Type Inference by Example, Part 7

Joakim Ahnfelt-Rønne
5 min readApr 18, 2020


At the end of this article, we can infer generic types like these.

Continuing where we left off in part 6, let’s add in mutually recursive functions with generics.

Consider the following example:

function even(x) { return x == 0 || odd(x - 1); }
function odd(x) { return x != 0 && even(x - 1); }

We say that these functions are mutually recursive: even calls odd and odd calls even. Because of that, we can’t infer the types of each function in isolation — we must consider both at the same time. Let’s add support for such mutually recursive functions to our syntax tree:

case class EFunctions(
functions : List[GenericFunction],
body : Expression
) extends Expression

The functions is a list of functions that may call each other, and they are in scope in body.

The next thing is that these functions may be generic — that is, they may have a type parameter list <…>, as in the classic map function:

function map<A, B>(array : Array<A>, f : A => B) : Array<B>

A generic function has a name and optionally a generic type. For the parameters and body, we’ll use the lambda function defined earlier in the series:

case class GenericFunction(
name : String,
typeAnnotation : Option[GenericType],
lambda : Expression

The GenericType is just a list of type parameters and an ordinary type:

case class GenericType(
generics : List[String],
uninstantiatedType : Type

It’s called uninstantiatedType because it may mention some of the type parameters from generics, and we’ll have to remember to instantiate that type, replacing the type parameters with fresh type variables on use.

Finally, when we use a variable, for example map, we need the ability to specify the type arguments, as in map<Int, String>(...), so we add generics to our variables:

case class EVariable(
name : String,
generics : List[Type] = List()
) extends Expression

Inferring the generic types

Earlier in the series, our environment looked like this: Map[String, Type]. We’re going to change Type to GenericType, so that variables can be generic in our environment. Now our infer function looks like this:

def infer(
environment : Map[String, GenericType],
expectedType : Type,
expression : Expression
) : Expression = expression match {

Whenever we encounter a variable, we’ll look up the variable in the environment and get its generic type back. We then replace each type parameter by a fresh type variable to get an ordinary type. If the user has specified the type arguments explicitly, we constrain each of these fresh type variables to be equal to the corresponding user supplied type.

case EVariable(name, generics) =>
val genericType = environment(name)
val newGenerics = genericType.generics.map(_ =>
val instantiation = genericType.generics.zip(newGenerics).toMap
val variableType = instantiate(
if(generics.nonEmpty) {
assert(generics.size == genericType.generics.size)
for((t, v) <- generics.zip(newGenerics)) {
+= CEquality(t, v)
typeConstraints += CEquality(expectedType, variableType)
EVariable(name, newGenerics)

Now we get to the mutually recursive, possibly generic functions. This is where we find out which type parameters our functions should have.

The first thing we need to do is to create an environment recursiveEnvironment that’s only for checking the mutually recursive functions. In this environment, the functions without a user supplied type annotation will get a fresh type variable instead. Then we use this environment to infer the type of each of the functions.

Now the functions — at least those without user supplied type annotations — have an ordinary type with type variables in them. We want to find out which of these type variables we can convert into type parameters to make the function generic. Before we do that, we solve all the outstanding type constraints to update our substitution, so we call solveConstraints().

After that, we call generalize on each of the functions that don’t have a user supplied type annotation. This finds out which type variables are safe to turn into type parameters. Now we have our GenericTypes for the functions.

The only thing that’s missing is to put these into an environment for inferring the body, where these functions are in scope.

case EFunctions(functions, body) =>  val recursiveEnvironment = 
environment ++ functions.map { function =>
function.name -> function.typeAnnotation.getOrElse(
GenericType(List(), freshTypeVariable())
val ungeneralizedFunctions = functions.map { function =>
val uninstantiatedType =
function.copy(lambda = infer(
solveConstraints() val newFunctions = ungeneralizedFunctions.map { function =>
if(function.typeAnnotation.nonEmpty) function else {
val functionType =
val (newTypeAnnotation, newLambda) = generalize(
typeAnnotation = Some(newTypeAnnotation),
lambda = newLambda
val newEnvironment =
environment ++ newFunctions.map { function =>
function.name -> function.typeAnnotation.get
val newBody = infer(newEnvironment, expectedType, body) EFunctions(newFunctions, newBody)

That’s it for the changes to the infer function.

} // the rest of this function remains the same

Instantiation and generalization

We’ve done some of the work to instantiate a GenericType to a Type already, so all we need to do in the instantiate function is to replace all the occurrences of the type parameter with the corresponding type:

def instantiate(
instantiation : Map[String, Type],
t : Type
) : Type = t match {
case _ if instantiation.isEmpty => t
case TVariable(i) if substitution(i) != TVariable(i) =>
instantiate(instantiation, substitution(i))
case TConstructor(name, generics) =>
instantiation.get(name).map { instantiationType =>
}.getOrElse {
TConstructor(name, generics.map(t =>
instantiate(instantiation, t)
case _ => t

The generalize function has the task of finding out which type variables in a type are safe to turn into type parameters.

So which ones are safe? Well, those type variables that won’t be constrained later in the code. The only way for the later code to get one of the type variables in the type to further constrain is if it occurs in the environment. That means we simply need exclude the type variables that occur in the environment.

In order to give these type parameters reasonable names, we consume from an iterator that produces names like A, B, C, etc.

We then replace those type variables with type parameters wherever they occur in the type and the expression. Along with that, we return a GenericType with the type parameters.

def generalize(
environment : Map[String, GenericType],
t : Type,
expression : Expression
) : (GenericType, Expression) = {
val genericTypeVariables =
freeInType(t) -- freeInEnvironment(environment)
val genericNames = genericTypeVariables.map(i =>
i -> genericParameterNames.next()
val localSubstitution = substitution.clone()
for((i, name) <- genericNames) {
localSubstitution(i) = TConstructor(name)
val newExpression =
substituteExpression(localSubstitution, expression)
val newType = substitute(localSubstitution, t)
genericNames.map { case (_, name) => name },
) -> newExpression

The freeInType function finds all the type variables that occur in a type. The freeInGenericType and freeInEnvironment does the same for generic types and the environment:

def freeInType(t : Type) : SortedSet[Int] = t match {
case TVariable(i) if substitution(i) != TVariable(i) =>
case TVariable(i) => SortedSet(i)
case TConstructor(_, generics) =>
generics.map(freeInType).fold(SortedSet()) { _ ++ _ }

def freeInGenericType(t : GenericType) : SortedSet[Int] = {

def freeInEnvironment(
environment : Map[String, GenericType]
) : SortedSet[Int] = {
fold(SortedSet[Int]()) { _ ++ _ }

And we’re done.


Now that we have the a version of our type inference that can infer generic types, it’s time to see it in action. View and run the code so far here.

Stay tuned for part 8, where we’ll take a look at overloading via type classes.



Joakim Ahnfelt-Rønne

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