# ploeh blog danish software design

## Full binary tree catamorphism

*The catamorphism for a full binary tree is a pair of functions.*

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 a full binary tree, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. 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.

A *full binary tree* (also known as a *proper* or *plane* binary tree) is a tree in which each node has either two or no branches.

The diagram shows an example of a tree of integers. The left branch contains two children, of which the right branch again contains two sub-branches. The rest of the nodes are leaf-nodes with no sub-branches.

### C# catamorphism #

As a C# representation of a full binary tree, I'll start with the `IBinaryTree<T>`

API from A Visitor functor. The catamorphism is the `Accept`

method:

TResult Accept<TResult>(IBinaryTreeVisitor<T, TResult> visitor);

So far in this article series, you've mostly seen Church-encoded catamorphisms, so a catamorphism represented as a Visitor may be too big of a cognitive leap. We know, however, from Visitor as a sum type that a Visitor representation is isomorphic to a Church encoding. Since these are isomorphic, it's possible to refactor `IBinaryTree<T>`

to a Church encoding. The GitHub repository contains a series of commits that demonstrates how that refactoring works. Once you're done, you arrive at this `Match`

method, which is the refactored `Accept`

method:

TResult Match<TResult>(Func<TResult, T, TResult, TResult> node, Func<T, TResult> leaf);

This method takes a pair of functions as arguments. The `node`

function deals with an internal node in the tree (the blue nodes in the above diagram), whereas the `leaf`

function deals with the leaf nodes (the green nodes in the diagram).

The `leaf`

function may be the easiest one to understand. A leaf node only contains a value of the type `T`

, so the only operation the function has to support is translating the `T`

value to a `TResult`

value. This is also the premise of the `Leaf`

class' implementation of the method:

private readonly T item; public TResult Match<TResult>(Func<TResult, T, TResult, TResult> node, Func<T, TResult> leaf) { return leaf(item); }

The `node`

function is more tricky. It takes three input arguments, of the types `TResult`

, `T`

, and `TResult`

. The roles of these are respectively *left*, *item*, and *right*. This is a typical representation of a binary node. Since there's always a left and a right branch, you put the node's value in the middle. As was the case with the tree catamorphism, the catamorphism function receives the branches as already-translated values; that is, both the left and right branch have already been translated to `TResult`

when `node`

is called. While it looks like magic, as always it's just the result of recursion:

private readonly IBinaryTree<T> left; private readonly T item; private readonly IBinaryTree<T> right; public TResult Match<TResult>(Func<TResult, T, TResult, TResult> node, Func<T, TResult> leaf) { return node(left.Match(node, leaf), item, right.Match(node, leaf)); }

This is the `Node<T>`

class implementation of the `Match`

method. It calls `node`

and returns whatever it returns, but notice that as the `left`

and `right`

arguments, if first, recursively, calls `left.Match`

and `right.Match`

. This is how it can call `node`

with the translated branches, as well as with the basic `item`

.

The recursion stops and unwinds on `left`

and `right`

whenever one of those are `Leaf`

instances.

### Examples #

You can use `Match`

to implement most other behaviour you'd like `IBinaryTree<T>`

to have. In the original article on the full binary tree functor you saw how to implement `Select`

with a Visitor, but now that the API is Church-encoded, you can derive `Select`

from `Match`

:

public static IBinaryTree<TResult> Select<TResult, T>( this IBinaryTree<T> tree, Func<T, TResult> selector) { if (tree == null) throw new ArgumentNullException(nameof(tree)); if (selector == null) throw new ArgumentNullException(nameof(selector)); return tree.Match( node: (l, x, r) => Create(l, selector(x), r), leaf: x => Leaf(selector(x))); }

In the `leaf`

case, the `Select`

method simply calls `selector`

with the `x`

value it receives, and puts the resulting `TResult`

object into a new `Leaf`

object.

In the `node`

case, the lambda expression receives three arguments: `l`

and `r`

are the *already-translated* left and right branches, so you only need to call `selector`

on `x`

and call the `Create`

helper method to produce a new `Node`

object.

You can also implement more specialised functionality, like calculating the sum of nodes, measuring the depth of the tree, and similar functions. You saw equivalent examples in the previous article.

For the examples in this article, I'll use the tree shown in the above diagram. Using static helper methods, you can write it like this:

var tree = BinaryTree.Create( BinaryTree.Create( BinaryTree.Leaf(42), 1337, BinaryTree.Create( BinaryTree.Leaf(2112), 5040, BinaryTree.Leaf(1984))), 2, BinaryTree.Leaf(90125));

To calculate the sum of all nodes, you can write a function like this:

public static int Sum(this IBinaryTree<int> tree) { return tree.Match((l, x, r) => l + x + r, x => x); }

The `leaf`

function just returns the value of the node, while the `node`

function adds the numbers together. It works for the above `tree`

:

> tree.Sum() 100642

To find the maximum value, you can write another extension method:

public static int Max(this IBinaryTree<int> tree) { return tree.Match((l, x, r) => Math.Max(Math.Max(l, r), x), x => x); }

Again, the `leaf`

function just returns the value of the node. The `node`

function receives the value of the current node `x`

, as well as the already-found maximum value of the left branch and the right branch; it then returns the maximum of these three values:

> tree.Max() 90125

As was also the case for trees, both of these operations are part of the standard repertoire available via a data structure's *fold*. That's not the case for the next two functions, which can't be implemented using a fold, but which can be defined with the catamorphism. The first is a function to count the leaves of a tree:

public static int CountLeaves<T>(this IBinaryTree<T> tree) { return tree.Match((l, _, r) => l + r, _ => 1); }

Since the `leaf`

function handles a leaf node, the number of leaf nodes in a leaf node is, by definition, *one*. Thus, that function can ignore the value of the node and always return `1`

. The `node`

function, on the other hand, receives the number of leaf nodes on the left-hand side (`l`

), the value of the current node, and the number of leaf nodes on the right-hand side (`r`

). Notice that since an internal node is never a leaf node, it doesn't count; instead, just add `l`

and `r`

together. Notice that, again, the value of the node itself is irrelevant.

How many leaf nodes does the above tree have?

> tree.CountLeaves() 4

You can also measure the maximum depth of a tree:

public static int MeasureDepth<T>(this IBinaryTree<T> tree) { return tree.Match((l, _, r) => 1 + Math.Max(l, r), _ => 0); }

Like in the previous article, I've arbitrarily decided that the depth of a leaf node is *zero*; therefore, the `leaf`

function always returns `0`

. The `node`

function receives the depth of the left and right branches, and returns the maximum of those two values, plus one, since the current node adds one level of depth.

> tree.MeasureDepth() 3

You may not have much need for working with full binary trees in your normal, day-to-day C# work, but I found it worthwhile to include this example for a couple of reasons. First, because the original of the API shows that a catamorphism may be hiding in a Visitor. Second, because binary trees are interesting, in that they're foldable functors, but not monads.

Where does the catamorphism come from, though? How can you trust that the `Match`

method is the catamorphism?

### Binary tree F-Algebra #

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

and `cata`

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

As always, start with the underlying endofunctor. You can think of this one as a specialisation of the rose tree from the previous article:

data FullBinaryTreeF a c = LeafF a | NodeF c a c deriving (Show, Eq, Read) instance Functor (FullBinaryTreeF a) where fmap _ (LeafF x) = LeafF x fmap f (NodeF l x r) = NodeF (f l) x (f r)

As usual, I've called the 'data' type `a`

and the carrier type `c`

(for *carrier*). The `Functor`

instance as usual translates the carrier type; the `fmap`

function has the type `(c -> c1) -> FullBinaryTreeF a c -> FullBinaryTreeF a c1`

.

As was the case when deducing the recent catamorphisms, Haskell isn't too happy about defining instances for a type like `Fix (FullBinaryTreeF a)`

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

wrapper:

newtype FullBinaryTreeFix a = FullBinaryTreeFix { unFullBinaryTreeFix :: Fix (FullBinaryTreeF a) } deriving (Show, Eq, Read)

You can define `Functor`

, `Foldable`

, and `Traversable`

instances (but not `Monad`

) 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 pair of helper functions make it easier to define `FullBinaryTreeFix`

values:

fbtLeafF :: a -> FullBinaryTreeFix a fbtLeafF = FullBinaryTreeFix . Fix . LeafF fbtNodeF :: FullBinaryTreeFix a -> a -> FullBinaryTreeFix a -> FullBinaryTreeFix a fbtNodeF (FullBinaryTreeFix l) x (FullBinaryTreeFix r) = FullBinaryTreeFix $ Fix $ NodeF l x r

In order to distinguish these helper functions from the ones that create `TreeFix a`

values, I prefixed them with `fbt`

(for *Full Binary Tree*). `fbtLeafF`

creates a leaf node:

Prelude Fix FullBinaryTree> fbtLeafF "fnaah" FullBinaryTreeFix {unFullBinaryTreeFix = Fix (LeafF "fnaah")}

`fbtNodeF`

is a helper function to create an internal node:

Prelude Fix FullBinaryTree> fbtNodeF (fbtLeafF 1337) 42 (fbtLeafF 2112) FullBinaryTreeFix {unFullBinaryTreeFix = Fix (NodeF (Fix (LeafF 1337)) 42 (Fix (LeafF 2112)))}

The `FullBinaryTreeFix`

type, or rather the underlying `FullBinaryTreeF a`

functor, is 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 (`FullBinaryTreeF a`

), and an object `c`

, but you still need to find a morphism `FullBinaryTreeF 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 articles, start by writing a function that will become the catamorphism, based on `cata`

:

fullBinaryTreeF = cata alg . unFullBinaryTreeFix where alg (LeafF x) = undefined alg (NodeF l x r) = undefined

While this compiles, with its `undefined`

implementation of `alg`

, 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 `alg`

? You could pass a function argument to the `fullBinaryTreeF`

function and use it with `x`

:

fullBinaryTreeF fl = cata alg . unFullBinaryTreeFix where alg (LeafF x) = fl x alg (NodeF l x r) = undefined

I called the function `fl`

for *function, leaf*, because we're also going to need a function for the `NodeF`

case:

fullBinaryTreeF :: (c -> a -> c -> c) -> (a -> c) -> FullBinaryTreeFix a -> c fullBinaryTreeF fn fl = cata alg . unFullBinaryTreeFix where alg (LeafF x) = fl x alg (NodeF l x r) = fn l x r

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 `FullBinaryTreeF`

, the compiler infers that the `alg`

function has the type `FullBinaryTreeF 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 a full binary tree. As always, it's not the only possible catamorphism, since you can easily reorder the arguments to both `fullBinaryTreeF`

, `fn`

, and `fl`

. These would all be isomorphic, though.

### Basis #

You can implement most other useful functionality with `treeF`

. Here's the `Functor`

instance:

instance Functor FullBinaryTreeFix where fmap f = fullBinaryTreeF (\l x r -> fbtNodeF l (f x) r) (fbtLeafF . f)

The `fl`

function first invokes `f`

, followed by `fbtLeafF`

. The `fn`

function uses the `fbtNodeF`

helper function to create a new internal node. `l`

and `r`

are already-translated branches, so you just need to call `f`

with the node value `x`

.

There's no `Monad`

instance for binary trees, because you can't flatten a binary tree of binary trees. You can, on the other hand, define a `Foldable`

instance:

instance Foldable FullBinaryTreeFix where foldMap f = fullBinaryTreeF (\l x r -> l <> f x <> r) f

The `f`

function passed to `foldMap`

has the type `Monoid m => (a -> m)`

, so the `fl`

function that handles leaf nodes simply calls `f`

with the contents of the node. The `fn`

function receives two branches already translated to `m`

, so it just has to call `f`

with `x`

and combine all the `m`

values using the `<>`

operator.

