# 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)

# Folding up a structure one layer at a time

Catamorphisms, or folds, model primitive recursion. cata tears down a fixpoint layer by layer, using an algebra function (or folding function) to process each layer. cata requires a Functor instance for the template type f.

cata :: Functor f => (f a -> a) -> Fix f -> a
cata f = f . fmap (cata f) . unFix

-- list example
foldr :: (a -> b -> b) -> b -> List a -> b
foldr f z = cata alg
    where alg Nil_ = z
          alg (Cons_ x acc) = f x acc

# Unfolding a structure one layer at a time

Anamorphisms, or unfolds, model primitive corecursion. ana builds up a fixpoint layer by layer, using a coalgebra function (or unfolding function) to produce each new layer. ana requires a Functor instance for the template type f.

ana :: Functor f => (a -> f a) -> a -> Fix f
ana f = Fix . fmap (ana f) . f

-- list example
unfoldr :: (b -> Maybe (a, b)) -> b -> List a
unfoldr f = ana coalg
    where coalg x = case f x of
                         Nothing -> Nil_
                         Just (x, y) -> Cons_ x y

Note that ana and cata are dual. The types and implementations are mirror images of one another.

# Unfolding and then folding, fused

It's common to structure a program as building up a data structure and then collapsing it to a single value. This is called a hylomorphism or refold. It's possible to elide the intermediate structure altogether for improved efficiency.

hylo :: Functor f => (a -> f a) -> (f b -> b) -> a -> b
hylo f g = g . fmap (hylo f g) . f  -- no mention of Fix!

Derivation:

hylo f g = cata g . ana f
         = g . fmap (cata g) . unFix . Fix . fmap (ana f) . f  -- definition of cata and ana
         = g . fmap (cata g) . fmap (ana f) . f  -- unfix . Fix = id
         = g . fmap (cata g . ana f) . f  -- Functor law
         = g . fmap (hylo f g) . f  -- definition of hylo

# Primitive recursion

Paramorphisms model primitive recursion. At each iteration of the fold, the folding function receives the subtree for further processing.

para :: Functor f => (f (Fix f, a) -> a) -> Fix f -> a
para f = f . fmap (\x -> (x, para f x)) . unFix

The Prelude's tails can be modelled as a paramorphism.

tails :: List a -> List (List a)
tails = para alg
    where alg Nil_ = cons nil nil  -- [[]]
          alg (Cons_ x (xs, xss)) = cons (cons x xs) xss  -- (x:xs):xss

# Primitive corecursion

Apomorphisms model primitive corecursion. At each iteration of the unfold, the unfolding function may return either a new seed or a whole subtree.

apo :: Functor f => (a -> f (Either (Fix f) a)) -> a -> Fix f
apo f = Fix . fmap (either id (apo f)) . f

Note that apo and para are dual. The arrows in the type are flipped; the tuple in para is dual to the Either in apo, and the implementations are mirror images of each other.

# Remarks

Functions mentioned here in examples are defined with varying degrees of abstraction in several packages, for example, data-fix and recursion-schemes (more functions here). You can view a more complete list by searching on Hayoo.