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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.

1class Noun {}2class City extends Noun {}3class 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

1type CovariantOf<X> = {2  +prop: X;3  getter(): X;4}

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:

1type CovariantOf<+X> = {2  +prop: X;3  getter(): X;4}

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

1type NonCovariantOf<X> = {2  prop: X;3  getter(): X;4};

Let's also declare a variable nonCovariantCity of type 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 write a Noun into prop, invalidating the original type. Flow catches this:

1class Noun {}2class City extends Noun {}3
4type NonCovariantOf<X> = {5  prop: X;6  getter(): X;7};8
9declare const nonCovariantCity: NonCovariantOf<City>;10const nonCovariantNoun: NonCovariantOf<Noun> = nonCovariantCity; // Error!incompatible-typeCannot assign nonCovariantCity to nonCovariantNoun because in type argument X [1]: Noun [2] is incompatible with City [3].

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:

1type InvariantOf<X> = {2  prop: X;3  getter(): X;4  setter(X): void;5};

Assuming a variable invariantCity of type 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 other 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:

1type ContravariantOf<-X> = {2  -prop: X;3  setter(X): void;4};

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.

Function parameter contravariance​

Function parameters are always in an input (contravariant) position. This means a function that accepts a more specific type cannot substitute for one that accepts a more general type. This commonly surprises people when passing callbacks with exact object types:

1type Exact = {foo: string};2type Inexact = {foo: string, ...};3
4declare function acceptsExact(item: Exact): void;5declare function takesCallback(cb: (item: Inexact) => void): void;6
7takesCallback(acceptsExact); // Error!incompatible-exactCannot call takesCallback with acceptsExact bound to cb because in the first parameter: inexact Inexact [1] is incompatible with exact Exact [2].

This error occurs because takesCallback may call cb with an object that has extra properties (since Inexact allows them). The callback acceptsExact only accepts objects with exactly {foo: string}, so passing an inexact object to it would be unsound. Even though passing an exact object directly to a function expecting an inexact one works (an exact type is a subtype of a compatible inexact type), the function types are flipped due to contravariance.

Input and Output Positions​

Flow's error messages refer to "input positions" and "output positions" when reporting variance errors. These terms correspond directly to the variance concepts described above:

  • An output position is a place where a value is read out of a type: return types, read-only properties, getter results. A type parameter marked with + (covariant) can only appear in output positions.
  • An input position is a place where a value is written into a type: function parameters, write-only properties, setter arguments. A type parameter marked with - (contravariant) can only appear in input positions.
  • A type parameter with no sigil (invariant) can appear in both input and output positions.

When you see an error like "Cannot use T in an input position because T is expected to occur only in output positions," it means you have a type parameter marked as covariant (+T) but you are using it somewhere that writes a value in, such as a function parameter:

1type Box<+T> = {2  get(): T;3  set(val: T): void; // Error: T is in an input position but is expected only in output positionsincompatible-varianceCannot use T [1] in an input position because T [1] is expected to occur only in output positions.4};

The fix depends on your intent: if the type genuinely needs to both read and write T, remove the + sigil to make T invariant. If the type should only produce values of type T (never accept them), remove the setter.

See Also​

  • Subtypes — the underlying subtyping relationships that variance builds on
  • Generics — variance sigils on generic type parameters
  • Arrays — ReadonlyArray (covariant) vs Array (invariant)
  • Interfaces — covariant and contravariant interface properties
  • Objects — read-only and write-only object properties