The `Traversable`

instance follows right on the heels of `Foldable`

:

instance Traversable FullBinaryTreeFix where sequenceA = fullBinaryTreeF (liftA3 fbtNodeF) (fmap fbtLeafF)

There are operations on binary trees that you can implement with a fold, but some that you can't. Consider the tree shown in the diagram at the beginning of the article. This is also the tree that the above C# examples use. In Haskell, using `FullBinaryTreeFix`

, you can define that tree like this:

tree = fbtNodeF (fbtNodeF (fbtLeafF 42) 1337 (fbtNodeF (fbtLeafF 2112) 5040 (fbtLeafF 1984))) 2 (fbtLeafF 90125)

Since `FullBinaryTreeFix`

is `Foldable`

, and that type class already comes with `sum`

and `maximum`

functions, no further work is required to repeat the first two of the above C# examples:

Prelude Fix FullBinaryTree> sum tree 100642 Prelude Fix FullBinaryTree> maximum tree 90125

Counting leaves, or measuring the depth of a tree, on the other hand, is impossible with the `Foldable`

instance, but can be implemented using the catamorphism:

countLeaves :: Num n => FullBinaryTreeFix a -> n countLeaves = fullBinaryTreeF (\l _ r -> l + r) (const 1) treeDepth :: (Ord n, Num n) => FullBinaryTreeFix a -> n treeDepth = fullBinaryTreeF (\l _ r -> 1 + max l r) (const 0)

The reasoning is the same as already explained in the above C# examples. The functions also produce the same results:

Prelude Fix FullBinaryTree> countLeaves tree 4 Prelude Fix FullBinaryTree> treeDepth tree 3

This, hopefully, illustrates that the catamorphism is more capable, and that the fold is just a (list-biased) specialisation.

### Summary #

The catamorphism for a full binary tree is a pair of functions. One function handles internal nodes, while the other function handles leaf nodes.

I thought it was interesting to show this example for two reasons: First, the original example was a Visitor implementation, and I think it's worth realising that a Visitor's `Accept`

method can also be viewed as a catamorphism. Second, a binary tree is an example of a data structure that has a fold, but isn't a monad.

All articles in the article series have, so far, covered data structures well-known from computer science. The next example will, on the other hand, demonstrate that even completely ad-hoc domain-specific data structures have catamorphisms.

**Next:** Payment types catamorphism.

## Composition Root location

*A Composition Root should be located near the point where user code first executes.*

Prompted by a recent Internet discussion, my DIPPP co-author Steven van Deursen wrote to me in order to help clarify the Composition Root pattern.

In the email, Steven ponders whether it's defensible to use an API that looks like a Service Locator from within a unit test. He specifically calls out my article that describes the Auto-mocking Container design pattern.

In that article, I show how to use Castle Windsor's `Resolve`

method from within a unit test:

[Fact] public void SutIsController() { var container = new WindsorContainer().Install(new ShopFixture()); var sut = container.Resolve<BasketController>(); Assert.IsAssignableFrom<IHttpController>(sut); }

Is the test using a Service Locator? If so, why is that okay? If not, why isn't it a Service Locator?

This article argues that that this use of `Resolve`

isn't a Service Locator.

### Entry points defined #

The original article about the Composition Root pattern defines a Composition Root as the place where you compose your object graph(s). It repeatedly describes how this ought to happen in, or as close as possible to, the application's entry point. I believe that this definition is compatible with the pattern description given in our book.

I do realise, however, that we may never have explicitly defined what an *entry point* is.

In order to do so, it may be helpful to establish a bit of terminology. In the following, I'll use the terms *user code* as opposed to *framework code*.

Much of the code you write probably runs within some sort of framework. If you're writing a web application, you're probably using a web framework. If you're writing a message-based application, you might be using some message bus, or actor, framework. If you're writing an app for a mobile device, you're probably using some sort of framework for that, too.

Even as a programmer, you're a *user* of frameworks.

As I usually do, I'll use Tomas Petricek's distinction between libraries and frameworks. A library is a collection of APIs that you can call. A framework is a software system that calls your code.

The reality is often more complex, as illustrated by the figure. While a framework will call your code, you can also invoke APIs afforded by the framework.

The point, however, is that *user code* is code that you write, while *framework code* is code that someone else wrote to develop the framework. The framework starts up first, and at some point in its lifetime, it calls your code.

**Definition:** The *entry point* is the user code that the framework calls first.

As an example, in ASP.NET Core, the (conventional) `Startup`

class is the first user code that the framework calls. (If you follow Tomas Petricek's definition to the letter, ASP.NET Core isn't a framework, but a library, because you have to write a `Main`

method and call `WebHost.CreateDefaultBuilder(args).UseStartup<Startup>().Build().Run()`

. In reality, though, you're supposed to configure the application from your `Startup`

class, making it the *de facto* entry point.)

### Unit testing endpoints #

Most .NET-based unit testing packages are frameworks. There's typically little explicit configuration. Instead, you just write a method and adorn it with an attribute:

[Fact] public async Task ReservationSucceeds() { var repo = new FakeReservationsRepository(); var sut = new ReservationsController(10, repo); var reservation = new Reservation( date: new DateTimeOffset(2018, 8, 13, 16, 53, 0, TimeSpan.FromHours(2)), email: "mark@example.com", name: "Mark Seemann", quantity: 4); var actual = await sut.Post(reservation); Assert.True(repo.Contains(reservation.Accept())); var expectedId = repo.GetId(reservation.Accept()); var ok = Assert.IsAssignableFrom<OkActionResult>(actual); Assert.Equal(expectedId, ok.Value); } [Fact] public async Task ReservationFails() { var repo = new FakeReservationsRepository(); var sut = new ReservationsController(10, repo); var reservation = new Reservation( date: new DateTimeOffset(2018, 8, 13, 16, 53, 0, TimeSpan.FromHours(2)), email: "mark@example.com", name: "Mark Seemann", quantity: 11); var actual = await sut.Post(reservation); Assert.False(reservation.IsAccepted); Assert.False(repo.Contains(reservation)); Assert.IsAssignableFrom<InternalServerErrorActionResult>(actual); }

With xUnit.net, the attribute is called `[Fact]`

, but the principle is the same in NUnit and MSTest, only that names are different.

Where's the entry point?

Each test is it's own entry point. The test is (typically) the first user code that the test runner executes. Furthermore, each test runs independently of any other.

For the sake of argument, you could write each test case in a new application, and run all your test applications in parallel. It would be impractical, but it oughtn't change the way you organise the tests. Each test method is, conceptually, a mini-application.

A test method is its own Composition Root; or, more generally, each test has its own Composition Root. In fact, xUnit.net has various extensibility points that enable you to hook into the framework before each test method executes. You can, for example, combine a `[Theory]`

attribute with a custom `AutoDataAttribute`

, or you can adorn your tests with a `BeforeAfterTestAttribute`

. This doesn't change that the test runner will run each test case independently of all the other tests. Those pre-execution hooks play the same role as middleware in real applications.

You can, therefore, consider the Arrange phase the Composition Root for each test.

Thus, I don't consider the use of an Auto-mocking Container to be a Service Locator, since its role is to resolve object graphs at the entry point instead of locating services from arbitrary locations in the code base.

### Summary #

A Composition Root is located at, or near, the *entry point*. An entry point is where *user code* is first executed by a framework. Each unit test method constitutes a separate, independent entry point. Therefore, it's consistent with these definitions to use an Auto-mocking Container in a unit test.

## Tree catamorphism

*The catamorphism for a tree is just a single function with a particular type.*

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 a tree, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. 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.

A tree is a general-purpose data structure where each node in a tree has an associated value. Each node can have an arbitrary number of branches, including none.

The diagram shows an example of a tree of integers. The left branch contains a sub-tree with only a single branch, whereas the right branch contains a sub-tree with three branches. Each of the leaf nodes are trees in their own right, but they all have zero branches.

In this example, each branch at the 'same level' has the same depth, but this isn't required.

### C# catamorphism #

As a C# representation of a tree, I'll use the `Tree<T>`

class from A Tree functor. The catamorphism is this instance method on `Tree<T>`

:

public TResult Cata<TResult>(Func<T, IReadOnlyCollection<TResult>, TResult> func) { return func(Item, children.Select(c => c.Cata(func)).ToArray()); }

Contrary to previous articles, I didn't call this method `Match`

, but simply `Cata`

(for *catamorphism*). The reason is that those other methods are called `Match`

for a particular reason. The data structures for which they are catamorphisms are all Church-encoded sum types. For those types, the `Match`

methods enable a syntax similar to pattern matching in F#. That's not the case for `Tree<T>`

. It's not a sum type, and it isn't Church-encoded.

The method takes a single function as an input argument. This is the first catamorphism in this article series that isn't made up of a pair of some sort. The Boolean catamorphism is a pair of values, the Maybe catamorphism is a pair made up of a value and a function, and the Either catamorphism is a pair of functions. The tree catamorphism, in contrast, is just a single function.

The first argument to the function is a value of the type `T`

. This will be an `Item`

value. The second argument to the function is a finite collection of `TResult`

values. This may take a little time getting used to, but it's a collection of already reduced sub-trees. When you supply such a function to `Cata`

, that function must return a single value of the type `TResult`

. Thus, the function must be able to digest a finite collection of `TResult`

values, as well as a `T`

value, to a single `TResult`

value.

The `Cata`

method accomplishes this by calling `func`

with the current `Item`

, as well as by recursively applying itself to each of the sub-trees. Eventually, `Cata`

will recurse into leaf nodes, which means that `children`

will be empty. When that happens, the lambda expression inside `children.Select`

never runs, and recursion stops and unwinds.

### Examples #

You can use `Cata`

to implement most other behaviour you'd like `Tree<T>`

to have. In the original article on the Tree functor you saw an ad-hoc implementation of `Select`

, but instead, you can derive `Select`

from `Cata`

:

public Tree<TResult> Select<TResult>(Func<T, TResult> selector) { return Cata<Tree<TResult>>((x, nodes) => new Tree<TResult>(selector(x), nodes)); }

The lambda expression receives `x`

, an object of the type `T`

, as well as `nodes`

, which is a finite collection of already translated sub-trees. It simply translates `x`

with `selector`

and returns a `new Tree<TResult>`

with the translated value and the already translated `nodes`

.

This works just as well as the ad-hoc implementation; it passes all the same tests as shown in the previous article.

If you have a tree of numbers, you can add them all together:

public static int Sum(this Tree<int> tree) { return tree.Cata<int>((x, xs) => x + xs.Sum()); }

This uses the built-in Sum method for `IEnumerable<T>`

to add all the partly calculated sub-trees together, and then adds the value of the current node. In this and remaining examples, I'll use the tree shown in the above diagram:

Tree<int> tree = Tree.Create(42, Tree.Create(1337, Tree.Leaf(-3)), Tree.Create(7, Tree.Leaf(-99), Tree.Leaf(100), Tree.Leaf(0)));

You can now calculate the sum of all these nodes:

> tree.Sum() 1384

Another option is to find the maximum value anywhere in a tree:

public static int Max(this Tree<int> tree) { return tree.Cata<int>((x, xs) => xs.Any() ? Math.Max(x, xs.Max()) : x); }

This method again utilises one of the LINQ methods available via the .NET base class library: Max. It is, however, necessary to first check whether the partially reduced `xs`

is empty or not, because the `Max`

extension method on `IEnumerable<int>`

doesn't know how to deal with an empty collection (it throws an exception). When `xs`

is empty that implies a leaf node, in which case you can simply return `x`

; otherwise, you'll first have to use the `Max`

method on `xs`

to find the maximum value there, and then use `Math.Max`

to find the maximum of those two. (I'll here remind the attentive reader that finding the maximum number forms a semigroup and that semigroups accumulate when collections are non-empty. It all fits together. Isn't maths lovely?)

Using the same `tree`

as before, you can see that this method, too, works as expected:

> tree.Max() 1337

So far, these two extension methods are just specialised *folds*. In Haskell, `Foldable`

is a specific type class, and `sum`

and `max`

are available for all instances. As promised in the introduction to the series, though, there are some functions on trees that you can't implement using a fold. One of these is to count all the leaf nodes. You can still derive that functionality from the catamorphism, though:

public int CountLeaves() { return Cata<int>((x, xs) => xs.Any() ? xs.Sum() : 1); }

Like `Max`

, the lambda expression used to implement `CountLeaves`

uses Any to detect whether or not `xs`

is empty, which is when `Any`

is `false`

. Empty `xs`

indicates that you've found a leaf node, so return `1`

. When `xs`

isn't empty, it contains a collection of `1`

values - one for each leaf node recursively found; add them together with `Sum`

.

