haskell - How do I use fix, and how does it work?

Question

I was a bit confused by the documentation for fix (although I think I understand what it's supposed to do now), so I looked at the source code. That left me more confused:

fix :: (a -> a) -> a
fix f = let x = f x in x

How exactly does this return a fixed point?

I decided to try it out at the command line:

Prelude Data.Function> fix id
...

And it hangs there. Now to be fair, this is on my old macbook which is kind of slow. However, this function can't be too computationally expensive since anything passed in to id gives that same thing back (not to mention that it's eating up no CPU time). What am I doing wrong?

Answer 1

You are doing nothing wrong. fix id is an infinite loop.

When we say that fix returns the least fixed point of a function, we mean that in the domain theory sense. So fix (x -> 2*x-1) is not going to return 1, because although 1 is a fixed point of that function, it is not the least one in the domain ordering.

I can't describe the domain ordering in a mere paragraph or two, so I will refer you to the domain theory link above. It is an excellent tutorial, easy to read, and quite enlightening. I highly recommend it.

For the view from 10,000 feet, fix is a higher-order function which encodes the idea of recursion. If you have the expression:

let x = 1:x in x

Which results in the infinite list [1,1..], you could say the same thing using fix:

fix (x -> 1:x)

(Or simply fix (1:)), which says find me a fixed point of the (1:) function, IOW a value x such that x = 1:x... just like we defined above. As you can see from the definition, fix is nothing more than this idea -- recursion encapsulated into a function.

It is a truly general concept of recursion as well -- you can write any recursive function this way, including functions that use polymorphic recursion. So for example the typical fibonacci function:

fib n = if n < 2 then n else fib (n-1) + fib (n-2)

Can be written using fix this way:

fib = fix (f -> 
 -> if n < 2 then n else f (n-1) + f (n-2))

Exercise: expand the definition of fix to show that these two definitions of fib are equivalent.

But for a full understanding, read about domain theory. It's really cool stuff.