Wednesday, 13 December 2017

Practical Awesome Recursion - Ch 02: Catamorphisms

Recursion schemes are awesome, and practical. This is chapter 2 in a series of blog posts about recursion schemes. This series uses Scala, and will focus on usage and applicability. It will be scarce in theory, and abundant in examples. The theory is valuable and fascinating, but I often find that knowing the theory alone is only half understanding. The other half of understanding comes from, and enables, practical application. I recommend you bounce back and forth between this series and theory. The internet is rich with blogs and videos on theory that explain it better than I would, so use those awesome resources.

Before you start...

If you don't know what I mean by any of the following, read or skim chapter 1 of this series.

  • Fix[_]
  • IntList / IntListF[_]
  • BinaryTree / BinaryTreeF[_]

Also, this series doesn't depend on, or emphasise, Matryoshka or any other library. If you'd like to understand why, I've explained here in the FAQ.

The Catamorphism

What is a catamorphism? It can be answered from a few perspectives.


It's often referred to as a fold over your data structure. Examples that sum or count values are very common. Conceptually speaking, an example using Scala stdlib would be List(1, 3, 7).foldLeft(0)(_ + _) == 11. As you'll see, folds and catamorphisms are capable of much more than calculating numbers.

The definition of catamorphism is:

def cata[F[_]: Functor, A, B](fAlgebra: F[A] => A)(f: Fix[F]): A =

The first argument fAlgebra is so-called because F[A] => A is known as an F-algebra. What it is, is your folding logic. You implement your folding logic as a function that processes a single level/layer of your structure without recursion. When I say level/layer, I mean in terms of recursive depth, example:


This is equivalent to List(1, 2, 3, 4, 5).

IntCons(1, _) ← Level 1
IntCons(2, _) ← Level 2
IntCons(3, _) ← Level 3
IntCons(4, _) ← Level 4
IntCons(5, _) ← Level 5
IntNil        ← Level 6

      Branch(_, "root", _)          ← Level 1
  |                       |
Leaf      Branch(_, "right", _)     ← Level 2
                 |           |
               Leaf         Leaf    ← Level 3


Look at the definition of cata and at the shape/type of the f-algebra. There's an interesting note about parametricity. The only means cata has of producing an A is to call fAlgebra... which requires an F[A]... so how does it put an A in the F[_] to call the function, if it can't produce As otherwise? Remember that the hole in F[_] represents the recursive case? In non-recursive cases (eg. Nil in a cons list) the type is a phantom-type, completely unused, or covariant and Nothing. For example:

case class Eg[T](int: Int)

def changeIt[X, Y](eg: Eg[X]): Eg[Y] =

When a type variable is unused you can replace it with anything you want, which is exactly what happens in Functor[F].

It's also important to understand the order in which things happen. Catamorphisms:

  1. start at the root (their input, the f: Fix[F])
  2. (computationally) move to the leaves
  3. calculate their way back to the root

Which means your folding logic is going to start executing against all the leaves first, then their parents, then their parents, etc.


I did say that this is a practical series. This is a good a place as any to mention that this cata definition, while correct, is inefficient.

Every time it recurses it has to create the same functions with the same logic over and over again. We can make it more efficient by creating what we need once and reusing it.

def cata2[F[_], A, B](algebra: F[A] => A)(f: Fix[F])(implicit F: Functor[F]): A = {
  var self: Fix[F] => A = null
  self = f => algebra(

In this new definition, we create the recursive function once and reuse it. Despite the null, it's 100% safe because we (provably) set it before it's used.

Let's measure it. How fast does it perform in comparison to the original? 105%? 110%?

[info] Benchmark         (size)  Mode  Cnt    Score   Error  Units
[info] RecursionBM.cata      10  avgt   10    0.274 ± 0.003  us/op
[info] RecursionBM.cata2     10  avgt   10    0.146 ± 0.001  us/op
[info] RecursionBM.cata     100  avgt   10    2.323 ± 0.020  us/op
[info] RecursionBM.cata2    100  avgt   10    1.555 ± 0.006  us/op
[info] RecursionBM.cata    1000  avgt   10   31.111 ± 0.720  us/op
[info] RecursionBM.cata2   1000  avgt   10   16.067 ± 0.187  us/op
[info] RecursionBM.cata   10000  avgt   10  326.443 ± 9.054  us/op
[info] RecursionBM.cata2  10000  avgt   10  165.470 ± 1.617  us/op

200%, it's twice as fast! Not bad for a tiny bit of one-off, hidden boilerplate.

