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Introduction to Higher-Order Categorical Logic by J. Lambek,P. J. Scott Pdf
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 by B. Jacobs Pdf
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.
This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.
Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.
IV Higher Order Workshop, Banff 1990 by Graham Birtwistle Pdf
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.
Theorem Proving in Higher Order Logics by Stefan Berghofer,Tobias Nipkow,Christian Urban,Makarius Wenzel Pdf
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.
Higher-Order Computability by John Longley,Dag Normann Pdf
This book offers a self-contained exposition of the theory of computability in a higher-order context, where 'computable operations' may themselves be passed as arguments to other computable operations. The subject originated in the 1950s with the work of Kleene, Kreisel and others, and has since expanded in many different directions under the influence of workers from both mathematical logic and computer science. The ideas of higher-order computability have proved valuable both for elucidating the constructive content of logical systems, and for investigating the expressive power of various higher-order programming languages. In contrast to the well-known situation for first-order functions, it turns out that at higher types there are several different notions of computability competing for our attention, and each of these has given rise to its own strand of research. In this book, the authors offer an integrated treatment that draws together many of these strands within a unifying framework, revealing not only the range of possible computability concepts but the relationships between them. The book will serve as an ideal introduction to the field for beginning graduate students, as well as a reference for advanced researchers
Categories and Types in Logic, Language, and Physics by Claudia Casadio,Bob Coecke,Michael Moortgat,Philip Scott Pdf
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.
Categories for the Working Philosopher by Elaine M. Landry Pdf
This is the first book on category theory for a broad philosophical readership. There is no other discussion of category theory comparable in its scope. It is designed to show the interest and significant of category theory for philosophers working in a range of areas, including mathematics, proof theory, computer science, ontology, physics, biology, cognition, mathematical modelling, the structure of scientific theories, and the structure of the world. Moreover, it does 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 fields, in a way that builds on the concepts already familiar to philosophers working in these areas. The book is split into two halves. The 'pure' chapters focus on the use of category theory for mathematical, foundational, and logical purposes, while the 'applied' chapters consider the use of category theory for representational purposes, investigating category theory as a framework for theories of physics and biology, for mathematical modelling more generally, and for the structure of scientific theories. Book jacket.
