Types: Never commit too early - Part 1: The Free Theorems

August 15, 2018

This article is part of a series:

I remember a year ago, I was arguing with my colleague: he wanted to use Task (from Monix) in some function because Task is great and offer tons of features. I wanted to use F[_]: Sync because I didn’t want to commit too early.

I may have trust issues, but I don’t like to commit too early when it’s not necessary.

Since a while now, the community talks about the tagless final encoding (great posts on typelevel, on scalac.io, on SoftwareMill) where you write an algebra and let the effect type be a F[_]. This is the same thing: you don’t want your algebra to depend on a specific effect implementation because it’s orthogonal to what your algebra is dealing with (a specific domain).

This algebra deals with what we can do with items, the effect does not matter, it can be Id, IO, Task, who cares (the caller):

trait ItemRepository[F[_]] {
  def findAll: F[List[Item]]
  def find(name: ItemName): F[Option[Item]]
  def save(item: Item): F[Unit]
  def remove(name: ItemName): F[Unit]

We won’t talk that much about tagless final but more about the general case: function encodings. We’ll take a tour about the typeclasses usage, advantages and downsides, and will make our way to un-stack monad-transformers stacks.

Committing to an implementation can have an impact on dependents applications: the need to refactor, the need to add libraries, add converters, lifting the types. It can also have an impact on performance, it can even break at runtime, depending on which implementation is used.


Two Pine Trees

I remember years ago, we were using Future in our codebase (yikes!) then we added a library working with Task from scalaz. Therefore we needed to convert them back and forth. We were not the only ones to have this issue: this project verizon/delorean deals with this very same problem.

The API is quite simple:

val f: Future[Foo] = ???
val t: Task[Foo] = f.toTask
val f2: Future[Foo] = t.unsafeToFuture

It provides an isomorphism between Task and Future. The implementation is quite straight-forward for experimented developers, but can be intimidating when you’re not fluent in Scala yet, hence this library was a good idea.

But the point is that because we had to deal with Task, we had to add another dependency, and add code here and there (or add some implicits) to make it work. This is not practical, can lead to bugs, and add overhead everywhere. We couldn’t just move all our code from Future to Task right? What if some other library were using Future?

The library was not about asynchrony, therefore it should not have expose us some scalaz’s Task (even if this was a good thing!) and let us work with Future because our codebase was using it: this concept (the effect used) was orthogonal to the library’s purpose.

Connascence of Type

When coding, we must always follow the principle of the least power.

The less you can do in a function, the easiest to implement it and to reason about it. Your mind is focused on the function parameters and nothing else. This is why environments which work with global variables and functions are the worst. They are not given as parameter of your function, but you can use and invoke them “by magic”.

You must have the world in your mind when coding a small function, it’s not reassuring and moreover, it’s difficult! You need to rely on documentation to know what is possible, understand where the parameters come from etc. We know documentations are always up-to-date right?

We want to reduce the connascence of type to the maximum. We want to decouple our code from implementations the most. Why code my function for a String if polymorphism is enough? (def f[A](a: A)). Why code my function with a Future if F[_] is enough? (def f(a: A): F[A])

Types are documentation —Tony Morris

People who implement libraries must commit only to use the bare minimum for their libraries to do their job. They should abstract upon what is not their core. We don’t want to fall into the case I described previously with Future and Task.

It’s not only reserved for library implementers: any application we develop should follow this principle. We all have colleagues (or our future-self) who are going to read and understand the code later on (and during the code review). Therefore their effort should be made easy the more we can.

Free Theorems

When you read a function signature, ask yourself this question: what this function can do?

def something[A](a: A): A = ???

There is only one possible implementation here.

Just look at the types and ask yourself. The simpler the types, the easiest to answer the question, without referring to the implementation. A simple type is a type with a few instances only (ie: has a low complexity).

Types Complexity

  • Boolean is a simple type: it can only be true or false.
  • String has a infinite complexity.
  • Co-products (sum-types) can be easy to reason about:
trait Furniture
case object Chair extends Furniture
case object Table extends Furniture

A Furniture has only 2 instances.

