Categorical Logic and Type Theory


Author: Bart Jacobs
Publisher: Gulf Professional Publishing
ISBN: 9780444508539
Category: Mathematics
Page: 760
View: 7376

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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: 7034

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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.

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: 4292

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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.

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: 4190

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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.

Categorical Logic


Author: Andrew M. Pitts
Publisher: N.A
ISBN: N.A
Category: Logic, Symbolic and mathematical
Page: 94
View: 937

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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."

Axiomatic Method and Category Theory


Author: Andrei Rodin
Publisher: Springer Science & Business Media
ISBN: 3319004042
Category: Philosophy
Page: 285
View: 610

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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: 9490

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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-

Handbook of the History of Logic: Sets and extensions in the twentieth century


Author: Dov M. Gabbay,John Hayden Woods,Akihiro Kanamori
Publisher: Elsevier
ISBN: 0444516212
Category: Reference
Page: 865
View: 6098

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"Starting at the very beginning with Aristotle's founding contributions, logic has been graced by several periods in which the subject has flourished, attaining standards of rigour and conceptual sophistication underpinning a large and deserved reputation as a leading expression of human intellectual effort. It is widely recognized that the period from the mid-nineteenth century until the three-quarter mark of the century just past marked one of these golden ages, a period of explosive creativity and transforming insights. It has been said that ignorance of our history is a kind of amnesia, concerning which it is wise to note that amnesia is an illness. It would be a matter for regret, if we lost contact with another of logic's golden ages, one that greatly exceeds in reach that enjoyed by mathematical symbolic logic. This is the period between the eleventh and sixteenth centuries, loosely conceived of as the Middle Ages. The logic of this period does not have the expressive virtues afforded by the symbolic resources of uninterpreted calculi, but mediaeval logic rivals in range, originality and intellectual robustness a good deal of the modern record. The range of logic in this period is striking, extending from investigation of quantifiers and logic consequence to enquiries into logical truth; from theories of reference to accounts of identity; from work on the modalities to the stirrings of the logic of relations, from theories of meaning to analyses of the paradoxes, and more. While the scope of mediaeval logic is impressive, of greater importance is that nearly all of it can be read by the modern logician with at least some prospect of profit. The last thing that mediaeval logic is, is a museum piece." -- Publisher's website.

Semantics of Type Theory

Correctness, Completeness, and Independence Results
Author: Thomas Streicher
Publisher: Birkhäuser
ISBN: 9783764335946
Category: Functional programming (Computer science)
Page: 298
View: 4007

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Typing plays an important role in software development. Types can be consid ered as weak specifications of programs and checking that a program is of a certain type provides a verification that a program satisfies such a weak speci fication. By translating a problem specification into a proposition in constructive logic, one can go one step further: the effectiveness and unifonnity of a con structive proof allows us to extract a program from a proof of this proposition. Thus by the "proposition-as-types" paradigm one obtains types whose elements are considered as proofs. Each of these proofs contains a program correct w.r.t. the given problem specification. This opens the way for a coherent approach to the derivation of provably correct programs. These features have led to a "typeful" programming style where the classi cal typing concepts such as records or (static) arrays are enhanced by polymor phic and dependent types in such a way that the types themselves get a complex mathematical structure. Systems such as Coquand and Huet's Calculus of Con structions are calculi for computing within extended type systems and provide a basis for a deduction oriented mathematical foundation of programming. On the other hand, the computational power and the expressive (impred icativity ) of these systems makes it difficult to define appropriate semantics."

Computer Science Logic

14th International Workshop, CSL 2000 Annual Conference of the EACSL Fischbachau, Germany, August 21-26, 2000 Proceedings
Author: Germany) Workshop on Computer Science Logic 2000 (Fischbachau,Peter Clote,European Association for Computer Science Logic. Conference
Publisher: Springer Science & Business Media
ISBN: 3540678956
Category: Computers
Page: 541
View: 4141

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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.

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: 1453

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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: 2717

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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.

Categories for the Working Philosopher


Author: Elaine Landry
Publisher: Oxford University Press
ISBN: 019874899X
Category: Mathematics
Page: 528
View: 6188

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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.

Introduction to Logic


Author: CTI Reviews
Publisher: Cram101 Textbook Reviews
ISBN: 1497025885
Category: Education
Page: 50
View: 392

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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.

Algebraic and Logic Programming

Third International Conference, Volterra, Italy, September 2-4, 1992 : Proceedings
Author: H. Kirchner,Giorgio Levi
Publisher: Springer Verlag
ISBN: N.A
Category: Computers
Page: 457
View: 3708

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"This volume contains the proceedings of the Third International Conference on Algebraic and Logic Programming, held in Pisa, Italy, September 2-4, 1992. Like the two previous conferences in Germany in 1988 and France in 1990, the third conference aims at strengthening the connections betweenalgebraic techniques and logic programming. On the one hand, logic programming has been very successful during the last decades and more and more systems compete in enhancing its expressive power. On the other hand, concepts like functions, equality theory, and modularity are particularly well handled in an algebraic framework. Common foundations of both approaches have recently been developed, and this conference is a forum for people from both areas to exchange ideas, results, and experiences. The book covers the following topics: semantics ofalgebraic and logic programming; integration of functional and logic programming; term rewriting, narrowing, and resolution; constraintlogic programming and theorem proving; concurrent features in algebraic and logic programming languages; and implementation issues."--PUBLISHER'S WEBSITE.

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: 8033

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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

6th International Conference, CTCS '95, Cambridge, United Kingdom, August 7 - 11, 1995. Proceedings
Author: David Pitt,David E. Rydeheard,Peter Johnstone
Publisher: Springer
ISBN: N.A
Category: Computer programming
Page: 252
View: 5557

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This book presents the proceedings of the Sixth International Conference on Category Theory and Computer Science, CTCS '95, held in Cambridge, UK in August 1995. The 15 revised full papers included in the volume document the exploitation of links between logic and category theory leading to a solid basis for much of the understanding of the semantics of computation. Notable amongst other advances is the introduction of linear logic and other substructural logics, providing a new approach to proof theory. Further aspects covered are semantics of lambda calculi and type theories, program specification and development, and domain theory.

Foundations of Software Science and Computation Structures

7th International Conference, FOSSACS 2004, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2004, Barcelona, Spain, March 29 - April 2, 2004, Proceedings
Author: Igor Walukiewicz
Publisher: Springer
ISBN: N.A
Category: Computer software
Page: 529
View: 8285

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This book constitutes the refereed proceedings of the 7th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2004, held in Barcelona, Spain in March/April 2004. The 34 revised full papers presented together with the abstracts of 2 invited talks were carefully reviewed and selected from over 130 submissions. Among the topics addressed are lambda calculus, cryptographic protocol analysis, graphs and grammar systems, decision theory, bisimulation, rewriting, normalization, specification, verification, process calculi, mobile code, automata, program semantics, dynamic logics, timed languages, security analysis, information-theoretical aspects.