Lectures on the Curry-Howard Isomorphism


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

Lectures on the Curry-Howard Isomorphism


Author: Morten Heine Sørensen,Paweł Urzyczyn
Publisher: Elsevier Science Limited
ISBN: 0444520775
Category: Computers
Page: 442
View: 3924

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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. P 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 reasoningP 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.-

Derivation and Computation

Taking the Curry-Howard Correspondence Seriously
Author: H. Simmons
Publisher: Cambridge University Press
ISBN: 9780521771733
Category: Mathematics
Page: 384
View: 3219

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Introduction to simple type theory; 200 exercises with complete solutions.

The Blind Spot

Lectures on Logic
Author: Jean-Yves Girard
Publisher: European Mathematical Society
ISBN: 9783037190883
Category: Mathematics
Page: 537
View: 9855

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A Short Introduction to Intuitionistic Logic


Author: Grigori Mints
Publisher: Springer Science & Business Media
ISBN: 0306469758
Category: Mathematics
Page: 131
View: 8388

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Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999.

Proofs and Types


Author: Jean-Yves Girard,Yves Lafont,Paul Taylor
Publisher: Cambridge University Press
ISBN: 9780521371810
Category: Computers
Page: 192
View: 4733

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This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will be essential reading for all those working in logic & computer science.

The Functional Interpretation of Logical Deduction


Author: Ruy J. G. B. de Queiroz,Anjolina G. de Oliveira,Dov M. Gabbay
Publisher: World Scientific
ISBN: 9814360953
Category: Computers
Page: 266
View: 3921

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This comprehensive book provides an adequate framework to establish various calculi of logical inference. Being an ‘enriched’ system of natural deduction, it helps to formulate logical calculi in an operational manner. By uncovering a certain harmony between a functional calculus on the labels and a logical calculus on the formulas, it allows mathematical foundations for systems of logic presentation designed to handle meta-level features at the object-level via a labelling mechanism, such as the D Gabbay's Labelled Deductive Systems. The book truly demonstrates that introducing ‘labels’ is useful to understand the proof-calculus itself, and also to clarify its connections with model-theoretic interpretations.

Computation and Reasoning

A Type Theory for Computer Science
Author: Zhaohui Luo
Publisher: Oxford University Press
ISBN: 0198538359
Category: Mathematics
Page: 228
View: 7487

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This book develops a new type theory and shows how it can be applied to computer science, in particular to the effective development of programs and proofs.

Introduction to Combinators and (lambda) Calculus


Author: J. R. Hindley,J. P. Seldin
Publisher: CUP Archive
ISBN: 9780521268967
Category: Mathematics
Page: 360
View: 7273

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Combinatory logic and lambda-conversion were originally devised in the 1920s for investigating the foundations of mathematics using the basic concept of 'operation' instead of 'set'. They have now developed into linguistic tools, useful in several branches of logic and computer science, especially in the study of programming languages. These notes form a simple introduction to the two topics, suitable for a reader who has no previous knowledge of combinatory logic, but has taken an undergraduate course in predicate calculus and recursive functions. The key ideas and basic results are presented, as well as a number of more specialised topics, and man), exercises are included to provide manipulative practice.

Types and Programming Languages


Author: Benjamin C. Pierce,Benjamin C. (Professor Pierce, University of Pennsylvania)
Publisher: MIT Press
ISBN: 9780262162098
Category: Computers
Page: 623
View: 9586

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Mathematical Preliminaries - Untyped Systems - Untyped Arithmetic Expressions - An ML Implementation of Arithmetic Expressions - The Untyped Lambda-Calculus - Nameless Representation of Terms - An ML Implementation of the Lambda-Calculus - Types Arithmetic Expressions - Simply Typed Lambda-Calculus - An ML Implementation of Simple Types - Simple Extensions - Normalization - Exceptions - Subtyping - Metatheory of Subtyping - An ML Implementation of Subtyping - Recursive Types - Metatheory of Recursive Types - Polymorphism - Type Reconstruction - Universal Types - Existential Types - An ML Implementation of System F - Bounded Quantification - Higher-Order Systems - Higher-Order Polymorphism - Higher-Order Subtyping.

