Categorical Logic and Type Theory

Author: Bart Jacobs
Publisher: Gulf Professional Publishing
ISBN: 9780444508539
Category: Mathematics
Page: 760
View: 6859

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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.

Introduction to Higher-Order Categorical Logic

Author: J. Lambek,P. J. Scott
Publisher: Cambridge University Press
ISBN: 9780521356534
Category: Mathematics
Page: 304
View: 7377

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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

Author: Andrew M. Pitts
Publisher: N.A
Category: Logic, Symbolic and mathematical
Page: 94
View: 1713

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Abstract: "This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists."

Foundations of Software Science and Computation Structures

19th International Conference, FOSSACS 2016, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016, Eindhoven, The Netherlands, April 2-8, 2016, Proceedings
Author: Bart Jacobs,Christof Löding
Publisher: Springer
ISBN: 3662496305
Category: Computers
Page: 550
View: 5085

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This book constitutes the proceedings of the 19th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2016, which took place in Eindhoven, The Netherlands, in April 2016, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016. The 31 full papers presented in this volume were carefully reviewed and selected from 85 submissions. They were organized in topical sections named: types; recursion and fixed-points; verification and program analysis; automata, logic, games; probabilistic and timed systems; proof theory and lambda calculus; algorithms for infinite systems; and monads.

From a Geometrical Point of View

A Study of the History and Philosophy of Category Theory
Author: Jean-Pierre Marquis
Publisher: Springer Science & Business Media
ISBN: 1402093845
Category: Science
Page: 310
View: 5245

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From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

Category Theory and Computer Science

Paris, France, September 3-6, 1991. Proceedings
Author: David H. Pitt,Pierre-Louis Curien,Samson Abramsky,Andrew Pitts,Axel Poigne,David E. Rydeheard
Publisher: Springer Science & Business Media
ISBN: 9783540544951
Category: Mathematics
Page: 304
View: 2422

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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.

Axiomatic Method and Category Theory

Author: Andrei Rodin
Publisher: Springer Science & Business Media
ISBN: 3319004042
Category: Philosophy
Page: 285
View: 4268

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This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

Categorical Quantum Models and Logics

Author: Chris Heunen
Publisher: Amsterdam University Press
ISBN: 9085550246
Category: Electronic books
Page: 214
View: 325

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This dissertation studies the logic behind quantum physics, using category theory as the principal tool and conceptual guide. To do so, principles of quantum mechanics are modeled categorically. These categorical quantum models are justified by an embedding into the category of Hilbert spaces, the traditional formalism of quantum physics. In particular, complex numbers emerge without having been prescribed explicitly. Interpreting logic in such categories results in orthomodular property lattices, and furthermore provides a natural setting to consider quantifiers. Finally, topos theory, incorporating categorical logic in a refined way, lets one study a quantum system as if it were classical, in particular leading to a novel mathematical notion of quantum-

Sets and Extensions in the Twentieth Century

Author: N.A
Publisher: Elsevier
ISBN: 0080930662
Category: Mathematics
Page: 880
View: 8481

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Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics that both proceeds with its own internal questions and is capable of contextualizing over a broad range, which makes set theory an intriguing and highly distinctive subject. This handbook covers the rich history of scientific turning points in set theory, providing fresh insights and points of view. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in mathematics, the history of philosophy, and any discipline such as computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration Serves as a singular contribution to the intellectual history of the 20th century Contains the latest scholarly discoveries and interpretative insights

Computer Science Logic

14th International Workshop, CSL 2000 Annual Conference of the EACSL Fischbachau, Germany, August 21-26, 2000 Proceedings
Author: Peter G. Clote,Helmut Schwichtenberg
Publisher: Springer
ISBN: 3540446222
Category: Computers
Page: 550
View: 6181

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This book constitutes the refereed proceedings of the 13th International Workshop on Computer Science Logic, CSL 2000, held in Fischbachau, Germany as the 8th Annual Conference of the EACSL in August 2000. The 28 revised full papers presented together with eight invited papers were carefully reviewed and selected by the program committee. Among the topics covered are automated deduction, theorem proving, categorical logic, term rewriting, finite model theory, higher order logic, lambda and combinatory calculi, computational complexity, logic programing, constraints, linear logic, modal logic, temporal logic, model checking, formal specification, formal verification, program transformation, etc.

