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Extensional Concepts In Intensional Type Theory
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Book Synopsis Extensional Concepts in Intensional Type Theory by : Martin Hofmann
Download or read book Extensional Concepts in Intensional Type Theory written by Martin Hofmann and published by . This book was released on 1995 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Theories of dependent types have been proposed as a foundation of constructive mathematics and as a framework in which to construct certified programs. In these applications an important role is played by identity types which internalise equality and therefore are essential for accommodating proofs and programs in the same formal system. This thesis attempts to reconcile the two different ways that type theories deal with identity types. In extensional type theory the propositional equality induced by the identity types is identified with definitional equality, i.e. conversion. This renders type-checking and well-formedness of propositions undecidable and leads to non-termination in the presence of universes. In intensional type theory propositional equality is coarser than definitional equality, the latter being confined to definitional expansion and normalisation. Then type-checking and well-formedness are decidable, and this variant is therefore adopted by most implementations. However, the identity type in intensional type theory is not powerful enough for formalisation of mathematics and program development. Notably, it does not identify pointwise equal functions (functional extensionality) and provides no means of redefining equality on a type as a given relation, i.e. quotient types. We call such capabilities extensional concepts. Other extensional concepts of interest are uniqueness of proofs and more specifically of equality proofs, subset types, and propositional extensionality--the identification of equivalent propositions. In this work we investigate to what extent these extensional concepts may be added to intensional type theory without sacrificing decidability and existence of canonical forms. The method we use is the translation of identity types into equivalence relations defined by induction on the type structure. In this way type theory with extensional concepts can be understood as a high-level language for working with equivalence relations instead of equality. Such translations of type theory into itself turn out to be best described using categorical models of type theory. We thus begin with a thorough treatment of categorical models with particular emphasis on the interpretation of type-theoretic syntax in such models. We then show how pairs of types and predicates can be organised into a model of type theory in which subset types are available and in which any two proofs of a proposition are equal. This model has applications in the areas of program extraction from proofs and modules for functional programs. For us its main purpose is to clarify the idea of syntactic translations via categorical model constructions. The main result of the thesis consists of the construction of two models in which functional extensionality and quotient types are available. In the first one types are modelled by types together with proposition-valued partial equivalence relations. This model is rather simple and in addition provides subset types and propositional extensionality. However, it does not furnish proper dependent types such as vectors or matrices. We try to overcome this disadvantage by using another model based on families of type-valued equivalence relations which is however much more complicated and validates certain conversion rules only up to propositional equality. We illustrate the use of these models by several small examples taken from both formalised mathematics and program development. We also establish various syntactic properties of propositional equality including a proof of the undecidability of typing in extensional type theory and a correspondence between derivations in extensional type theory and terms in intensional type theory with extensional concepts added. Furthermore we settle affirmatively the hitherto open question of the independence of unicity of equality proofs in intensional type theory which implies that the addition of pattern matching to intensional type theory does not yield a conservative extension.
Book Synopsis Extensional Constructs in Intensional Type Theory by : Martin Hofmann
Download or read book Extensional Constructs in Intensional Type Theory written by Martin Hofmann and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: Extensional Constructs in Intensional Type Theory presents a novel approach to the treatment of equality in Martin-Loef type theory (a basis for important work in mechanised mathematics and program verification). Martin Hofmann attempts to reconcile the two different ways that type theories deal with identity types. The book will be of interest particularly to researchers with mainly theoretical interests and implementors of type theory based proof assistants, and also fourth year undergraduates who will find it useful as part of an advanced course on type theory.
Book Synopsis Treatise on Intuitionistic Type Theory by : Johan Georg Granström
Download or read book Treatise on Intuitionistic Type Theory written by Johan Georg Granström and published by Springer Science & Business Media. This book was released on 2011-06-02 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: Intuitionistic type theory can be described, somewhat boldly, as a partial fulfillment of the dream of a universal language for science. This book expounds several aspects of intuitionistic type theory, such as the notion of set, reference vs. computation, assumption, and substitution. Moreover, the book includes philosophically relevant sections on the principle of compositionality, lingua characteristica, epistemology, propositional logic, intuitionism, and the law of excluded middle. Ample historical references are given throughout the book.
Book Synopsis Categorical Logic and Type Theory by : B. Jacobs
Download or read book Categorical Logic and Type Theory written by B. Jacobs and published by Gulf Professional Publishing. This book was released on 2001-05-10 with total page 784 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Intuitionistic Type Theory by : Per Martin-Löf
Download or read book Intuitionistic Type Theory written by Per Martin-Löf and published by . This book was released on 1984 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Types for Proofs and Programs by : Stefano Berardi
Download or read book Types for Proofs and Programs written by Stefano Berardi and published by Springer Science & Business Media. This book was released on 1996-10-02 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains a refereed selection of revised full papers chosen from the contributions presented during the Third Annual Workshop held under the auspices of the ESPRIT Basic Research Action 6453 Types for Proofs and Programs. The workshop took place in Torino, Italy, in June 1995. Type theory is a formalism in which theorems and proofs, specifications and programs can be represented in a uniform way. The 19 papers included in the book deal with foundations of type theory, logical frameworks, and implementations and applications; all in all they constitute a state-of-the-art survey for the area of type theory.