Side-note: I ran the benchmark on a i7-6700HQ and all results are under 1ms, even at structure size of 1x10000 (length x depth), which means that either implementation is going to be fine on a fast CPU in a non-high-throughput solution. It'd be interesting to know what the measurements would be in prod, on real GCP/AWS VMs; the savings of the optimisation would be more significant because the VCPUs are slower.


This isn't necessary but I'm also going to add a type alias.

type FAlgebra[F[_], A] = F[A] => A

and tweak the catamorphism definition to

def cata2[F[_], A, B](algebra: FAlgebra[F, A])(f: Fix[F])(implicit F: Functor[F]): A = {
  var self: Fix[F] => A = null
  self = f => algebra(

Type aliases don't exist at runtime, the compilation process dealiases them completely. You can also use either definition interchangably.

def useAlias[F[_], A](f: F[A] => A): FAlgebra[F, A] =

def removeAlias[F[_], A](f: FAlgebra[F, A]): F[A] => A =

There are two advantages to having and using a type alias.

  1. Readability. The A and the F are separated; the A doesn't repeat; it clarifies intent the more you get used to recursion schemes.
  2. There are cases in which it helps type inference.

Simple Examples

Let's start with some basic examples: the usual blah -> Int stuff:

List Sum

Let's sum a list:

val listSum: FAlgebra[IntListF, Int] = {
  case IntListF.Cons(h, t) => h + t
  case IntListF.Nil        => 0

How does it work?

val listSumVerbose: IntListF[Int] => Int = {
  case IntListF.Cons(h, t) => h + t
  //                 |  |
  // Int by definition  |
  //                    Sum of tail (Int)
  case IntListF.Nil => 0

Notice this is an algebra, to actually use it in you need to call cata:

def sumThisListPlease(list: IntList): Int =

For reasons that will become obvious later, when using this stuff in your own project or libraries, the algebra itself is the unit that you'll be exposing most often. Instead of creating functions that take fixed-point data (or codata) structures and return a result, you create algebras and leave it to users to call cata themselves. More on this later but the point is, from the next example onwards, I'll show just the algebras.

List Length

Counting elements in a list is similar:

val listLength: FAlgebra[IntListF, Int] = {
  case IntListF.Cons(_, t) => 1 + t
  case IntListF.Nil        => 0

How does it work?

val listLengthVerbose: IntListF[Int] => Int = {
  case IntListF.Cons(_, t) => 1 + t
  //                    |     |
  //                    |     Add 1 for this element
  // Length of tail (Int)
  case IntListF.Nil => 0

BinaryTree Algebras

Here's a few algebras for BinaryTree:

val binaryTreeNodeCount: FAlgebra[BinaryTreeF[Any, ?], Int] = {
  case BinaryTreeF.Node(left, _, right) => left + 1 + right
  case BinaryTreeF.Leaf                 => 0

val binaryTreeMaxDepth: FAlgebra[BinaryTreeF[Any, ?], Int] = {
  case BinaryTreeF.Node(left, _, right) => left.max(right) + 1
  case BinaryTreeF.Leaf                 => 0

def binaryTreeSum[N](implicit N: Numeric[N]): FAlgebra[BinaryTreeF[N, ?], N] = {
  case BinaryTreeF.Node(left, n, right) =>,, right))
  case BinaryTreeF.Leaf                 =>

Pretty straight-forward. Each recursive slot (left & right) already has the computed value for that subtree.


Let's take a JSON value in our JSON ADT, and turn it into a JSON string that we can send out the door.

val jsonToString: FAlgebra[JsonF, String] = {
  case JsonF.Null        => "null"
  case JsonF.Bool(b)     => b.toString
  case JsonF.Num(n)      => n.toString
  case JsonF.Str(s)      => escapeString(s)
  case JsonF.Arr(values) => values.mkString("[", ",", "]")
  case JsonF.Obj(fields) => { case (k, v) => k + ":" + v }.mkString("{", ",", "}")

Is that easy or what? The array values and object fields are already all strings, we just mindlessly combine them using array/object notation.

What if, instead of the slower String concatenation, we wanted to use StringBuilder? Don't let the mutability discourage you; it's the same concept, we'll just replace String in the algebra type signature with StringBuilder => Unit. Executing a StringBuilder => Unit is mutable but the function itself is immutable, referentially transparent and pure. Descriptions of side-effects are safe.

val jsonToStringSB: FAlgebra[JsonF, StringBuilder => Unit] = {
  case JsonF.Null        => _ append "null"
  case JsonF.Bool(b)     => _ append b.toString
  case JsonF.Num(n)      => _ append n.toString
  case JsonF.Str(s)      => _ append escapeString(s)
  case JsonF.Arr(values) => sb => {
    sb append '['
    for (v <- values) v(sb)
    sb append ']'
  case JsonF.Obj(fields) => sb => {
    sb append '{'
    for ((k, v) <- fields) {
      sb append k
      sb append ':'
    sb append '}'

To be clear, usage would look like this:

def jsonToStringBuilderUsage(json: Json): String = {
  val sbToUnit = cata(jsonToStringSB)(json)
  val sb = new StringBuilder

A File System

Let's look at a more interesting example: a file system.

