X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;h=a80cc58e340154930fc4114cb61e1eb62d3028ee;hb=5cab83962241676d710c788561ac107a3563a3e8;hp=1e3edbcb3d24cadcc5ad0d9f212f3fed26fb86d6;hpb=69cefc9d2ec23b63ee6ae14d6b0c2b0ddabfd68e;p=lambda.git
diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 1e3edbcb..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 | Ï1 -> Ï2 | â'a. Ï
- expressions e ::= x | λx:Ï. e | e1 e2 | Î'a. e | e [Ï]
+ types Ï ::= c | α | Ï1 -> Ï2 | â'a. Ï
+ expressions e ::= x | λx:Ï. e | e1 e2 | Îα. 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`. "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.). "`Ï1 -> Ï2`" is the type of a function from expressions of
-type `Ï1` to expressions of type `Ï2`. And "`â'a. Ï`" is called a
+type `Ï1` to expressions of type `Ï2`. And "`âα. 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 `âα. Ï`, the type `Ï` 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: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: "`Îα. 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
@@ -70,9 +70,9 @@ 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)
+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
----------------