Version 19 (modified by RyanGlScott, 2 years ago) (diff)

Roles for type families


The idea of roles comes from the paper Generative Type Abstraction and Type-level Computation, published at POPL 2011. The implementation of roles in GHC, however, is somewhat different than stated in that paper. This page focuses on the user-visible features of roles.

Role annotations are enabled using {-# LANGUAGE RoleAnnotations #-}.

See also


Use Keyword = Roles to ensure that a ticket ends up on these lists.

Open Tickets:

support for deriving Vector/MVector instances
Can't eta-reduce representational coercions
Constraint vs *
Role ranges (allow decomposition on newtypes)
Coercing between constraints of newtypes
Solve Coercible constraints over type constructors
Allow: Coercing (a:~:b) to (b:~:a)
GHC doesn't allow Coercion between partly-saturated type constructors
Incompleteness in the Coercible constraint solver
Data type with phantoms using TypeInType isn't coercible
More liberally kinded coercions for newtypes

Closed Tickets:

Role annotations does not allow the use of parenthesis
Coercible constraint solver misses one
Emit quantified Coercible constraints in GeneralizedNewtypeDeriving
Experiment with a dedicated solver for Coercible
Incorrect number of parameters in "role" errors
No run-time exception for deferred type errors when error is in a phantom role position
type synonyms confuse generalized newtype deriving role checking
Type synonyms can make roles too conservative
GHC doesn't use the fact that Coercible is symmetric
:type hangs on coerce
Unused "foralls" prevent types from being Coercible
Coercible and Existential types don't play nicely

The problem we wish to solve

GHC has had a hole in its type system for several years, as documented in #1496, #4846, #5498, and #7148. The common cause behind all of this is the magic behind -XGeneralizedNewtypeDeriving (GND). Here is an example:

newtype Age = MkAge { unAge :: Int }

type family Inspect x
type instance Inspect Age = Int
type instance Inspect Int = Bool

class BadIdea a where
  bad :: a -> Inspect a

instance BadIdea Int where
  bad = (> 0)

deriving instance BadIdea Age

This code is accepted by GHC 7.6.3. Yet, it goes wrong when you say bad (MkAge 5) -- we see the internal encoding of Bool! Let's trace what is happening here.

A newtype is a new algebraic datatype that wraps up exactly one field (in our example, of type Int). Yet, the semantics of Haskell makes a guarantee that wrapping and unwrapping a value (with MkAge or unAge) has no runtime cost. Thus, internally, we must consider Age to be wholly equivalent to Int.

The problem with this idea comes with type families. (There are other ways to tickle the bug, but one example is enough here.) A type family can branch on Haskell type, and of course, in Haskell (unlike in the internals of a compiler), Age is most certainly not Int. (If it were, newtypes would be useless for controlling instance selection, a very popular use case.) So, in our example, we see that Inspect Age is Int, but Inspect Int is Bool. Now, note the type of bad, the method in class BadIdea. When passed an Int, bad will return a Bool. When passed an Age, bad will return an Int. What happens on the last line above, when we use GND? Internally, we take the existing instance for Int and just transform it into an instance for Age. But, this transformation is very dumb -- because Age and Int are the same, internally, the code for the Age instance and the code for the Int instance are the same. This means that when we call bad (MkAge 5), we run 5 through the existing implementation for bad, which produces a Bool. But, of course, the type of bad (MkAge 5) is Int, and so we have effectively converted a Bool to an Int. Yuck.

The solution

What to do? It turns out we need a subtler definition of type equality than what we have had. Specifically, we must differentiate between nominal equality and representational equality. Nominal equality (called C in the paper cited above) is the Haskell equality we all know and love. If two types have the same name, they are nominally equal. If they don't have the same name (expanding type synonyms), they are not nominally equal. Representational equality, on the other hand, shows that two types have the same representation. This is the equality that newtypes produce -- Age is representationally equal to Int, but they are not nominally equal.

Datatypes, classes, and type synonyms can be parametric in their type arguments or not. By "parametric", I mean that they do not inspect the type argument. A non-parametric type variable is inspect. Here are some examples:

data List a = Nil | Cons a (List a)    -- parametric
data GADT a where                      -- non-parametric
  GAge :: GADT Age
  GInt :: GADT Int

class C1 a where                       -- parametric
  foo :: a -> List a

class C2 a where                       -- non-parametric
  bar :: a -> GADT a

class BadIdea a where                  -- non-parametric
  bad :: a -> Inspect a

In the terminology here, non-parametric types and classes care, in some fundamental way, what type parameter they are given. Parametric ones don't. We can generalize this idea a bit further to label each type variable as either parametric or not. For example,

data Mixed a b where
  MInt :: a -> Mixed a Int
  MAge :: a -> Mixed a Age

is parametric in its first parameter but not its second. We say that a parametric type variable has a representational role and a non-parametric one has a nominal role.


