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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Simply typed lambda calculus ： ウィキペディア英語版 Simply typed lambda calculus The simply typed lambda calculus ($\backslash lambda^\backslash to$), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor: $\backslash to$ that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties. The term ''simple type'' is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered ''simply typed''. The former are still considered ''simple'' because the Church encodings of such structures can be done using only $\backslash to$ and suitable type variables, while polymorphism and dependency cannot. == Syntax == In this article, we use $\backslash sigma$ and $\backslash tau$ to range over types. Informally, the ''function type'' $\backslash sigma\; \backslash to\; \backslash tau$ refers to the type of functions that, given an input of type $\backslash sigma$, produce an output of type $\backslash tau$. By convention, $\backslash to$ associates to the right: we read $\backslash sigma\backslash to\backslash tau\backslash to\backslash rho$ as $\backslash sigma\backslash to(\backslash tau\backslash to\backslash rho)$. To define the types, we begin by fixing a set of ''base types'', $B$. These are sometimes called ''atomic types'' or ''type constants''. With this fixed, the syntax of types is: :$\backslash tau\; ::=\; \backslash tau\; \backslash to\; \backslash tau\; \backslash mid\; T\; \backslash quad\; \backslash mathrm\; \backslash quad\; T\; \backslash in\; B$. For example, if $B\; =\; \backslash$, then $a$, $b$, $a\; \backslash to\; a$, $a\; \backslash to\; b$, and $a\; \backslash to\; b\; \backslash to\; a$ are all valid types (among others). We also fix a set of ''term constants'' for the base types. For example, we might assume a base type nat, and the term constants could be the natural numbers. In the original presentation, Church used only two base types: $o$ for "the type of propositions" and $\backslash iota$ for "the type of individuals". The type $o$ has no term constants, whereas $\backslash iota$ has one term constant. Frequently the calculus with only one base type, usually $o$, is considered. The syntax of the simply typed lambda calculus is essentially that of the lambda calculus itself. We write $x\backslash mathbin\backslash tau$ to denote that the variable $x$ is of type $\backslash tau$. The term syntax is then: :$e\; ::=\; x\; \backslash mid\; \backslash lambda\; x\backslash mathbin\backslash tau.e\; \backslash mid\; e\; \backslash ,\; e\; \backslash mid\; c$ where $c$ is a term constant. That is, ''variable reference'', ''abstractions'', ''application'', and ''constant''. A variable reference $x$ is ''bound'' if it is inside of an abstraction binding $x$. A term is ''closed'' if there are no unbound variables. Compare this to the syntax of untyped lambda calculus: :$e\; ::=\; x\; \backslash mid\; \backslash lambda\; x.e\; \backslash mid\; e\; \backslash ,\; e$ We see that in typed lambda calculus every function (''abstraction'') must specify the type of its argument. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Simply typed lambda calculus」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.