This also works for the same `tree`

as before:

> tree.CountLeaves() 4

You can also measure the maximum depth of a tree:

public int MeasureDepth() { return Cata<int>((x, xs) => xs.Any() ? 1 + xs.Max() : 0); }

This implementation considers a leaf node to have no depth:

> Tree.Leaf("foo").MeasureDepth() 0

This is a discretionary definition; you could also argue that, by definition, a leaf node ought to have a depth of one. If you think so, you'll need to change the `0`

to `1`

in the above `MeasureDepth`

implementation.

Once more, you can use `Any`

to detect leaf nodes. Whenever you find a leaf node, you return its depth, which, by definition, is `0`

. Otherwise, you find the maximum depth already found among `xs`

, and add `1`

, because `xs`

contains the maximum depths of all immediate sub-trees.

Using the same `tree`

again:

> tree.MeasureDepth() 2

The above `tree`

has the same depth for all sub-trees, so here's an example of a tilted tree:

> Tree.Create(3, . Tree.Create(1, . Tree.Leaf(0), . Tree.Leaf(0)), . Tree.Leaf(0), . Tree.Leaf(0), . Tree.Create(2, . Tree.Create(1, . Tree.Leaf(0)))) . .MeasureDepth() 3

To make it easier to understand, I've labelled all the leaf nodes with `0`

, because that's their depth. I've then labelled the other nodes with the maximum number 'under' them, plus one. That's the algorithm used.

### Tree F-Algebra #

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

and `cata`

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

As always, start with the underlying endofunctor. I've taken some inspiration from `Tree a`

from `Data.Tree`

, but changed some names:

data TreeF a c = NodeF { nodeValue :: a, nodes :: ListFix c } deriving (Show, Eq, Read) instance Functor (TreeF a) where fmap f (NodeF x ns) = NodeF x $ fmap f ns

Instead of using Haskell's standard list (`[]`

) for the sub-forest, I've used `ListFix`

from the article on list catamorphism. This should, hopefully, demonstrate how you can build on already established definitions derived from first principles.

As usual, I've called the 'data' type `a`

and the carrier type `c`

(for *carrier*). The `Functor`

instance as usual translates the carrier type; the `fmap`

function has the type `(c -> c1) -> TreeF a c -> TreeF a c1`

.

As was the case when deducing the recent catamorphisms, Haskell isn't too happy about defining instances for a type like `Fix (TreeF a)`

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

wrapper:

newtype TreeFix a = TreeFix { unTreeFix :: Fix (TreeF 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 pair of helper functions make it easier to define `TreeFix`

values:

leafF :: a -> TreeFix a leafF x = TreeFix $ Fix $ NodeF x nilF nodeF :: a -> ListFix (TreeFix a) -> TreeFix a nodeF x = TreeFix . Fix . NodeF x . fmap unTreeFix

`leafF`

creates a leaf node:

Prelude Fix List Tree> leafF "ploeh" TreeFix {unTreeFix = Fix (NodeF "ploeh" (ListFix (Fix NilF)))}

`nodeF`

is a helper function to create a non-leaf node:

Prelude Fix List Tree> nodeF 4 (consF (leafF 9) nilF) TreeFix {unTreeFix = Fix (NodeF 4 (ListFix (Fix (ConsF (Fix (NodeF 9 (ListFix (Fix NilF)))) (Fix NilF)))))}

Even with helper functions, construction of `TreeFix`

values is cumbersome, but keep in mind that the code shown here isn't meant to be used in practice. The goal is only to deduce catamorphisms from more basic universal abstractions, and you now have all you need to do that.

### Haskell catamorphism #

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

), and an object `c`

, but you still need to find a morphism `TreeF 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 articles, start by writing a function that will become the catamorphism, based on `cata`

:

treeF = cata alg . unTreeFix where alg (NodeF x ns) = undefined

While this compiles, with its `undefined`

implementation of `alg`

, 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 `alg`

? You could pass a function argument to the `treeF`

function and use it with `x`

and `ns`

:

treeF :: (a -> ListFix c -> c) -> TreeFix a -> c treeF f = cata alg . unTreeFix where alg (NodeF x ns) = f x ns

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 `TreeF`

, the compiler infers that the `alg`

function has the type `TreeF 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 a tree. So far in this article series, all previous catamorphisms have been pairs, but this one is just a single function. It's still not the only possible catamorphism, since you could trivially flip the arguments to `f`

.

I've chosen the representation shown here because it's isomorphic to the `foldTree`

function from Haskell's built-in `Data.Tree`

module, which explicitly documents that the function "is also known as the catamorphism on trees." `foldTree`

is defined using Haskell's standard list type (`[]`

), so the type is simpler: `(a -> [b] -> b) -> Tree a -> b`

. The two representations of trees, `TreeFix`

and `Tree`

are, however, isomorphic, so `foldTree`

is equivalent to `treeF`

. Notice how both of these functions are also equivalent to the above C# `Cata`

method.

### Basis #

You can implement most other useful functionality with `treeF`

. Here's the `Functor`

instance:

instance Functor TreeFix where fmap f = treeF (nodeF . f)

`nodeF . f`

is just the point-free version of `\x ns -> nodeF (f x) ns`

, which follows the exact same implementation logic as the above C# `Select`

implementation.

The `Applicative`

instance is, I'm afraid, the most complex code you've seen so far in this article series:

instance Applicative TreeFix where pure = leafF ft <*> xt = treeF (\f ts -> addNodes ts $ f <$> xt) ft where addNodes ns (TreeFix (Fix (NodeF x xs))) = TreeFix (Fix (NodeF x (xs <> (unTreeFix <$> ns))))

I'd be surprised if it's impossible to make this terser, but I thought that it was just complicated enough that I needed to make one of the steps explicit. The `addNodes`

helper function has the type `ListFix (TreeFix a) -> TreeFix a -> TreeFix a`

, and it adds a list of sub-trees to the top node of a tree. It looks worse than it is, but it really just peels off the wrappers (`TreeFix`

, `Fix`

, and `NodeF`

) to access the data (`x`

and `xs`

) of the top node. It then concatenates `xs`

with `ns`

, and puts all the wrappers back on.

I have to admit, though, that the `Applicative`

and `Monad`

instance in general are mind-binding to me. The `<*>`

operation, particularly, takes a *tree of functions* and has to combine it with a *tree of values*. What does that even mean? How does one do that?

Like the above, apparently. I took the `Applicative`

behaviour from `Data.Tree`

and made sure that my implementation is isomorphic. I even have a property to make 'sure' that's the case:

testProperty "Applicative behaves like Data.Tree" $ do xt :: TreeFix Integer <- fromTree <$> resize 10 arbitrary ft :: TreeFix (Integer -> String) <- fromTree <$> resize 5 arbitrary let actual = ft <*> xt let expected = toTree ft <*> toTree xt return $ expected === toTree actual

The `Monad`

instance looks similar to the `Applicative`

instance:

instance Monad TreeFix where t >>= f = treeF (\x ns -> addNodes ns $ f x) t where addNodes ns (TreeFix (Fix (NodeF x xs))) = TreeFix (Fix (NodeF x (xs <> (unTreeFix <$> ns))))

The `addNodes`

helper function is the same as for `<*>`

, so you may wonder why I didn't extract that as a separate, reusable function. I decided, however, to apply the rule of three, and since, ultimately, `addNodes`

appear only twice, I left them as the implementation details they are.

Fortunately, the `Foldable`

instance is easier on the eyes:

instance Foldable TreeFix where foldMap f = treeF (\x xs -> f x <> fold xs)

Since `f`

is a function of the type `a -> m`

, where `m`

is a `Monoid`

instance, you can use `fold`

and `<>`

to accumulate everything to a single `m`

value.

The `Traversable`

instance is similarly terse:

instance Traversable TreeFix where sequenceA = treeF (\x ns -> nodeF <$> x <*> sequenceA ns)

Finally, you can implement conversions to and from the `Tree`

type from `Data.Tree`

, using `ana`

as the dual of `cata`

:

toTree :: TreeFix a -> Tree a toTree = treeF (\x ns -> Node x $ toList ns) fromTree :: Tree a -> TreeFix a fromTree = TreeFix . ana coalg where coalg (Node x ns) = NodeF x (fromList ns)

This demonstrates that `TreeFix`

is isomorphic to `Tree`

, which again establishes that `treeF`

and `foldTree`

are equivalent.

### Relationships #

In this series, you've seen various examples of catamorphisms of structures that have no folds, catamorphisms that coincide with folds, and catamorphisms that are more general than the fold. The introduction to the series included this diagram:

The Either catamorphism is another example of a catamorphism that is more general than the fold, but that one turned out to be identical to the *bifold*. That's not the case here, because `TreeFix`

isn't a `Bifoldable`

instance at all.

There are operations on trees that you can implement with a fold, but some that you can't. Consider the tree in shown in the diagram at the beginning of the article. This is also the tree that the above C# examples use. In Haskell, using `TreeFix`

, you can define that tree like this:

tree = nodeF 42 (consF (nodeF 1337 (consF (leafF (-3)) nilF)) $ consF (nodeF 7 (consF (leafF (-99)) $ consF (leafF 100) $ consF (leafF 0) nilF)) nilF)

Yes, that almost looks like some Lisp dialect...

Since `TreeFix`

is `Foldable`

, and that type class already comes with `sum`

and `maximum`

functions, no further work is required to repeat the first two of the above C# examples:

*Tree Fix List Tree> sum tree 1384 *Tree Fix List Tree> maximum tree 1337

Counting leaves, or measuring the depth of a tree, on the other hand, is impossible with the `Foldable`

instance, but can be implemented using the catamorphism:

countLeaves :: Num n => TreeFix a -> n countLeaves = treeF (\_ xs -> if null xs then 1 else sum xs) treeDepth :: (Ord n, Num n) => TreeFix a -> n treeDepth = treeF (\_ xs -> if null xs then 0 else 1 + maximum xs)

The reasoning is the same as already explained in the above C# examples. The functions also produce the same results:

*Tree Fix List Tree> countLeaves tree 4 *Tree Fix List Tree> treeDepth tree 2

This, hopefully, illustrates that the catamorphism is more capable, and that the fold is just a (list-biased) specialisation.

### Summary #

The catamorphism for a tree is just a single function, which is recursively evaluated. It enables you to translate, traverse, and reduce trees in many interesting ways.

You can use the catamorphism to implement a (list-biased) fold, including enumerating all nodes as a flat list, but there are operations (such as counting leaves) that you can implement with the catamorphism, but not with the fold.

**Next:** Rose tree catamorphism.

## Either catamorphism

*The catamorphism for Either is a generalisation of its fold. The catamorphism enables operations not available via 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 Either (also known as *Result*), as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. 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.

*Either* is a data container that models two mutually exclusive results. It's often used to return values that may be either correct (*right*), or incorrect (*left*). In statically typed functional programming with a preference for total functions, Either offers a saner, more reasonable way to model success and error results than throwing exceptions.

### C# catamorphism #

This article uses the Church encoding of Either. The catamorphism for Either is the `Match`

method:

T Match<T>(Func<L, T> onLeft, Func<R, T> onRight);

Until this article, all previous catamorphisms have been a pair made from an initial value and a function. The Either catamorphism is a generalisation, since it's a pair of functions. One function handles the case where there's a *left* value, and the other function handles the case where there's a *right* value. Both functions must return the same, unifying type, which is often a string or something similar that can be shown in a user interface:

> IEither<TimeSpan, int> e = new Left<TimeSpan, int>(TimeSpan.FromMinutes(3)); > e.Match(ts => ts.ToString(), i => i.ToString()) "00:03:00" > IEither<TimeSpan, int> e = new Right<TimeSpan, int>(42); > e.Match(ts => ts.ToString(), i => i.ToString()) "42"

You'll often see examples like the above that turns both left and right cases into strings or something that can be represented as a unifying response type, but this is in no way required. If you can come up with a unifying type, you can convert both cases to that type:

> IEither<Guid, string> e = new Left<Guid, string>(Guid.NewGuid()); > e.Match(g => g.ToString().Count(c => 'a' <= c), s => s.Length) 12 > IEither<Guid, string> e = new Right<Guid, string>("foo"); > e.Match(g => g.ToString().Count(c => 'a' <= c), s => s.Length) 3

In the two above examples, you use two different functions that both reduce respectively `Guid`

and `string`

values to a number. The function that turns `Guid`

values into a number counts how many of the hexadecimal digits that are greater than or equal to A (10). The other function simply returns the length of the `string`

, if it's there. That example makes little sense, but the `Match`

method doesn't care about that.

