X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_system_F.mdwn;fp=topics%2F_week5_system_F.mdwn;h=f7c38eb1b86d69bbceae417336099e0b9d6e8c1b;hp=a80cc58e340154930fc4114cb61e1eb62d3028ee;hb=6c8edb34886abac6afe327d50ebfefeb19c85d4c;hpb=5cab83962241676d710c788561ac107a3563a3e8
diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index a80cc58e..f7c38eb1 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -117,13 +117,13 @@ however. Here is one way, coded in
System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
relevant evaluator is called "fullpoly"):
- N = All X . (X->X)->X->X;
+ N = â α . (α->α)->α->α;
Pair = (N -> N -> N) -> N;
- let zero = lambda X . lambda s:X->X . lambda z:X. z in
+ let zero = lambda α . lambda s:α->α . lambda z:α. z in
let fst = lambda x:N . lambda y:N . x in
let snd = lambda x:N . lambda y:N . y in
let pair = lambda x:N . lambda y:N . lambda z:N->N->N . z x y in
- let suc = lambda n:N . lambda X . lambda s:X->X . lambda z:X . s (n [X] s z) in
+ let suc = lambda n:N . lambda α . lambda s:α->α . lambda z:α . s (n [α] s z) in
let shift = lambda p:Pair . pair (suc (p fst)) (p fst) in
let pre = lambda n:N . n [Pair] shift (pair zero zero) snd in
@@ -138,7 +138,7 @@ lambda).
The key to the extra expressive power provided by System F is evident
in the typing imposed by the definition of `pre`. The variable `n` is
-typed as a Church number, i.e., as `All X . (X->X)->X->X`. The type
+typed as a Church number, i.e., as `â α . (α->α)->α->α`. The type
application `n [Pair]` instantiates `n` in a way that allows it to
manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`. In
other words, the instantiation turns a Church number into a
@@ -151,8 +151,8 @@ that the ordered pairs we need here are pairs of numbers. If we tried
to replace the type for Church numbers with a concrete (simple) type,
we would have to replace each `X` with the type for Pairs, `(N -> N ->
N) -> N`. But then we'd have to replace each of these `N`'s with the
-type for Church numbers, `(X -> X) -> X -> X`. And then we'd have to
-replace each of these `X`'s with... ad infinitum. If we had to choose
+type for Church numbers, `(α -> α) -> α -> α`. And then we'd have to
+replace each of these `α`'s with... ad infinitum. If we had to choose
a concrete type built entirely from explicit base types, we'd be
unable to proceed.
@@ -165,19 +165,19 @@ Typing ω
In fact, unlike in the simply-typed lambda calculus,
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
+ω = lambda x:(â α. α->α) . x [â α . α->α] x
In order to see how this works, we'll apply ω to the identity
function.
ω id ==
- (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+ (lambda x:(â α . α->α) . x [â α . α->α] x) (lambda α . lambda x:α . x)
-Since the type of the identity function is `(All X . X->X)`, it's the
+Since the type of the identity function is `(â α . α->α)`, 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
+variable `α` to the universal type `â α . α->α`. Instantiating the
identity function in this way results in an identity function whose
type is (in some sense, only accidentally) the same as the original
fully polymorphic identity function.
@@ -229,9 +229,9 @@ uses. Can we capture this using polymorphic types?
With these basic types, we want to say something like this:
and:t->t->t = lambda l:t . lambda r:t . l r false
- and = lambda 'a . lambda 'b .
- lambda l:'a->'b . lambda r:'a->'b .
- lambda x:'a . and:'b (l x) (r x)
+ and = lambda α . lambda β .
+ lambda l:α->β . lambda r:α->β .
+ lambda x:α . and:β (l x) (r x)
The idea is that the basic *and* conjoins expressions of type `t`, and
when *and* conjoins functional types, it builds a function that
@@ -258,7 +258,7 @@ argument of that type. We would like to instantiate the recursive use
of *and* in the definition by using the result type. But fully
instantiating the definition as given requires type application to a
pair of types, not to just a single type. We want to somehow
-guarantee that 'b will always itself be a complex type.
+guarantee that β will always itself be a complex type.
So conjunction and disjunction provide a compelling motivation for
polymorphism in natural language, but we don't yet have the ability to