A more complete ADT with mix of products and co-products can be more difficult to reason about, for the same reason as mentioned:

trait Furniture
case class Chair(size: (Double, Double)) extends Furniture
case class Table(height: Double, chairs: List[Chair]) extends Furniture

The complexity increases because Chair and Table are products.

Polymorphism helps

The polymorphism of a function helps understanding what it does: its parametricity tells us what the function can and cannot do:

def something[A](in: List[A]): List[A]

This function cannot create new element A. It works at the container level. It can only return an empty list or a list with some or all the elements of the original list (like reverse them). It cannot do anything else, it cannot manipulate the elements themselves: A has no method associated with!

Compared this to:

def something(in: List[String]): List[String]

This function can do anything beyond our imagination. It create elements, alter the existing elements, split each items by words: we have no idea!

Polymorphism is a way to limit what functions can do. It can be complicated to read, because it can be so abstract.

This is an example of Parallel.parTraverse in cats:

def parTraverse[T[_]: Traverse, M[_], F[_], A, B]
  (ta: T[A])(f: A => M[B])(implicit P: Parallel[M, F]): M[T[B]] = {
    val gtb: F[T[B]] = Traverse[T].traverse(ta)(f andThen P.parallel.apply)(P.applicative)

It take 5 type parameters and combine the whole thing to return a M[T[B]].

  • It has no idea what A and B are: it doesn’t care.
  • It knows T has the Traverse capability (Traverse[T].traverse)
  • It knows it exists an instance of Parallel[M, F].

That’s it. You can’t do much with that right? But thanks to this, anyone satisfying those constraints (just 2) can call this function: it will work! The compiler will make sure of it.

Would you have prefered the function to be the following, all types explicit?

def parTraverse[A, B](ta: List[A])(f: A => Option[B]): Option[List[B]] = {
  val p = Parallel.identity[Option]
  val gtb: Option[List[B]] = ta.traverse(f andThen p.parallel.apply)(Applicative[Option])

parTraverse(List(1, 2, 3))(Some(_)) // Some(List(1, 2, 3))

Yes, it works! … But only for a List and Option. It’s not very open, I won’t use it.

Whereas I can use the original with anything:

// Try and Task
// Success(good)

// List and Either[Int, String]
val x: Either[Int, List[String]] = Parallel.parTraverse(List(1, 2, 3))(_.toString.asRight[Int])
// Right(List(1, 2, 3))

My advice: write a version without polymorphism first, then try to generalize it by adding more polymorphism (later, directly go to the polymorphism step, less to type!). You’ll probably stumbled upon functions you were calling you can’t anymore on F[_]: this is where you switch to typeclasses.


Philip Wadler studied the types in Theorems for free! (also see Free Theorems Involving Type Constructor Classes).

From a given function, you can deduce theorems on what it can do according to the types (input and output) used.

In any language, relying on types is the best way to have a strong understanding of a program: function names and comments are easily out of date, we should rarely rely on them. This is why it’s complicated to understand programs in dynamic-typed languages. :troll:

Unfortunately, Scala (the JVM) has some features that allows us to create edge cases going against some theorems. Think about null or throw: we could return null or throw an exception in our previous example:

def something[A](in: List[A]): List[A]

The types don’t show it, but it’s a possibility: subtle edge cases to handle.

In Scala, the null case is never even considered. It’s never checked in code that the argument or result are not null. It’s expected that we never ever use null in Scala (Option exists to expose the fact something is nullable).