In practical use, Either is often used for error handling. The article on the Church encoding of Either contains an example.

### Either F-Algebra #

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

and `cata`

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

While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. You already saw an example of that in the articles about Boolean catamorphism and Maybe catamorphism. The difference between e.g. Maybe values and Either is that both cases of Either carry a value. You can model this as a `Functor`

with both a carrier type and two type arguments for the data that Either may contain:

data EitherF l r c = LeftF l | RightF r deriving (Show, Eq, Read) instance Functor (EitherF l r) where fmap _ (LeftF l) = LeftF l fmap _ (RightF r) = RightF r

I chose to call the 'data types' `l`

(for *left*) and `r`

(for *right*), and the carrier type `c`

(for *carrier*). As was also the case with `BoolF`

and `MaybeF`

, the `Functor`

instance ignores the map function because the carrier type is missing from both the `LeftF`

case and the `RightF`

case. Like the `Functor`

instances for `BoolF`

and `MaybeF`

, it'd seem that nothing happens, but at the type level, this is still a translation from `EitherF l r c`

to `EitherF l r c1`

. Not much of a function, perhaps, but definitely an *endofunctor*.

As was also the case when deducing the Maybe and List catamorphisms, Haskell isn't too happy about defining instances for a type like `Fix (EitherF l r)`

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

wrapper:

newtype EitherFix l r = EitherFix { unEitherFix :: Fix (EitherF l r) } 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 pair of helper functions make it easier to define `EitherFix`

values:

leftF :: l -> EitherFix l r leftF = EitherFix . Fix . LeftF rightF :: r -> EitherFix l r rightF = EitherFix . Fix . RightF

With those functions, you can create `EitherFix`

values:

Prelude Data.UUID Data.UUID.V4 Fix Either> leftF <$> nextRandom EitherFix {unEitherFix = Fix (LeftF e65378c2-0d6e-47e0-8bcb-7cc29d185fad)} Prelude Data.UUID Data.UUID.V4 Fix Either> rightF "foo" EitherFix {unEitherFix = Fix (RightF "foo")}

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 (`EitherF l r`

), and an object `c`

, but you still need to find a morphism `EitherF l r 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 types' `l`

and `r`

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

and `r`

, as you'll see.

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

:

eitherF = cata alg . unEitherFix where alg (LeftF l) = undefined alg (RightF r) = 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 `LeftF`

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

function:

eitherF fl = cata alg . unEitherFix where alg (LeftF l) = fl l alg (RightF r) = undefined

While you could, technically, pass an argument of the type `c`

to `eitherF`

and then return that value from the `LeftF`

case, that would mean that you would ignore the `l`

value. This would be incorrect, so instead, make the argument a function and call it with `l`

. Likewise, you can deal with the `RightF`

case in the same way:

eitherF :: (l -> c) -> (r -> c) -> EitherFix l r -> c eitherF fl fr = cata alg . unEitherFix where alg (LeftF l) = fl l alg (RightF r) = fr r

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 `EitherF`

, the compiler infers that the `alg`

function has the type `EitherF l r 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 Either. As has been consistent so far, it's a pair, but now made from two functions. As you've seen repeatedly, this isn't the only possible catamorphism, since you can, for example, trivially flip the arguments to `eitherF`

.

I've chosen the representation shown here because it's isomorphic to the `either`

function from Haskell's built-in `Data.Either`

module, which calls the function the "Case analysis for the `Either`

type". Both of these functions (`eitherF`

and `either`

) are equivalent to the above C# `Match`

method.

### Basis #

You can implement most other useful functionality with `eitherF`

. Here's the `Bifunctor`

instance:

instance Bifunctor EitherFix where bimap f s = eitherF (leftF . f) (rightF . s)

From that instance, the `Functor`

instance trivially follows:

instance Functor (EitherFix l) where fmap = second

On top of `Functor`

you can add `Applicative`

:

instance Applicative (EitherFix l) where pure = rightF f <*> x = eitherF leftF (<$> x) f

Notice that the `<*>`

implementation is similar to to the `<*>`

implementation for `MaybeFix`

. The same is the case for the `Monad`

instance:

instance Monad (EitherFix l) where x >>= f = eitherF leftF f x

Not only is `EitherFix`

`Foldable`

, it's `Bifoldable`

:

instance Bifoldable EitherFix where bifoldMap = eitherF

Notice, interestingly, that `bifoldMap`

is identical to `eitherF`

.

The `Bifoldable`

instance enables you to trivially implement the `Foldable`

instance:

instance Foldable (EitherFix l) where foldMap = bifoldMap mempty

You may find the presence of `mempty`

puzzling, since `bifoldMap`

(or `eitherF`

; they're identical) takes as arguments two functions. Is `mempty`

a function?

Yes, `mempty`

can be a function. Here, it is. There's a `Monoid`

instance for any function `a -> m`

, where `m`

is a `Monoid`

instance, and `mempty`

is the identity for that monoid. That's the instance in use here.

Just as `EitherFix`

is `Bifoldable`

, it's also `Bitraversable`

:

instance Bitraversable EitherFix where bitraverse fl fr = eitherF (fmap leftF . fl) (fmap rightF . fr)

You can comfortably implement the `Traversable`

instance based on the `Bitraversable`

instance:

instance Traversable (EitherFix l) where sequenceA = bisequenceA . first pure

Finally, you can implement conversions to and from the standard `Either`

type, using `ana`

as the dual of `cata`

:

toEither :: EitherFix l r -> Either l r toEither = eitherF Left Right fromEither :: Either a b -> EitherFix a b fromEither = EitherFix . ana coalg where coalg (Left l) = LeftF l coalg (Right r) = RightF r

This demonstrates that `EitherFix`

is isomorphic to `Either`

, which again establishes that `eitherF`

and `either`

are equivalent.

### Relationships #

In this series, you've seen various examples of catamorphisms of structures that have no folds, catamorphisms that coincide with folds, and now a catamorphism that is more general than the fold. The introduction to the series included this diagram:

This shows that Boolean values and Peano numbers have catamorphisms, but no folds, whereas for Maybe and List, the fold and the catamorphism is the same. For Either, however, the fold is a special case of the catamorphism. The fold for Either 'pretends' that the left side doesn't exist. Instead, the left value is interpreted as a missing right value. Thus, in order to fold Either values, you must supply a 'fallback' value that will be used in case an Either value isn't a *right* value:

Prelude Fix Either> e = rightF LT :: EitherFix Integer Ordering Prelude Fix Either> foldr (const . show) "" e "LT" Prelude Fix Either> e = leftF 42 :: EitherFix Integer Ordering Prelude Fix Either> foldr (const . show) "" e ""

In a GHCi session like the above, you can create two Either values of the same type. The *right* case is an `Ordering`

value, while the *left* case is an `Integer`

value.

With `foldr`

, there's no way to access the *left* case. While you can access and transform the right `Ordering`

value, the number `42`

is simply ignored during the fold. Instead, the default value `""`

is returned.

Contrast this with the catamorphism, which can access both cases:

Prelude Fix Either> e = rightF LT :: EitherFix Integer Ordering Prelude Fix Either> eitherF show show e "LT" Prelude Fix Either> e = leftF 42 :: EitherFix Integer Ordering Prelude Fix Either> eitherF show show e "42"

In a session like this, you recreate the same values, but using the catamorphism `eitherF`

, you can now access and transform both the *left* and the *right* cases. In other words, the catamorphism enables you to perform operations not possible with the fold.

It's interesting, however, to note that while the fold is a specialisation of the catamorphism, the *bifold* is identical to the catamorphism.

### Summary #

The catamorphism for Either is a pair of functions. One function transforms the *left* case, while the other function transforms the *right* case. For any Either value, only one of those functions will be used.

When I originally encountered the concept of a *catamorphism*, I found it difficult to distinguish between catamorphism and fold. My problem was, I think, that the tutorials I ran into mostly used linked lists to demonstrate how, in that case, the fold *is* the catamorphism. It turns out, however, that this isn't always the case. A catamorphism is a general abstraction. A fold, on the other hand, seems to me to be mostly related to collections.

In this article you saw the first example of a catamorphism that can do more than the fold. For Either, the fold is just a special case of the catamorphism. You also saw, however, how the catamorphism was identical to the *bifold*. Thus, it's still not entirely clear how these concepts relate. Therefore, in the next article, you'll get an example of a container where there's no bifold, and where the catamorphism is, indeed, a generalisation of the fold.

**Next:** Tree catamorphism.

## List catamorphism

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

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!

`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 the next 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:** Either catamorphism.

## Maybe catamorphism

*The catamorphism for Maybe is just a simplification of its fold.*

This article presents the catamorphism for Maybe, as well as how to identify it. The beginning of this article presents the catamorphism in C#, with examples. 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.

*Maybe* is a data container that models the absence or presence of a value. Contrary to null, Maybe has a type, so offers a sane and reasonable way to model that situation.

### C# catamorphism #

This article uses Church-encoded Maybe. Other, alternative implementations of Maybe are possible. The catamorphism for Maybe is the `Match`

method:

TResult Match<TResult>(TResult nothing, Func<T, TResult> just);

Like the Peano catamorphism, the Maybe catamorphism is a pair of a value and a function. The `nothing`

value corresponds to the absence of data, whereas the `just`

function handles the presence of data.

Given, for example, a Maybe containing a number, you can use `Match`

to get the value out of the Maybe:

> IMaybe<int> maybe = new Just<int>(42); > maybe.Match(0, x => x) 42 > IMaybe<int> maybe = new Nothing<int>(); > maybe.Match(0, x => x) 0

The functionality is, however, more useful than a simple *get-value-or-default* operation. Often, you don't have a good default value for the type potentially wrapped in a Maybe object. In the core of your application architecture, it may not be clear how to deal with, say, the absence of a `Reservation`

object, whereas at the boundary of your system, it's evident how to convert both absence and presence of data into a unifying type, such as an HTTP response:

Maybe<Reservation> maybe = // ... return maybe .Select(r => Repository.Create(r)) .Match<IHttpActionResult>( nothing: Content( HttpStatusCode.InternalServerError, new HttpError("Couldn't accept.")), just: id => Ok(id));

This enables you to avoid special cases, such as null `Reservation`

objects, or magic numbers like `-1`

to indicate the absence of `id`

values. At the boundary of an HTTP-based application, you know that you must return an HTTP response. That's the unifying type, so you can return `200 OK`

with a reservation ID in the response body when data is present, and `500 Internal Server Error`

when data is absent.

### Maybe F-Algebra #

`Fix`

and `cata`

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

While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. You already saw an example of that in the article about Boolean catamorphism. The difference between Boolean values and Maybe is that the *just* case of Maybe carries a value. You can model this as a `Functor`

with both a carrier type and a type argument for the data that Maybe may contain:

data MaybeF a c = NothingF | JustF a deriving (Show, Eq, Read) instance Functor (MaybeF a) where fmap _ NothingF = NothingF fmap _ (JustF x) = JustF x

I chose to call the 'data type' `a`

and the carrier type `c`

(for *carrier*). As was also the case with `BoolF`

, the `Functor`

instance ignores the map function because the carrier type is missing from both the `NothingF`

case and the `JustF`

case. Like the `Functor`

instance for `BoolF`

, it'd seem that nothing happens, but at the type level, this is still a translation from `MaybeF a c`

to `MaybeF a c1`

. Not much of a function, perhaps, but definitely an *endofunctor*.