Reasoned Programming


Author: Krysia Broda,Susan Eisenbach
Publisher: Prentice Hall Direct
ISBN: 9780130988317
Category: Computers
Page: 296
View: 2045

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This text is for use by advanced undergraduate/graduate students of computer science.

Constructive Adpositional Grammars

Foundations of Constructive Linguistics
Author: Federico Gobbo,Marco Benini
Publisher: Cambridge Scholars Publishing
ISBN: 144383128X
Category: Language Arts & Disciplines
Page: 280
View: 1970

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This book presents a new paradigm of natural language grammar analysis, based on adposition as the key concept, considered a general connection between two morphemes – or group of morphemes. The adpositional paradigm considers the morpheme as the basic unit to represent morphosyntax, taken as a whole, in terms of constructions, while semantics and pragmatics are treated accordingly. All linguistic observations within the book can be described through the methods and tools of Constructive Mathematics, so that the modelling becomes formally feasible. A full description in category-theoretic terms of the formal model is provided in the Appendix. A lot of examples taken from natural languages belonging to different typological areas are offered throughout the volume, in order to explain and validate the modeling – with special attention given to ergativity. Finally, a first real-world application of the paradigm is given, i.e., conversational analysis of the transcript of therapeutic settings in terms of constructive speech acts. The main goal of this book is to broaden the scope of Linguistics by including Constructive Mathematics in order to deal with known topics such as grammaticalization, children’s speech, language comparison, dependency and valency from a different perspective. It primarily concerns advanced students and researchers in the field of Theoretical and Mathematical Linguistics but the audience can also include scholars interested in applications of Topos Theory in Linguistics.

Real World OCaml

Functional programming for the masses
Author: Yaron Minsky,Anil Madhavapeddy,Jason Hickey
Publisher: "O'Reilly Media, Inc."
ISBN: 1449324754
Category: Computers
Page: 510
View: 8352

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This fast-moving tutorial introduces you to OCaml, an industrial-strength programming language designed for expressiveness, safety, and speed. Through the book’s many examples, you’ll quickly learn how OCaml stands out as a tool for writing fast, succinct, and readable systems code. Real World OCaml takes you through the concepts of the language at a brisk pace, and then helps you explore the tools and techniques that make OCaml an effective and practical tool. In the book’s third section, you’ll delve deep into the details of the compiler toolchain and OCaml’s simple and efficient runtime system. Learn the foundations of the language, such as higher-order functions, algebraic data types, and modules Explore advanced features such as functors, first-class modules, and objects Leverage Core, a comprehensive general-purpose standard library for OCaml Design effective and reusable libraries, making the most of OCaml’s approach to abstraction and modularity Tackle practical programming problems from command-line parsing to asynchronous network programming Examine profiling and interactive debugging techniques with tools such as GNU gdb

Type Theory and Formal Proof

An Introduction
Author: Rob Nederpelt,Herman Geuvers
Publisher: Cambridge University Press
ISBN: 1316061086
Category: Computers
Page: N.A
View: 397

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Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.

Lambda Calculus with Types


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

Categorical Logic and Type Theory


Author: Bart Jacobs
Publisher: Gulf Professional Publishing
ISBN: 9780444508539
Category: Mathematics
Page: 760
View: 7751

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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.

Interactive Theorem Proving and Program Development

Coq’Art: The Calculus of Inductive Constructions
Author: Yves Bertot,Pierre Castéran
Publisher: Springer Science & Business Media
ISBN: 366207964X
Category: Mathematics
Page: 472
View: 8025

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A practical introduction to the development of proofs and certified programs using Coq. An invaluable tool for researchers, students, and engineers interested in formal methods and the development of zero-fault software.

Basic Proof Theory


Author: A. S. Troelstra,H. Schwichtenberg
Publisher: Cambridge University Press
ISBN: 9780521779111
Category: Computers
Page: 417
View: 3750

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Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.

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: 3918

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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.