From: Chris Date: Mon, 23 Feb 2015 21:40:54 +0000 (-0500) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=cd52b6efa26b90210609bc266ade1fea3953b6ce;ds=sidebyside edits --- diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn index 2c37ae34..559135e0 100644 --- a/topics/_week5_system_F.mdwn +++ b/topics/_week5_system_F.mdwn @@ -25,49 +25,55 @@ continuations.) System F enhances the simply-typed lambda calculus with abstraction over types. In order to state System F, we'll need to adopt the -notational convention that "`x:α`" represents a -expression whose type is `α`. +notational convention that "`x:α`" represents an +expression `x` whose type is `α`. Then System F can be specified as follows (choosing notation that will match up with usage in O'Caml, whose type system is based on System F): - 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] In the definition of the types, "`c`" is a type constant (e.g., `e` or -`t`). "`'a`" is a type variable (the tick mark just indicates that -the variable ranges over types rather than values). "`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 universal type, since it -universally quantifies over the type variable `'a`. - -In the definition of the expressions, we have variables "`x`". +`t`, or in arithmetic contexts, `N` or `Int`). "`'a`" is a type +variable (the tick mark just indicates that the variable ranges over +types rather than over values). "`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 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.) + +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 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 a type -abstraction, and "`e [Ï]`" is a type application. The idea is that -`Λ` is a capital `λ`. Just like -the lower-case `λ`, `Λ` binds -variables in its body; unlike `λ`, -`Λ` binds type variables. So in the expression +additional ways of forming expressions: "`Î'a. 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 +variables in its body, except that unlike `λ`, +`Λ` binds type variables instead of expression +variables. So in the expression `Λ 'a (λ x:'a . x)` the `Λ` binds the type variable `'a` that occurs in the `λ` abstract. This expression is a polymorphic -version of the identity function. It says that this one general -identity function can be adapted for use with expressions of any -type. In order to get it ready to apply to, say, a variable of type -boolean, just do this: +version of the identity function. It defines one general identity +function that can be adapted for use with expressions of any type. In order +to get it ready to apply to, say, a variable of type boolean, just do +this: `(Λ 'a (λ x:'a . x)) [t]` -The type application (where `t` is a type constant for Boolean truth -values) specifies the value of the type variable `α`, which is -the type of the variable bound in the `λ` expression. Not +This type application (where `t` is a type constant for Boolean truth +values) specifies the value of the type variable α, which is +the type of the variable bound in the λ expression. Not surprisingly, the type of this type application is a function from Booleans to Booleans: @@ -84,13 +90,16 @@ instantiated as a function from expresions of type `'a` to expressions of type `'a`. In general, then, the type of the unapplied (polymorphic) identity function is -```(Λ 'a (λ x:'a . x)): (\forall 'a . 'a -> 'a) - - +(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a) +Pred in System F +---------------- -## +We saw that the predecessor function couldn't be expressed in the +simply-typed lambda calculus. It can be expressed in System F, however. +[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT +Press, pp. 350--353, for `tail` for lists in System F.] Types in OCaml ```