Control.Monad.Fix

Monads having fixed points with a 'knot-tying' semantics. Instances of MonadFix should satisfy the following laws:

Purity

mfix (return . h) = return (fix h)

Left shrinking (or Tightening)

mfix (\x -> a >>= \y -> f x y) = a >>= \y -> mfix (\x -> f x y)

Sliding

mfix (liftM h . f) = liftM h (mfix (f . h))

, for strict

h

.

Nesting

mfix (\x -> mfix (\y -> f x y)) = mfix (\x -> f x x)

This class is used in the translation of the recursive do notation supported by GHC and Hugs.

fix f is the least fixed point of the function f, i.e. the least defined x such that f x = x.

When f is strict, this means that because, by the definition of strictness, f ⊥ = ⊥ and such the least defined fixed point of any strict function is ⊥.

Examples

We can write the factorial function using direct recursion as

>>> let fac n = if n <= 1 then 1 else n * fac (n-1) in fac 5120

This uses the fact that Haskell’s let introduces recursive bindings. We can rewrite this definition using fix,

Instead of making a recursive call, we introduce a dummy parameter rec; when used within fix, this parameter then refers to fix’s argument, hence the recursion is reintroduced.

>>> fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5120

Using fix, we can implement versions of repeat as fix . (:) and cycle as fix . (++)

>>> take 10 $ fix (0:)[0,0,0,0,0,0,0,0,0,0]

>>> map (fix (\rec n -> if n < 2 then n else rec (n - 1) + rec (n - 2))) [1..10][1,1,2,3,5,8,13,21,34,55]

Implementation Details

The current implementation of fix uses structural sharing

fix f = let x = f x in x

A more straightforward but non-sharing version would look like

fix f = f (fix f)