blog-static/content/blog/haskell_catamorphisms.md

---
title: "Generalizing Folds in Haskell"
date: 2022-04-22T12:19:22-07:00
tags: ["Haskell"]
---

Have you encountered Haskell's `foldr` function? Did you know that you can use it to express any 
function on a list? What's more, there's a way to derive similar functions for
{{< sidenote "right" "positive-note" "a large class of data types in Haskell." >}}
Specifically, this is the class of <a href="https://en.wikipedia.org/wiki/Inductive_type">inductive types</a>.
{{< /sidenote >}}
This is precisely the focus of this post.
Before we get into the details, it's good to review the underlying concepts in a more familiar setting: functions.

### Recursive Functions
Let's start off with a little bit of a warmup, and take a look at a simple recursive function:
`length`. Here's a straightforward definition:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 4 6 >}}

Haskell is nice because it allows for clean definitions of recursive functions; `length` can
just reference itself in its definition, and everything works out in the end. In the underlying
lambda calculus, though, a function definition doesn't come with a name -- you only get anonymous
functions via the lambda abstraction. There's no way for such functions to just refer to themselves by
their name in their body. But the lambda calculus is Turing complete, so something is making recursive
definitions possible.

The trick is to rewrite your recursive function in such a way that instead of calling itself by its name
(which, with anonymous functions, is hard to come by), it receives a reference to itself as an argument.
As a concrete example:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 8 10 >}}

This new function can easily me anonymous; if we enable the `LambdaCase` extension,
we can write it using only lambda functions as:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 12 14 >}}

This function is not equivalent to `length`, however. It expects "itself", or a function
which has type `[a] -> Int`, to be passed in as the first argument. Once fed this `rec`
argument, though, `lengthF` returns a length function. Let's try feed it something, then!

```Haskell
lengthF _something
```

But if `lengthF` produces a length function when given this _something_, why can't we feed
this newly-produced length function back to it?

```Haskell
lengthF (lengthF _something)
```

And again:

```Haskell
lengthF (lengthF (lengthF _something))
```

If we kept going with this process infinitely, we'd eventually have what we need:

{{< latex >}}
\text{length} = \text{lengthF}(\text{lengthF}(\text{lengthF}(...)))
{{< /latex >}}

But hey, the stuff inside the first set of parentheses is still an infinite sequence of applications
of the function \(\text{lengthF}\), and we have just defined this to be \(\text{length}\). Thus,
we can rewrite the above equation as:

{{< latex >}}
\text{length} = \text{lengthF}(\text{length})
{{< /latex >}}

What we have just discovered is that the actual function that we want, `length`, is a [fixed point](https://mathworld.wolfram.com/FixedPoint.html)
of the non-recursive function `lengthF`. Fortunately, Haskell comes with a function that can find
such a fixed point. It's defined like this:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 16 16 >}}

This definition is as declarative as can be; `fix` returns the \(x\) such that \(x = f(x)\). With
this, we finally write:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 18 18 >}}

Loading up the file in GHCi, and running the above function, we get exactly the expected results.

```
ghci> Main.length' [1,2,3]
3
```

You may be dissatisfied with the way we handled `fix` here; we went through and pretended that we didn't
have recursive function definitions, but then used a recursive `let`-expression in the body `fix`!
This is a valid criticism, so I'd like to briefly talk about how `fix` is used in the context of the
lambda calculus.

In the untyped typed lambda calculus, we can just define a term that behaves like `fix` does. The
most common definition is the \(Y\) combinator, defined as follows:

{{< latex >}}
Y = \lambda f. (\lambda x. f (x x)) (\lambda x. f (x x ))
{{< /latex >}}

When applied to a function, this combinator goes through the following evaluation steps:

{{< latex >}}
Y f = f (Y f) = f (f (Y f)) =\ ...
{{< /latex >}}

This is the exact sort of infinite series of function applications that we saw above with \(\text{lengthF}\).

### Recursive Data Types
We have now seen how we can rewrite a recursive function as a fixed point of some non-recursive function.
Another cool thing we can do, though, is to transform recursive __data types__ in a similar manner!
Let's start with something pretty simple.

