week3 tweaks

Signed-off-by: Jim Pryor <profjim@jimpryor.net>

diff --git a/week3.mdwn b/week3.mdwn
index 40bcb99..613981d 100644 (file)
--- a/week3.mdwn
+++ b/week3.mdwn
@@ -146,11 +146,88 @@ But functions like the Ackermann function require us to develop a more general t
 
 ##How to do recursion with lower-case omega##
 
 
 ##How to do recursion with lower-case omega##
 

-[TODO]
+Recall our initial, abortive attempt above to define the `get_length` function in the lambda calculus. We said "What we really want to do is something like this:
+
+       \lst. (isempty lst) zero (add one (... (extract-tail lst)))
+
+where this very same formula occupies the `...` position."
+
+We are not going to exactly that, at least not yet. But we are going to do something close to it.
+
+Consider a formula of the following form (don't worry yet about exactly how we'll fill the `...`s):
+
+       \h \lst. (isempty lst) zero (add one (... (extract-tail lst)))
+
+Call that formula `H`. Now what would happen if we applied `H` to itself? Then we'd get back:
+
+       \lst. (isempty lst) zero (add one (... (extract-tail lst)))
+
+where any occurrences of `h` inside the `...` were substituted with `H`. Call this `F`. `F` looks pretty close to what we're after: a function that takes a list and returns zero if it's empty, and so on. And `F` is the result of applying `H` to itself. But now inside `F`, the occurrences of `h` are substituted with the very formula `H` we started with.So if we want to get `F` again, all we have to do is apply `h` to itself---since as we said, the self-application of `H` is how we created `F` in the first place.
+
+So, the way `F` should be completed is:
+
+       \lst. (isempty lst) zero (add one ((h h) (extract-tail lst)))
+
+and our original `H` is:
+
+       \h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst)))
+
+The self-application of `H` will give us `F` with `H` substituted in for its free variable `h`.
+
+Instead of writing out a long formula twice, we could write:
+
+       (\x. x x) LONG-FORMULA
+
+and the initial `(\x. x x)` is just what we earlier called the <code>&omega;</code> combinator (lower-case omega, not the non-termination <code>&Omega;</code>). So the self-application of `H` can be written:
+
+<pre><code>&omega; (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))</code></pre>
+
+and this will indeed implement the recursive function we couldn't earlier figure out how to define.
+
+In broad brush-strokes, `H` is half of the `get_length` function we're seeking, and H has the form:
+
+       \h other-arguments. ... (h h) ...
+
+We get the whole `get_length` function by applying `H` to itself. Then `h` is replaced by the half `H`, and when we later apply `h` to itself, we re-create the whole `get_length` again.
+
+Now suppose you wanted to wrap this up in a pretty interface, so that the programmer didn't need to write `(h h)` but could just write `g` for some function `g`. How could you do it?
+
+Now the `F`-like expression we'd be aiming for---call it `F*`---would look like this:
+
+       \lst. (isempty lst) zero (add one (g (extract-tail lst)))
+
+or, abbreviating:
+
+       \lst. ...g...
+
+Here we have just a single `g` instead of `(h h)`. We'd want `F*` to be the result of self-applying some `H*`, and then binding to `g` that very self-application of `H*`. We'd get this if `H*` had the form:
+
+       \h. (\g lst. ...g...) (h h)
+
+The self-application of `H*` would be:
+
+       (\h. (\g lst. ...g...) (h h)) (\h. (\g lst. ...g...) (h h))
+
+or:
+
+       (\f. (\h. f (h h)) (\h. f (h h))) (\g lst. ...g...)
+
+The left-hand side of this is known as **the Y-combinator** and so this could be written more compactly as:
+
+       Y (\g lst. ...g...)
+
+or, replacing the abbreviated bits:
+
+       Y (\g lst. (isempty lst) zero (add one (g (extract-tail lst))))
+
+So this is another way to implement the recursive function we couldn't earlier figure out how to define.
+

 
 ##Generalizing##
 
 
 ##Generalizing##
 

-In general, a **fixed point** of a function f is a value *x* such that f<em>x</em> is equivalent to *x*. For example, what is a fixed point of the function from natural numbers to their squares? What is a fixed point of the successor function?
+Let's step back and fill in some theory to help us understand why these tricks work.
+
+In general, we call a **fixed point** of a function f any value *x* such that f <em>x</em> is equivalent to *x*. For example, what is a fixed point of the function from natural numbers to their squares? What is a fixed point of the successor function?

 
 In the lambda calculus, we say a fixed point of an expression `f` is any formula `X` such that:
 
 
 In the lambda calculus, we say a fixed point of an expression `f` is any formula `X` such that:
 


@@ -174,13 +251,13 @@ Moreover, the recipes that enable us to name fixed points for any given formula
 
 OK, so how do we make use of this?
 
 
 OK, so how do we make use of this?
 

-Recall our initial, abortive attempt above to define the `get_length` function in the lambda calculus. We said "What we really want to do is something like this:
+Recall again our initial, abortive attempt above to define the `get_length` function in the lambda calculus. We said "What we really want to do is something like this:

 
        \lst. (isempty lst) zero (add one (... (extract-tail lst)))
 
 where this very same formula occupies the `...` position."
 
 
        \lst. (isempty lst) zero (add one (... (extract-tail lst)))
 
 where this very same formula occupies the `...` position."
 

-Now, what if we *were* somehow able to get ahold of this formula, as an additional argument? We could take that argument and plug it into the `...` position. Something like this:
+If we could somehow get ahold of this formula, as an additional argument, then we could take the argument and plug it into the `...` position. Something like this:

 
        \self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) )
 
 
        \self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) )