In the previous articles, it was possible to work directly with the fixed points of both functors; i.e. `Fix BoolF`

and `Fix NatF`

. Haskell isn't happy about attempts to define various instances for `Fix (MaybeF a)`

, so in order to make this easier, you can define a `newtype`

wrapper:

newtype MaybeFix a = MaybeFix { unMaybeFix :: Fix (MaybeF a) } deriving (Show, Eq, Read)

In order to make it easier to work with `MaybeFix`

you can add helper functions to create values:

nothingF :: MaybeFix a nothingF = MaybeFix $ Fix NothingF justF :: a -> MaybeFix a justF = MaybeFix . Fix . JustF

You can now create `MaybeFix`

values to your heart's content:

Prelude Fix Maybe> justF 42 MaybeFix {unMaybeFix = Fix (JustF 42)} Prelude Fix Maybe> nothingF MaybeFix {unMaybeFix = Fix NothingF}

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 (`MaybeF`

), and an object `a`

, but you still need to find a morphism `MaybeF 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`

:

maybeF = cata alg . unMaybeFix where alg NothingF = undefined alg (JustF x) = 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 `NothingF`

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

function:

maybeF n = cata alg . unMaybeFix where alg NothingF = n alg (JustF x) = undefined

The `JustF`

case, contrary to `NothingF`

, already contains a value, and it'd be incorrect to ignore it. On the other hand, `x`

is a value of type `a`

, and you need to return a value of type `c`

. You'll need a function to perform the conversion, so pass such a function as an argument to `maybeF`

as well:

maybeF :: c -> (a -> c) -> MaybeFix a -> c maybeF n f = cata alg . unMaybeFix where alg NothingF = n alg (JustF x) = f x

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 `MaybeF`

, the compiler infers that the `alg`

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

, which is just what you need!

`c`

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

Notice that `maybeF`

, like the above C# `Match`

method, takes as arguments a pair of a value and a function (together with the Maybe value itself). Those are two representations of the same idea. Furthermore, this is nearly identical to the `maybe`

function in Haskell's `Data.Maybe`

module. I found if fitting, therefore, to name the function `maybeF`

.

### Basis #

You can implement most other useful functionality with `maybeF`

. Here's the `Functor`

instance:

instance Functor MaybeFix where fmap f = maybeF nothingF (justF . f)

Since `fmap`

should be a structure-preserving map, you'll have to map the *nothing* case to the *nothing* case, and *just* to *just*. When calling `maybeF`

, you must supply a value for the *nothing* case and a function to deal with the *just* case. The *nothing* case is easy to handle: just use `nothingF`

.

In the *just* case, first apply the function `f`

to map from `a`

to `b`

, and then use `justF`

to wrap the `b`

value in a new `MaybeFix`

container to get `MaybeFix b`

.

`Applicative`

is a little harder, but not much:

instance Applicative MaybeFix where pure = justF f <*> x = maybeF nothingF (<$> x) f

The `pure`

function is just *justF* (pun intended). The *apply* operator `<*>`

is more complex.

Both `f`

and `x`

surrounding `<*>`

are `MaybeFix`

values: `f`

is `MaybeFix (a -> b)`

, and `x`

is `MaybeFix a`

. While it's becoming increasingly clear that you can use a catamorphism like `maybeF`

to implement most other functionality, to which `MaybeFix`

value should you apply it? To `f`

or `x`

?

Both are possible, but the code looks (in my opinion) more readable if you apply it to `f`

. Again, when `f`

is *nothing*, return `nothingF`

. When `f`

is *just*, use the functor instance to map `x`

(using the infix `fmap`

alias `<$>`

).

The `Monad`

instance, on the other hand, is almost trivial:

instance Monad MaybeFix where x >>= f = maybeF nothingF f x

As usual, map *nothing* to *nothing* by supplying `nothingF`

. `f`

is already a function that returns a `MaybeFix b`

value, so just use that.

The `Foldable`

instance is likewise trivial (although, as you'll see below, you can make it even more trivial):

instance Foldable MaybeFix where foldMap = maybeF mempty

The `foldMap`

function must return a `Monoid`

instance, so for the *nothing* case, simply return the identity, *mempty*. Furthermore, `foldMap`

takes a function `a -> m`

, but since the `foldMap`

implementation is point-free, you can't 'see' that function as an argument.

Finally, for the sake of completeness, here's the `Traversable`

instance:

instance Traversable MaybeFix where sequenceA = maybeF (pure nothingF) (justF <$>)

In the *nothing* case, you can put `nothingF`

into the desired `Applicative`

with `pure`

. In the *just* case you can take advantage of the desired `Applicative`

being also a `Functor`

by simply mapping the inner value(s) with `justF`

.

Since the `Applicative`

instance for `MaybeFix`

equals `pure`

to `justF`

, you could alternatively write the `Traversable`

instance like this:

instance Traversable MaybeFix where sequenceA = maybeF (pure nothingF) (pure <$>)

I like this alternative less, since I find it confusing. The two appearances of `pure`

relate to two different types. The `pure`

in `pure nothingF`

has the type `MaybeFix a -> f (MaybeFix a)`

, while the `pure`

in `pure <$>`

has the type `a -> MaybeFix a`

!

Both implementations work the same, though:

Prelude Fix Maybe> sequenceA (justF ("foo", 42)) ("foo",MaybeFix {unMaybeFix = Fix (JustF 42)})

Here, I'm using the `Applicative`

instance of `(,) String`

.

Finally, you can implement conversions to and from the standard `Maybe`

type, using `ana`

as the dual of `cata`

:

toMaybe :: MaybeFix a -> Maybe a toMaybe = maybeF Nothing return fromMaybe :: Maybe a -> MaybeFix a fromMaybe = MaybeFix . ana coalg where coalg Nothing = NothingF coalg (Just x) = JustF x

This demonstrates that `MaybeFix`

is isomorphic to `Maybe`

, which again establishes that `maybeF`

and `maybe`

are equivalent.

### Alternatives #

As usual, the above `maybeF`

isn't the only possible catamorphism. A trivial variation is to flip its arguments, but other variations exist.

It's a recurring observation that a catamorphism is just a generalisation of a *fold*. In the above code, the `Foldable`

instance already looked as simple as anyone could desire, but another variation of a catamorphism for Maybe is this gratuitously embellished definition:

maybeF :: (a -> c -> c) -> c -> MaybeFix a -> c maybeF f n = cata alg . unMaybeFix where alg NothingF = n alg (JustF x) = f x n

This variation redundantly passes `n`

as an argument to `f`

, thereby changing the type of `f`

to `a -> c -> c`

. There's no particular motivation for doing this, apart from establishing that this catamorphism is exactly the same as the fold:

instance Foldable MaybeFix where foldr = maybeF

You can still implement the other instances as well, but the rest of the code suffers in clarity. Here's a few examples:

instance Functor MaybeFix where fmap f = maybeF (const . justF . f) nothingF instance Applicative MaybeFix where pure = justF f <*> x = maybeF (const . (<$> x)) nothingF f instance Monad MaybeFix where x >>= f = maybeF (const . f) nothingF x

I find that the need to compose with `const`

does nothing to improve the readability of the code, so this variation is mostly, I think, of academic interest. It does show, though, that the catamorphism of Maybe is isomorphic to its fold, as the diagram in the overview article indicated:

You can demonstrate that this variation, too, is isomorphic to `Maybe`

with a set of conversion:

toMaybe :: MaybeFix a -> Maybe a toMaybe = maybeF (const . return) Nothing fromMaybe :: Maybe a -> MaybeFix a fromMaybe = MaybeFix . ana coalg where coalg Nothing = NothingF coalg (Just x) = JustF x

Only `toMaybe`

has changed, compared to above; the `fromMaybe`

function remains the same. The only change to `toMaybe`

is that the arguments have been flipped, and `return`

is now composed with `const`

.

Since (according to Conceptual Mathematics) isomorphisms are transitive this means that the two variations of `maybeF`

are isomorphic. The latter, more complex, variation of `maybeF`

is identical `foldr`

, so we can consider the simpler, more frequently encountered variation a simplification of *fold*.

### Summary #

The catamorphism for Maybe is the same as its Church encoding: a pair made from a default value and a function. In Haskell's base library, this is simply called `maybe`

. In the above C# code, it's called `Match`

.

This function is total, and you can implement any other functionality you need with it. I therefore consider it the canonical representation of Maybe, which is also why it annoys me that most Maybe implementations come equipped with partial functions like `fromJust`

, or F#'s `Option.get`

. Those functions shouldn't be part of a sane and reasonable Maybe API. You shouldn't need them.

In this series of articles about catamorphisms, you've now seen the first example of catamorphism and fold coinciding. In the next article, you'll see another such example - probably the most well-known catamorphism example of them all.

**Next:** List catamorphism.

## Peano catamorphism

*The catamorphism for Peano numbers involves a base value and a successor function.*

This article presents the catamorphism for natural numbers, as well as how to identify it. The beginning of the article presents the catamorphism in C#, with examples. The rest of the article describes how I deduced 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.

### C# catamorphism #

In this article, I model natural numbers using Peano's model, and I'll reuse the Church-encoded implementation you've seen before. The catamorphism for `INaturalNumber`

is:

public static T Cata<T>(this INaturalNumber n, T zero, Func<T, T> succ) { return n.Match(zero, p => p.Cata(succ(zero), succ)); }

Notice that this is an extension method on `INaturalNumber`

, taking two other arguments: a `zero`

argument which will be returned when the number is *zero*, and a successor function to return the 'next' value based on a previous value.

The `zero`

argument is the easiest to understand. It's simply passed to `Match`

so that this is the value that `Cata`

returns when `n`

is *zero*.

The other argument to `Match`

must be a `Func<INaturalNumber, T>`

; that is, a function that takes an `INaturalNumber`

as input and returns a value of the type `T`

. You can supply such a function by using a lambda expression. This expression receives a predecessor `p`

as input, and has to return a value of the type `T`

. The only function available in this context, however, is `succ`

, which has the type `Func<T, T>`

. How can you make that work?

As is often the case when programming with generics, it pays to *follow the types*. A `Func<T, T>`

requires an input of the type `T`

. Do you have any `T`

objects around?

The only available `T`

object is `zero`

, so you could call `succ(zero)`

to produce another `T`

value. While you could return that immediately, that'd ignore the predecessor `p`

, so that's not going to work. Another option, which is the one that works, is to recursively call `Cata`

with `succ(zero)`

as the `zero`

value, and `succ`

as the second argument.

What this accomplishes is that `Cata`

keeps recursively calling itself until `n`

is *zero*. The `zero`

object, however, will be the result of repeated applications of `succ(zero)`

. In other words, `succ`

will be called as many times as the natural number. If `n`

is 7, then `succ`

will be called seven times, the first time with the original `zero`

value, the next time with the result of `succ(zero)`

, the third time with the result of `succ(succ(zero))`

, and so on. If the number is 42, `succ`

will be called 42 times.

### Arithmetic #

You can implement all the functionality you saw in the article on Church-encoded natural numbers. You can start gently by converting a Peano number into a normal C# `int`

:

public static int Count(this INaturalNumber n) { return n.Cata(0, x => 1 + x); }

You can play with the functionality in *C# Interactive* to get a feel for how it works:

> NaturalNumber.Eight.Count() 8 > NaturalNumber.Five.Count() 5

The `Count`

extension method uses `Cata`

to count the level of recursion. The `zero`

value is, not surprisingly, `0`

, and the successor function simply adds one to the previous number. Since the successor function runs as many times as encoded by the Peano number, and since the initial value is `0`

, you get the integer value of the number when `Cata`

exits.

A useful building block you can write using `Cata`

is a function to increment a natural number by one:

public static INaturalNumber Increment(this INaturalNumber n) { return n.Cata(One, p => new Successor(p)); }

This, again, works as you'd expect:

> NaturalNumber.Zero.Increment().Count() 1 > NaturalNumber.Eight.Increment().Count() 9

With the `Increment`

method and `Cata`

, you can easily implement addition:

public static INaturalNumber Add(this INaturalNumber x, INaturalNumber y) { return x.Cata(y, p => p.Increment()); }

The trick here is to use `y`

as the `zero`

case for `x`

. In other words, if `x`

is *zero*, then `Add`

should return `y`

. If `x`

isn't *zero*, then `Increment`

it as many times as the number encodes, but starting at `y`

. In other words, start with `y`

and `Increment`

`x`

times.

The catamorphism makes it much easier to implement arithmetic operation. Just consider multiplication, which wasn't the simplest implementation in the previous article. Now, it's as simple as this:

public static INaturalNumber Multiply(this INaturalNumber x, INaturalNumber y) { return x.Cata(Zero, p => p.Add(y)); }

Start at `0`

and simply `Add(y)`

`x`

times.

