Haskell Language Recursion Schemes Fixed points


Fix takes a "template" type and ties the recursive knot, layering the template like a lasagne.

newtype Fix f = Fix { unFix :: f (Fix f) }

Inside a Fix f we find a layer of the template f. To fill in f's parameter, Fix f plugs in itself. So when you look inside the template f you find a recursive occurrence of Fix f.

Here is how a typical recursive datatype can be translated into our framework of templates and fixed points. We remove recursive occurrences of the type and mark their positions using the r parameter.

{-# LANGUAGE DeriveFunctor #-}

-- natural numbers
-- data Nat = Zero | Suc Nat
data NatF r = Zero_ | Suc_ r deriving Functor
type Nat = Fix NatF

zero :: Nat
zero = Fix Zero_
suc :: Nat -> Nat
suc n = Fix (Suc_ n)

-- lists: note the additional type parameter a
-- data List a = Nil | Cons a (List a)
data ListF a r = Nil_ | Cons_ a r deriving Functor
type List a = Fix (ListF a)

nil :: List a
nil = Fix Nil_
cons :: a -> List a -> List a
cons x xs = Fix (Cons_ x xs)

-- binary trees: note two recursive occurrences
-- data Tree a = Leaf | Node (Tree a) a (Tree a)
data TreeF a r = Leaf_ | Node_ r a r deriving Functor
type Tree a = Fix (TreeF a)

leaf :: Tree a
leaf = Fix Leaf_
node :: Tree a -> a -> Tree a -> Tree a
node l x r = Fix (Node_ l x r)