We'll start with a typical representation with hard-coded recursion.

sealed trait Entry
final case class Dir(files: Map[String, Entry]) extends Entry
final case class File(size: Long) extends Entry

This is an example inhabitant.

// Example of 3 files:
// 1. /usr/bin/find
// 2. /usr/bin/ls
// 3. /tmp/example.tmp
val example =
    "usr" -> Dir(Map(
      "bin" -> Dir(Map(
        "find" -> File(197360),
        "ls" -> File(133688))))),
    "tmp" -> Dir(Map(
      "example.tmp" -> File(12)))))

Now let's create an API:

def totalFileSize(e: Entry): Long = e match {
  case File(s) => s
  case Dir(fs) => fs.values.foldLeft(0L)(_ + totalFileSize(_))

def countFiles(e: Entry): Int = e match {
  case File(_) => 1
  case Dir(fs) => fs.values.foldLeft(0)(_ + countFiles(_))

def countDirs(e: Entry): Int = e match {
  case File(_) => 0
  case Dir(fs) => fs.values.foldLeft(1)(_ + countDirs(_))

Looks great! SHIP IT! Ok so now people are using our super-awesome file system and associated API. One day a user wants to collect a bunch of stats and writes the following code:

final case class Stats(totalSize: Long, files: Int, dirs: Int)

def stats(e: Entry): Stats =
  Stats(totalFileSize(e), countFiles(e), countDirs(e))

The user then complains that their stats method takes 3 times as long as other operations. This is because each stat is produced by traversing the entire file system. 3 stats = 3 traversals. Now obviously, with the pure definitions given above, the extra time is going to be negligible but imagine it's a real file system here, maybe even one distributed over the network, all that drive/network/hardware cost to traverse the file system is likely to be very noticable and very significant when it's repeated 3 times.

What can the user do? Well... nothing. The only recourse they have is to raise an issue and complain. They have no control or power in this situation. They're at the mercy of the decisions made by the library authors.

After a few years of complaints, the authors of the super-awesome file system library do a big rewrite and create a new API that looks a little something like this...

final case class Stats(totalSize: Long, files: Int, dirs: Int)

def stats(e: Entry): Stats = e match {

  case File(fileSize) =>
    Stats(fileSize, 1, 0)

  case Dir(fs) =>
    fs.values.foldLeft(Stats(0, 0, 0)) { (statsAcc, entry) =>
      val b = stats(entry)
        statsAcc.totalSize + b.totalSize,
        statsAcc.files + b.files,
        statsAcc.dirs + b.dirs)

def totalFileSize(e: Entry): Long =

def countFiles(e: Entry): Int =

def countDirs(e: Entry): Int =

KICK-ASS! Now all stats are gathered in a single traversal; issue: resolved. Except there are two problems now where there once was one. First, over the next few versions of the library more and more stats (permissions, ownership, etc.) get added, and the stats function gets bigger, more complex and more wild. Second, now new complaints start coming in, complaints like "totalFileSize() significantly slower since new update", "dirCount() 4x slower than in v1.8.3". What's going on? Well, now in every traversal we're spending more time fetching more data that we don't need and end up throwing away. If a user only wants a directory count, the library spends oodles of time looking up attributes, group names, etc. only to discard them all at the end.

What can the user do? Well... again: nothing. The only recourse they have is to raise an issue and complain. They have no control or power in this situation. They're yet again at the mercy of the decisions made by the library authors.

