From: chris Date: Thu, 26 Feb 2015 02:04:38 +0000 (-0500) Subject: (no commit message) X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=5cab83962241676d710c788561ac107a3563a3e8 --- diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 4afb43ba..a80cc58e 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -39,22 +39,22 @@ match up with usage in O'Caml, whose type system is based on System F): System F: --------- - types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ - expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ] + types τ ::= c | α | τ1 -> τ2 | ∀'a. τ + expressions e ::= x | λx:τ. e | e1 e2 | Λα. e | e [τ] In the definition of the types, "`c`" is a type constant. Type constants play the role in System F that base types play in the simply-typed lambda calculus. So in a lingusitics context, type -constants might include `e` and `t`. "`'a`" is a type variable. The +constants might include `e` and `t`. "α" is a type variable. The tick mark just indicates that the variable ranges over types rather than over values; in various discussion below and later, type variable can be distinguished by using letters from the greek alphabet (α, β, etc.), or by using capital roman letters (X, Y, etc.). "`τ1 -> τ2`" is the type of a function from expressions of -type `τ1` to expressions of type `τ2`. And "`∀'a. τ`" is called a +type `τ1` to expressions of type `τ2`. And "`∀α. τ`" is called a universal type, since it universally quantifies over the type variable -`'a`. You can expect that in `∀'a. τ`, the type `τ` will usually -have at least one free occurrence of `'a` somewhere inside of it. +`'a`. You can expect that in `∀α. τ`, the type `τ` will usually +have at least one free occurrence of `α` somewhere inside of it. In the definition of the expressions, we have variables "`x`" as usual. Abstracts "`λx:τ. e`" are similar to abstracts in the simply-typed lambda @@ -62,7 +62,7 @@ calculus, except that they have their shrug variable annotated with a type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus. In addition to variables, abstracts, and applications, we have two -additional ways of forming expressions: "`Λ'a. e`" is called a *type +additional ways of forming expressions: "`Λα. e`" is called a *type abstraction*, and "`e [τ]`" is called a *type application*. The idea is that Λ is a capital λ: just like the lower-case λ, Λ binds @@ -72,7 +72,7 @@ variables. So in the expression Λ Î± (λ x:α . x) -the Λ binds the type variable `'a` that occurs in +the Λ binds the type variable `α` that occurs in the λ abstract. Of course, as long as type variables are carefully distinguished from expression variables (by tick marks, Grecification, or capitalization), there is no need to @@ -85,27 +85,27 @@ be adapted for use with expressions of any type. In order to get it ready to apply this identity function to, say, a variable of type boolean, just do this: -(Λ 'a (λ x:'a . x)) [t] +(Λ Î± (λ x:α . x)) [t] This type application (where `t` is a type constant for Boolean truth -values) specifies the value of the type variable `'a`. Not +values) specifies the value of the type variable `α`. Not surprisingly, the type of this type application is a function from Booleans to Booleans: -((Λ 'a (λ x:'a . x)) [t]): (b -> b) +((Λ Î± (λ x:α . x)) [t]): (b -> b) Likewise, if we had instantiated the type variable as an entity (base type `e`), the resulting identity function would have been a function of type `e -> e`: -((Λ 'a (λ x:'a . x)) [e]): (e -> e) +((Λ Î± (λ x:α . x)) [e]): (e -> e) -Clearly, for any choice of a type `'a`, the identity function can be -instantiated as a function from expresions of type `'a` to expressions -of type `'a`. In general, then, the type of the uninstantiated +Clearly, for any choice of a type `α`, the identity function can be +instantiated as a function from expresions of type `α` to expressions +of type `α`. In general, then, the type of the uninstantiated (polymorphic) identity function is -(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a) +(Λ Î± (λ x:α . x)): (∀ α . α -> α) Pred in System F ----------------