A blog about functional programming

by Phil Freeman on 2012/02/02

In the following, I will write a polykinded version of the combinators `fold`

and `unfold`

, along with three examples: folds for regular datatypes (specialized to kind `*`

), folds for nested datatypes (specialized to kind `* -> *`

), and folds for mutually recursive data types (specialized to the product kind `(*, *)`

). The approach should generalise easily enough to things such as types indexed by another kind (e.g. by specializing to kind `Nat -> *`

, using the `DataKinds`

extension), or higher order nested datatypes (e.g. by specializing to kind `(* -> *) -> (* -> *)`

).

The following will compile in the new GHC 7.4.1 release. We require the following GHC extensions:

```
{-# LANGUAGE GADTs #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE ScopedTypeVariables #-}
```

The basic `fold`

and `unfold`

combinators can be written as follows:

```
-- fold phi = phi . fmap (fold phi) . out
-- unfold psi = in . fmap (unfold psi) . psi
```

The idea now is to generalize these combinators by working over different categories. We can capture the basic operations in a category with a typeclass:

```
class Category hom where
ident :: hom a a
compose :: hom a b -> hom b c -> hom a c
```

A category has two operations: an identity morphism for every object, and for every two compatible morphisms, the composition of those morphisms.

In earlier versions of GHC, the type hom would have been specialized to kind `* -> * -> *`

, but with the new `PolyKinds`

extension, hom is polykinded, and the `Category`

typeclass can be instantiated to `k -> k -> *`

for any kind `k`

. This means that in addition to all of the Category instances that we could have written before, we can now write instances of `Category`

for type constructors, type constructor constructors, etc.

Here is the instance for the category Hask of Haskell types. Objects are Haskell types and morphisms are functions between types. The identity is the regular polymorphic identity function `id`

, and composition is given by the (flipped) composition operator `(.)`

```
instance Category (->) where
ident = id
compose = flip (.)
```

Another example is the category of type constructors and natural transformations. A natural transformation is defined as follows:

```
newtype Nat f g = Nat { nu :: (forall a. f a -> g a) }
```

Here is the Category instance for natural transformations. This time the type hom is inferred to have kind `(* -> *) -> (* -> *) -> *`

. Identity and composition are both defined pointwise.

```
instance Category (Nat :: (* -> *) -> (* -> *) -> *) where
ident = Nat id
compose f g = Nat (nu g . nu f)
```

Let's define a type class which will capture the idea of a fixed point in a category. This generalizes the idea of recursive types in `Hask`

:

```
class Rec hom f t where
_in :: hom (f t) t
out :: hom t (f t)
```

The class `Rec`

defines two morphisms: `_in`

, which is the constructor of the fixed point type `t`

, and `out`

, its destructor.

The final piece is the definition of a higher order functor, which generalizes the typeclass `Functor`

:

```
class HFunctor hom f where
hmap :: hom a b -> hom (f a) (f b)
```

Note the similarity with the type signature of the function `fmap :: (Functor f) => (a -> b) -> f a -> f b`

. Indeed, specializing hom to `(->)`

in the definition of `HFunctor`

gives back the type signature of `fmap`

.

Finally, we can define folds and unfolds in a category. The definitions are as before, but with explicit composition, constructors and destructors replaced with the equivalent type class methods, and `hmap`

in place of `fmap`

:

```
fold :: (Category hom, HFunctor hom f, Rec hom f rec) => hom (f t) t -> hom rec t
fold phi = compose out (compose (hmap (fold phi)) phi)
unfold :: (Category hom, HFunctor hom f, Rec hom f rec) => hom t (f t) -> hom t rec
unfold phi = compose phi (compose (hmap (unfold phi)) _in)
```

Now for some examples.

The first example is a regular recursive datatype of binary leaf trees. The functor `FTree`

is the base functor of this recursive type:

```
data FTree a b = FLeaf a | FBranch b b
data Tree a = Leaf a | Branch (Tree a) (Tree a)
```

An instance of `Rec`

shows the relationship between the defining functor and the recursive type itself:

```
instance Rec (->) (FTree a) (Tree a) where
_in (FLeaf a) = Leaf a
_in (FBranch a b) = Branch a b
out (Leaf a) = FLeaf a
out (Branch a b) = FBranch a b
```

`FTree`

is indeed a functor, so it is also a `HFunctor`

:

```
instance HFunctor (->) (FTree a) where
hmap f (FLeaf a) = FLeaf a
hmap f (FBranch a b) = FBranch (f a) (f b)
```

These instances are enough to define folds and unfolds for this type. The following fold calculates the depth of a tree:

```
depth :: Tree a -> Int
depth = (fold :: (FTree a Int -> Int) -> Tree a -> Int) phi where
phi :: FTree a Int -> Int
phi (FLeaf a) = 1
phi (FBranch a b) = 1 + max a b
```

The second example is a fold for the nested (or non-regular) datatype of complete binary leaf trees.

