At some point, everyone wakes up in the middle of the night, in a cold sweat of panic that they don't truly understand how to derive the Y-combinator. Well maybe not everyone, but at least me. (Note that I'm talking about the higher order function, not the startup incubator.) I ended up reading through quite a few web pages, all of which presupposed a slightly different background, before I finally understood. This post distills my understanding, expressed in clojure, which happens to be what I'm into now. It can now be one of the pages that someone else finds not quite adequate for understanding this concept.

Having read the synopsis, we know that the point here is that higher order functions, being functions, can have fixed points - i.e. f(g)=g, and that if we were able to find that fixed point we would be able to implement recursion in a language that doesn't have it. But it's best to forget that for the moment and just convince yourself that the following steps follow from each other.

Here is a standard definition of the factorial function:

(defn fact [n] (if (= 0 n) 1 (* n (fact (- n 1)))))
(assert (= (fact 5) 120))

Following the usual path, we now do something that seems pointless. Rather than explicitly call the function recursively, we pass in the function to call.

(defn fact2 [fact n] (if (= 0 n) 1 (* n (fact fact (- n 1)))))
(assert (= (fact2 fact2 5) 120))

This fact2 thing is no longer explicitly recursive, but it is of course not particularly useful as it presupposes its own existence. We're going to try to make it more useful.

First, we're going to curry it, so we only have to deal with functions of one argument.

(defn fact3 [fact]
  (fn [n] (if (= 0 n) 1 (* n ((fact fact) (- n 1)) ))))
(assert (= ((fact3 fact3) 5) 120))

What we're edging towards is something where the middle bit looks as much like a normal factorial function as possible, so I'm going to pull the (fact fact) bit out, passing it in as an argument to an inner function:

(defn fact4 [fact] (fn [n] 
                     (let [f (fn [g n] (if (= 0 n) 1 (* n (g (- n 1))))) ]
                       (f (fact fact) n)
(assert (= ((fact4 fact4) 5) 120))

Now f no longer has a (fact fact) in it, and we'll make it even prettier by currying it:

(defn fact5 [fact] (fn [n] 
                     (let [f (fn [g] (fn [n] (if (= 0 n) 1 (* n (g (- n 1))))))]
                       ((f (fact fact)) n)
;(assert (= ((fact5 fact5) 5) 120))

The exciting news is that (fn [g] (fn [n] (if (= 0 n) 1 (* n (g (- n 1)))))) in the middle is self contained, normal function. It's not a closure, and it looks a lot like the original factorial. In fact, it's almost exactly like fact2. Let's pull it out and give it an evocative name

(def fact-maker (fn [g]
           (fn [n] (if (= 0 n) 1 (* n (g (- n 1)))))))

suggesting that it might be used to make factorial functions, in concert with another function to which we pass it as an argument:

(defn fact6 [f]
  (fn [fact] (fn [n] ((f (fact fact)) n))))
(assert (= (((fact6 fact-maker) (fact6 fact-maker)) 5) 120))

Things have started to get cool. We've broken a complicated expression that doesn't know how to do anything but make factorial functions into two simpler functions, of which fact-maker defines the mathematics of a factorial, and fact6 has nothing to do with factorials and could potentially be used to make anything-maker into a recursive function.

Now let's make spruce things up a bit, so the person invoking this doesn't have to type fact6 twice.

(defn fact7 [f]
  (let [g (fn [fact] (fn [n] ((f (fact fact)) n)))] (g g)))
(assert (= ((fact7 fact-maker) 5) 120))

And we'll make it look like more vanilla lisp, by getting rid of the let:

(defn fact8 [f]
  ((fn [g] (g g))
   (fn [fact] (fn [n] ((f (fact fact)) n)))))
(assert (= ((fact8 fact-maker) 5) 120))

Sensing victory, we now make the variable names short and pretty.

(defn Y [f]
  ((fn [g] (g g))
   (fn [h] (fn [n] ((f (h h)) n)))))
(assert (= ((Y fact-maker) 5) 120))

And we're done. Before getting philosophical, let's verify that our lovely combinator can be used to make other recursive functions, irrespective of the type of argument or return value. Here's an example using a function that returns a list.

(defn range-maker [f] (fn [n] (if (= 0 n) () (conj (f (- n 1)) n))))
(assert (= ((Y range-maker) 5) (list 5 4 3 2 1)))

Remember that the point of Y is that it finds fixed points. In infix notion, it's like we were able to vary a function g until

    (f(g))(x) = g(x)   for all x

with Y(f) being a shortcut to g. If that doesn't sound impressive enough, imagine you had a machine that, given something you want but don't know how to make, will make an exact copy, and then it somehow figures out how to make it without being told

To verify and emphasize that what we've got here is a fixed point, we can explicitly pass the output of the combinator back into the maker function.

(assert (= ((fact-maker (Y fact-maker)) 5) 120))

This might be magic.


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