{{< codelines "Haskell" "catamorphisms/Cata.hs" 20 20 >}}

Just like we did with functions, we can extract the recursive occurrences of `MyList` into a parameter.

{{< codelines "Haskell" "catamorphisms/Cata.hs" 21 21 >}}

Just like `lengthF`, `MyListF` isn't really a list. We can't write a function `sum :: MyListF -> Int`.
`MyListF` requires _something_ as an argument, and once given that, produces a type of integer lists.
Once again, let's try feeding it:

```
MyListF a
```

From the definition, we can clearly see that `a` is where the "rest of the list" is in the original
`MyList`. So, let's try fill `a` with a list that we can get out of `MyListF`:

```
MyListF (MyListF a)
```

And again:

```
MyListF (MyListF (MyListF a))
```

Much like we used a `fix` function to turn our `lengthF` into `length`, we need a data type,
which we'll call `Fix` (and which has been [implemented before](https://hackage.haskell.org/package/data-fix-0.3.2/docs/Data-Fix.html)). Here's the definition:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 23 23 >}}

Looking past the constructors and accessors, we might write the above in pseudo-Haskell as follows:

```Haskell
newtype Fix f = f (Fix f)
```

This is just like the lambda calculus \(Y\) combinator above! Unfortunately, we _do_ have to
deal with the cruft induced by the constructors here. Thus, to write down the list `[1,2,3]`
using `MyListF`, we'd have to produce the following:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 25 26 >}}

This is actually done in practice when using some approaches to help address the
[expression problem](https://en.wikipedia.org/wiki/Expression_problem); however,
it's quite unpleasant to write code in this way, so we'll set it aside.

Let's go back to our infinite chain of type applications. We've a similar pattern before, 
with \(\text{length}\) and \(\text{lengthF}\). Just like we did then, it seems like
we might be able to write something like the following:

{{< latex >}}
\begin{aligned}
    & \text{MyList} = \text{MyListF}(\text{MyListF}(\text{MyListF}(...))) \\
    \Leftrightarrow\ & \text{MyList} = \text{MyListF}(\text{MyList})
\end{aligned}
{{< /latex >}}

In something like Haskell, though, the above is not quite true. `MyListF` is a
non-recursive data type, with a different set of constructors to `MyList`; they aren't
_really_ equal. Instead of equality, though, we use the next-best thing: isomorphism.

{{< latex >}}
    \text{MyList} \cong \text{MyListF}(\text{MyList})
{{< /latex >}}

Two types are isomorphic when there exist a
{{< sidenote "right" "fix-isomorphic-note" "pair of functions, \(f\) and \(g\)," >}}
Let's a look at the types of <code>Fix</code> and <code>unFix</code>, by the way.
Suppose that we <em>did</em> define <code>MyList</code> to be <code>Fix MyListF</code>.
Let's specialize the <code>f</code> type parameter of <code>Fix</code> to <code>MyListF</code>
for a moment, and check:<br><br>
In one direction, <code>Fix :: MyListF MyList -> MyList</code><br>
And in the other, <code>unFix :: MyList -> MyListF MyList</code><br><br>
The two mutual inverses \(f\) and \(g\) fall out of the definition of the <code>Fix</code>
data type! If we didn't have to deal with the constructor cruft, this would be more
ergonomic than writing our own <code>myIn</code> and <code>myOut</code> functions.
{{< /sidenote >}}
that take you from one type to the other (and vice versa), such that applying \(f\) after \(g\),
or \(g\) after \(f\), gets you right back where you started. That is, \(f\) and \(g\)
need to be each other's inverses. For our specific case, let's call the two functions `myOut`
and `myIn` (I'm matching the naming in [this paper](https://maartenfokkinga.github.io/utwente/mmf91m.pdf)). They are not hard to define:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 28 34 >}}

By the way, when a data type is a fixed point of some other, non-recursive type constructor,
this second type constructor is called a __base functor__. We can verify that `MyListF` is a functor
by providing an instance (which is rather straightforward):

{{< codelines "Haskell" "catamorphisms/Cata.hs" 36 38 >}}

### Recursive Functions with Base Functors
One neat thing you can do with a base functor is define recursive functions on the actual data type!

