Support for deriving Functor, Foldable, and Traversable instances

GHC 6.12.1 introduces an extension to the deriving mechanism allowing for automatic derivation of Functor, Foldable, and Traversable instances using the DeriveFunctor, DeriveFoldable, and DeriveTraversable extensions, respectively. Twan van Laarhoven first proposed this feature in 2007, and opened a related GHC ticket in 2009.

Example

The derived code would look something like this:

Algorithm description

DeriveFunctor, DeriveFoldable, and DeriveTraversable all operate using the same underlying mechanism. GHC inspects the arguments of each constructor and derives some operation to perform on each argument, which depends of the type of the argument itself. In a Functor instance, for example fmap would be applied to occurrences of the last type parameter, but id would be applied to other type parameters. Typically, there are five cases to consider. (Suppose we have a data type data A a = ....)

1. Terms whose type does not mention a
2. Terms whose type mentions a
3. Occurrences of a
4. Tuple values
5. Function values

After this is done, the new terms are combined in some way. For instance, Functor instances combine terms in a derived fmap definition by applying the appropriate constructor to all terms, whereas in Foldable instances, a derived foldMap definition would mappend the terms together.

DeriveFunctor

A comment in TcGenDeriv.hs lays out the basic structure of DeriveFunctor, which derives an implementation for fmap.

DeriveFunctor is special in that it can recurse into function types, whereas DeriveFoldable and DeriveTraversable cannot (see the section on covariant and contravariant positions).

DeriveFoldable

Another comment in TcGenDeriv.hs reveals the underlying mechanism behind DeriveFoldable:

In addition to foldr, DeriveFoldable also generates a definition for foldMap as of GHC 7.8.1 (addressing #7436 (closed)). The pseudo-definition for $(foldMap) would look something like this:

DeriveTraversable

From TcGenDeriv.hs:

Covariant and contravariant positions

One challenge of deriving Functor instances for arbitrary data types is handling function types. To illustrate this, note that these all can have derived Functor instances:

but none of these can:

In CovFun1, CovFun2, and CovFun3, all occurrences of the type variable a are in covariant positions (i.e., the a values are produced), whereas in ContraFun1, ContraFun2, and ContraFun3, all occurrences of a are in contravariant positions (i.e., the a values are consumed). If we have a function f :: a -> b, we can't apply f to an a value in a contravariant position, which precludes a Functor instance.

Most type variables appear in covariant positions. Functions are special in that the lefthand side of a function arrow reverses variance. If a function type a -> b appears in a covariant position (e.g., CovFun1 above), then a is in a contravariant position and b is in a covariant position. Similarly, if a -> b appears in a contravariant position (e.g., CovFun2 above), then a is in a covariant position and b is in a contravariant position.

If we annotate covariant positions with p (for positive) and contravariant positions with n (for negative), then we can examine the above examples with the following pseudo-type signatures:

Since ContraFun1, ContraFun2, and ContraFun3 all use the last type parameter in at least one n position, GHC would reject a derived Functor instance for each of them.

Requirements for legal instances

This mechanism cannot derive Functor, Foldable, or Traversable instances for all data types. Currently, GHC checks if a data type meets the following criteria:

1. The data type has at least one type parameter. (For example, data NoArg = NoArg cannot have a Functor instance.)
2. The data type's last type parameter cannot be used contravariantly. (see the section on covariant and contravariant positions.)
3. The data type's last type parameter cannot be used in the "wrong place" in any constructor's data arguments. For example, in data Right a = Right [a] (Either Int a), the type parameter a is only ever used as the last type argument in [] and Either, so both [a] and Either Int a values can be fmapped. However, in data Wrong a = Wrong (Either a a), the type variable a appears in a position other than the last, so trying to fmap an Either a a value would not typecheck.

Note that there are two exceptions to this rule: tuple and function types.

1. The data type's last type variable cannot used in a -XDatatypeContexts constraint. For example, data Ord a => O a = O a deriving Functor would be rejected.

In addition, GHC performs checks for certain classes only:

1. For derived Foldable and Traversable instances, a data type cannot use function types. This restriction does not apply to derived Functor instances, however.
2. For derived Functor and Traversable instances, the data type's last type variable must be truly universally quantified, i.e., it must not have any class or equality constraints. This means that the following is legal:

but the following is not legal:

This restriction does not apply to derived Foldable instances. See the following section for more details.

