# List catamorphism by Mark Seemann

*The catamorphism for a list is the same as its fold.*

This article is part of an article series about catamorphisms. A catamorphism is a universal abstraction that describes how to digest a data structure into a potentially more compact value.

This article presents the catamorphism for (linked) lists, and other collections in general. It also shows how to identify it. The beginning of this article presents the catamorphism in C#, with an example. The rest of the article describes how to deduce the catamorphism. This part of the article presents my work in Haskell. Readers not comfortable with Haskell can just read the first part, and consider the rest of the article as an optional appendix.

The C# part of the article will discuss `IEnumerable<T>`

, while the Haskell part will deal specifically with linked lists. Since C# is a less strict language anyway, we have to make some concessions when we consider how concepts translate. In my experience, the functionality of `IEnumerable<T>`

closely mirrors that of Haskell lists.

### C# catamorphism #

The .NET base class library defines this Aggregate overload:

public static TAccumulate Aggregate<TSource, TAccumulate>( this IEnumerable<TSource> source, TAccumulate seed, Func<TAccumulate, TSource, TAccumulate> func);

This is the catamorphism for linked lists (and, I'll conjecture, for `IEnumerable<T>`

in general). The introductory article already used it to show several motivating examples, of which I'll only repeat the last:

> new[] { 42, 1337, 2112, 90125, 5040, 7, 1984 } . .Aggregate(Angle.Identity, (a, i) => a.Add(Angle.FromDegrees(i))) [{ Angle = 207° }]

In short, the catamorphism is, similar to the previous catamorphisms covered in this article series, a pair made from an initial value and a function. This has been true for both the Peano catamorphism and the Maybe catamorphism. An initial value is just a value in all three cases, but you may notice that the function in question becomes increasingly elaborate. For `IEnumerable<T>`

, it's a function that takes two values. One of the values are of the type of the input list, i.e. for `IEnumerable<TSource>`

it would be `TSource`

. By elimination you can deduce that this value must come from the input list. The other value is of the type `TAccumulate`

, which implies that it could be the `seed`

, or the result from a previous call to `func`

.

### List F-Algebra #

As in the previous article, I'll use `Fix`

and `cata`

as explained in Bartosz Milewski's excellent article on F-Algebras. The `ListF`

type comes from his article as well, although I've renamed the type arguments:

data ListF a c = NilF | ConsF a c deriving (Show, Eq, Read) instance Functor (ListF a) where fmap _ NilF = NilF fmap f (ConsF a c) = ConsF a $ f c

Like I did with `MaybeF`

, I've named the 'data' type argument `a`

, and the carrier type `c`

(for *carrier*). Once again, notice that the `Functor`

instance maps over the carrier type `c`

; not over the 'data' type `a`

.

As was also the case when deducing the Maybe catamorphism, Haskell isn't too happy about defining instances for a type like `Fix (ListF a)`

. To address that problem, you can introduce a `newtype`

wrapper:

newtype ListFix a = ListFix { unListFix :: Fix (ListF a) } deriving (Show, Eq, Read)

You can define `Functor`

, `Applicative`

, `Monad`

, etc. instances for this type without resorting to any funky GHC extensions. Keep in mind that ultimately, the purpose of all this code is just to figure out what the catamorphism looks like. This code isn't intended for actual use.

A few helper functions make it easier to define `ListFix`

values:

nilF :: ListFix a nilF = ListFix $ Fix NilF consF :: a -> ListFix a -> ListFix a consF x = ListFix . Fix . ConsF x . unListFix

With those functions, you can create `ListFix`

linked lists:

Prelude Fix List> nilF ListFix {unListFix = Fix NilF} Prelude Fix List> consF 42 $ consF 1337 $ consF 2112 nilF ListFix {unListFix = Fix (ConsF 42 (Fix (ConsF 1337 (Fix (ConsF 2112 (Fix NilF))))))}

The first example creates an empty list, whereas the second creates a list of three integers, corresponding to `[42,1337,2112]`

.

That's all you need to identify the catamorphism.

### Haskell catamorphism #

At this point, you have two out of three elements of an F-Algebra. You have an endofunctor (`ListF`

), and an object `a`

, but you still need to find a morphism `ListF a c -> c`

. Notice that the algebra you have to find is the function that reduces the functor to its *carrier type* `c`

, not the 'data type' `a`

. This takes some time to get used to, but that's how catamorphisms work. This doesn't mean, however, that you get to ignore `a`

, as you'll see.