Let's go back to the very basics. When we write recursive functions, we try to think of it as solving
a problem, assuming that we are given solutions to the sub-problems that make it up. In the more
specific case of recursive functions on data types, we think of it as performing a given operation,
assuming that we know how to perform this operation on the smaller pieces of the data structure.
Some quick examples:

1. When writing a `sum` function on a list, we assume we know how to find the sum of
   the list's tail (`sum xs`), and add to it the current element (`x+`). Of course,
   if we're looking at a part of a data structure that's not recursive, we don't need to
   perform any work on its constituent pieces.
   ```Haskell
   sum [] = 0
   sum (x:xs) = x + sum xs
   ```
2. When writing a function to invert a binary tree, we assume that we can invert the left and right
   children of a non-leaf node. We might write:
   ```Haskell
   invert Leaf = Leaf
   invert (Node l r) = Node (invert r) (invert l)
   ```

What does this have to do with base functors? Well, recall how we arrived at `MyListF` from
`MyList`: we replaced every occurrence of `MyList` in the definition with a type parameter `a`.
Let me reiterate: wherever we had a sub-list in our definition, we replaced it with `a`.
The `a` in `MyListF` marks the locations where we _would_ have to use recursion if we were to
define a function on `MyList`.

What if instead of a stand-in for the list type (as it was until now), we use `a` to represent
the result of the recursive call on that sub-list? To finish computing the sum of the list, then,
the following would suffice:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 40 42 >}}

Actually, this is enough to define the whole `sum` function. First things first, let's use `myOut`
to unpack one level of the `Mylist` type:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 28 28 >}}

We know that `MyListF` is a functor; we can thus use `fmap sum` to compute the sum of the
remaining list:

```Haskell
fmap mySum :: MyListF MyList -> MyListF Int
```

Finally, we can use our `mySumF` to handle the last addition:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 40 40 >}}

Let's put all of these together:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 44 45 >}}

Notice, though, that the exact same approach would work for _any_ function
with type:

```Haskell
MyListF a -> a
```

We can thus write a generalized version of `mySum` that, instead of using `mySumF`,
uses some arbitrary function `f` with the aforementioned type:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 47 48 >}}

Let's use `myCata` to write a few other functions:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 50 60 >}}

#### It's just a `foldr`!
When you write a function with the type `MyListF a -> a`, you are actually
providing two things: a "base case" element of type `a`, for when you match `MyNilF`,
and a "combining function" with type `Int -> a -> a`, for when you match `MyConsF`.
We can thus define:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 64 66 >}}

We could also go in the opposite direction, by writing:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 68 69 >}}

Hey, what was it that we said about types with two functions between them, which
are inverses of each other? That's right, `MyListF a -> a` and `(a, Int -> a -> a)`
are isomorphic. The function `myCata`, and the "traditional" definition of `foldr`
are equivalent!

#### Base Functors for All!
We've been playing with `MyList` for a while now, but it's kind of getting boring:
it's just a list of integers! Furthermore, we're not _really_ getting anything
out of this new "generalization" procedure -- `foldr` is part of the standard library,
and we've just reinvented the wheel.

But you see, we haven't quite. This is because, while we've only been working with `MyListF`,
the base functor for `MyList`, our approach works for _any recursive data type_, provided
an `out` function. Let's define a type class, `Cata`, which pairs a data type `a` with
its base functor `f`, and specifies how to "unpack" `a`:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 71 72 >}}

We can now provide a more generic version of our `myCata`, one that works for all types
with a base functor:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 74 75 >}}

Clearly, `MyList` and `MyListF` are one instance of this type class:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 77 78 >}}

We can also write a base functor for Haskell's built-in list type, `[a]`:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 80 84 >}}

We can use our `cata` function for regular lists to define a generic `sum`:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 86 89 >}}

It works perfectly:

```
ghci> Main.sum [1,2,3]
6
ghci> Main.sum [1,2,3.0]
6.0
ghci> Main.sum [1,2,3.0,-1]
5.0
```

What about binary trees, which served as our second example of a recursive data structure?
We can do that, too:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 91 96 >}}

Given this, here's an implementation of that `invert` function we mentioned earlier:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 98 101 >}}

#### Degenerate Cases

Actually, the data types we consider don't have to be recursive. We can apply the same
procedure of replacing recursive occurrences in a data type's definition with a new type parameter
to `Maybe`; the only difference is that now the new parameter will not be used!

