Blog tagged generics

Intersections and variance

Warning: the following post is for people who enjoy thinking about types, and want to understand how the Ceylon compiler reasons about them. This post is partly an attempt to demystify §3.7 of the spec. To understand the following, you first need to understand how variance works in Ceylon.

Snap quiz!


MutableList<Topic> | MutableList<Queue> list;
  1. What is the type of list.first?
  2. What is the type of the parameter of list.add()?


One of the key goals of Ceylon's type system is to make subtyping and generics play nicely together. Almost every object-oriented language since C++ and Eiffel has tried this, and more recent languages have been getting gradually closer to a really convincing solution, without, in my opinion, completely nailing it. Ceylon's contribution, of course, is to introduce first-class union and intersection types to the mix. In this post we'll see one of the ways in which they help.

Four useful identities

To begin to fully understand how the Ceylon typechecker reasons about generic types, you need to grasp the central importance of the following identities:

  1. Given a covariant type List<Element>, the union List<String>|List<Integer> is a subtype of List<String|Integer>.
  2. Given a contravariant type ListMutator<Element>, the union ListMutator<Queue>|ListMutator<Topic> is a subtype of ListMutator<Queue&Topic>.
  3. Given a covariant type List<Element>, the intersection List<Queue>&List<Topic> is a subtype of List<Queue&Topic>.
  4. Given a contravariant type ListMutator<Element>, the intersection ListMutator<String>&ListMutator<Integer> is a subtype of ListMutator<String|Integer>.

Note that in each identity, we take a union or intersection of two instantiations of the same generic type, and produce an instantiation of the generic type that is a supertype of the union or intersection.

These identities are actually quite intuitive. No? OK then, let's consider them one at a time, and convince ourselves that they make sense:

  1. If we have a list that, when iterated, either produces Strings or produces Integers then clearly every object it produces is either a String or an Integer.
  2. If we have a list in which we can either insert Queues or insert Topics, then we clearly have a list in which we can insert something that is both a Queue and a Topic.
  3. If we have a list that, when iterated produces only Queues, and, when iterated, produces only Topics, then clearly every object it produces must be both a Queue and a Topic.
  4. If we have a list in which we can both insert a String and insert an Integer, then we clearly have a list in which we can insert something that we know is either a String or an Integer.

Satisfied? Good.

Generic supertypes and principal instantiations

So, how are these identities useful? Well, imagine that we have a type T whose supertypes include multiple distinct instantiations of a generic type. For example, imagine that T has the supertypes List<String> and List<Integer>. And suppose we want to determine the type of one of the members inherited by T from the generic type, say List.first.

Then we would first need to form a principal instantiation of the generic type List. In this case the principal instantiation would be List<String|Integer>, according to identity 1. It's a "principal" instantiation because every other instantiation of List that is a supertype of T is also a supertype of List<String|Integer>. That it is even possible to form a principal instantiation like this is one of the things that makes Ceylon's type system special. Now we can determine the type of first by substituting String|Integer for the type parameter of List. For the record, the result is the type String|Integer|Null.

A type like T arises when we explicitly write down an intersection like List<Queue>&List<Topic>, or a union like List<String>|List<Integer>, but it also arises through the use of inheritance.

Principal instantiation inheritance

In Java, a type can only inherit a single instantiation of a supertype. A class can't inherit (either directly or indirectly) both List<Queue> and List<Topic> in Java. We call this inheritance model single instantiation inheritance. Ceylon features a more flexible model called principal instantiation inheritance1, where this restriction does not apply. I'm allowed to write:

interface Queues satisfies List<Queue> { ... }
interface Topics satisfies List<Topic> { ... }

class Endpoints() satisfies Queues & Topics { ... }

In which case, Endpoints is a subtype of List<Queue>&List<Topic>. Then the typechecker uses identity 3 above to automatically infer that Endpoints is a List<Queue&Topic>. Other languages can't do this.

Problem: invariant types

Now notice something about the identities above: they don't say anything about invariant types. Unfortunately, I simply can't form a principal instantiation of MutableList that is a supertype of MutableList<Queue>&MutableList<Topic>. Or at least I can't within Ceylon's type system as it exists today.

Caveat: I'm not a fan of use-site variance. I've been too burned by it in Java. However, if we do ever decide to introduce use-site variance in Ceylon, which does remain a real possibility, what I've just said will no longer apply, since this instantiation:

MutableList<in Queue&Topic out Queue|Topic> 

would be a principal supertype for MutableList<Queue>&MutableList<Topic>. But who the hell wants to have to deal with nasty-looking types like that?


The significance of this is that we should, where reasonable, be careful with how we use invariant types in Ceylon. To be sure, we still need invariant types, especially for concrete implementations like the class Endpoints above, but we should try to declare public APIs on covariant and contravariant types.

Consider the case of MutableList. What I'm saying is that its interesting operations should all be inherited from the covariant List and the contravariant ListMutator. It doesn't matter much whether we declare MutableList as a mixin interface like this:

shared interface MutableList<Element> 
        satisfies List<Element> & 
                  ListMutator<Element> { ... }

Or just as a type alias, like this:

shared alias MutableList<Element> 
        => List<Element> & ListMutator<Element>;

In either case, the following code2 is accepted by the typechecker, assuming the interesting operations first and add() are defined on List and ListMutator respectively:

void doSomeStuff(MutableList<Topic>|MutableList<Queue> list) {
    Topic|Queue|Null topicOrQueue = list.first;
    if (!topicOrQueue exists) {
        Topic&Queue topicAndQueue = .... ;

Here we're calling both "producer" and "consumer" operations of the invariant type MutableList on an intersection of distinct instantiations of MutableList. The typechecker determined that the principal instantiation of the covariant supertype is List<Topic|Queue> and that the principal instantiation of the contravariant supertype is ListMutator<Topic&Queue>, and that therefore the code is sound.

1The notion of principal instantiation inheritance is mainly based on Ross Tate's research. As far as I know, the idea of combining principal instantiation inheritance with union and intersection types is original to Ceylon.

2Thanks for Riener Zwitzerloot for this example and for inspiring this post.