relevant evaluator is called "fullpoly"):
N = All X . (X->X)->X->X;
- Pair = All X . (N -> N -> X) -> X;
+ Pair = (N -> N -> N) -> N;
let zero = lambda X . lambda s:X->X . lambda z:X. 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 X . lambda z:N->N->X . z x 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 shift = lambda p:Pair . p [Pair] (lambda a:N . lambda b:N . pair (suc a) a) in
- let pre = lambda n:N . n [Pair] shift (pair zero zero) [N] snd 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
pre (suc (suc (suc zero)));
lower case (for ordinary lambda) or upper case (for type-level
lambda).
-The key to the extra flexibility provided by System F is that we can
-instantiate the `pair` function to return a number, as in the
-definition of `pre`, or we can instantiate it to return an ordered
-pair, as in the definition of the `shift` function. Because we don't
-have to choose a single type for all uses of the pair-building
-function, we aren't forced into a infinite regress of types.
-
+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
+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
+pair-manipulating function, which is the heart of the strategy for
+this version of predecessor.
+
+But of course, the type `Pair` (in this simplified example) is defined
+in terms of Church numbers. If we tried to replace the type for
+Church numbers with a concrete (simple) type, we would have to replace
+each `X` with `(N -> N -> N) -> N`. But then we'd have to replace
+each `N` with `(X -> X) -> X -> X`. And then replace each `X`
+with... ad infinitum. If we had to choose a concrete type built
+entirely from explicit base types, we'd be unable to proceed.
+
[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
-Press, pp. 350--353, for `tail` for lists in System F.]
+Press, chapter 23.]
Typing ω
--------------
-In fact, it is even possible to give a type for ω in System F.
+In fact, unlike in the simply-typed lambda calculus,
+it is even possible to give a type for ω in System F.
<code>ω = lambda x:(All X. X->X) . x [All X . X->X] x</code>
In order to see how this works, we'll apply ω to the identity
function.
-<code>ω [All X . X -> X] id ==</code>
+<code>ω id ==</code>
(lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
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.
+type is (in some sense, only accidentally) 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!
+possible for a function to apply to itself!
Does this mean that we can implement recursion in System F? Not at
all. In fact, despite its differences with the simply-typed lambda
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.)
+ω. Furthermore, 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