The libraries with GHC 7.8 offer a new class

class Coercible a b where
  coerce :: a -> b

The idea is that a Coercible instance exists allowing coercions between any two types that are representationally equal. A programmer can then use coerce to get between the types. The instances themselves are magically generated as necessary; it is not allowed for programmers to declare their own Coercible instances. So, we have Coercible Age Int but never Coercible Bool Int.

The reason we need roles is to describe how these representational equalities (or, equivalently, Coercible instances) "lift" through other types. For example, is [Age] representationally equal to [Int]? Sure. But, is GADT Age representationally equal to GADT Int? I hope not!

The rule is this: we have instance Coercible a b => Coercible (T a) (T b) if and only if the first parameter has a representational role. Thus, we have instance Coercible a b => Coercible [a] [b] but not instance Coercible a b => Coercible (GADT a) (GADT b). This generalizes straightforwardly when there are multiple parameters, and it's worth noting that Coercible is always reflexive, even when nominal roles are involved.

GeneralizedNewtypeDeriving implemented using coerce

Now that we have all of this Coercible machinery, we can define the behavior of GND in terms of it -- we simply coerce each method of the derived class. For example:

newtype RestrictedIO a = MkRIO { unRIO :: IO a }
  deriving Monad


instance Monad RestrictedIO where
  return = coerce (return :: a -> IO a) :: forall a. a -> RestrictedIO a
  (>>=) = coerce ((>>=) :: IO a -> (a -> IO b) -> IO b) :: forall a b. RestrictedIO a -> (a -> RestrictedIO b) -> RestrictedIO b
  fail = coerce (fail :: String -> IO a) :: forall a. String -> RestrictedIO a

Note that each of these is just a call to coerce over the method in the instance for the newtype's representation type (in this case, IO a). All those type annotations are necessary to make sure that the type checker does the right conversion (and that scoped type variables are bound appropriately).

Putting all of this together, GND works exactly when each of the methods being derived is Coercible into the new type.

Phantom parameters

It turns out that a third role is also useful (though unnecessary for type soundness): the phantom role. It is often the case that programmers use type variables simply to constrain the type checker, not to make any statement about the runtime representation of a type. For example data Phant a = MkPhant Int. Because a doesn't appear on the right-hand side, we say that a is at role phantom. Why is this nice? Because it allows us to say that, say, Phant Int and Phant Bool are representationally equal, because they really do have the same representation. Thus, there would be instance Coercible (Phant a) (Phant b) for any a and b.

Role inference

How do we know what role a type parameter should have? We use role inference! We start with a few base facts: (->) has two representational parameters; (~) has two nominal parameters; and all type families' parameters are nominal. Then, we just propagate the information. By defaulting parameters to role phantom, any parameters unused in the right-hand side (or used only in other types in phantom positions) will be phantom. Whenever a parameter is used in a representational position (that is, used as a type argument to a constructor whose corresponding variable is at role representational), we raise its role from phantom to representational. Similarly, when a parameter is used in a nominal position, its role is upgraded to nominal. We never downgrade a role from nominal to phantom or representational, or from representational to phantom. In this way, we infer the most-general role for each parameter.

The exception to the above algorithm is for classes: all parameters for a class default to a nominal role. This is because we generally consider, say, Ord Age and Ord Int to be quite distinct, even if their representation is the same under the hood. Changing the behavior of type classes is a major use case for newtypes, and we wouldn't want to subvert that!

Role annotations

As we have learned with type and kind inference, sometimes the programmer wants to constrain the inference process. For example, the base library contains the following definition:

data Ptr a = Ptr Addr#

The idea is that a should really be a representational parameter, but role inference assigns it to phantom. This makes some level of sense: a pointer to an Int really is representationally the same as a pointer to a Bool. But, that's not at all how we want to use Ptrs! So, we want to be able to say

type role Ptr representational
data Ptr a = Ptr Addr#

The type role annotation forces the parameter a to be at role representational, not role phantom. We, then, of course, need to check the user-supplied roles to make sure they don't break any promises. It would be bad if the user could make BadIdea's role be representational!

If Ptr were to have multiple type parameter we would have used multiple nominal/representational annotations

type role Foo representational representational
data Foo a b = Foo Int

The other place where role annotations may be necessary are in .hs-boot files, where the right-hand sides of definitions can be omitted. As usual, the types/classes declared in an .hs-boot file must match up with the definitions in the .hs file, including down to the roles. The default role is representational in hs-boot files, corresponding to the common use case. Note that this will break code. But, the change is necessary to close the type-safety hole discussed above.

Role annotations are allowed on type variables in data, newtype, and class, declarations. They will not be allowed on type/data family declarations or in explicit foralls in function type signatures.

