Type Variance

Variance is a topic that comes up fairly often in type systems. It is used to determine how type parameters behave with respect to subtyping.

First we'll setup a couple of classes that extend one another.

class Noun {}
class City extends Noun {}
class SanFrancisco extends City {}

We saw in the section on generic types that it is possible to use variance sigils to describe when a type parameter is used in an output position, when it is used in an input position, and when it is used in either one.

Here we'll dive deeper into each one of these cases.

Covariance

Consider for example the type

type CovariantOf<X> = {
  +prop: X;
  getter(): X;
}

Here, X appears strictly in output positions: it is used to read out information from objects o of type CovariantOf<X>, either through property accesses o.prop, or through calls to o.getter().

Notably, there is no way to input data through the reference to the object o, given that prop is a readonly property.

When these conditions hold, we can use the sigil + to annotate X in the definition of CovariantOf:

type CovariantOf<+X> = {
  +prop: X;
  getter(): X;
}

These conditions have important implications on the way that we can treat an object of type CovariantOf<T> with respect to subtyping. As a reminder, subtyping rules help us answer the question: "given some context that expects values of type T, is it safe to pass in values of type S?" If this is the case, then S is a subtype of T.

Using our CovariantOf definition, and given that City is a subtype of Noun, it is also the case that CovariantOf<City> is a subtype of CovariantOf<Noun>. Indeed

• it is safe to read a property prop of type City when a property of type Noun is expected, and
• it is safe to return values of type City when calling getter(), when values of type Noun are expected.

Combining these two, it will always be safe to use CovariantOf<City> whenever a CovariantOf<Noun> is expected.

A commonly used example where covariance is used is $ReadOnlyArray<T>. Just like with the prop property, one cannot use a $ReadOnlyArray reference to write data to an array. This allows more flexible subtyping rules: Flow only needs to prove that S is a subtype of T to determine that $ReadOnlyArray<S> is also a subtype of $ReadOnlyArray<T>.

Invariance

Let's see what happens if we try to relax the restrictions on the use of X and make, for example, prop be a read-write property. We arrive at the type definition

type NonCovariantOf<X> = {
  prop: X;
  getter(): X;
};

Let's also declare a variable nonCovariantCity of type NonCovariantOf<City>

declare const nonCovariantCity: NonCovariantOf<City>;

Now, it is not safe to consider nonCovariantCity as an object of type NonCovariantOf<Noun>. Were we allowed to do this, we could have the following declaration:

const nonCovariantNoun: NonCovariantOf<Noun> = nonCovariantCity;

This type permits the following assignment:

nonCovariantNoun.prop = new Noun;

which would invalidate the original type for nonCovariantCity as it would now be storing a Noun in its prop field.

What distinguishes NonCovariantOf from the CovariantOf definition is that type parameter X is used both in input and output positions, as it is being used to both read and write to property prop. Such a type parameter is called invariant and is the default case of variance, thus requiring no prepending sigil:

type InvariantOf<X> = {
  prop: X;
  getter(): X;
  setter(X): void;
};

Assuming a variable

declare const invariantCity: InvariantOf<City>;

it is not safe to use invariantCity in a context where:

• an InvariantOf<Noun> is needed, because we should not be able to write a Noun to property prop.
• an InvariantOf<SanFrancisco> is needed, because reading prop could return a City which may not be SanFrancisco.

In orther words, InvariantOf<City> is neither a subtype of InvariantOf<Noun> nor a subtype of InvariantOf<SanFrancisco>.

Contravariance

When a type parameter is only used in input positions, we say that it is used in a contravariant way. This means that it only appears in positions through which we write data to the structure. We use the sigil - to describe this kind of type parameters:

type ContravariantOf<-X> = {
  -prop: X;
  setter(X): void;
};

Common contravariant positions are write-only properties and "setter" functions.

An object of type ContravariantOf<City> can be used whenever an object of type ContravariantOf<SanFrancisco> is expected, but not when a ContravariantOf<Noun> is. In other words, ContravariantOf<City> is a subtype of ContravariantOf<SanFrancisco>, but not ContravariantOf<Noun>. This is because it is fine to write SanFrancisco into a property that can have any City written to, but it is not safe to write just any Noun.