{{< codelines "Haskell" "catamorphisms/Cata.hs" 103 107 >}}

And then we can define a function on `Maybe` using `cata`:

{{< codelines "Haskell" "catamorphisms/Cata.hs" 109 112 >}}

This isn't _really_ useful, since we're still pattern matching on a type that looks
identical to `Maybe` itself. There is one reason that I bring it up, though. Remember
how `foldr` was equivalent to `cata` for `MyList`, because defining a function `MyListF a -> a` was
the same as providing a base case `a` and a "combining function" `Int -> a -> a`? Well,
defining a function `MaybeF x a -> a` is the same as providing a base case `a` (for `NothingF`)
and a handler for the contained value, `x -> a`. So we might imagine the `foldr` function for `Maybe`
to have type:

```Haskell
maybeFold :: a -> (x -> a) -> Maybe x -> a
```

This is exactly the function [`maybe` from `Data.Maybe`](https://hackage.haskell.org/package/base-4.16.1.0/docs/Data-Maybe.html#v:maybe)! Hopefully you can follow a similar process in your head to arrive at "fold"
functions for `Either` and `Bool`. Indeed, there are functions that correspond to these data types
in the Haskell standard library, named [`either`](https://hackage.haskell.org/package/base-4.16.1.0/docs/Data-Either.html#v:either) and [`bool`](https://hackage.haskell.org/package/base-4.16.1.0/docs/Data-Bool.html#v:bool).
Much like `fold` can be used to represent any function on lists, `maybe`, `either`, and `bool` can be
used to represent any function on their corresponding data types. I think that's neat.

#### What About `Foldable`?
If you've been around the Haskell ecosystem, you may know the `Foldable` type class.
Isn't this exactly what we've been working towards here? No, not at all. Take
a look at how the documentation describes [`Data.Foldable`](https://hackage.haskell.org/package/base-4.16.1.0/docs/Data-Foldable.html):

> The Foldable class represents data structures that can be reduced to a summary value one element at a time.

One at a time, huh? Take a look at the signature of `foldMap`, which is sufficient for
an instance of `Foldable`:

```Haskell
foldMap :: Monoid m => (a -> m) -> t a -> m
```

A `Monoid` is just a type with an associative binary operation that has an identity element.
Then, `foldMap` simply visits the data structure in order, and applies this binary operation
pairwise to each monoid produced via `f`. Alas,
this function is not enough to be able to implement something like inverting a binary tree;
there are different configurations of binary tree that, when visited in-order, result in the
same sequence of elements. For example:

```
ghci> fold (Node "Hello" Leaf (Node ", " Leaf (Node "World!" Leaf Leaf)))
"Hello, World!"
ghci> fold (Node "Hello" (Node ", " Leaf Leaf) (Node "World!" Leaf Leaf))
"Hello, World!"
```

As far as `fold` (which is just `foldMap id`) is concerned, the two trees are equivalent. They
are very much not equivalent for the purposes of inversion! Thus, whereas `Foldable` helps
us work with list-like data types, the `Cata` type class lets us express _any_ function on
a recursive data type similarly to how we'd do it with `foldr` and lists.

#### Catamorphisms
Why is the type class called `Cata`, and the function `cata`? Well, a function that
performs a computation by recursively visiting the data structure is called a catamorphism.
Indeed, `foldr f b`, for function `f` an "base value" `b` is an example of a list catamorophism.
It's a fancy word, and there are some fancier descriptions of what it is, especially when
you step into category theory (check out the [Wikipedia entry](https://en.wikipedia.org/wiki/Catamorphism)
if you want to know what I mean). However, for our purposes, a catamorphism is just a generalization
of `foldr` from lists to any data type!