How to reinvent the Y combinator

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I don’t believe in the doctrine “Don’t reinvent the wheels”. I believe that one can never learn the essence of anything without reinventing it (intentionally or not). Once you reinvent a thing, you can never forget it — because otherwise you can just reinvent it again.

Today I found that I forgot how to derive the definition of the Y combinator. I learned it several years ago from an online article, but now the search term “Y combinator” only brings news about startups (sigh). I attempted it for two hours, but still couldn’t make the leap from the “poor man’s Y” to “Y”. Finally, I opened my good old friend The Little Schemer. Alas. Chapter 9 tells me exactly how to reinvent Y. Thank you Dan Friedman and Matthias Felleisen!

To prevent myself from forgetting how to derive Y again, I made a slide to record my understanding of it. It is slightly different from the ways of the recent version of The Little Schemer, but I think it’s easier to understand. I hope it can help some people (including the future me). So here it is.

Exercise: The Y combinator derived from this tutorial only works for direct recursion, try to derive the Y combinator for mutual recursive functions, for example the following two functions even and odd.

(define even
(lambda (x)
[(zero? x) #t]
[(= 1 x) #f]
[else (odd (sub1 x))])))

(define odd
(lambda (x)
[(zero? x) #f]
[(= 1 x) #t]
[else (even (sub1 x))])))