The higher order functor `FCTree`

defines the type constructor `CTree`

as its fixed point:

```
data FCTree f a = FCLeaf a | FCBranch (f (a, a))
data CTree a = CLeaf a | CBranch (CTree (a, a))
```

Again, we define type class instances for HFunctor and Rec:

```
instance HFunctor Nat FCTree where
hmap (f :: Nat (f :: * -> *) (g :: * -> *)) = Nat ff where
ff :: forall a. FCTree f a -> FCTree g a
ff (FCLeaf a) = FCLeaf a
ff (FCBranch a) = FCBranch (nu f a)
instance Rec Nat FCTree CTree where
_in = Nat inComplete where
inComplete (FCLeaf a) = CLeaf a
inComplete (FCBranch a) = CBranch a
out = Nat outComplete where
outComplete(CLeaf a) = FCLeaf a
outComplete(CBranch a) = FCBranch a
```

Morphisms between type constructors are natural transformations, so we need a type constructor to act as the target of the fold. For our purposes, a constant functor will do:

```
data K a b = K a
```

And finally, the following fold calculates the depth of a complete binary leaf tree:

```
cdepth :: CTree a -> Int
cdepth c = let (K d) = nu (fold (Nat phi)) c in d where
phi :: FCTree (K Int) a -> K Int a
phi (FCLeaf a) = K 1
phi (FCBranch (K n)) = K (n + 1)
```

The final example is a fold for the pair of mutually recursive datatype of lists of even and odd lengths. The fold will take a list of even length and produce a list of pairs.

We cannot express type constructors in Haskell whose return kind is anything other than `*`

, so we cheat a little and emulate the product kind using an arrow kind `Choice -> *`

, where `Choice`

is a two point kind, lifted using the `DataKinds`

extension:

```
data Choice = Fst | Snd
```

A morphism of pairs of types is just a pair of morphisms. For technical reasons, we represent this using a Church-style encoding, along with helper methods, as follows:

```
newtype PHom h1 h2 p1 p2 = PHom { runPHom :: forall r. (h1 (p1 Fst) (p2 Fst) -> h2 (p1 Snd) (p2 Snd) -> r) -> r }
mkPHom f g = PHom (\h -> h f g)
fstPHom p = runPHom p (\f -> \g -> f)
sndPHom p = runPHom p (\f -> \g -> g)
```

Now, `PHom`

allows us to take two categories and form the product category:

```
instance (Category h1, Category h2) => Category (PHom h1 h2) where
ident = mkPHom ident ident
compose p1 p2 = mkPHom (compose (fstPHom p1) (fstPHom p2)) (compose (sndPHom p1) (sndPHom p2))
```

We can define the types of lists of even and odd length as follows. Note that the kind annotation indicates the appearance of the kind `Choice -> *`

:

```
data FAlt :: * -> (Choice -> *) -> Choice -> * where
FZero :: FAlt a p Fst
FSucc1 :: a -> (p Snd) -> FAlt a p Fst
FSucc2 :: a -> (p Fst) -> FAlt a p Snd
data Alt :: * -> Choice -> * where
Zero :: Alt a Fst
Succ1 :: a -> Alt a Snd -> Alt a Fst
Succ2 :: a -> Alt a Fst -> Alt a Snd
```

Again, we need to define instances of `Rec`

and `HFunctor`

:

```
instance Rec (PHom (->) (->)) (FAlt a) (Alt a) where
_in = mkPHom f g where
f FZero = Zero
f (FSucc1 a b) = Succ1 a b
g (FSucc2 a b) = Succ2 a b
out = mkPHom f g where
f Zero = FZero
f (Succ1 a b) = FSucc1 a b
g (Succ2 a b) = FSucc2 a b
instance HFunctor (PHom (->) (->)) (FAlt a) where
hmap p = mkPHom hf hg where
hf FZero = FZero
hf (FSucc1 a x) = FSucc1 a (sndPHom p x)
hg (FSucc2 a x) = FSucc2 a (fstPHom p x)
```

As before, we create a target type for our fold, and this time a type synonym as well:

```
data K2 :: * -> * -> Choice -> * where
K21 :: a -> K2 a b Fst
K22 :: b -> K2 a b Snd
type PairUpResult a = K2 [(a, a)] (a, [(a, a)])
```

At last, here is the fold `pairUp`

, taking even length lists to lists of pairs:

```
pairUp :: Alt a Fst -> [(a, a)]
pairUp xs = let (K21 xss) = (fstPHom (fold (mkPHom phi psi))) xs in xss where
phi FZero = K21 []
phi (FSucc1 x1 (K22 (x2, xss))) = K21 ((x1, x2):xss)
psi (FSucc2 x (K21 xss)) = K22 (x, xss)
test = Succ1 0 $ Succ2 1 $ Succ1 2 $ Succ2 3 $ Succ1 4 $ Succ2 5 Zero
```

Copyright Phil Freeman 2010-2016