Practical Foundations of Mathematics

Author: Paul Taylor
Publisher: Cambridge University Press
ISBN: 9780521631075
Category: Mathematics
Page: 572
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Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

Categories for the Working Philosopher

Author: Elaine Landry
Publisher: Oxford University Press
ISBN: 019874899X
Category: Mathematics
Page: 528
View: 6440

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This is the first volume on category theory for a broad philosophical readership. It is designed to show the interest and significance of category theory for a range of philosophical interests: mathematics, proof theory, computation, cognition, scientific modelling, physics, ontology, the structure of the world. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, in an accessible waythat builds on the concepts that are already familiar to philosophers working in these areas.

Computer Science Logic

20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL, Szeged, Hungary, September 25-29, 2006, Proceedings
Author: European Association for Computer Science Logic. Conference
Publisher: Springer Science & Business Media
ISBN: 3540454586
Category: Computers
Page: 626
View: 1097

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This book constitutes the refereed proceedings of the 20th International Workshop on Computer Science Logic, CSL 2006, held as the 15th Annual Conference of the EACSL in Szeged, Hungary in September 2006.The 37 revised full papers presented together with 4 invited contributions were carefully reviewed and selected from 132 submissions. All current aspects of logic in computer science are addressed, including automated deduction and interactive theorem proving, constructive mathematics and type theory, equational logic and term rewriting, automata and formal logics, modal and temporal logic, model checking, logical aspects of computational complexity, finite model theory, computational proof theory, logic programming and constraints, lambda calculus and combinatory logic, categorical logic and topological semantics, domain theory, database theory, specification, extraction and transformation of programs, logical foundations of programming paradigms, verification of security protocols, linear logic, higher-order logic, nonmonotonic reasoning, as well as logics and type systems for biology.

Logical Foundations of Computer Science

International Symposium, LFCS 2013, San Diego, CA, USA, January 6-8, 2013. Proceedings
Author: Sergei Artemov,Anil Nerode
Publisher: Springer
ISBN: 3642357229
Category: Mathematics
Page: 415
View: 3099

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This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2013, held in San Diego, CA, USA in January 2013. The volume presents 29 revised refereed papers carefully selected by the program committee. The scope of the Symposium is broad and includes constructive mathematics and type theory; logic, automata and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logic; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple agent system logics; logics of proof and justification; nonmonotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; and other logics in computer science.

Categorical Methods in Computer Science

With Aspects from Topology
Author: Hartmut Ehrig,Horst Herrlich,Hans-Jörg Kreowski,Gerhard Preuß
Publisher: Springer Science & Business Media
ISBN: 9783540517221
Category: Computers
Page: 354
View: 7597

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This volume contains selected papers of the International Workshop on "Categorical Methods in Computer Science - with Aspects from Topology" and of the "6th International Data Type Workshop" held in August/September 1988 in Berlin. The 23 papers of this volume are grouped into three parts: Part 1 includes papers on categorical foundations and fundamental concepts from category theory in computer science. Part 2 presents applications of categorical methods to algebraic specification languages and techniques, data types, data bases, programming, and process specifications. Part 3 comprises papers on categorial aspects from topology which mainly concentrate on special adjoint situations like cartesian closeness, Galois connections, reflections, and coreflections which are of growing interest in categorical topology and computer science.

Introduction to Logic

Author: CTI Reviews
Publisher: Cram101 Textbook Reviews
ISBN: 1497025885
Category: Education
Page: 50
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Facts101 is your complete guide to Introduction to Logic. In this book, you will learn topics such as Unit Three: Truth-Functional Logic, Unit Four: Predicate Logic, Unit Five: Informal and Inductive Logic, and Unit Six: Modal Logic plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.