Fixing our FileSystem

Start with the usual changes: type param, Functor, Fix:

sealed trait EntryF[+F]
final case class Dir[F](files: Map[String, F]) extends EntryF[F]
final case class File(size: Long) extends EntryF[Nothing]

object EntryF {
  implicit val functor: Functor[EntryF] = new Functor[EntryF] {
    override def map[A, B](fa: EntryF[A])(f: A => B): EntryF[B] = fa match {
      case f: File => f
      case Dir(fs) => Dir( { case (k, v) => (k, f(v)) })

type Entry = Fix[EntryF]

Now let's add a little DSL and re-create our sample value:

object Entry {
  def apply(f: EntryF[Entry]): Entry = Fix(f)
  def file(s: Long): Entry = apply(File(s))
  def dir(es: (String, Entry)*): Entry = apply(Dir(es.toMap))

// Example of 3 files:
// 1. /usr/bin/find
// 2. /usr/bin/ls
// 3. /tmp/example.tmp
val example =
    "usr" -> Entry.dir(
      "bin" -> Entry.dir(
        "find" -> Entry.file(197360),
        "ls" -> Entry.file(133688))),
    "tmp" -> Entry.dir(
      "example.tmp" -> Entry.file(12)))

Next up are the queries. Small, reusable, independent functions are good practice for good reason; let's recreate what we had in the initial version:

val totalFileSize: FAlgebra[EntryF, Long] = {
  case File(s) => s
  case Dir(fs) => fs.values.sum

val countFiles: FAlgebra[EntryF, Int] = {
  case File(_) => 1
  case Dir(fs) => fs.values.sum

val countDirs: FAlgebra[EntryF, Int] = {
  case File(_) => 0
  case Dir(fs) => fs.values.sum + 1

If you compare this to the first attempt, it's just as simple from the outside, and even simpler in its internal implementation!

Now what happens when a user wants to get all three stats? F-Algebras compose. Here's a simple, reusable snippet to combine two F-algebras that share the same F:

def algebraZip[F[_], A, B](fa: FAlgebra[F, A],
                           fb: FAlgebra[F, B])
                          (implicit F: Functor[F]): FAlgebra[F, (A, B)] =
  fab => {
    val a = fa(
    val b = fb(
    (a, b)

This is all the user needs to create their own stats gathering algebra:

val statsAlg: FAlgebra[EntryF, (Long, (Int, Int))] =
  algebraZip(totalFileSize, algebraZip(countFiles, countDirs))

The (Long, (Int, Int)) is a little ugly, let's clean it up a bit:

final case class Stats(totalSize: Long, files: Int, dirs: Int)

def stats(e: Entry): Stats = {
  val (totalSize, (files, dirs)) = cata(statsAlg)(e)
  Stats(totalSize, files, dirs)

Hoorah. Without any dependencies on the library authors, our user was able to choose the 3 stats they were interested in, and retrieve them all in a single file system traversal. Whilst not parallelism (wait for the next episode in this series), they've created their own concurrency!

Now contrast this with the previous incarnation. This time around, the user has the control, the user has the power. Library authors don't need to decide tradeoffs for everyone.

These are the advantages of providing algebras instead of higher-purpose functions. F-algebras (like val statsAlg) allow you to accrue power; that power is spent when you apply it (like def stats).

Personally speaking, when using all this stuff in my own work projects, I often find it nice to provide two sets of interfaces:

  1. A low-level ecosystem of algebras, raw data types, logic, etc. Usually an entire package.
  2. A high-level DSL that very cleanly exposes common high-level functions and hides all the algebra composition, calls to cata and other morphisms, etc. Usually a single object.

This approach allows all devs to get the most common types of work done without any knowledge of recursion schemes or theory. They just see a small, high-level DSL that's easy to skim, understand and use. It also serves as a good means of teaching recursion schemes because devs unfamiliar with it can then click through into the lower-level code and see real-world examples in a domain they're familiar with. This makes for a nice experience on teams with differing skill levels, both in terms of code quality and a path to gradually up-skill the team.

To be continued...

That's plenty for now.

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  1. Super nice post. I think that your code could benefit from using something like foldMap (or cataMap if you like) that merges fold (or cata if you like) with map. But, maybe, that's the subjext if a subsequent post ...

  2. Thanks for the series, it's really great!
    I have one question: how do you make binary tree algebras, that have Any as an element type (e.g. binaryTreeNodeCount) work with more precisely typed trees (say BinaryTree[Int])?
    Fix is invariant, so it doesn't allow to put BinaryTree[Int] where BinaryTree[Any] is required. I could only make it work by making algebra a def with explicit element type parameter:

    def binaryTreeNodeCount[X]: FAlgebra[BinaryTreeF[Any, X], Int] = ...

    1. Sorry, correct version of the last line is:

      def binaryTreeNodeCount[X]: FAlgebra[BinaryTreeF[X, ?], Int] = ...

  3. You have a minor typo in a code sample that demonstrates old school technique you don't ultimately support:
    `fs.values.foldLeft(Stats(0, 0, 0)) { (statsAcc, entry) =>`
    should be changed to add 1 for the directory count
    `fs.values.foldLeft(Stats(0, 0, 1)) { (statsAcc, entry) =>`


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