XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_simply_typed_lambda.mdwn;h=14e21729d3c573984875fcdd4dca223008622f1a;hp=a7917b28f4e0190cc4e326dc425a636e82c22582;hb=c70ea8653fd391ba094c8a64419f3532af2076ce;hpb=a95160473037089ea4deb34eceff948391b0cee4
diff git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index a7917b28..14e21729 100644
 a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ 220,7 +220,7 @@ ways to suit present purposes:
Let's see how far we can get typing these terms. `zero` is the Church
encoding of zero. Using `N` as the type for Church numbers (i.e.,
N == (σ > σ) > σ > σ
for some
+N ≡ (σ > σ) > σ > σ
for some
σ, `zero` has type `N`. `snd` takes two numbers, and returns
the second, so `snd` has type `N > N > N`. Then the type of `pair`
is `N > N > (type(snd)) > N`, that is, `N > N > (N > N > N) >
@@ 230,24 +230,24 @@ namely, pair ≡ (N > N > N) > N
. So far so good.
The problem is the way in which `pred` puts these parts together. In
particular, `pred` applies its argument, the number `n`, to the
`collect` function. Since `n` is a number, its type is `(σ >
σ) > σ > σ`. This means that the type of
`collect` has to match `σ > σ`. But we concluded above
that the type of `collect` also had to be `pair > pair`. Putting
these constraints together, it appears that `σ` must be the type
of a pair of numbers. But we already decided that the type of a pair
of numbers is `(N > N > N) > N`. Here's the difficulty: `N` is
shorthand for a type involving `σ`. If `σ` turns out to
depend on `N`, and `N` depends in turn on `σ`, then `σ` is a proper
subtype of itself, which is not allowed in the simplytyped lambda
calculus.

The way we got here is that the pred function relies on the rightfold
structure of the Church numbers to recursively walk down the spine of
its argument. In order to do that, the argument number had to take
the operation in question as its first argument. And the operation
required in order to build up the predecessor must be the sort of
operation that manipulates numbers, and the infinite regress is
+`collect` function. Since `n` is a number, its type is (σ
+> σ) > σ > σ
. This means that the type of
+`collect` has to match σ > σ
. But we
+concluded above that the type of `collect` also had to be `pair >
+pair`. Putting these constraints together, it appears that
+σ
must be the type of a pair of numbers. But we
+already decided that the type of a pair of numbers is `(N > N > N)
+> N`. Here's the difficulty: `N` is shorthand for a type involving
+σ
. If σ
turns out to depend on
+`N`, and `N` depends in turn on σ
, then
+σ
is a proper subtype of itself, which is not
+allowed in the simplytyped lambda calculus.
+
+The way we got here is that the `pred` function relies on the builtin
+rightfold structure of the Church numbers to recursively walk down
+the spine of its argument. In order to do that, the argument had to
+apply to the `collect` operation. And since `collect` had to be the
+sort of operation that manipulates numbers, the infinite regress is
established.
Now, of course, this is only one of myriad possible implementations of
@@ 255,3 +255,12 @@ the predecessor function in the lambda calculus. Could one of them
possibly be simplytypeable? It turns out that this can't be done.
See the works cited by Oleg for details.
+Because lists are (in effect) a generalization of the Church numbers,
+computing the tail of a list is likewise beyond the reach of the
+simplytyped lambda calculus.
+
+This result is surprising. It illustrates how recursion is built into
+the structure of the Church numbers (and lists). Most importantly for
+the discussion of the simplytyped lambda calculus, it demonstrates
+that even fairly basic recursive computations are beyond the reach of
+a simplytyped system.