ω (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))
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: \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)) then this is a fixed-point combinator: 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