`x:α`

" represents an
-expression `x` whose type is `α`

.
+over types. Normal lambda abstraction abstracts (binds) an expression
+(a term); type abstraction abstracts (binds) a type.
+
+In order to state System F, we'll need to adopt the
+notational convention (which will last throughout the rest of the
+course) 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):
@@ -36,15 +40,19 @@ match up with usage in O'Caml, whose type system is based on System F):
types Ï ::= c | 'a | Ï1 -> Ï2 | 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`, 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). "`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 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 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
+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
+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
@@ -63,17 +71,21 @@ 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 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:
+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
+distinguish expression abstraction from type abstraction by also
+changing the shape of the lambda.
+
+This expression is a polymorphic 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 this
+identity function to, say, a variable of type boolean, just do this:
`(Λ 'a (λ x:'a . x)) [t]`

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
+values) specifies the value of the type variable `'a`. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
@@ -87,7 +99,7 @@ of type `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 unapplied
+of type `'a`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
`(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)`

@@ -96,11 +108,11 @@ 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,
+simply-typed lambda calculus. It *can* be expressed in System F,
however. Here is one way, coded in
[[Benjamin Pierce's type-checker and evaluator for
System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
-part you want is called "fullpoly"):
+relevant evaluator is called "fullpoly"):
N = All X . (X->X)->X->X;
Pair = All X . (N -> N -> X) -> X;
@@ -133,15 +145,40 @@ Press, pp. 350--353, for `tail` for lists in System F.]
Typing ω
--------------
-In fact, it is even possible to give a type for &omeage; in System F.
+In fact, it is even possible to give a type for ω in System F.
+
+`ω = lambda x:(All X. X->X) . x [All X . X->X] x`

+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+`ω [All X . X -> X] id ==`

+
+ (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+
+Since the type of the identity function is `(All X . X->X)`, it's the
+right type to serve as the argument to ω. The definition of
+ω instantiates the identity function by binding the type
+variable `X` to the universal type `All X . X->X`. Instantiating the
+identity function in this way results in an identity function whose
+type is the same as the original fully polymorphic identity function.
+
+So in System F, unlike in the simply-typed lambda calculus, it *is*
+possible for a function (in this case, the identity function) to apply
+to itself!
- omega = lambda x:(All X. X->X) . x [All X . X->X] x in
- omega;
+Does this mean that we can implement recursion in System F? Not at
+all. In fact, despite its differences with the simply-typed lambda
+calculus, one important property that System F shares with the
+simply-typed lambda calculus is that they are both strongly
+normalizing: *every* expression in either system reduces to a normal
+form in a finite number of steps.
-Each time the internal application is performed, the type of the head
-is chosen anew. And each time, we choose the same type as before, the
-type of a function that takes an argument of any type and returns a
-result of the same type...
+Not only does a fixed-point combinator remain out of reach, we can't
+even construct an infinite loop. This means that although we found a
+type for ω, there is no general type for Ω ≡ ω
+ω. (It turns out that no Turing complete system can be strongly
+normalizing, from which it follows that System F is not Turing complete.)
Types in OCaml