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
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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: 950

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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.
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Applied Proof Theory: Proof Interpretations and their Use in Mathematics

Author: Ulrich Kohlenbach

Publisher: Springer Science & Business Media

ISBN: 3540775331

Category: Mathematics

Page: 536

View: 6466

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This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.
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Types for Proofs and Programs

LIPIcs Proceedings 19th International Conference TYPES 2013, April 22–26, 2013, Toulouse, France

Author: Ralph Matthes,Aleksy Schubert

Publisher: BoD – Books on Demand

ISBN: 3735756867

Category: Computers

Page: 280

View: 3982

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In this LIPIcs proceedings one can find research papers on the following topics: analysis of the classical principles in intuitionistic calculi, type isomorphisms for intersection types, monads and their semantics in functional programming languages, realizability, extensions of type theory, extensions of linear logic, models of type theory, control operators in type systems, formal verification of programs, program extraction, compiler formalization and modelling of natural language features. All papers obtained at least two reviews, and up to six reviews, counting a second round of review.
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Advances in Computational Intelligence

17th Mexican International Conference on Artificial Intelligence, MICAI 2018, Guadalajara, Mexico, October 22–27, 2018, Proceedings

Author: Ildar Batyrshin,María de Lourdes Martínez-Villaseñor,Hiram Eredín Ponce Espinosa

Publisher: Springer

ISBN: 3030044971

Category: Computers

Page: 371

View: 8297

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The two-volume set LNAI 11288 and 11289 constitutes the proceedings of the 17th Mexican International Conference on Artificial Intelligence, MICAI 2018, held in Guadalajara, Mexico, in October 2018. The total of 62 papers presented in these two volumes was carefully reviewed and selected from 149 submissions. The contributions are organized in topical as follows: Part I: evolutionary and nature-inspired intelligence; machine learning; fuzzy logic and uncertainty management. Part II: knowledge representation, reasoning, and optimization; natural language processing; and robotics and computer vision.
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Foundation of Software Science and Computation Structures

Third International Conference, FOSSACS 2000 Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2000 Berlin, Germany, March 25 - April 2, 2000 Proceedings

Author: Jerzy Tiuryn

Publisher: Springer

ISBN: N.A

Category: Computer software

Page: 391

View: 7668

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This volume constitutes the refereed proceedings of the Third International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2000, held as part of ETAPS 2000 in Berlin, Germany in March/April 2000. The 25 revised full papers presented were carefully reviewed and selected from a total of 68 submissions. Among the topics covered are rewriting, pi-calculus, lambda-calculus, formal languages, proof systems, type theory, mobile processes, automata theory, predicate logic, etc.
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