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 m

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 -> \n -> 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.