Zygomorphism is the high-falutin' mathsy name we give to folds built from two semi-mutually recursive functions. I'll give an example.

Imagine a function pm :: [Int] -> Int (for plus-minus) which intersperses + and - alternately through a list of numbers, such that pm [v,w,x,y,z] = v - (w + (x - (y + z))) . You can write it out using primitive recursion:

lengthEven :: [a] -> Bool lengthEven = even . length pm0 [] = 0 pm0 (x:xs) = if lengthEven xs then x - pm0 xs else x + pm0 xs

Clearly pm0 is not compositional - you need to inspect the length of the whole list at each position to determine whether you're adding or subtracting. Paramorphism models primitive recursion of this sort, when the folding function needs to traverse the whole subtree at each iteration of the fold. So we can at least rewrite the code to conform to an established pattern.

paraL :: (a -> [a] -> b -> b) -> b -> [a] -> b paraL f z [] = z paraL f z (x:xs) = f x xs (paraL f z xs) pm1 = paraL (\x xs acc -> if lengthEven xs then x - acc else x + acc) 0

But this is inefficient. lengthEven traverses the whole list at each iteration of the paramorphism resulting in an O(n2) algorithm.

We can make progress by noting that both lengthEven and para can be expressed as a catamorphism with foldr ...

cataL = foldr lengthEven' = cataL (\_ p -> not p) True paraL' f z = snd . cataL (\x (xs, acc) -> (x:xs, f x xs acc)) ([], z)

... which suggests that we may be able to fuse the two operations into a single pass over the list.

pm2 = snd . cataL (\x (isEven, total) -> (not isEven, if isEven then x - total else x + total)) (True, 0)

We had a fold which depended on the result of another fold, and we were able to fuse them into one traversal of the list. Zygomorphism captures exactly this pattern.

zygoL :: (a -> b -> b) -> -- a folding function (a -> b -> c -> c) -> -- a folding function which depends on the result of the other fold b -> c -> -- zeroes for the two folds [a] -> c zygoL f g z e = snd . cataL (\x (p, q) -> (f x p, g x p q)) (z, e)

On each iteration of the fold, f sees its answer from the last iteration as in a catamorphism, but g gets to see both functions' answers. g entangles itself with f .

We'll write pm as a zygomorphism by using the first folding function to count whether the list is even or odd in length and the second one to calculate the total.

pm3 = zygoL (\_ p -> not p) (\x isEven total -> if isEven then x - total else x + total) True 0

This is classic functional programming style. We have a higher order function doing the heavy lifting of consuming the list; all we had to do was plug in the logic to aggregate results. The construction evidently terminates (you need only prove termination for foldr ), and it's more efficient than the original hand-written version to boot.

Aside: @AlexR points out in the comments that zygomorphism has a big sister called mutumorphism, which captures mutual recursion in all its glory. mutu generalises zygo in that both the folding functions are allowed to inspect the other's result from the previous iteration. mutuL :: (a -> b -> c -> b) -> (a -> b -> c -> c) -> b -> c -> [a] -> c mutuL f g z e = snd . cataL (\x (p, q) -> (f x p q, g x p q)) (z, e) You recover zygo from mutu simply by ignoring the extra argument. zygoL f = mutuL (\x p q -> f x p)

Of course, all of these folding patterns generalise from lists to the fixpoint of an arbitrary functor:

newtype Fix f = Fix { unFix :: f (Fix f) } cata :: Functor f => (f a -> a) -> Fix f -> a cata f = f . fmap (cata f) . unFix para :: Functor f => (f (Fix f, a) -> a) -> Fix f -> a para f = snd . cata (\x -> (Fix $ fmap fst x, f x)) zygo :: Functor f => (f b -> b) -> (f (b, a) -> a) -> Fix f -> a zygo f g = snd . cata (\x -> (f $ fmap fst x, g x)) mutu :: Functor f => (f (b, a) -> b) -> (f (b, a) -> a) -> Fix f -> a mutu f g = snd . cata (\x -> (f x, g x))

Compare the definition of zygo with that of zygoL . Also note that zygo Fix = para , and that the latter three folds can be implemented in terms of cata . In foldology everything is related to everything else.

You can recover the list version from the generalised version.