Relaxed universality check for DeriveFoldable

DeriveFunctor and DeriveTraversable cannot be used with data types that use existential constraints, since the type signatures of fmap and traverse make this impossible. However, Foldable instances are unique in that they do not produce constraints, but only consume them. Therefore, it is permissible to derive Foldable instances for constrained data types (e.g., GADTs).

For example, consider the following GADT:

In the type signatures for fmap :: Functor t => (a -> b) -> t a -> t b and traverse :: (Applicative f, Traversable t) => (a -> f b) -> t a -> f (t b), the t parameter appears both in an argument and the result type, so pattern-matching on a value of t must not impose any constraints, as neither fmap nor traverse would typecheck.

Foldable, however, only mentions t in argument types:

Therefore, a derived Foldable instance for T typechecks:

Deriving Foldable instances for GADTs with equality constraints could become murky, however. Consider this GADT:

All four E constructors have the same "shape" in that they all take an argument of type a (or Int, to which a is constrained to be equal). Does that mean all four constructors would have their arguments folded over? While it is possible to derive perfectly valid code which would do so:

it is much harder to determine which arguments are equivalent to a. Also consider this case:

For all we know, it may be that a ~ Int => Mystery a. Does this mean that the Int argument in UC should be folded over?

To avoid these thorny edge cases, we only consider constructor arguments (1) whose types are syntactically equivalent to the last type parameter and (2) in cases when the last type parameter is a truly universally polymorphic. In the above E example, only E1 fits the bill, so the derived Foldable instance is actually:

To expound more on the meaning of criterion (2), we want not only to avoid cases like E2 :: Int -> E Int, but also something like this:

In this example, the last type variable is instantiated with f a, which contains one type variable f applied to another type variable a. We would not fold over the argument of type f a in this case, because the last type variable should be simple, i.e., contain only a single variable without any application.

For the original discussion on this proposal, see #10447 (closed).

Alternative strategy for deriving Foldable and Traversable

We adapt the algorithms for -XDeriveFoldable and -XDeriveTraversable based on that of -XDeriveFunctor. However, there is an important difference between deriving the former two typeclasses and the latter one (as of GHC 8.2, addressing Trac #11174), which is best illustrated by the following scenario:

The generated code for the Functor instance is straightforward:

But if we use too similar of a strategy for deriving the Foldable and Traversable instances, we end up with this code:

This is unsatisfying for two reasons:

1. The Traversable instance doesn't typecheck! Int# is of kind #, but pure expects an argument whose type is of kind *. This effectively prevents Traversable from being derived for any datatype with an unlifted argument type (see Trac #11174).

2. The generated code contains superfluous expressions. By the Monoid laws, we can reduce f a <> mempty to f a, and by the Applicative laws, we can reduce fmap WithInt (f a) <*> pure i to fmap (\b -> WithInt b i) (f a).

We can fix both of these issues by incorporating a slight twist to the usual algorithm that we use for -XDeriveFunctor. The differences can be summarized as follows:

1. In the generated expression, we only fold over arguments whose types mention the last type parameter. Any other argument types will simply produce useless memptys or pures, so they can be safely ignored.

2. In the case of -XDeriveTraversable, instead of applying ConName, we apply \b_i ... b_k -> ConName a_1 ... a_n, where

• ConName has n arguments

• {b_i, ..., b_k} is a subset of {a_1, ..., a_n} whose indices correspond to the arguments whose types mention the last type parameter. As a consequence, taking the difference of {a_1, ..., a_n} and {b_i, ..., b_k} yields the all the argument values of ConName whose types do not mention the last type parameter. Note that [i, ..., k] is a strictly increasing—but not necessarily consecutive—integer sequence.

  For example, the datatype

would generate the following Traversable instance:

Technically, this approach would also work for -XDeriveFunctor as well, but we decide not to do so because:

1. There's not much benefit to generating, e.g., (\b -> WithInt b i) (f a) instead of WithInt (f a) i.

2. There would be certain datatypes for which the above strategy would generate Functor code that would fail to typecheck. For example:

   With the conventional algorithm, it would generate something like:

   which typechecks. But with the strategy mentioned above, it would generate:

   which does not typecheck, since GHC cannot unify the rank-2 type variables in the types of b and fmap f a.

