(Available by ftp from )/pub/csreports/papers/smith/hlcs.ps.gz Smith, “Martin-Lof’s Type Theory.” A chapter in Handbook of Logic in Computer Sciencewritten in 1994, to appear. Smith Programming in Martin-Lof’s Type Theory An InroductionMonographs on Computer Science 7, Oxford University Press, 1990.ī. Martin-Lof Intuitionistic Type TheoryBibliopolis, Napoli, 1984.ī. Scott Introduction to higher order categorical logicCambridge Studies in Advanced Mathematics 7, Cambridge University Press, 1986. Taylor Proofs and TypesCambridge Tracts in Theoretical Computer Science 7, Cambridge University Press, 1989. Gallier, “Constructive Logics, Part I: A tutorial on proof systems and typed A-calculi” Theoretical Computer Science 110249–339, 1991. 31 in APIC studies in Data Processing, pp. Gallier, “On Girard’s ”Candidats de Reductibilité“, in Logic and Computer Scienceed. Barendregt The Lambda CalculusNorth Holland Publishing Co., Amsterdam. A common type included in type theories is the Natural numbers, often notated as " N. A term and its type are often written together as " term : type". In type theory, every term is associated with a type. This article is not an exhaustive categorization of every type theory, but an introduction covering major approaches. ( July 2023) ( Learn how and when to remove this template message) See Wikipedia's guide to writing better articles for suggestions. This section's tone or style may not reflect the encyclopedic tone used on Wikipedia. Type theories are an area of active research, as demonstrated by homotopy type theory. Another is Thierry Coquand's calculus of constructions, which is used as the foundation by Coq, Lean, and other "proof assistants" (computerized proof writing programs). One influential system is Per Martin-Löf's intuitionistic type theory, which was proposed as a foundation for constructive mathematics. The phrase "type theory" now generally refers to a typed system based around lambda calculus. Church demonstrated that it could serve as a foundation of mathematics and it was referred to as a higher-order logic. Church's theory of types helped the formal system avoid the Kleene–Rosser paradox that afflicted the original untyped lambda calculus. The most famous early example is Church's simply typed lambda calculus. Types did gain a hold when used with one particular logic, Alonzo Church's lambda calculus. There were other techniques to avoid Russell's paradox. Russell's theory of types, therefore, ruled out the possibility of a set being a member of itself. Entities of a given type are built exclusively of subtypes of that type, thus preventing an entity from being defined using itself. This system avoided such set theoretical antinomies suggested in Russell's paradox by creating a hierarchy of types and then assigning each concrete mathematical entity to a specific type. By 1908 Russell arrived at a "ramified" theory of types together with an " axiom of reducibility", both of which featured prominently in Whitehead and Russell's Principia Mathematica published in 1910, 1912, and 1913. Between 19, Bertrand Russell proposed various "theories of type" to fix the problem. Russell's paradox, which was pinpointed by Bertrand Russell in Gottlob Frege's renown work The Foundations of Arithmetic, existed because a set could be defined using "all possible sets", which included itself. Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic.
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