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 Ï ::= c | 'a | Ï1 -> Ï2 | â'a. Ï - expressions e ::= x | Î»x:Ï. e | e1 e2 | Î'a. e | e [Ï] + types Ï ::= c | Î± | Ï1 -> Ï2 | â'a. Ï + expressions e ::= x | Î»x:Ï. e | e1 e2 | ÎÎ±. e | e [Ï] 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`. "Î±" 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.). "`Ï1 -> Ï2`" is the type of a function from expressions of -type `Ï1` to expressions of type `Ï2`. And "`â'a. Ï`" is called a +type `Ï1` to expressions of type `Ï2`. And "`âÎ±. Ï`" is called a universal type, since it universally quantifies over the type variable -`'a`. You can expect that in `â'a. Ï`, the type `Ï` will usually -have at least one free occurrence of `'a` somewhere inside of it. +`'a`. You can expect that in `âÎ±. Ï`, the type `Ï` will usually +have at least one free occurrence of `Î±` somewhere inside of it. In the definition of the expressions, we have variables "`x`" as usual. Abstracts "`Î»x:Ï. 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: "`Î'a. e`" is called a *type +additional ways of forming expressions: "`ÎÎ±. e`" is called a *type abstraction*, and "`e [Ï]`" 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:Î± . x)` -the `Λ` binds the type variable `'a` that occurs in +the `Λ` binds the type variable `Î±` 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:Î± . 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 `Î±`. 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:Î± . 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:Î± . 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 `Î±`, the identity function can be +instantiated as a function from expresions of type `Î±` to expressions +of type `Î±`. 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:Î± . x)): (∀ Î± . Î± -> Î±)` Pred in System F ----------------