Documentation

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.

For example, 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 5
120

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

>>> fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5
120

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.