Saturday, May 08, 2010

Scheme: My Y Combinator

I've been reading The Little Schemer, and it posed an interesting question: how can you create a recursive function without having the ability to "define" a name for it? For instance, in Scheme, how can you create a simple, recursive function to calculate the length of a list without having the ability to use "define"?

Here's my approach:
((lambda (my-length l)
(my-length my-length l))
(lambda (my-length l)
((null? l) 0)
(else (add1 (my-length my-length (cdr l))))))
'(1 2 3 4 5))
The outer function is "(lambda (my-length l) ...)". It takes a reference to a function that it calls my-length. It calls that function "(my-length my-length l)". Hence, that function, which I call "my-length", receives the list as well as a reference to itself, "(lambda (my-length l) ...)". Since it receives a reference to itself, it's able to call itself recursively.

It turns out the real answer is called the Y Combinator. It's written as:
Y = λf·(λx·f (x x)) (λx·f (x x))
It was created by Haskell Curry. To use the Y Combinator to solve the above problem, you'd write:
((lambda (le)
((lambda (mk-length)
(mk-length mk-length))
(lambda (mk-length)
(le (lambda (x)
((mk-length mk-length) x))))))
(lambda (length)
(lambda (l)
((null? l) 0)
(else (add1 (length (cdr l)))))))
Notice that my version is shorter, but the Y Combinator version is more elegant because the actual length function isn't burdened with having to recursively pass a reference to itself.

If you didn't understand any of the above, don't sweat it. I read it twice, and I still barely get it ;) However, I think I now understand why Paul Graham used that name for his startup incubator, Y Combinator. Paul Graham is a Lisp guy, and his Y Combinator is a startup meant to recursively launch other startups ;)


slobin said...

Y = lambda f: (lambda g: f (lambda h: g(g)(h))) \
(lambda g: f (lambda h: g(g)(h)))

fact = lambda f: lambda n: 1 if n == 0 else n * f(n-1)

print Y(fact)(100)

Shannon -jj Behrens said...

> slobin said...

You said "Y =" and "fact =". I think I said, "how can you create a recursive function without having the ability to "define" a name for it?" I'm assuming you can't make any definitions.

slobin said...

The problem is not "naming" by itself (after all, naming is just a handy shortcut), but using the name inside it's own definition. If you insist on reading your conditions literally, just add parentheses around each of two expressions, and concatenate them:

>>> (lambda f: (lambda g: f (lambda h: g(g)(h)))(lambda g: f (lambda h: g(g)(h))
))(lambda f: lambda n: 1 if n == 0 else n * f(n-1))(100)

(Sorry for the long line, it is a copy of real python session ;-). The Y combinator is inside the first outermost pair of parentheses, the factorial definition is inside the second, and the numeric argument is inside the third. In my original comment this is just a bit more readable, but the internal mechanic is the same.

slobin said...

P.S. And, by the way, the analog of your first solution is also possible:

(lambda f,n: f(f,n))(lambda f,n: 1 if n < 2 else n*f(f,n-1),100)

And I do 100% agree with you that "combinator version is more elegant" albeit textually longer.