Roles and Traversable

Though a minor issue in the overall scheme, work on Roles had led to an interesting interaction with Traversable, excerpted here:

class Traversable t where
  traverse :: Applicative f => (a -> f b) -> t a -> f (t b)

According to the rules for roles, the parameter t must be at role nominal, as it is used as a parameter to the type variable f. We must account for the possibility that f will be instantiated with a type whose last parameter is at role nominal, so we force t to be at role nominal as well.

This means that GND no longer works with Traversable. But, DeriveTraversable does still work. However, GHC previously preferred using GND over DeriveTraversable when both were available, which assumedly produced the same code. How is this all possible? If GND doesn't work anymore, is there something wrong with DeriveTraversable? The answer is that GND and DeriveTraversable make different instances, contrary to expectations. The reason is that DeriveTraversable has to use fmap to cast the result of traverse from the representation type back to the newtype. According to the Functor laws, fmapping this cast should be a no-op (the law of interest is fmap id == id). But, if that law is not obeyed, fmapping the cast may change the result of the traverse. Contrast this with a GND instance, which magically casts the result, without using fmap. If the Functor law is not obeyed, these two instances have different behavior.

Despite this, I believe that using GND with Traversable is indeed type-safe. Why? Because of the parametricity guaranteed in Functor and Applicative. The reason GND is prohibited with Traversable is that we are worried f's last parameter will be at role nominal. While it is possible to write Functor and Applicative instances for such a type, the methods of those classes can't really use the any constructors that force the role to be nominal. For example, consider this:

data G a where
  GInt :: a -> G Int
  Ga   :: a -> G a

instance Functor G where
  fmap f (GInt _) = error "urk"  -- no way out here
  fmap f (Ga a)   = Ga (f a)

instance Applicative G where
  pure a = Ga a
  (Ga f) <*> (Ga a) = Ga (f a)
  _ <*> _ = error "urk" -- no way out here, either

There's no way to usefully interact with the GInt constructor and get the code to type-check. Thus, I believe (but haven't yet proved), that using GND with Traversable is safe, because the f in traverse can't ever do bad things with its argument. If you, the reader, have more insight into this (or a counterexample!), please write!

Changing default role to nominal

In GHC 7.8, unannotated datatype parameters default to phantom. This means that most normal parameters are given a representational role. It has been argued that perhaps nominal is a better (safer) default, and that users should specify representational when they want it. The problem with a nominal default is that it breaks all current usages of GND by default. Furthering the problem, when a user is unable to use GND it's the library that has to change, not the user's code.

On Mar 31, 2014, Dominique Devriese writes the following suggestion:

What I was wondering about is if the dilemma could be solved by choosing nominal-by-default in the long term for the role inference (so that library writers cannot accidentally leave abstraction holes open by forgetting to add role annotations) and use them in the long-term-supported SafeNewtypeDeriving extension, but provide a deprecated not-quite-as-safe GND extension for helping out users of libraries that have not yet added role annotations. I would fancy that this not-quite-as-safe GND could use unsafeCoerce wherever the safe one would give an error about annotated roles.

Proposal: roles for type families

Currently, the type constructors for all type families and data families all conservatively assign role nominal to all their parameters. This is a safe choice, but a restrictive one, as it rules out some useful, coercion-safe programs. In this section, I propose a way to allow type families to have parameters with phantom and representational roles.

Examples we cannot write today

This example (courtesy of glguy) will not typecheck:

-- | Family of N-ary operator types.
type family Op n a b where
  Op 'Z     a b = b
  Op ('S n) a b = a -> Op n a b

coerceOp :: Coercible a b => Op n a c -> Op n b c
coerceOp = coerce

Since the role signature for Op is type role Op nominal nominal nominal. But in an ideal world, the role signature for Op would be inferred as type role Op nominal representational representational. After all, neither a nor b is "scrutinized" in any sense, so it feels perfectly safe to coerce them freely.

Another example (courtesy of int-index) is:

-- represents effect methods for some monad `m`
data family EffDict (eff :: k) (m :: Type -> Type)

-- Note this matches on `eff`, not `m`
data StateEff s
data instance EffDict (StateEff s) m =
    { _state :: forall a . (s -> (a,s)) -> m a,
      _get :: m s,
      _put :: s -> m () }

-- composition of monad transformers
newtype TComp t1 t2 m a = TComp (t1 (t2 m) a)

coerceDict :: EffDict eff (t1 (t2 m)) -> EffDict eff (TComp t1 t2 m)
coerceDict = coerce

Again, coerceDict will not typecheck due to the role of m in EffDict being nominal. But there's no good reason why this must be the case—we ought to be able to tell GHC to allow m to have representational role. (Of course, this would prevent any EffDict instance from using m at a nominal role, but them's the breaks.)

