory - Generic Haskell: Applications - Generic Properties of Datatypes - Basic Category Theory for Models of Syntax

Higher Topos Theory Am 170

Author : Jacob Lurie
ISBN : 9781400830558
Genre : Mathematics
File Size : 25. 18 MB
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Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Theorem Proving In Higher Order Logics

Author : Stefan Berghofer
ISBN : 9783642033599
Genre : Computers
File Size : 45. 49 MB
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This book constitutes the refereed proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics, TPHOLs 200, held in Munich, Germany, in August 2009. The 26 revised full papers presented together with 1 proof pearl, 4 tool presentations, and 3 invited papers were carefully reviewed and selected from 55 submissions. The papers cover all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification such as formal semantics of specification, modeling, and programming languages, specification and verification of hardware and software, formalization of mathematical theories, advances in theorem prover technology, as well as industrial application of theorem provers.

Iv Higher Order Workshop Banff 1990

Author : Graham Birtwistle
ISBN : 9781447131823
Genre : Mathematics
File Size : 28. 63 MB
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It is many years since Landin, Burge and others showed us how to apply higher order techniques and thus laid some foundations for modern functional programming. The advantage of higher order descriptions - that they can be very succinct and clear - has been percolating through ever since. Current research topics range from the design, implementation and use of higher order proof assistants and theorem provers, through program specification and verification, and programming language design, to its applications in hardware description and verification. The papers in this book represent the presentations made at a workshop held at Banff, Canada, September 10-14 1990 and organised by the Computer Science Department of the University of Calgary. The workshop gathered together researchers interested in applying higher order techniques to a range of problems. The workshop format had a few (but fairly long) presentations per day. This left ample time for healthy discussion and argument, many of which continued on into the small hours. With so much to choose from, the program had to be selective. This year's workshop was divided into five parts: 1. Expressing and reasoning about concurrency: Warren Burton and Ken Jackson, John Hughes, and Faron Moller. 2. Reasoning about synchronous circuits: Geraint Jones and Mary Sheeran (with a bonus on the fast Fourier transform from Geraint). 3. Reasoning about asynchronous circuits: Albert Camilleri, Jo Ebergen, and Martin Rem. 4. Categorical concepts for programming languages: Robin Cockett, Barry Jay, and Andy Pitts.

Logic And Computation

Author : Lawrence C. Paulson
ISBN : 0521395607
Genre : Computers
File Size : 34. 97 MB
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Logic and Computation is concerned with techniques for formal theorem-proving, with particular reference to Cambridge LCF (Logic for Computable Functions). Cambridge LCF is a computer program for reasoning about computation. It combines methods of mathematical logic with domain theory, the basis of the denotational approach to specifying the meaning of statements in a programming language. This book consists of two parts. Part I outlines the mathematical preliminaries: elementary logic and domain theory. They are explained at an intuitive level, giving references to more advanced reading. Part II provides enough detail to serve as a reference manual for Cambridge LCF. It will also be a useful guide for implementors of other programs based on the LCF approach.

Basic Category Theory

Author : Tom Leinster
ISBN : 9781107044241
Genre : Mathematics
File Size : 51. 85 MB
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A short introduction ideal for students learning category theory for the first time.

Categories And Modules With K Theory In View

Author : A. J. Berrick
ISBN : 0521632765
Genre : Mathematics
File Size : 21. 16 MB
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This book, first published in 2000, develops aspects of category theory fundamental to the study of algebraic K-theory. Ring and module theory illustrates category theory which provides insight into more advanced topics in module theory. Starting with categories in general, the text then examines categories of K-theory. This leads to the study of tensor products and the Morita theory. The categorical approach to localizations and completions of modules is formulated in terms of direct and inverse limits, prompting a discussion of localization of categories in general. Finally, local-global techniques which supply information about modules from their localizations and completions and underlie some interesting applications of K-theory to number theory and geometry are considered. Many useful exercises, concrete illustrations of abstract concepts placed in their historical settings and an extensive list of references are included. This book will help all who wish to work in K-theory to master its prerequisites.

Galois Theories

Author : Francis Borceux
ISBN : 0521803098
Genre : Mathematics
File Size : 36. 88 MB
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Develops Galois theory in a more general context, emphasizing category theory.

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