> NaturalNumber.Nine.Multiply(NaturalNumber.Four).Count() 36

Finally, you can also implement some common predicates:

public static IChurchBoolean IsZero(this INaturalNumber n) { return n.Cata<IChurchBoolean>(new ChurchTrue(), _ => new ChurchFalse()); } public static IChurchBoolean IsEven(this INaturalNumber n) { return n.Cata<IChurchBoolean>(new ChurchTrue(), b => new ChurchNot(b)); } public static IChurchBoolean IsOdd(this INaturalNumber n) { return new ChurchNot(n.IsEven()); }

Particularly `IsEven`

is elegant: It considers `zero`

even, so simply uses a `new ChurchTrue()`

object for that case. In all other cases, it alternates between *false* and *true* by negating the predecessor.

```
> NaturalNumber.Three.IsEven().ToBool()
false
```

It seems convincing that we can use `Cata`

to implement all the other functionality we need. That seems to be a characteristic of a catamorphism. Still, how do we know that `Cata`

is, in fact, the catamorphism for natural numbers?

### Peano 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 `NatF`

type comes from his article as well:

data NatF a = ZeroF | SuccF a deriving (Show, Eq, Read) instance Functor NatF where fmap _ ZeroF = ZeroF fmap f (SuccF x) = SuccF $ f x

You can use the fixed point of this functor to define numbers with a shared type. Here's just the first ten:

zeroF, oneF, twoF, threeF, fourF, fiveF, sixF, sevenF, eightF, nineF :: Fix NatF zeroF = Fix ZeroF oneF = Fix $ SuccF zeroF twoF = Fix $ SuccF oneF threeF = Fix $ SuccF twoF fourF = Fix $ SuccF threeF fiveF = Fix $ SuccF fourF sixF = Fix $ SuccF fiveF sevenF = Fix $ SuccF sixF eightF = Fix $ SuccF sevenF nineF = Fix $ SuccF eightF

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 (`NatF`

), and an object `a`

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

.

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

:

natF = cata alg where alg ZeroF = undefined alg (SuccF predecessor) = 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 `a`

from the `ZeroF`

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

function:

natF z = cata alg where alg ZeroF = z alg (SuccF predecessor) = undefined

In the `SuccF`

case, `predecessor`

is already of the polymorphic type `a`

, so instead of returning a constant value, you can supply a function as an argument to `natF`

and use it in that case:

natF :: a -> (a -> a) -> Fix NatF -> a natF z next = cata alg where alg ZeroF = z alg (SuccF predecessor) = next predecessor

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 `NatF`

, the compiler infers that the `alg`

function has the type `NatF a -> a`

, which is just what you need!

For good measure, I should point out that, as usual, the above `natF`

function isn't the only possible catamorphism. Trivially, you can flip the order of the arguments, and this would also be a catamorphism. These two alternatives are isomorphic.

The `natF`

function identifies the Peano number catamorphism, which is equivalent to the C# representation in the beginning of the article. I called the function `natF`

, because there's a tendency in Haskell to name the 'case analysis' or catamorphism after the type, just with a lower-case initial letter.

### Basis #

A catamorphism can be used to implement most (if not all) other useful functionality, like all of the above C# functionality. In fact, I wrote the Haskell code first, and then translated my implementations into the above C# extension methods. This means that the following functions apply the same reasoning:

evenF :: Fix NatF -> Fix BoolF evenF = natF trueF notF oddF :: Fix NatF -> Fix BoolF oddF = notF . evenF incF :: Fix NatF -> Fix NatF incF = natF oneF $ Fix . SuccF addF :: Fix NatF -> Fix NatF -> Fix NatF addF x y = natF y incF x multiplyF :: Fix NatF -> Fix NatF -> Fix NatF multiplyF x y = natF zeroF (addF y) x

Here are some GHCi usage examples:

Prelude Boolean Nat> evenF eightF Fix TrueF Prelude Boolean Nat> toNum $ multiplyF sevenF sixF 42

The `toNum`

function corresponds to the above `Count`

C# method. It is, again, based on `cata`

. You can use `ana`

to convert the other way:

toNum :: Num a => Fix NatF -> a toNum = natF 0 (+ 1) fromNum :: (Eq a, Num a) => a -> Fix NatF fromNum = ana coalg where coalg 0 = ZeroF coalg x = SuccF $ x - 1

This demonstrates that `Fix NatF`

is isomorphic to `Num`

instances, such as `Integer`

.

### Summary #

The catamorphism for Peano numbers is a pair consisting of a zero value and a successor function. The most common description of catamorphisms that I've found emphasise how a catamorphism is like a *fold;* an operation you can use to reduce a data structure like a list or a tree to a single value. This is what happens here, but even so, the `Fix NatF`

type isn't a `Foldable`

instance. The reason is that while `NatF`

is a polymorphic type, its fixed point `Fix NatF`

isn't. Haskell's `Foldable`

type class requires foldable containers to be polymorphic (what C# programmers would call 'generic').

When I first ran into the concept of a *catamorphism*, it was invariably described as a 'generalisation of fold'. The examples shown were always how the catamorphism for linked list is the same as its *fold*. I found such explanations unhelpful, because I couldn't understand how those two concepts differ.

The purpose with this article series is to show just how much more general the abstraction of a catamorphism is. In this article you saw how an infinitely recursive data structure like Peano numbers have a catamorphism, even though it isn't a parametrically polymorphic type. In the next article, though, you'll see the first example of a polymorphic type where the catamorphism coincides with the fold.

**Next:** Maybe catamorphism.

## Boolean catamorphism

*The catamorphism for Boolean values is just the common ternary operator.*

This article presents the catamorphism for Boolean values, as well as how you identify it. The beginning of this article presents the catamorphism in C#, with a simple example. The rest of the article describes how I deduced the catamorphism. That 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.

### C# catamorphism #

The catamorphism for Boolean values is the familiar ternary conditional operator:

> DateTime.Now.Day % 2 == 0 ? "Even date" : "Odd date" "Odd date"

Given a Boolean expression, you basically provide two values: one to use in case the Boolean expression is *true*, and one to use in case it's *false*.

For Church-encoded Boolean values, the catamorphism looks like this:

T Match<T>(T trueCase, T falseCase);

This is an instance method where you must, again, supply two alternatives. When the instance represents *true*, you'll get the left-most value `trueCase`

; otherwise, when the instance represents *false*, you'll get the right-most value `falseCase`

.

The catamorphism turns out to be the same as the Church encoding. This seems to be a recurring pattern.

### Alternatives #

To be accurate, there's more than one catamorphism for Boolean values. It's only by convention that the value corresponding to *true* goes on the left, and the *false* value goes to the right. You could flip the arguments, and it would still be a catamorphism. This is, in fact, what Haskell's `Data.Bool`

module does:

Prelude Data.Bool> bool "Odd date" "Even date" $ even date "Odd date"

The module documentation calls this the *"Case analysis for the Bool type"*, instead of a catamorphism, but the two representations are isomorphic:

```
"This is equivalent to
````if p then y else x`

; that is, one can think of it as an if-then-else construct with its arguments reordered."

This is another recurring result. There's typically more than one catamorphism, but the alternatives are isomorphic. In this article series, I'll mostly present the alternative that strikes me as the one you'll encounter most frequently.
### Fix #

In this and future articles, I'll derive the catamorphism from an F-Algebra. For an introduction to F-Algebras and fixed points, I'll refer you to Bartosz Milewski's excellent article on the topic. In it, he presents a generic data type for a fixed point, as well as polymorphic functions for catamorphisms and anamorphisms. While they're available in his article, I'll repeat them here for good measure:

newtype Fix f = Fix { unFix :: f (Fix f) } cata :: Functor f => (f a -> a) -> Fix f -> a cata alg = alg . fmap (cata alg) . unFix ana :: Functor f => (a -> f a) -> a -> Fix f ana coalg = Fix . fmap (ana coalg) . coalg

This should be recognisable from Bartosz Milewski's article. With one small exception, this is just a copy of the code shown there.

### Boolean F-Algebra #

While F-Algebras and fixed points are mostly used for recursive data structures, you can also define an F-Algebra for a non-recursive data structure. As data types go, they don't get much simpler than Boolean values, which are just two mutually exclusive cases. In order to make a `Functor`

out of the definition, though, you can equip it with a *carrier type:*

data BoolF a = TrueF | FalseF deriving (Show, Eq, Read) instance Functor BoolF where fmap _ TrueF = TrueF fmap _ FalseF = FalseF

The `Functor`

instance simply ignores the carrier type and just returns `TrueF`

and `FalseF`

, respectively. It'd seem that nothing happens, but at the type level, this is still a translation from `BoolF a`

to `BoolF b`

. Not much of a function, perhaps, but definitely an *endofunctor*.

Another note that may be in order here, as well as for all future articles in this series, is that you'll notice that most types and custom functions come with the `F`

suffix. This is simply a suffix I've added to avoid conflicts with built-in types, values, and functions, such as `Bool`

, `True`

, `and`

, and so on. The `F`

is for *F-Algebra*.

You can lift these values into `Fix`

in order to make it fit with the `cata`

function:

trueF, falseF :: Fix BoolF trueF = Fix TrueF falseF = Fix FalseF

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 (`BoolF`

), and an object `a`

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

. At first glance, this seems impossible, because neither `TrueF`

nor `FalseF`

actually contain a value of the type `a`

. How, then, can you conjure an `a`

value out of thin air?

The `cata`

function has the answer.

What you can do is to start writing the function that will become the catamorphism, basing it on `cata`

:

boolF = cata alg where alg TrueF = undefined alg FalseF = 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 `a`

from the `TrueF`

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

function:

boolF x = cata alg where alg TrueF = x alg FalseF = undefined

That seems promising, so do that for the `FalseF`

case as well:

boolF :: a -> a -> Fix BoolF -> a boolF x y = cata alg where alg TrueF = x alg FalseF = y

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 `BoolF`

, the compiler infers that the `alg`

function has the type `BoolF a -> a`

, which is just what you need!

The `boolF`

function identifies the Boolean catamorphism, which is equivalent to representations in the beginning of the article. I called the function `boolF`

, because there's a tendency in Haskell to name the 'case analysis' or catamorphism after the type, just with a lower-case initial letter.

You can use the `boolF`

function just like the above ternary operator:

Prelude Boolean Nat> boolF "Even date" "Odd date" $ evenF dateF "Odd date"

Here, I've also used `evenF`

from the `Nat`

module shown in the next article in the series.

From the above definition of `boolF`

, it should also be clear that you can arrive at the alternative catamorphism defined by `Data.Bool.bool`

by simply flipping `x`

and `y`

.

### Basis #

A catamorphism can be used to implement most (if not all) other useful functionality. For Boolean values, that would be the standard Boolean operations *and*, *or*, and *not:*

andF :: Fix BoolF -> Fix BoolF -> Fix BoolF andF x y = boolF y falseF x orF :: Fix BoolF -> Fix BoolF -> Fix BoolF orF x y = boolF trueF y x notF :: Fix BoolF -> Fix BoolF notF = boolF falseF trueF

They work as you'd expect them to work:

Prelude Boolean> andF trueF falseF Fix FalseF Prelude Boolean> orF trueF falseF Fix TrueF Prelude Boolean> orF (notF trueF) falseF Fix FalseF

You can also implement conversion to and from the built-in `Bool`

type:

toBool :: Fix BoolF -> Bool toBool = boolF True False fromBool :: Bool -> Fix BoolF fromBool = ana coalg where coalg True = TrueF coalg False = FalseF

This demonstrates that `Fix BoolF`

is isomorphic to `Bool`

.

### Summary #

The catamorphism for Boolean values is a function, method, or operator akin to the familiar ternary conditional operator. The most common descriptions of catamorphisms that I've found emphasise how a catamorphism is like a *fold;* an operation you can use to reduce a data structure like a list or a tree to a single value. In that light, it may be surprising that something as simple as Boolean values have an associated catamorphism.

Since `Fix BoolF`

is isomorphic to `Bool`

, you may wonder what the point is. Why define this data type, and implement functions like `andF`

, `orF`

, and `notF`

?

The code presented here is nothing but an analysis tool. It's a way to identify the catamorphism for Boolean values.

**Next:** Peano catamorphism.

## Catamorphisms

*A catamorphism is a general abstraction that enables you to handle multiple values, for example in order to reduce them to a single value.*

This article series is part of an even larger series of articles about the relationship between design patterns and category theory. In another article series in this big series of articles, you learned about functors, applicatives, and other types of data containers.

