Type Inference by Example, Part 4

Joakim Ahnfelt-Rønne
4 min readApr 12, 2020


Using unification to turn each equality constraint into a substitution.

Continuing where we left off in part 3, let’s take a look at unification.

The first thing we’ll need is a representation for types. Let’s look at the types we’ve seen so far:

  • Int — a plain type
  • Array<Int> — a generic type
  • (Int, Int) => Int — a function type
  • $1 — a type variable

The Array thing up there is called a type constructor — not to be confused with constructors in object oriented programming. A type constructor is something that takes type parameters — Array is a type constructor, and when given a type parameter, it becomes a type: Array<Int>.

The function type => is also a type constructor. When given its type parameters, it becomes a type: (Int, Int) => Int. To make this more clear, we’ll name function types FunctionN internally, where N is the number of arguments it takes, e.g. Function2<Int, Int, Int>. The last Int here is the return type of the function.

And if we squint a bit, we can say that Int is a nullary type constructor. Given no type parameters at all, it becomes a type: Int. Thus we arrive at the following representation:

sealed abstract class Typecase class TConstructor(
name : String,
generics : List[Type] = List()
) extends Type
case class TVariable(
index : Int
) extends Type

This is Scala code. The sealed modifier says that there are no subclasses apart from those listed here. A case class can be used as a dumb carrier for data, where each field is public and immutable by default. Let’s see how our types look in this internal representation:

  • IntTConstructor("Int")
  • Array<Int>TConstructor("Array", List(TConstructor("Int")))
  • (Int, Int) => IntTConstructor("Function2", List(TConstructor("Int"), TConstructor("Int"), TConstructor("Int")))
  • $1TVariable(1)

Back to unification.

What is unification? Unification is an algorithm that takes two types and finds a substitution that makes them equal, if such a substitution exists.

While inferring types, the substitution will grow as we unify types. Recalling that a substitution maps type variables to types, and type variables are identified by integers, we’ll use an expanding array for this:

val substitution = ArrayBuffer[Type]()

We’ll choose a substitution where each type variable is initially substituted with itself, i.e.substitution(x) == TypeVariable(x).

The unification itself is simple:

  1. Given two TConstructors, check that their name is equal and that they have the same number of type parameters. Then do a pointwise unification of the type arguments.
  2. Given a TVariable and another type, check if the type variable has been bound to something else than itself in the substitution. If so, unify whatever it’s bound to with the other type. Otherwise, update the unification by binding the type variable to the other type.

When binding a type variable, we must perform an occurs check to avoid constructing an infinite type such as $1 := Array<$1>.

The unification algorithm in code.

The unify function takes in two types and pattern matches on them.

def unify(t1 : Type, t2 : Type) : Unit = (t1, t2) match {

If we have two TConstructors, we check that their name is equal and that they have the same number of type parameters. Then we do a pointwise unification of the type arguments.

case (TConstructor(name1,generics1),TConstructor(name2,generics2))=>
assert(name1 == name2)
assert(generics1.size == generics2.size)
for((t1, t2) <- generics1.zip(generics2)) unify(t1, t2)

If both sides are the same type variable, do nothing.

case (TVariable(i), TVariable(j)) if i == j => // do nothing

If one of the types is a type variable that’s bound in the substitution, use unify with that type instead.

case (TVariable(i), _) if substitution(i) != TVariable(i) =>
unify(substitution(i), t2)
case (_, TVariable(i)) if substitution(i) != TVariable(i) =>
unify(t1, substitution(i))

Otherwise, if one of the types is an unbound type variable, bind it to the other type. Remember to do an occurs check to avoid constructing infinite types.

case (TVariable(i), _) =>
assert(!occursIn(i, t2))
substitution(i) = t2
case (_, TVariable(i)) =>
assert(!occursIn(i, t1))
substitution(i) = t1

The assertions should be replaced with proper error reporting, but that is a topic for another day.

} // That's it for unification

We’ll need the occursIn method as well, which simply recurses into the type and checks if the type variable index in question occurs:

def occursIn(index : Int, t : Type) : Boolean = t match {
case TVariable(i) if substitution(i) != TVariable(i) =>
occursIn(index, substitution(i))
case TVariable(i) =>
i == index
case TConstructor(_, generics) =>
generics.exists(t => occursIn(index, t))

And we’re done.

Stay tuned for part 5, where we’ll finish up the first version of the type inference.



Joakim Ahnfelt-Rønne

MSc Computer Science, working with functional programming in the industry — github.com/ahnfelt