What is (Y Y), the Y-combinator applied to itself?

Question

In Chapter 9 of The Little Schemer, the authors introduce the Y-combinator and the penultimate question asks: "What is (Y Y)". They answer: "Who knows, but it works very hard."

I tried reasoning about it and had a hard time. I assume this just causes an infinite loop, but maybe I am missing something. Are higher-order fixed-point combinators even interesting?

The book defines the Y-combinator as:

(define Y
  (lambda(le)
    ((lambda(f) (f f))
     (lambda(f)
       (le (lambda(x) ((f f) x)))))))

However, using a let-binding is clearer, for me:

(define Y (lambda(exp)
            (let ([a (lambda(f)
                       (exp (lambda(x) ( (f f) x))))])
        
      (a a))))

and simulating the length function works fine.

((Y (lambda(s)
     (lambda(lat)
       (cond
         ((null? lat) 0)
         (else (add1 (s (cdr lat)))))))) '(1 2 4))

=> 3

---

When I try to evaluate (Y Y) in the Dr Racket IDE, the program crashes and runs out of memory. How could I step through and reason about (Y Y) where I define Y using the let-binding above?

Answer 1

Maybe writing it in λ-calculus can help: Y = λf.(λx.f(xx))(λx.f(xx)). It has the following β-reduction (aka execution):

Yf → f(Yf) → f(f(Yf)) → ...

Thus YY → Y(YY) → (YY)(Y(YY)) and you can keep on reducing, making the size of the term explode, as well as the number of arguments. Basically, you recursively apply to itself something that expects a function and applies it recursively...

Answer 2

This is much easier to follow when throwing the combinators notation into the mix. Using

U x = x x       ; U = ^x. x x

your

(define Y
  (lambda (le)
    ((lambda(f) (f f))
     (lambda(f)
       (le (lambda(x) ((f f) x)))))))

becomes

Y g = U (^f. g (^x. (f f) x))

and if we define

     H g f = g (^x. (f f) x)    = g (U f) = B g U f     ; H = CBU

we can then write it down as

Y g = U (H g)                                  ; Y = BUH = BU(CBU)
    = H g (H g)                                ; Y = SHH 
    = g (^x. ((H g) (H g)) x) 
    = g (^x. (Y g) x)

Y Y = Y (^x. (Y Y) x)
    = (^x. (Y Y) x) (^x. (Y (^x. (Y Y) x)) x)
    =      (Y Y)    (^x. (Y (^x. (Y Y) x)) x)
    = (^x. (Y Y) x) (^x. (Y (^x. (Y Y) x)) x) (^x. (Y (^x. (Y Y) x)) x)
    = ...

so, writing Q for (^x. (Y (^x. (Y Y) x)) x) we see that it's

Y Y = Y Y Q
    = Y Y Q Q
    = Y Y Q Q Q
    = Y Y Q Q Q Q
    = ...

and the reduction sequence never stops.