As in the previous article, start by writing a function that will become the catamorphism, based on `cata`

:

listF = cata alg . unListFix where alg NilF = undefined alg (ConsF h t) = undefined

While this compiles, with its `undefined`

implementations, it obviously doesn't do anything useful. I find, however, that it helps me think. How can you return a value of the type `c`

from the `NilF`

case? You could pass an argument to the `listF`

function:

listF n = cata alg . unListFix where alg NilF = n alg (ConsF h t) = undefined

The `ConsF`

case, contrary to `NilF`

, contains both a head `h`

(of type `a`

) and a tail `t`

(of type `c`

). While you could make the code compile by simply returning `t`

, it'd be incorrect to ignore `h`

. In order to deal with both, you'll need a function that turns both `h`

and `t`

into a value of the type `c`

. You can do this by passing a function to `listF`

and using it:

listF :: (a -> c -> c) -> c -> ListFix a -> c listF f n = cata alg . unListFix where alg NilF = n alg (ConsF h t) = f h t

This works. Since `cata`

has the type `Functor f => (f a -> a) -> Fix f -> a`

, that means that `alg`

has the type `f a -> a`

. In the case of `ListF`

, the compiler infers that the `alg`

function has the type `ListF a c -> c`

, which is just what you need!

You can now see what the carrier type `c`

is for. It's the type that the algebra extracts, and thus the type that the catamorphism returns.

This, then, is the catamorphism for lists. As has been consistent so far, it's a pair made from an initial value and a function. Once more, this isn't the only possible catamorphism, since you can, for example, trivially flip the arguments to `listF`

:

listF :: c -> (a -> c -> c) -> ListFix a -> c listF n f = cata alg . unListFix where alg NilF = n alg (ConsF h t) = f h t

You can also flip the arguments of `f`

:

listF :: c -> (c -> a -> c) -> ListFix a -> c listF n f = cata alg . unListFix where alg NilF = n alg (ConsF h t) = f t h

These representations are all isomorphic to each other, but notice that the last variation is similar to the above C# `Aggregate`

overload. The initial `n`

value is the `seed`

, and the function `f`

has the same shape as `func`

. Thus, I consider it reasonable to conjecture that that `Aggregate`

overload is the catamorphism for `IEnumerable<T>`

.

### Basis #

You can implement most other useful functionality with `listF`

. The rest of this article uses the first of the variations shown above, with the type `(a -> c -> c) -> c -> ListFix a -> c`

. Here's the `Semigroup`

instance:

instance Semigroup (ListFix a) where xs <> ys = listF consF ys xs

The initial value passed to `listF`

is `ys`

, and the function to apply is simply the `consF`

function, thus 'consing' the two lists together. Here's an example of the operation in action:

Prelude Fix List> consF 42 $ consF 1337 nilF <> (consF 2112 $ consF 1 nilF) ListFix {unListFix = Fix (ConsF 42 (Fix (ConsF 1337 (Fix (ConsF 2112 (Fix (ConsF 1 (Fix NilF))))))))}

With a `Semigroup`

instance, it's trivial to also implement the `Monoid`

instance:

instance Monoid (ListFix a) where mempty = nilF

While you *could* implement `mempty`

with `listF`

(`mempty = listF (const id) nilF nilF`

), that'd be overcomplicated. Just because you can implement all functionality using `listF`

, it doesn't mean that you should, if a simpler alternative exists.

You can, on the other hand, use `listF`

for the `Functor`

instance:

instance Functor ListFix where fmap f = listF (\h l -> consF (f h) l) nilF

You could write the function you pass to `listF`

in a point-free fashion as `consF . f`

, but I thought it'd be easier to follow what happens when written as an explicit lambda expression. The function receives a 'current value' `h`

, as well as the part of the list which has already been translated `l`

. Use `f`

to translate `h`

, and `consF`

the result unto `l`

.

You can add `Applicative`

and `Monad`

instances in a similar fashion:

instance Applicative ListFix where pure x = consF x nilF fs <*> xs = listF (\f acc -> (f <$> xs) <> acc) nilF fs instance Monad ListFix where xs >>= f = listF (\x acc -> f x <> acc) nilF xs

What may be more interesting, however, is the `Foldable`

instance:

instance Foldable ListFix where foldr = listF

The demonstrates that `listF`

and `foldr`

is the same.

Next, you can also add a `Traversable`

instance:

instance Traversable ListFix where sequenceA = listF (\x acc -> consF <$> x <*> acc) (pure nilF)

Finally, you can implement conversions to and from the standard list `[]`

type, using `ana`

as the dual of `cata`

:

toList :: ListFix a -> [a] toList = listF (:) [] fromList :: [a] -> ListFix a fromList = ListFix . ana coalg where coalg [] = NilF coalg (h:t) = ConsF h t

This demonstrates that `ListFix`

is isomorphic to `[]`

, which again establishes that `listF`

and `foldr`

are equivalent.

### Summary #

The catamorphism for lists is a pair made from an initial value and a function. One variation is equal to `foldr`

. Like Maybe, the catamorphism is the same as the fold.

In C#, this function corresponds to the `Aggregate`

extension method identified above.

You've now seen two examples where the catamorphism coincides with the fold. You've also seen examples (Boolean catamorphism and Peano catamorphism) where there's a catamorphism, but no fold at all. In a subsequent article, you'll see an example of a container that has both catamorphism and fold, but where the catamorphism is more general than the fold.

**Next:** NonEmpty catamorphism.