Author :Hendrik Pieter Barendregt Publisher :Springer Science & Business Media ISBN 13 :9783540580850 Total Pages :404 pages Book Rating :4.5/5 (88 download)
Book Synopsis Types for Proofs and Programs by : Hendrik Pieter Barendregt
Download or read book Types for Proofs and Programs written by Hendrik Pieter Barendregt and published by Springer Science & Business Media. This book was released on 1994-05-20 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains thoroughly refereed and revised full papers selected from the presentations at the first workshop held under the auspices of the ESPRIT Basic Research Action 6453 Types for Proofs and Programs in Nijmegen, The Netherlands, in May 1993. As the whole ESPRIT BRA 6453, this volume is devoted to the theoretical foundations, design and applications of systems for theory development. Such systems help in designing mathematical axiomatisation, performing computer-aided logical reasoning, and managing databases of mathematical facts; they are also known as proof assistants or proof checkers.
Book Synopsis Types for Proofs and Programs by : Paul Callaghan
Download or read book Types for Proofs and Programs written by Paul Callaghan and published by Springer. This book was released on 2003-08-03 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed post-proceedings of the International Workshop of the TYPES Working Group, TYPES 2000, held in Durham, UK in December 2000. The 15 revised full papers presented were carefully reviewed and selected during two rounds of refereeing and revision. All current issues on type theory and type systems and their applications to programming, systems design, and proof theory are addressed.
Book Synopsis Homotopy Type Theory: Univalent Foundations of Mathematics by :
Download or read book Homotopy Type Theory: Univalent Foundations of Mathematics written by and published by Univalent Foundations. This book was released on with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Types for Proofs and Programs by : Jean-Christophe Filliatre
Download or read book Types for Proofs and Programs written by Jean-Christophe Filliatre and published by Springer. This book was released on 2006-01-20 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 17 revised full papers presented here cover all current issues of formal reasoning and computer programming based on type theory are addressed; in particular languages and computerised tools for reasoning, and applications in several domains such as analysis of programming languages, certified software, formalisation of mathematics and mathematics education.
Book Synopsis Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics by : Claudia Casadio
Download or read book Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics written by Claudia Casadio and published by Springer Nature. This book was released on 2021-04-21 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is dedicated to the life and work of the mathematician Joachim Lambek (1922–2014). The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.
Book Synopsis Proof and System-Reliability by : Helmut Schwichtenberg
Download or read book Proof and System-Reliability written by Helmut Schwichtenberg and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: As society comes to rely increasingly on software for its welfare and prosperity there is an urgent need to create systems in which it can trust. Experience has shown that confidence can only come from a more profound understanding of the issues, which in turn can come only if it is based on logically sound foundations. This volume contains contributions from leading researchers in the critical disciplines of computing and information science, mathematics, logic, and complexity. All contributions are self-contained, aiming at comprehensibility as well as comprehensiveness. The volume also contains introductory hints to technical issues, concise surveys, introductions, and various fresh results and new perspectives.
Book Synopsis Types for Proofs and Programs by : Herman Geuvers
Download or read book Types for Proofs and Programs written by Herman Geuvers and published by Springer Science & Business Media. This book was released on 2003-04-28 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed post-proceedings of the Second International Workshop of the TYPES Working Group, TYPES 2002, held in Berg en Dal, The Netherlands in April 2002. The 18 revised full papers presented were carefully selected during two rounds of reviewing and improvement. All current issues in type theory and type systems and their applications to programming, systems design, and proof theory are addressed. Among the systems dealt with are Coq and Isar/HOL.
Book Synopsis Computation and Logic in the Real World by : S. Barry Cooper
Download or read book Computation and Logic in the Real World written by S. Barry Cooper and published by Springer Science & Business Media. This book was released on 2007-06-11 with total page 842 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Third International Conference on Computability in Europe, CiE 2007, held in Sienna, Italy, in June 2007. The 50 revised full papers presented together with 36 invited papers were carefully reviewed and selected from 167 submissions.
Book Synopsis Higher-Order Logic and Type Theory by : John L. Bell
Download or read book Higher-Order Logic and Type Theory written by John L. Bell and published by Cambridge University Press. This book was released on 2022-03-31 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory (also known as higher-order intuitionistic logic), an important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined.
Book Synopsis Semantics and Logics of Computation by : Andrew M. Pitts
Download or read book Semantics and Logics of Computation written by Andrew M. Pitts and published by Cambridge University Press. This book was released on 1997-01-30 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this volume is to present modern developments in semantics and logics of computation in a way that is accessible to graduate students. The book is based on a summer school at the Isaac Newton Institute and consists of a sequence of linked lecture course by international authorities in the area. The whole set have been edited to form a coherent introduction to these topics, most of which have not been presented pedagogically before.
Book Synopsis Types for Proofs and Programs by : Thorsten Altenkirch
Download or read book Types for Proofs and Programs written by Thorsten Altenkirch and published by Springer. This book was released on 2003-06-29 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the strictly refereed post-workshop proceedings of the International Workshop on Types for Proofs and Programs, TYPES '98, held under the auspices of the ESPRIT Working Group 21900. The 14 revised full papers presented went through a thorough process of reviewing and revision and were selected from a total of 25 candidate papers. All current aspects of type theory and type systems and their relation to proof theory are addressed.