Pure Functional Programming

This is where we encounter the Pure Functional Programming (PFP) principles, where functions must be:

  • Total: the behavior should be explicit just by looking at the types:
// Those are NOT total

def f[A](a: A): Boolean = a match { case String => true }
// What happens on f(5) ???
// It's not visible in the function signature!

sealed abstract class Option[+A] {
    def get: A
None.get // AH! Are we getting an A?
  • Deterministic: the same call should return the same result:
// Those are NOT deterministic

def f(a: Int): Boolean = if (Math.random() > 0.5) true else false
// is f(1) true or false?
  • Side-effects free: nothing outside of the scope of the function should be altered:
// Those are NOT side-effect free

def f(a: Int): Boolean = { println(s"a=$a"); a > 0 }

f(1) returns true AND alter stdout. But it’s not explicit in the types!

var c = 0
def f(): Boolean = { c += 1; c }
// f()
// => 1
// f()
// => 2

The same call returns different values and alter the outer scope. It’s difficult to reason about. It’s not referentially transparent:


// has a different outcome than:

val a = f()

If we follow those 3 rules, we unblock some achievements:

  • it’s easier to reason about the program, about the functions: the types don’t hide anything.
  • it’s easier to compose: small primitives goes into bigger and so on. All the types must be composed: nothing can be lost along the way, nothing can be hidden.
  • it provides bug-free refactoring: the behavior can’t be altered if everything is referentially transparent.
  • it allows for better concurrent programming: no shared state.

Those constraints make it difficult to work with Scala/Java/the JVM because we have “features” which do not follow those principles. Tom Morris suggested a subset of features to remove in Scala. It’s implemented through scalaz/scalazzi which will fail the compilation if non-PFP features are used: null, throw, mutable collections, side-effect functions not wrapped into IO

Automatic implementation generation

If a program follows PFP principles, we can rely on the Curry-Howard correspondence: types are theorems, programs are proofs.

It’s powerful and beyond my comprehension. A well-known talk is the one from Philip Wadler: Propositions as Types where it explains the links between logic, mathematics, and type theory.

In a more pragmatic way, there is a library based on this correspondence, which provides a macro to generate the implementation of any function, just by following its types: chymyst/curryhoward.

def f[X, Y](x: X, y: Y): X = implement
def g[X, Y](x: X, y: Y): Y = implement
f(5, "foo") // 5
g(5, "foo") // "foo"

It works for more complex examples:

case class Person(name: String, friends: List[Person])

def f(p: String): Person = implement
// Person(john, List())

// we can look at the implementation:
// f.lambdaTerm.prettyPrint
// a ⇒ Person2(a, (0 + Nil()))

But it’s not magic, it can’t be sure what’s you’re thinking about:

def addFriend(p: Person, p2: String): Person = implement
addFriend(f("john"), "henry")
// Person(henry, List())

As we said, we can never rely on function names!

If we declare the function as a value, we stumble upon more troubles:

val addFriend: (Person2, String) => Person2 = implement

Error:(28, 49) type (Person2, <c>String) ⇒ Person2 can be implemented in 2 inequivalent ways:
 a ⇒ a._1 [score: (0,10000,0,0,0)];
 a ⇒ Person2(a._2, (0 + Nil())) [score: (0,10000,0,0,0)].

The resolver find two ways (he’s right about that! but it also misses other ones) and can’t decide (it’s based on heuristics). TIMTOWTDI.

On a last note, we can use it to implement some typeclasses without thinking twice:

case class Person[A](id: A)

implicit val fperson = new Monad[Person] {
  override def flatMap[A, B](fa: Person[A])(f: A => Person[B]): Person[B] = implement
  override def pure[A](x: A): Person[A] = implement
  override def tailRecM[A, B](a: A)(f: A => Person[Either[A, B]]): Person[B] = ???
for {
  id <- 5.pure[Person]
  p <- Person(id + 100)
} yield p
// Person(105)

It’s not perfect, but it’s a very nice proof-of-concept. For instance, here, it couldn’t implement tailRecM (it’s not an easy one).

All that to say: provide simple types to your function, it makes it easier to know what it does and how it’s implemented.

Part 2: How typeclasses save us

Let’s take a look at our deus-ex machina: typeclasses!

We’ll take a look at some cats-effect typeclasses to see how it goes in real programs, but all the reasonings are valid for any typeclasses: Part 2: How typeclasses save us

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Me, Stéphane Derosiaux

Software & Data Engineer, I live in France. Enthousiast and eager to learn, I'm always seeking for challenges. Find out more about me.

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