Lectures on the Curry-Howard Isomorphism

Author: Morten Heine Sørensen,Pawel Urzyczyn
Publisher: Elsevier
ISBN: 9780080478920
Category: Mathematics
Page: 456
View: 9587

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The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning

Enzyklopädie Philosophie und Wissenschaftstheorie

Bd. 4: Ins–Loc
Author: Jürgen Mittelstraß
Publisher: Springer-Verlag
ISBN: 3476001369
Category: Philosophy
Page: 595
View: 6685

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Das ganze Wissen der Philosophie und Wissenschaftstheorie. Lückenlos belegt das größte allgemeine Lexikon zur Philosophie in deutscher Sprache den heutigen Kenntnisstand. Erweitert auf acht Bände dokumentiert die 2. Auflage insbesondere die jüngsten Entwicklungen in Logik, Erkenntnis- und Wissenschaftstheorie sowie Sprachphilosophie. Jetzt liegt der vierte Band in Neuauflage vor mit über 100 zusätzlichen Einträgen, u. a. zu Intelligenz, Interdisziplinarität, Isotropie, Kognitionswissenschaft, Komplexitätstheorie, Konvention, Lebenswissenschaften und einer Vielzahl neuer Personenartikel.

Foundations of Algebraic Specification and Formal Software Development

Author: Donald Sannella,Andrzej Tarlecki
Publisher: Springer Science & Business Media
ISBN: 3642173365
Category: Computers
Page: 584
View: 9554

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This book provides foundations for software specification and formal software development from the perspective of work on algebraic specification, concentrating on developing basic concepts and studying their fundamental properties. These foundations are built on a solid mathematical basis, using elements of universal algebra, category theory and logic, and this mathematical toolbox provides a convenient language for precisely formulating the concepts involved in software specification and development. Once formally defined, these notions become subject to mathematical investigation, and this interplay between mathematics and software engineering yields results that are mathematically interesting, conceptually revealing, and practically useful. The theory presented by the authors has its origins in work on algebraic specifications that started in the early 1970s, and their treatment is comprehensive. This book contains five kinds of material: the requisite mathematical foundations; traditional algebraic specifications; elements of the theory of institutions; formal specification and development; and proof methods. While the book is self-contained, mathematical maturity and familiarity with the problems of software engineering is required; and in the examples that directly relate to programming, the authors assume acquaintance with the concepts of functional programming. The book will be of value to researchers and advanced graduate students in the areas of programming and theoretical computer science.

Lambda Calculus with Types

Author: Henk Barendregt,Wil Dekkers,Richard Statman
Publisher: Cambridge University Press
ISBN: 1107276349
Category: Mathematics
Page: N.A
View: 3524

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This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

Applied Semantics

International Summer School, APPSEM 2000, Caminha, Portugal, September 9-15, 2000. Advanced Lectures
Author: Gilles Barthe,Peter Dybjer,Luis Pinto,João Saraiva
Publisher: Springer
ISBN: 3540456996
Category: Language Arts & Disciplines
Page: 536
View: 4778

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This book is based on material presented at the international summer school on Applied Semantics that took place in Caminha, Portugal, in September 2000. We aim to present some recent developments in programming language research, both in semantic theory and in implementation, in a series of graduate-level lectures. The school was sponsored by the ESPRIT Working Group 26142 on Applied Semantics(APPSEM),whichoperatedbetweenApril1998andMarch2002.The purpose of this working group was to bring together leading reseachers, both in semantic theory and in implementation, with the speci?c aim of improving the communication between theoreticians and practitioners. TheactivitiesofAPPSEMwerestructuredintonineinterdisciplinarythemes: A: Semantics for object-oriented programming B: Program structuring C: Integration of functional languages and proof assistants D: Veri?cation methods E: Automatic program transformation F: Games, sequentiality, and abstract machines G: Types and type inference in programming H: Semantics-based optimization I: Domain theory and real number computation These themes were identi?ed as promising for pro?table interaction between semantic theory and practice, and were chosen to contribute to the following general topics: – description of existing programming language features; – design of new programming language features; – implementation and analysis of programming languages; – transformation and generation of programs; – veri?cation of programs. The chapters in this volume give examples of recent developments covering a broad range of topics of interest to APPSEM.

Mathematical Knowledge Management

4th International Conference, MKM 2005, Bremen, Germany, July 15-17, 2005 ; Revised Selected Papers
Author: Michael Kohlhase
Publisher: N.A
Category: Computers
Page: 403
View: 5770

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ACL 2007

proceedings of the 45th annual meeting of the Association for Computational Linguistics, June 23-30, 2007, Prague, Czech Republic
Author: Association for Computational Linguistics. Meeting,Association for Computational Linguistics
Publisher: N.A
Category: Computational linguistics
Page: N.A
View: 7371

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Logic and the Modalities in the Twentieth Century

Author: Dov M. Gabbay,John Woods
Publisher: Elsevier
ISBN: 9780080463032
Category: Mathematics
Page: 732
View: 7378

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Logic and the Modalities in the Twentieth Century is an indispensable research tool for anyone interested in the development of logic, including researchers, graduate and senior undergraduate students in logic, history of logic, mathematics, history of mathematics, computer science and artificial intelligence, linguistics, cognitive science, argumentation theory, philosophy, and the history of ideas. This volume is number seven in the eleven volume Handbook of the History of Logic. It concentrates on the development of modal logic in the 20th century, one of the most important undertakings in logic’s long history. Written by the leading researchers and scholars in the field, the volume explores the logics of necessity and possibility, knowledge and belief, obligation and permission, time, tense and change, relevance, and more. Both this volume and the Handbook as a whole are definitive reference tools for students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, artificial intelligence, for whom the historical background of his or her work is a salient consideration. · Detailed and comprehensive chapters covering the entire range of modal logic. · Contains the latest scholarly discoveries and interpretative insights that answer many questions in the field of logic.

Die Grundlagen der Mathematik

Author: David Hilbert
Publisher: Springer-Verlag
ISBN: 3663161021
Category: Mathematics
Page: 29
View: 7742

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.


Author: Luitzen Egbertus Jan Brouwer
Publisher: N.A
ISBN: 9783411153718
Category: Intuitionistic mathematics
Page: 161
View: 2605

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