Future work

There are more classes in base that we could derive!

In particular, the Bifunctor class (born from the bifunctors library) was added to base in GHC 7.10, and the Bifoldable and Bitraversable classes (also from bifunctors) were added to base in GHC 8.2. All three classes could be derived in much the same way as their cousins Functor, Foldable, and Traversable. The existing algorithms would simply need to be adapted to accommodate two type parameters instead of one. The Data.Bifunctor.TH module from the bifunctors library demonstrates an implementation of the following proposal using Template Haskell.

In GHC 8.0, higher-order versions of the Eq, Ord, Read, and Show typeclasses were added to base in the Data.Functor.Classes module (which originally lived in the transformers library). These classes are generalized to work over datatypes indexed by one type parameter (for Eq1, Ord1, Read1, and Show1) or by two type parameters (Eq2, Ord2, Read2, and Show2). Haskell programmers have been able to derive Eq, Ord, Read, and Show for a long time, so it wouldn't be hard at all to envision a deriving mechanism for Eq1, Eq2, and friends which takes advantage of tricks that DeriveFunctor uses. The deriving-compat library demonstrates proofs-of-concept for deriving Eq1/2, Ord1/2, Read1/2, and Show1/2 using Template Haskell.

Classes

The Bifunctor, Bifoldable, and Bitraversable classes are defined as follows:

Each class contains further methods, but they can be defined in terms of the above ones. Therefore, we need only derive implementations for them. This also mirrors how the algorithms currently work in the one-parameter cases, as they only implement fmap, foldMap, foldr, and traverse.

The typeclasses in Data.Functor.Classes are defined as follows:

Algorithms

A pseudo-code algorithm for generating a bimap implementation is:

This algorithm isn't terribly different from the one above for generating an fmap implementation, and that's the point. It's simply generalizing the same ideas to work over a typeclass of kind * -> * -> *. The algorithms for generating foldMap/foldr and traverse can be generalized to generate bifoldMap/bifoldr and bitraverse, respectively. For example, here's what the algorithm for bifoldMap would look like:

(The caveats in https://gitlab.haskell.org/trac/ghc/wiki/Commentary/Compiler/DeriveFunctor#AlternativestrategyforderivingFoldableandTraversable apply.)

There's one part of the bifoldMap algorithm that deserves futher discussion: the overlapping cases for T c1 c1 c3. Whenever an argument to a constructor has a type where each of the last two type variables mention a or b, we opt to generate bifoldMap instead of foldMap. We could go the other way, though. For instance, the following is a valid implementation of Bifoldable for newtype T a b = T (Either a b):

But this is unsatisfying for a couple of reasons, though. One obvious issue is that this definition blatantly ignores the first argument to bifoldMap, preventing users from folding over the a type parameter. Another problem is that doing this would be inconsistent with how bimap and bitraverse are generated. Unlike with bifoldMap, parametricity forces there to be one definition for bimap and bitraverse (see https://gitlab.haskell.org/trac/ghc/wiki/Commentary/Compiler/DeriveFunctor#RelaxeduniversalitycheckforDeriveFoldable for more info):

Therefore, it feels far more natural to generate this Bifoldable instance:

This also ensures that bifoldMapDefault gives the same result as bifoldMap.

Corner case: GADTs

Consider the following code:

What should be the definition of bifoldMap for Both? We have a choice, since both the function argument of type (a -> m) and of type (b -> m) can be applied to either argument. In such a scenario, the second fold function takes precedence over the first fold function, so the derived Bifoldable instance would be:

This definition ensures that bifoldMap id = foldMap for a derived Foldable instance for Both.

Data.Functor.Classes

Deriving Eq1/2, Ord1/2, and Show1/2 could be done in a very similar way to deriving Foldable/Bifoldable. For instance, you can substitute in liftEq/liftEq2 in place of foldMap/bifoldMap in the above algorithm and have a sensible way to generate liftEq/liftEq2 expressions. In addition, Eq1/2, Ord1/2, and Show1/2 can all be derived for GADTs as well.

Read1/2 could not be derived for GADTs, since it mentions type variables in the return types of its methods, but it could still be derived using the same principles as deriving Foldable/Bifoldable.