Additionally, we might like to have roles for associated type families. For instance, consider this example (courtesy of dmcclean):

data Variant = DQuantity | DUnit Prefixability
data Dimension

class KnownVariant (var :: Variant) where
  data Dimensional var :: Dimension -> * -> *

instance KnownVariant DQuantity where
  newtype Dimensional DQuantity d v = Quantity' v

instance KnownVariant (DUnit p) where
  data Dimensional (DUnit p) d v = Unit' UnitName v

type Quantity = Dimensional DQuantity
coerceQuantity :: Coercible v w => Quantity d v -> Quantity d w
coerceQuantity = coerce

Once again, coerceQuantity is ill typed, simply because of the conservative nominal role that the last type parameter of Dimensional has. Associated type families are an interesting case, since they can have extra type parameters (and thus extra roles) that the parent class does not have.


Implementing roles for type families would not require too many changes to the syntax of the language, as most of the required pieces are already there. The biggest current restriction is the fact that one cannot declare role annotations for type families, e.g.,

type role F nominal
type family F a

But this is a restriction that is easily overcome. In addition, the parser does not currently recognize role annotations for associated type families:

{{#!hs class C a where

type role Assoc nominal nominal type Assoc a b


But this could be added without much difficulty.

Role inference for type families

Regardless of whether we're dealing with a closed, open, or associated type family, GHC will need to infer the most permissive roles possible for every type family, and possibly check these roles against a user-provided role signature. This section describes how role inference will operate.


Consider this type family:

type family F (e :: *) (f :: *) (g :: *) (h :: *) :: k where
  F Int       b c d = c
  F (Maybe a) b a d = Maybe b
  F a         b c d = a

There are five type parameters for F: k, e, f, g, and h. What should be the roles for each one? We will start off by assuming each parameter has role phantom, and then walk the structure of the type family, progressively marking parameters with more restrictive roles.

The type family kind

First, we gather all of the free variables in the type family's kind and mark each as nominal. This is under the observation that only type variables can be at role phantom or nominal, never kind variables. Therefore, k would be marked as nominal.

The type family equations

Next, we descend into each defining equation of the type family and inspect the left-hand and right-hand sides. The right-hand sides are analyzed just like the fields of a data constructor; see the Role inference section above for more details. From the right-hand sides, we learn that the roles of e, f, and g should be (at least) representational.

The more interesting analysis comes when inspecting the left-hand sides. We want to mark any type variable that is scrutinized as nominal. By "scrutinized", we mean a variable that is being used in a non-parametric fashion. For instance, we want to rule out scenarios like this one:

type family Inspect x where
  Inspect Bool = Int
  Inspect Int  = Bool

coerceInspect :: Coercible a b => Inspect a -> Inspect b
coerceInspect = coerce

unsafeBoolToInt :: Bool -> Int
unsafeBoolToInt = (coerceInspect :: Inspect Int -> Inspect Age)

To accomplish this, we check for any occurences of the either of the following sorts of scrutinization:

  1. A type pattern that is not a single type variable. For instance, all of these equations provde examples of type patterns which do scrutinize a particular type variable:
type family Inspect x where
  Inspect Int          = Bool
  Inspect (Either a b) = a
  Inspect (f a)        = a

Any type variable that is scrutinized in this fashion (x in the above example) is marked as nominal.

  1. Type patterns that are syntactically equal are all marked as nominal. For instance:
type family Eq w x y z where
  Eq a b (Either b a) c = a

Here, we have two variable names that are used in multiple places: a and b. As a result, the type variables which they inhabit (w, x, and y) are all marked as nominal.

Returning to the earlier F example, we would learn that e and g should be marked nominal, as they are both scrutinized. Therefore, the final inferred roles for k, e, f, g, and h are nominal, nominal, representational, nominal, and phantom.

Role checking for type families

Users can also specify role annotations for type families that should be checked against the inferred roles. For instance:

type role G nominal nominal
type family G a b where
  G a b = Either a b

If the user hadn't written the role annotation for G, its role signature would have been inferred to be type role G representational representational. However, role checking overrides the inferred roles and assigns the more conservative roles of type role G nominal nominal.

Note that while writing role annotations for closed type families is purely optional, it is somewhat more important for open type families. For instance, what should be the roles for this type family?

type family Open a b

Here, we have a choice to make. We could decide to make the roles for open type families default to, say, representational. While this would give us the freedom to coerce values of type Open a b more freely, it simultaneously restricts the instances we can give for Open, since every type instance must be checked to ensure that neither a nor b is used at a nominal role.

For the sake of backwards compatibility and the principle of least surprise, roles for open type families default to nominal. This allows more instances to be written, but makes it harder to coerce them. If a user wishes to coerce open type families, the onus is on her to write a role annotation, e.g.,

type role Open representational representational
type family Open a b

Type family roles and hs-boot files

Just like we default roles for open type families to nominal, we do the same for type families declared in hs-boot files.