Can someone explain this snippet: (#[#] &)[#[#][#] &]

Question

A friend showed me this snippet of code today:

(#[#] &)[#[#][#] &]

I can sort of see that we're constructing an anonymous function that takes a function as an argument and calls that function passing it itself. Then we're calling that anonymous function, passing another, and I get lost...

A cogent explanation of this code would be great.

Apologies if this is well known, I tried to google for it, but its nature makes searching for it hard.

Answer 1

Going out on a limb here, but the exhibited expression looks like a brave but flawed attempt to implement the Y-combinator extremely concisely.

The Y-combinator is a technical trick used to implement recursion in the lambda calculus. Here is an implementation that stoops to using some symbols:

Y[f_] := #[#]&[Function[n, f[#[#]][n]]&]

... and here is an example of its use to calculate factorials recursively:

fac[r_] := If[# < 2, 1, # * r[# - 1]]&

Y[fac][10]

3628800

Of course, in Mathematica there is no need to engage in such gymnastics since explicit recursion is supported directly. But it is a nice brain-teaser: can Y be expressed using no symbols? (Ideally using nothing other than special input form #, the postfix operator &, the matchfix operator [...] and parentheses -- just like the original expression.)

The Obscurity Continues

Since we are exploring obscure corners of Mathematica function syntax, here is another version of the Y-combinator that uses the rarely seen \[Function] syntax, ESCfnESC:

Y = f ↦ (g ↦ g[g])[h ↦ n ↦ f[h[h]][n]]

(* but copy this instead to get the correct Mathematica character:
ClearAll[Y]
Y = f \[Function] (g \[Function] g[g])[h \[Function] n \[Function] f[h[h]][n]]
*)

Answer 2

(#[#] &)[#[#][#] &]

You have a function that applies its argument to itself

(#[#] &)[8] --> 8[8]

We have, on the other hand, a function that applies to itself to make the head it applies to itself

(#[#][#]&)[8] --> (8[8])[8]

8[8] is the head of 8[8][8]

If we apply the first function to the second one, we get the second function applied to the second function. We then evaluate that and get the second function applied to the second function as a head, with argument the second function... Then you expand the head again, and you got a terrible infinite recursion. As David said in the comment, nonsense snippet of code.

Answer 3

Here's what it looks like if we limit the recursion level to something smaller:

Block[{$RecursionLimit = 20}, 
 Function[f, f[f]][Function[g, g[g][g]]]]

(* 
   $RecursionLimit::reclim :  "Recursion depth of 20 exceeded.

   $RecursionLimit::reclim :  "Recursion depth of 20 exceeded.
*)

(*
==> Hold[Function[g, g[g][g]][Function[g, g[g][g]]]][
                  Hold[Function[g, g[g][g]]]][Function[g, g[g][g]]][
                Function[g, g[g][g]]][Function[g, g[g][g]]][
              Function[g, g[g][g]]][Function[g, g[g][g]]][
            Function[g, g[g][g]]][Function[g, g[g][g]]][
          Function[g, g[g][g]]][Function[g, g[g][g]]][
        Function[g, g[g][g]]][Function[g, g[g][g]]][
      Function[g, g[g][g]]][Function[g, g[g][g]]][
    Function[g, g[g][g]]][Function[g, g[g][g]]][Function[g, g[g][g]]][
 Function[g, g[g][g]]]
*)