You may have heard about *map-reduce* architectures. Much software can be designed around two general types of operations: those that *map* data, and those that *reduce* data. A functor is a container of data that supports structure-preserving maps. Thus, you can think of functors as the general abstraction for map operations (also sometimes called *projections*). Does a similar universal abstraction exist for operations that reduce data?

Yes, that abstraction is called a *catamorphism*.

### Aggregation #

*Catamorphism* is an intimidating word, so let's start with an example. You often have a collection of values that you'd like to reduce to a single value. Such a collection can contain arbitrarily complex objects, but I'll keep it simple and start with a collection of numbers:

`new[] { 42, 1337, 2112, 90125, 5040, 7, 1984 };`

This particular list of numbers is an array, but that's not important. What comes next works for any `IEnumerable<T>`

, including arrays. I only chose an array because the C# syntax for array creation is more compact than for other collection types.

How do you reduce those seven numbers to a single number? That depends on what you want that number to tell you. One option is to add the numbers together. There's a specific, built-in function for that:

```
> new[] { 42, 1337, 2112, 90125, 5040, 7, 1984 }.Sum();
100647
```

The Sum extension method is a one of many built-in functions that enable you to reduce a list of numbers to a single number: Average, Max, Count, and so on.

What do you do, though, if you need to reduce many values to one, and there's no existing function for that? What if, for example, you need to add all the numbers using modulo 360 addition?

In that case, you use Aggregate:

```
> new[] { 42, 1337, 2112, 90125, 5040, 7, 1984 }.Aggregate((x, y) => (x + y) % 360)
207
```

The way to interpret this result is that the initial array represents a sequence of rotations (measured in degrees), and the result is the final angle after all the rotations have completed.

In other (functional) languages, such a 'reduce' operation is called a *fold*. The metaphor, I suppose, is that you fold multiple values together, two by two.

A *fold* is a catamorphism, but a catamorphism is a more general abstraction. For some data structures, the catamorphism is more powerful than the fold, but for collections, there's no difference.

There's one edge case we need to be aware of, though. What if the collection is empty?

### Aggregation of empty containers #

What happens if you attempt to aggregate an empty collection?

> new int[0].Aggregate((x, y) => (x + y) % 360) Sequence contains no elements + System.Linq.Enumerable.Aggregate<TSource>(IEnumerable<TSource>, Func<TSource, TSource, TSource>)

The `Aggregate`

method throws an exception because it doesn't know how to deal with empty collections. The lambda expression you supply tells the `Aggregate`

method how to combine two values into one. This is, for instance, how semigroups accumulate.

The lambda expression handles all cases where you have two or more values. If you have only a single value, then that's no problem either:

```
> new[] { 1337 }.Aggregate((x, y) => (x + y) % 360)
1337
```

In that case, the lambda expression isn't involved at all, because the single value is simply returned without modification. In this example, this could even be interpreted as being incorrect, since you'd expect the result to be 257 (`1337 % 360`

).

It's safer to use the `Aggregate`

overload that takes a *seed* value:

> new int[0].Aggregate(0, (x, y) => (x + y) % 360) 0

Not only does that gracefully handle empty collections, it also gives you a 'better' result for a single value:

```
> new[] { 1337 }.Aggregate(0, (x, y) => (x + y) % 360)
257
```

This works better because the method always starts with the *seed* value, which means that even if there's only a single value (`1337`

), the lambda expression still runs (`(0 + 1337) % 360`

).

This overload of `Aggregate`

has a different type, though:

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

Notice that the `func`

doesn't require the accumulator to have the same type as elements from the `source`

collection. This enables you to translate on the fly, so to speak. You can still use binary operations like the above modulo 360 addition, because that just implies that both `TSource`

and `TAccumulate`

are `int`

.

With this overload, you could, for example, use the Angle class to perform the work:

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

Now the `seed`

argument is `Angle.Identity`

, which implies that `TAccumulate`

is `Angle`

. The `source`

is still a collection of numbers, so `TSource`

is `int`

. Hence, I called the angle `a`

and the integer `i`

in the lambda expression. The output is an `Angle`

object that represents 207°.

That `Aggregate`

overload is the catamorphism for collections. It reduces a collection to an object.

### Catamorphisms and folds #

Is *catamorphism* just an intimidating word for *aggregate*, *accumulate*, *fold*, or *reduce?*

It took me a long time to be able to tell the difference, because in many cases, it seems that there's no difference. The purpose of this article series is to make the distinction clearer. In short, a catamorphism is a more general concept.

For some data structures, such as Boolean values, or Peano numbers, the catamorphism is all there is; no fold exists. For other data structures, such as Maybe or collections, the catamorphism and the fold coincide. Still other data structures, such as Either and trees, support folding, but the fold is based on the catamorphism. For those types, there are operations you can do with the catamorphism that are impossible to implement with the *fold* function. One example is that a tree's catamorphism enables you to count its leaves; you can't do that with its *fold* function.

You'll see plenty of examples in this article series:

- Boolean catamorphism
- Peano catamorphism
- Maybe catamorphism
- List catamorphism
- Either catamorphism
- Tree catamorphism
- Rose tree catamorphism
- Full binary tree catamorphism
- Payment types catamorphism

Each of these articles will contain a fair amount of Haskell code, but even if you're an object-oriented programmer who doesn't read Haskell, you should still scan them, as I'll start each with some C# examples. The Haskell code, by the way, is available on GitHub.

### Greek #

When encountering a word like *catamorphism*, your reaction might be:

"Catamorphism?! What does that even mean? It's all Greek to me."Indeed, it's Greek, as is so much of mathematical terminology. The

*cata*prefix means 'down'; lots of words start with

*cata*, like

*catastrophe*,

*catalogue*,

*catatonia*,

*catacomb*, etc.

The *morph* suffix generally means 'shape'. While the *cata* prefix appears in common words like *catastrophe*, the *morph* suffix mostly appears in more academic contexts. Programmers will probably have encountered *polymorphism* and *skeuomorphism*, not to mention isomorphism. While *morphism* is heavily used in mathematics, other sciences use the suffix too, like *dimorphism* in biology.

In category theory, a *morphism* is basically just an arrow that points from one object to another. Think of it as a function.

If a morphism is just a function, why don't we just call it that, then? Is it really necessary with this intimidating terminology? Yes and no.

If someone had originally figured all of this out in the context of mainstream programming, he or she would probably have used friendlier names, like *condense*, *reduce*, *fold*, and so on. This would have been more encouraging, although I'm not sure it would have been better.

In software architecture we use many overloaded terms. For example, what's a *service*, or a *client?* What does *tier* mean? Is it the same as a *layer*, or is it something different? What's the difference between a library and a framework?

At least a word like *catamorphism* is concise. It's not in common use, so isn't overloaded and vague.

Another, more pragmatic, concern is that whether you like it or not, the terminology is already established. Mathematicians decided to name the concept *catamorphism*. While the name may seem intimidating, I prefer to teach concepts like these using established terminology. This means that if my articles are unclear, you can do further research with other resources. That's the benefit of established terminology, whether you like the specific words or not.

### Summary #

You can compose entire applications based on the abstractions of *map* and *reduce*. You can see one example of such a system in my A Functional Architecture with F# Pluralsight course.

The terms *map* and *reduce* may, however, not be helpful, because it may not be clear exactly what types of data you can map, and what types you can reduce. One of the most important goals of this overall article series about universal abstractions is to help you identify when such software architectures apply. This is more often that you think.

What sort of data can you map? You can map *functors*. While hardly finite, there's a catalogue of well-known functors, of which I've covered some, but not all. That catalogue contains data containers like Maybe, Tree, lazy computations, tasks, and perhaps a score more. The catalogue of (actually useful) functors has, in my experience, a manageable size.

Likewise you could ask: What sort of data can you reduce? How do you implement that reduction? Again, there's a compact set of well-known catamorphisms. How do you reduce a collection? You use its catamorphism (which is equal to a fold). How do you reduce a tree? You use its catamorphism. How do you reduce an Either object? You use its catamorphism.

When we learn new programming languages, new libraries, new frameworks, we gladly invest time in learning hundreds, if not thousands, of keywords, APIs, extensibility points, and so on. May I offer, for your consideration, that your mental resources are better spent learning only a handful of universal abstractions?

**Next:** Boolean catamorphism.

## Applicative monoids

*An applicative functor containing monoidal values itself forms a monoid.*

This article is an instalment in an article series about applicative functors. An applicative functor is a data container that supports combinations. If an applicative functor contains values of a type that gives rise to a monoid, then the functor itself forms a monoid.

In a previous article you learned that lazy computations of monoids remain monoids. Furthermore, a lazy computation is an applicative functor, and it turns out that the result generalises. The result regarding lazy computation is just a special case.

### Monap #

Since version 4.11 of Haskell's *base* library, `Monoid`

is a subset of `Semigroup`

, so in order to create a `Monoid`

instance, you must first define a `Semigroup`

instance.

In order to escape the need for flexible contexts, you'll have to define a wrapper `newtype`

that'll be the instance. What should you call it? It's going to be an applicative functor of monoids, so perhaps something like *ApplicativeMonoid?* Nah, that's too long. *AppMon*, then? Sure, but how about flipping the terms: *MonApp?* That's better. Let's drop the last *p* and dispense with the Pascal case: *Monap*.

*Monap* almost looks like *Monad*, only with the last letter rotated half a revolution. This should allow for maximum confusion.

To be clear, I normally don't advocate for droll word play when writing production code, but I occasionally do it in articles and presentations. The *Monap* in this article exists only to illustrate a point. It's not intended to be used. Furthermore, this article doesn't discuss monads at all, so the risk of confusion should, hopefully, be minimised. I may, however, regret this decision...

### Applicative semigroup #

First, introduce the wrapper `newtype`

:

newtype Monap f a = Monap { runMonap :: f a } deriving (Show, Eq)

This only states that there's a type called `Monap`

that wraps some higher-kinded type `f a`

; that is, a container `f`

of values of the type `a`

. The intent is that `f`

is an applicative functor, hence the use of the letter *f*, but the type itself doesn't constrain `f`

to any type class.

The `Semigroup`

instance does, though:

instance (Applicative f, Semigroup a) => Semigroup (Monap f a) where (Monap x) <> (Monap y) = Monap $ liftA2 (<>) x y

This states that when `f`

is a `Applicative`

instance, and `a`

is a `Semigroup`

instance, then `Monap f a`

is also a `Semigroup`

instance.

Here's an example of combining two applicative semigroups:

λ> Monap (Just (Max 42)) <> Monap (Just (Max 1337)) Monap {runMonap = Just (Max {getMax = 1337})}

This example uses the `Max`

semigroup container, and `Maybe`

as the applicative functor. For `Max`

, the `<>`

operator returns the value that contains the highest value, which in this case is 1337.

It even works when the applicative functor in question is `IO`

:

λ> runMonap $ Monap (Sum <$> randomIO @Word8) <> Monap (Sum <$> randomIO @Word8) Sum {getSum = 165}

This example uses `randomIO`

to generate two random values. It uses the `TypeApplications`

GHC extension to make `randomIO`

generate `Word8`

values. Each random number is projected into the `Sum`

container, which means that `<>`

will add the numbers together. In the above example, the result is 165, but if you evaluate the expression a second time, you'll most likely get another result:

λ> runMonap $ Monap (Sum <$> randomIO @Word8) <> Monap (Sum <$> randomIO @Word8) Sum {getSum = 246}

You can also use linked list (`[]`

) as the applicative functor. In this case, the result may be surprising (depending on what you expect):

λ> Monap [Product 2, Product 3] <> Monap [Product 4, Product 5, Product 6] Monap {runMonap = [Product {getProduct = 8},Product {getProduct = 10},Product {getProduct = 12}, Product {getProduct = 12},Product {getProduct = 15},Product {getProduct = 18}]}

Notice that we get all the combinations of products: *2* multiplied with each element in the second list, followed by *3* multiplied by each of the elements in the second list. This shouldn't be that startling, though, since you've already, previously in this article series, seen several examples of how an applicative functor implies combinations.

### Applicative monoid #

With the `Semigroup`

instance in place, you can now add the `Monoid`

instance:

instance (Applicative f, Monoid a) => Monoid (Monap f a) where mempty = Monap $ pure $ mempty

This is straightforward: you take the identity (`mempty`

) of the monoid `a`

, promote it to the applicative functor `f`

with `pure`

, and finally put that value into the `Monap`

wrapper.

