javascript中的y-combinator

Question

I have built a y-combinator in js like this

const y = f => { const g = self => x => f(self(self))(x); return g(g);}

and I simplified this code like this

 const y = f => { const g = self => f(self(self)); return g(g);}

this get an infinite recursion. What's the difference between these two versions?

Answer 1

If you don't understand the difference between the two, I would be surprised that you actually built them. Nevertheless, perhaps the best way to demonstrate the difference between the two, is to follow their evaluation

const y = f => {
  const g = self => x => f(self(self))(x)
  return g(g)
}

y (z) ...
// (self => x => z(self(self))(x)) (self => x => z(self(self))(x)) ...
// returns:
// x => z((self => x1 => z(self(self))(x1))(self => x2 => z(self(self))(x2)))(x)

Ok, so y(z) (where z is some function, it doesn't matter) returns a function x => ... . Until we apply that function, evaluation stops there.

Now let's compare that to your second definition

const y = f => {
  const g = self => f(self(self))
  return g(g)
}

y (z) ...
// (self => z(self(self))) (self => z(self(self)))
// z((self => z(self(self)))(self => z(self(self)))) ...
// z(z((self => z(self(self)))(self => z(self(self))))) ...
// z(z(z((self => z(self(self)))(self => z(self(self)))))) ...
// z(z(z(z((self => z(self(self)))(self => z(self(self))))))) ...
// ... and on and on

So y (z) never terminates – at least in JavaScript which uses applicative order evaluation – where function arguments are evaluated before the called function is applied

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Alternate Y-combinators

Here we can build a Y-combinator from the ground up

// standard definition
const Y = f => f (Y (f))

// prevent immediate infinite recursion in applicative order language (JS)
const Y = f => f ( Y (f) )

// remove reference to self using U combinator
const U = f => f (f)
const Y =  f => f (x =>  (f) (x))

Let's test it out

 const U = f => f (f) const Y = U (h => f => f (x => h (h) (f) (x))) // range :: Int -> Int -> [Int] const range = Y (f => acc => x => y => x > y ? acc : f ([...acc,x]) (x + 1) (y)) ([]) // fibonacci :: Int -> Int const fibonacci = Y (f => a => b => x => x === 0 ? a : f (b) (a + b) (x - 1)) (0) (1) console.log(range(0)(10).map(fibonacci)) // [ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ] 

Or my recent favourite

 // simplified Y const Y = f => x => f (Y (f)) (x) // range :: Int -> Int -> [Int] const range = Y (f => acc => x => y => x > y ? acc : f ([...acc,x]) (x + 1) (y)) ([]) // fibonacci :: Int -> Int const fibonacci = Y (f => a => b => x => x === 0 ? a : f (b) (a + b) (x - 1)) (0) (1) console.log(range(0)(10).map(fibonacci)) // [ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ]