Coercions and Roles for Dummies

haskell, beginner

A discussion about roles broke out yesterday on fpchat which is a great place to hang out and talk about nerdy functional programming stuff if you don't already know about it. Go sign up! Anyway, people suggested I clean up my little soliloquy about roles and turn it into a blog post, so here we are.

You've heard of the type system, which makes sure your terms are sane. Maybe you're also aware of the kind system, whose job it is is to make sure your types are reasonable! But did you know Haskell has an even more obscure system than these? It's called the role system, and its purpose in life is to prevent you from shooting yourself in the foot when dealing with coercions.

Coercions and roles have been around since 2014, but there's been surprisingly little discussion about them in the blogosphere. In short, if two types have the same representation at runtime, then it should be safe to coerce a value of one into a value of the other. The role system is used to describe under what circumstances such a coercion is legal.

To illustrate the point, let's talk about newtypes. Consider the following:

newtype AnInt = AnInt Int

The promise of a newtype in Haskell is that it is zero-overhead; at runtime, AnInt is exactly identical to Int. Newtypes are often used for adding type-safety; it's nice if you have a newtype Temperature = Temperature Int and a newtype Money = Money Int because the extra type wrappers ensure you can't accidentally add the weather to your bank account, even if at the end of the day they are both just integers.

AnInt and Int are not literally the same type, but they don't actually differ at runtime. This property is known as being representationally equal. If two types are representationally equal, we should be able to do the equivalent of C++'s reinterpret_cast and just pretend like a value of one is in fact a value of the other. Since these types correspond exactly at runtime, this is usually a safe thing to do.

If AnInt and Int are the same type at runtime, it means we should be able to coerce :: AnInt -> Int (and backwards) freely between the two types without any problems. Morally, this coerce function is just id, because we're not actually doing any work to the value.

Consider now the slightly more interesting type:

newtype Identity a = Identity a

Again, because Identity is a newtype, we should expect Identity a to be representationally equal to a. Since this is true, we expect that Identity AnInt also be representationally equal to Identity Int, via Identity AnInt --> AnInt --> Int --> Identity Int. And thus, we should be able to coerce :: Identity AnInt -> Identity Int. We can see that Identity a preserves the coercion relationship of its type parameter a, and this property is known as the a having role representational.

More generally, if the type parameter a in F a has role representational, then F X is representationally equal to F Y whenever X is representationally equal to Y. This works whether F be a data or a newtype.

However, not all type parameters have role representational! Consider Data.Map.Map k v which has keys k and values v. Because Map is implemented as a balanced tree, it uses the Ord k instance to figure out where to store a kv-pair.

One of the reasons we write newtypes is to give a different typeclass instance than the underlying type has. For example, newtype ZipList a = ZipList [a] has a different Applicative instance than [] does. In general, we have no reason to expect that a newtype and its underlying type have instances that agree with one another.

Which leads us to a problem. Because a value of Map k v is a balanced tree which depends on the Ord k instance, we can't simply swap in SomeOtherK and expect everything to work hunky-dory. They have different Ord instances, and things would go screwy at runtime. All of this is to say that we do not want to be able to coerce :: Map AnInt v -> Map Int v because it's likely to crash at runtime.

However, it is still fine to coerce :: Map k AnInt -> Map k Int, because the values don't have this implicit dependency on any typeclass instances. There are no invariants to maintain on the v parameter, and so we are free to coerce to our hearts' content.

The role system is what describes this difference between the k and v type parameters of Data.Map.Map. While v is still role representational, k has role nominal.

nominal coercions of the form coerce :: a -> b are allowed iff you already have a proof that a ~ b, which is to say that a and b are literally the same type.

There's also a third role, phantom, which, you guessed it, is given to phantom type parameters (eg. the a in data Const x a = Const x.) Because phantom types are by-definition not referenced in the data definition of a type, we are always free to coerce a phantom type to any other type.

All of this cashes out in the form of Data.Coerce's coerce :: Coercible a b => a -> b. GHC will automatically provide instances of Coercible a b whenever a and b are representationally coercible. That means you get these instances (and all of their symmetries):

  • Given a ~ b, Coercible a b
  • If NT is a newtype over T, Coercible NT T
  • If p in F p has role phantom, Coercible (F a) (F b)
  • If r in F r has role representational, Coercible a b => Coercible (F a) (F b)
  • If n in F n has role nominal, (a ~ b) => Coercible (F a) (F b)

GHC is pretty clever, and has a role-inference mechanism. It works by knowing that (->) has two representational roles, that (~) has two nominal roles, and propagates from there. Every type parameter is assumed to have role phantom until it is used, whence it gets upgraded to the more restrictive role corresponding to the position it was used in. For example, if a is used in a ~ Bool, a gets role nominal since both of (~)'s parameters have nominal roles.

GADTs are syntactic sugar on top of (~) so expect GADTs to have nominal role type parameters. Furthermore, any parameters of a type family that are scrutinized will also have role nominal (the motivated reader will be able to find an interesting implementation of unsafeCoerce :: forall a b. a -> b if this were not the case.)

This inference mechanism will give you the most permissible roles that don't obviously destroy the type system, but sometimes it's necessary to explicitly give a role annotation, like in the Data.Map.Map example. Role annotations can be given by eg. type role Map nominal representational after turning on {-# LANGUAGE RoleAnnotations #-}. It's worth pointing out that you can only give less permissive roles than GHC has inferred; there's no fighting with it on this one.


At the end of the day, why is any of this stuff useful? Besides being nice to know, custom role annotations can provide type-safety ala Data.Map.Map. But we can also get asymptotic performance gains out of coerce:

If f :: Coercible a b => a -> b (common if f is a newtype un/wrapper, or composition of such), then fmap f O(n) is equivalent to coerce O(1) in most cases1. In fact, mpickering has written a GHC source plugin that will tell you if you can apply this transformation. Cool!


  1. Assuming f's type parameter a is not nominal role.

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