This works fine as well:

λ> mempty :: Monap Maybe (Sum Integer) Monap {runMonap = Just (Sum {getSum = 0})} λ> mempty :: Monap [] (Product Word8) Monap {runMonap = [Product {getProduct = 1}]}

The identity laws also seem to hold:

λ> Monap (Right mempty) <> Monap (Right (Sum 2112)) Monap {runMonap = Right (Sum {getSum = 2112})} λ> Monap ("foo", All False) <> Monap mempty Monap {runMonap = ("foo",All {getAll = False})}

The last, right-identity example is interesting, because the applicative functor in question is a tuple. Tuples are `Applicative`

instances when the first, or left, element is a `Monoid`

instance. In other words, `f`

is, in this case, `(,) String`

. The `Monoid`

instance that `Monap`

sees as `a`

, on the other hand, is `All`

.

Since tuples of monoids are themselves monoids, however, I can get away with writing `Monap mempty`

on the right-hand side, instead of the more elaborate template the other examples use:

λ> Monap ("foo", All False) <> Monap ("", mempty) Monap {runMonap = ("foo",All {getAll = False})}

or perhaps even:

λ> Monap ("foo", All False) <> Monap (mempty, mempty) Monap {runMonap = ("foo",All {getAll = False})}

Ultimately, all three alternatives mean the same.

### Associativity #

As usual, I'm not going to do the work of formally proving that the monoid laws hold for the `Monap`

instances, but I'd like to share some QuickCheck properties that indicate that they do, starting with a property that verifies associativity:

assocLaw :: (Eq a, Show a, Semigroup a) => a -> a -> a -> Property assocLaw x y z = (x <> y) <> z === x <> (y <> z)

This property is entirely generic. It'll verify associativity for any `Semigroup a`

, not only for `Monap`

. You can, however, run it for various `Monap`

types, as well. You'll see how this is done a little later.

### Identity #

Likewise, you can write two properties that check left and right identity, respectively.

leftIdLaw :: (Eq a, Show a, Monoid a) => a -> Property leftIdLaw x = x === mempty <> x rightIdLaw :: (Eq a, Show a, Monoid a) => a -> Property rightIdLaw x = x === x <> mempty

Again, this is entirely generic. These properties can be used to test the identity laws for any monoid, including `Monap`

.

### Properties #

You can run each of these properties multiple time, for various different functors and monoids. As `Applicative`

instances, I've used `Maybe`

, `[]`

, `(,) Any`

, and `Identity`

. As `Monoid`

instances, I've used `String`

, `Sum Integer`

, `Max Int16`

, and `[Float]`

. Notice that a list (`[]`

) is both an applicative functor as well as a monoid. In this test set, I've used it in both roles.

tests = [ testGroup "Properties" [ testProperty "Associativity law, Maybe String" (assocLaw @(Monap Maybe String)), testProperty "Left identity law, Maybe String" (leftIdLaw @(Monap Maybe String)), testProperty "Right identity law, Maybe String" (rightIdLaw @(Monap Maybe String)), testProperty "Associativity law, [Sum Integer]" (assocLaw @(Monap [] (Sum Integer))), testProperty "Left identity law, [Sum Integer]" (leftIdLaw @(Monap [] (Sum Integer))), testProperty "Right identity law, [Sum Integer]" (rightIdLaw @(Monap [] (Sum Integer))), testProperty "Associativity law, (Any, Max Int8)" (assocLaw @(Monap ((,) Any) (Max Int8))), testProperty "Left identity law, (Any, Max Int8)" (leftIdLaw @(Monap ((,) Any) (Max Int8))), testProperty "Right identity law, (Any, Max Int8)" (rightIdLaw @(Monap ((,) Any) (Max Int8))), testProperty "Associativity law, Identity [Float]" (assocLaw @(Monap Identity [Float])), testProperty "Left identity law, Identity [Float]" (leftIdLaw @(Monap Identity [Float])), testProperty "Right identity law, Identity [Float]" (rightIdLaw @(Monap Identity [Float])) ] ]

All of these properties pass.

### Summary #

It seems that any applicative functor that contains monoidal values itself forms a monoid. The `Monap`

type presented in this article only exists to demonstrate this conjecture; it's not intended to be *used*.

If it holds, I think it's an interesting result, because it further enables you to reason about the properties of complex systems, based on the properties of simpler systems.

**Next: ** Bifunctors.

## Comments

It seems that any applicative functor that contains monoidal values itself forms a monoid.

Is it necessary for the functor to be applicative? Do you know of a functor that contains monoidal values for which itself does *not* form a monoid?

Tyson, thank you for writing. Yes, it's necessary for the functor to be applicative, because you need the applicative combination operator `<*>`

in order to implement the combination. In C#, you'd need an `Apply`

method as shown here.

Technically, the monoidal `<>`

operator for `Monap`

is, as you can see, implemented with a call to `liftA2`

. In Haskell, you can implement an instance of `Applicative`

by implementing either `liftA2`

or `<*>`

, as well as `pure`

. You usually see `Applicative`

described by `<*>`

, which is what I've done in my article series on applicative functors. If you do that, you can define `liftA2`

by a combination of `<*>`

and `fmap`

(the `Select`

method that defines functors).

If you want to put this in C# terms, you need both `Select`

and `Apply`

in order to be able to lift a monoid into a functor.

Is there a functor that contains monoidal values that itself doesn't form a monoid?

Yes, indeed. In order to answer that question, we 'just' need to identify a functor that's *not* an applicative functor. Tuples are good examples.

A tuple forms a functor, but in general nothing more than that. Consider a tuple where the first element is a `Guid`

. It's a functor, but can you implement the following function?

public static Tuple<Guid, TResult> Apply<TResult, T>( this Tuple<Guid, Func<T, TResult>> selector, Tuple<Guid, T> source) { throw new NotImplementedException("What would you write here?"); }

You can pull the `T`

value out of `source`

and project it to a `TResult`

value with `selector`

, but you'll need to put it back in a `Tuple<Guid, TResult>`

. Which `Guid`

value are you going to use for that tuple?

There's no clear answer to that question.

More specifically, consider `Tuple<Guid, int>`

. This is a functor that contains monoidal values. Let's say that we want to use the addition monoid over integers. How would you implement the following method?

public static Tuple<Guid, int> Add(this Tuple<Guid, int> x, Tuple<Guid, int> y) { throw new NotImplementedException("What would you write here?"); }

Again, you run into the issue that while you can pull the integers out of the tuples and add them together, there's no clear way to figure out which `Guid`

value to put into the tuple that contains the sum.

The issue particularly with tuples is that there's no general way to combine the leftmost values of the tuples. If there is - that is, if leftmost values form a monoid - then the tuple is also an applicative functor. For example, `Tuple<string, int>`

is applicative and forms a monoid over addition:

public static Tuple<string, TResult> Apply<TResult, T>( this Tuple<string, Func<T, TResult>> selector, Tuple<string, T> source) { return Tuple.Create( selector.Item1 + source.Item1, selector.Item2(source.Item2)); } public static Tuple<string, int> Add(this Tuple<string, int> x, Tuple<string, int> y) { return Tuple.Create(x.Item1 + y.Item1, x.Item2 + y.Item2); }

You can also implement `Add`

with `Apply`

, but you're going to need two `Apply`

overloads to make it work.

Incidentally, unbeknownst to me, the `Ap`

wrapper was added to Haskell's `Data.Monoid`

module 12 days before I wrote this article. In all but name, it's equivalent to the `Monap`

wrapper presented here.

...if leftmost values form a monoid - then the tuple is also an applicative functor. For example,

`Tuple<string, int>`

is applicative...

I want to add some prepositional phrases to our statements like I commented here to help claify things. I don't think that `Tuple<string, int>`

can be applicative because there are no type parameters in which it could be applicative. Were you trying to say that `Tuple<string, B>`

is applicative in `B`

? This seems to match your `Apply`

function, which doesn't depend on `int`

.

Tyson, you're quite right; good catch. My wording was incorrect (I was probably either tired or in a hurry when I wrote that), but fortunately, the code looks like it's correct.

That you for pointing out my error.

...

`Tuple<string, int>`

is applicative and forms a monoid over addition...

I do agree with the monoid part, where "addition" means string concatenation for the first item and integer addition for the second item.

`Tuple<string, B>`

is applicative in`B`

Now I am trying to improve my understanding of this statement. In Haskell, my understanding the definition of the Applicative type class applied to `Tuple<string, B>`

requires a function `pure`

from `B`

to `Tuple<string, B>`

. What it the definition of this funciton? Does it use the empty string in order to make an instance of `Tuple<string, B>`

? If so, what is the justification for this? Or maybe my reasoning here is mistaken.

Tyson, thank you for writing. In Haskell, it's true that applicative functors must also define `pure`

. In this article series, I've glosssed over that constraint, since I'm not aware of any data containers that can lawfully implement `<*>`

or `liftA2`

, but *can't* define `pure`

.

The applicative instance for tuples is, however, constrained:

Monoid a => Applicative ((,) a)

The construct `((,) a)`

means any tuple where the first element has the generic type `a`

. The entire expression means that tuples are applicative functors when the first element forms a monoid; that's the restriction on `a`

. The definition of `pure`

, then, is:

pure x = (mempty, x)

That is, use the monoidal identity (`mempty`

) for the first element, and use `x`

as the second element. For strings, since the identity for string concatenation is the empty string, yes, it does exactly mean that `pure`

for `Tuple<string, B>`

would return a tuple with the empty string, and the input argument as the second element:

public static Tuple<string, T> Pure<T>(this T x) { return Tuple.Create("", x); }

That's the behaviour you get from Haskell as well:

Prelude Data.Monoid> pure 42 :: (String, Int) ("",42)

## Comments

My impression is that part of the functional programming style is to avoid function overloading. Consistent with that is the naming used by Language Ext for these concepts. In Language Ext, the function with type (in F# notation)

`seq<'a> -> ('a -> 'a -> 'a) -> 'a`

is called Reduce and the function with type (in F# notation)`seq<'a> -> 'b -> ('b -> 'a -> 'b) -> 'b`

is called Fold.I don't know the origin of these two names, but I remember the difference by thinking about preparing food. In cooking, reduction increases the concentration of a liquid by boiling away some of its water. I think of the returned

`'a`

as being a highly concentrated form of the input sequence since every sequence element (and only those elements) was used to create that return value. In baking, folding is a technique that carefully combines two mixtures into one. This reminds me of how the seed value`'b`

and the sequence of`'a`

are (typically) two different types and are combined by the given function. They are not perfect analogies, but they work for me.On a related note, I dislike that Reduce returns

`'a`

instead of`Option<'a>`

.Tyson, thank you for writing. As you may know, my book liberally employs cooking analogies, but I admit that I've never thought about

reductionandfoldin that light before. Good analogies, although perhaps a bitstrained(pun intended).They do work well, though, for the reasons you give.

As far as I can tell, this has more to do with the combination of Hindley–Milner type inference and currying you encounter in Haskell and ML-derived languages than it has to do with functional programming in itself. If I recall correctly, Clojure makes copious use of overloading.The problem with overloading in a language like F# is that if you imagine that the function you refer to as

`fold`

was also called`reduce`

, a partially applied expression like this would be ambiguous:What is

`bar`

, here? If`reduce`

is overloaded, is it a function, or is it a 'seed value'?As far as I can tell, the compiler can't infer that, so instead of compromising on type inference, the languages in question disallow function overloading.

Notice, additionally, that F# does allow

methodoverloading, for the part of the language that enables interoperability with the rest of .NET. In that part of the language, type inference rarely works anyway. I'm not an expert in how the F# compiler works, but I've always understood this to indicate that the interop part of F# isn't based on Hindley-Milner. I don't see how it could be, since the .NET/IL type system isn't a Hindley-Milner type system.The

`reduce`

function you refer to is, by the way, based on a semigroup instance. More specifically, it's simply how semigroups accumulate. I agree that`reduce`

is partial, and therefore not as pretty as one could wish, but I think a more appropriate solution is to define it on`NotEmptyCollection<T>`

, instead of on`IEnumerable<T>`

, as shown in that article.In other words, I don't think

`reduce`

belongs on`IEnumerable<T>`

at all. I know it's in both F# and Haskell, but my personal opinion is that it shouldn't be, just like Haskell's`head`

function ought not to exist either...