`x`_{n}

and `zero`, where `x`_{n}

is the last element of the list. This gives us `successor zero`, or `one`. That's the value we've accumuluted "so far." Then we go apply the function `\x sofar. successor sofar` to the two arguments `x`_{n-1}

and the value `one` that we've accumulates "so far." This gives us `two`. We continue until we get to the start of the list. The value we've then built up "so far" will be the length of the list.
+ What's happening here? We start with the value zero, then we apply the function `\x sofar. successor sofar` to the two arguments `x`_{n}

and `zero`, where `x`_{n}

is the last element of the list. This gives us `successor zero`, or `one`. That's the value we've accumuluted "so far." Then we go apply the function `\x sofar. successor sofar` to the two arguments `x`_{n-1}

and the value `one` that we've accumulated "so far." This gives us `two`. We continue until we get to the start of the list. The value we've then built up "so far" will be the length of the list.
We can use similar techniques to define many recursive operations on lists and numbers. The reason we can do this is that our "version 3," fold-based implementation of lists, and Church's implementations of numbers, have a internal structure that *mirrors* the common recursive operations we'd use lists and numbers for.
@@ -116,7 +126,10 @@ With sufficient ingenuity, a great many functions can be defined in the same way
##However...##
-Some computable functions are just not definable in this way. The simplest function that *simply cannot* be defined using the resources we've so far developed is the Ackermann function:
+Some computable functions are just not definable in this way. We can't, for example, define a function that tells us, for whatever function `f` we supply it, what is the smallest integer `x` where `f x` is `true`.
+
+Neither do the resources we've so far developed suffice to define the
+[[!wikipedia Ackermann function]]:
A(m,n) =
| when m == 0 -> n + 1
@@ -124,9 +137,9 @@ Some computable functions are just not definable in this way. The simplest funct
| else -> A(m-1, A(m,n-1))
A(0,y) = y+1
- A(1,y) = y+2
- A(2,y) = 2y + 3
- A(3,y) = 2^(y+3) -3
+ A(1,y) = 2+(y+3) - 3
+ A(2,y) = 2(y+3) - 3
+ A(3,y) = 2^(y+3) - 3
A(4,y) = 2^(2^(2^...2)) [where there are y+3 2s] - 3
...
@@ -136,11 +149,90 @@ But functions like the Ackermann function require us to develop a more general t
##How to do recursion with lower-case omega##
-...
+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 `ω`

combinator (lower-case omega, not the non-terminating `Ω`

). So the self-application of `H` can be written:
+
+`ω (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))`

+
+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.
+
+##Neat! Can I make it easier to use?##
+
+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 that 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##
-In general, a **fixed point** of a function f is a value *x* such that f`Ψ`

. That is, some function that returns, for any expression `f` we give it as argument, a fixed point for `f`. In other words:
+
+`Ψ f <~~> f (Ψ f)`

+
+Then applying `Ψ`

to the "starting formula" displayed above would give us our fixed point `X` for the starting formula:
+
+`Ψ (\self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) ))`

+
+And this is the fully general strategy for
+defining recursive functions in the lambda calculus. You begin with a "body formula":
+
+ ...self...
+
+containing free occurrences of `self` that you treat as being equivalent to the body formula itself. In the case we're considering, that was:
+
+ \lst. (isempty lst) zero (add one (self (extract-tail lst)))
+
+You bind the free occurrence of `self` as: `\self. BODY`. And then you generate a fixed point for this larger expression:
+
+`Ψ (\self. BODY)`

+
+using some fixed-point combinator `Ψ`

.
+
+Isn't that cool?
+
+##Okay, then give me a fixed-point combinator, already!##
+
+Many fixed-point combinators have been discovered. (And some fixed-point combinators give us models for building infinitely many more, non-equivalent fixed-point combinators.)
+
+Two of the simplest:
+
+```
Θ′ ≡ (\u f. f (\n. u u f n)) (\u f. f (\n. u u f n))
+Y′ ≡ \f. (\u. f (\n. u u n)) (\u. f (\n. u u n))
```

+
+`Θ′`

has the advantage that `f (Θ′ f)`

really *reduces to* `Θ′ f`

. Whereas `f (Y′ f)`

is only *convertible with* `Y′ f`

; that is, there's a common formula they both reduce to. For most purposes, though, either will do.
+
+You may notice that both of these formulas have eta-redexes inside them: why can't we simplify the two `\n. u u f n` inside `Θ′`

to just `u u f`? And similarly for `Y′`

?
+
+Indeed you can, getting the simpler:
+
+```
Θ ≡ (\u f. f (u u f)) (\u f. f (u u f))
+Y ≡ \f. (\u. f (u u)) (\u. f (u u))
```

+
+I stated the more complex formulas for the following reason: in a language whose evaluation order is *call-by-value*, the evaluation of `Θ (\self. BODY)`

and `Y (\self. BODY)` will in general not terminate. But evaluation of the eta-unreduced primed versions will.
+
+Of course, if you define your `\self. BODY` stupidly, your formula will never terminate. For example, it doesn't matter what fixed point combinator you use for `Ψ`

in:
+
+`Ψ (\self. \n. self n)`

+
+When you try to evaluate the application of that to some argument `M`, it's going to try to give you back:
+
+ (\n. self n) M
+
+where `self` is equivalent to the very formula `\n. self n` that contains it. So the evaluation will proceed:
+
+ (\n. self n) M ~~>
+ self M ~~>
+ (\n. self n) M ~~>
+ self M ~~>
+ ...
+
+You've written an infinite loop!
+
+However, when we evaluate the application of our:
+
+`Ψ (\self (\lst. (isempty lst) zero (add one (self (extract-tail lst))) ))`

+
+to some list `L`, we're not going to go into an infinite evaluation loop of that sort. At each cycle, we're going to be evaluating the application of:
+
+ \lst. (isempty lst) zero (add one (self (extract-tail lst)))
+
+to *the tail* of the list we were evaluating its application to at the previous stage. Assuming our lists are finite (and the implementations we're using don't permit otherwise), at some point one will get a list whose tail is empty, and then the evaluation of that formula to that tail will return `zero`. So the recursion eventually bottoms out in a base value.
+
+##Fixed-point Combinators Are a Bit Intoxicating##
+
+![tatoo](/y-combinator.jpg)
+
+There's a tendency for people to say "Y-combinator" to refer to fixed-point combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Y-combinator is only one of many fixed-point combinators.
+
+I used `Ψ`

above to stand in for an arbitrary fixed-point combinator. I don't know of any broad conventions for this. But this seems a useful one.
+
+As we said, there are many other fixed-point combinators as well. For example, Jan Willem Klop pointed out that if we define `L` to be:
-[Explain in terms of an arbitrary fixed point combinator Ψ.]
+ \a b c d e f g h i j k l m n o p q s t u v w x y z r. (r (t h i s i s a f i x e d p o i n t c o m b i n a t o r))
-[Give some examples: first, versions of Y and Θ usable with call-by-value. Then do the internal eta-reductions and say these work for call-by-name only.]
+then this is a fixed-point combinator:
-[Explain how what we've done relates to the version using lower-case ω.]
+ L L L L L L L L L L L L L L L L L L L L L L L L L L