Y Combinator in Emacs Lisp

I talked today with Prabhakar Ragde, who is a resident at the Recurse Center until Oct15, and I was enlightened by lambda calculus as we figured out how to deduce the Y Combinator that is used a lot to make recursive functions.

Here I plan to post the implementation and derivation of the Y Combinator in Emacs Lisp.

:DISCLAIMER_BEGIN:

You will need to have lexical binding enabled for most of the examples to work, it is a buffer-local variable so make sure you paste everything on the same buffer.

( setq lexical-binding t )

:DISCLAIMER_END:

We'll take factorial as it's good example of a recursive function

( defun factorial ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall #'factorial ( 1- number ) ) ) ) ) ( factorial 5 ) ( factorial 4 )

The recursion looks trivial right? Now lets try to implement it without referencing the name of the function, should be easy as well.

( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall fact? ( 1- number ) ) ) ) )

We find that we cannot reference back to the same lambda function as we don't have a name for it, that's why we use fact? we'll have to figure out what to put in there. Lets say that we have another function that looks like factorial .

( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall fun ( 1- number ) ) ) ) ) )

This is great! now we have a function that can take another function and that returns a function that looks like factorial . There is one little detail though, the function that proto-factorial accepts only takes a number, so we could say that.

( let ( ( proto-factorial ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall fun ( 1- number ) ) ) ) ) ) ) ) ( eq ( funcall 'factorial 4 ) ( funcall ( funcall proto-factorial 'factorial ) 4 ) ) )

This holds true but it's not what we want. We show that applying proto-factorial to factorial gives us back factorial which is cool, but we need to not use factorial at all, let's see if using proto-factorial on itself might work

( let ( ( proto-factorial ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall fun ( 1- number ) ) ) ) ) ) ) ) ( funcall ( funcall proto-factorial proto-factorial ) 4 ) )

It didn't T_T. What this tells us is that we have not provided enough arguments, and that's true, because proto-factorial, the function we send, takes a function and and returns a function that takes number, and the outer funcall just takes another number but no function. this is tricky, but we can have a way of doing it, let's call the function on itself as an argument before passing another number!

( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall ( funcall fun fun ) ( 1- number ) ) ) ) ) )

This looks more promising! meta-factorial takes a function and returns a function that depends on a number, but in this inner number-dependent function instead of applying fun to (1- number) we apply the function to itself, and then apply that result to (1- number) , this way we got what we wanted, a recursive function!. But still looks ugly, lets arrange it in a better way

( let ( ( meta-factorial ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall ( funcall fun fun ) ( 1- number ) ) ) ) ) ) ) ) ( eq ( funcall 'factorial 4 ) ( funcall ( funcall meta-factorial meta-factorial ) 4 ) ) )

You might be thinking: "Wait what? did we just did a recursion by calling two lambdas?" yes we did, it might not look like so here, but if you replace each meta-fact with the lambda definition, we eliminate references to functions and are only left with lambda functions but the recursion still works! and we haven't used a reference to a function from within itself. If you don't believe me, then this is the ugly version:

( funcall ( funcall #' ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall ( funcall fun fun ) ( 1- number ) ) ) ) ) ) #' ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall ( funcall fun fun ) ( 1- number ) ) ) ) ) ) ) 4 )

This still returns what we wanted which is 4! . So what we have now is that meta-factorial evaluated on itself returns a function that takes a number and that function is equivalent to factorial .

We want a way to transform from proto-factorial function to factorial because in that way we can get recursion, out of a simpler function the only things that we did was to take out the function call and call it on itself.

Let's see if we can define one such a function that takes as a parameter the proto-factorial and will output the meta-factorial version because in that case we would only need to apply that function twice and we would get our so desired factorial.

( lambda ( fun ) ( funcall #' ( lambda ( fun2 ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall fun2 ( 1- number ) ) ) ) ) ) ( funcall fun fun ) ) )

What we just did was to wrap the inner number-dependent function so that the (funcall fun fun) could be abstracted. This is called composing functions, kinda like g(x^2) could be written as g(f(x)) where f(x) = x^2 and we find that this inner thing is actually, proto-factorial !! This is great!, now we can write meta factorial in terms of proto factorial . We just needed to wrap the fun2 into a call to itself.

( let ( ( meta-factorial ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall ( funcall fun fun ) ( 1- number ) ) ) ) ) ) ) ( proto-factorial ( lambda ( fun ) ( lambda ( number ) ( if ( eq number 0 ) 1 ( * number ( funcall fun ( 1- number ) ) ) ) ) ) ) ) ( eq ( funcall 'factorial 4 ) ( funcall ( funcall #' ( lambda ( r ) ( funcall proto-factorial ( funcall r r ) ) ) #' ( lambda ( r ) ( funcall proto-factorial ( funcall r r ) ) ) ) 4 ) ) )

This is basically what we've been going for so much, if we could only make one more abstraction so that in case of proto-factorial we could pass any function wouldn't it be nice? well it is possible and it has a name! it's called the Y combinator, and it looks like this:

( defun YCombinator ( fun ) ( funcall #' ( lambda ( r ) ( funcall fun ( funcall r r ) ) ) #' ( lambda ( r ) ( funcall fun ( funcall r r ) ) ) ) )

Lets try it with our factorial function!

( funcall ( YCombinator #' ( lambda ( f ) #' ( lambda ( n ) ( if ( eq n 0 ) 1 ( * n ( funcall f ( 1- n ) ) ) ) ) ) ) 5 )

This should work, but it just goes into an infinite recursive loop, :(.

Fortunately enough, I had been warned about this type of behavior due to the non-laziness from elisp, its eagerness it's not letting the code keep on going without evaluating stuff that shouldn't be evaluated yet, this doesn't happen in other languages such as haskell, but in emacs lisp the YCombinator should look a little bit different, like this:

( defun YCombinator ( f ) ( funcall #' ( lambda ( x ) ( funcall f #' ( lambda ( y ) ( funcall ( funcall x x ) y ) ) ) ) #' ( lambda ( x ) ( funcall f #' ( lambda ( y ) ( funcall ( funcall x x ) y ) ) ) ) ) )

There! now we have a working YCombinator that doesn't go into an infinite loop :) lets try it

( funcall ( YCombinator #' ( lambda ( f ) #' ( lambda ( n ) ( if ( eq n 0 ) 1 ( * n ( funcall f ( 1- n ) ) ) ) ) ) ) 8 )