From 6c8edb34886abac6afe327d50ebfefeb19c85d4c Mon Sep 17 00:00:00 2001
From: chris
Date: Wed, 25 Feb 2015 21:18:04 -0500
Subject: [PATCH]
---
topics/_week5_system_F.mdwn | 28 ++++++++++++++--------------
1 file changed, 14 insertions(+), 14 deletions(-)
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 Î± . (Î±->Î±)->Î±->Î±;
Pair = (N -> N -> N) -> N;
- let zero = lambda X . lambda s:X->X . lambda z:X. z in
+ let zero = lambda Î± . lambda s:Î±->Î± . lambda z:Î±. 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 Î± . lambda s:Î±->Î± . lambda z:Î± . s (n [Î±] 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 Î± . (Î±->Î±)->Î±->Î±`. 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, `(Î± -> Î±) -> Î± -> Î±`. And then we'd have to
+replace each of these `Î±`'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 Î±. Î±->Î±) . x [â Î± . Î±->Î±] 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 Î± . Î±->Î±) . x [â Î± . Î±->Î±] x) (lambda Î± . lambda x:Î± . 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 Î± . Î±->Î±)`, 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 `Î±` to the universal type `â Î± . Î±->Î±`. 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 Î± . lambda Î² .
+ lambda l:Î±->Î² . lambda r:Î±->Î² .
+ lambda x:Î± . and:Î² (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 Î² 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
--
2.11.0