Version 10 (modified by 4 years ago) (diff) | ,
---|
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 Trac ticket in 2009.
Example
{-# LANGUAGE DeriveFunctor, DeriveFoldable, DeriveTraversable #-} data Example a = Ex a Char (Example a) (Example Char) deriving (Functor, Foldable, Traversable)
The derived code would look something like this:
instance Functor Example where fmap f (Ex a1 a2 a3 a4) = Ex (f a1) a2 (fmap f a3) a4 instance Foldable Example where foldr f z (Ex a1 a2 a3 a4) = f a1 (foldr f z a3) foldMap f (Ex a1 a2 a3 a4) = mappend (f a1) (mappend mempty (mappend (foldMap f a3) mempty)) instance Traversable Example where traverse f (Ex a1 a2 a3 a4) = Ex <$> (f a) <*> pure a2 <*> traverse f a3 <*> pure a4
Algorithm description
DeriveFunctor
, DeriveFoldable
, and DeriveTraversable
all operate using the same underlying mechanism. GHC inspects the arguments of each constructor and derives some operation to perform on each argument, which depends of the type of the argument itself. In a Functor
instance, for example fmap
would be applied to occurrences of the last type parameter, but id
would be applied to other type parameters. Typically, there are five cases to consider. (Suppose we have a data type data A a = ...
.)
- Terms whose type does not mention
a
- Terms whose type mentions
a
- Occurrences of
a
- Tuple values
- Function values
After this is done, the new terms are combined in some way. For instance, Functor
instances combine terms in a derived fmap
definition by applying the appropriate constructor to all terms, whereas in Foldable
instances, a derived foldMap
definition would mappend
the terms together.
DeriveFunctor
A comment in TcGenDeriv.hs lays out the basic structure of DeriveFunctor
, which derives an implementation for fmap
.
For the data type: data T a = T1 Int a | T2 (T a) We generate the instance: instance Functor T where fmap f (T1 b1 a) = T1 b1 (f a) fmap f (T2 ta) = T2 (fmap f ta) Notice that we don't simply apply 'fmap' to the constructor arguments. Rather - Do nothing to an argument whose type doesn't mention 'a' - Apply 'f' to an argument of type 'a' - Apply 'fmap f' to other arguments That's why we have to recurse deeply into the constructor argument types, rather than just one level, as we typically do. What about types with more than one type parameter? In general, we only derive Functor for the last position: data S a b = S1 [b] | S2 (a, T a b) instance Functor (S a) where fmap f (S1 bs) = S1 (fmap f bs) fmap f (S2 (p,q)) = S2 (a, fmap f q) However, we have special cases for - tuples - functions More formally, we write the derivation of fmap code over type variable 'a for type 'b as ($fmap 'a 'b). In this general notation the derived instance for T is: instance Functor T where fmap f (T1 x1 x2) = T1 ($(fmap 'a 'b1) x1) ($(fmap 'a 'a) x2) fmap f (T2 x1) = T2 ($(fmap 'a '(T a)) x1) $(fmap 'a 'b) = \x -> x -- when b does not contain a $(fmap 'a 'a) = f $(fmap 'a '(b1,b2)) = \x -> case x of (x1,x2) -> ($(fmap 'a 'b1) x1, $(fmap 'a 'b2) x2) $(fmap 'a '(T b1 b2)) = fmap $(fmap 'a 'b2) -- when a only occurs in the last parameter, b2 $(fmap 'a '(b -> c)) = \x b -> $(fmap 'a' 'c) (x ($(cofmap 'a 'b) b)) For functions, the type parameter 'a can occur in a contravariant position, which means we need to derive a function like: cofmap :: (a -> b) -> (f b -> f a) This is pretty much the same as $fmap, only without the $(cofmap 'a 'a) case: $(cofmap 'a 'b) = \x -> x -- when b does not contain a $(cofmap 'a 'a) = error "type variable in contravariant position" $(cofmap 'a '(b1,b2)) = \x -> case x of (x1,x2) -> ($(cofmap 'a 'b1) x1, $(cofmap 'a 'b2) x2) $(cofmap 'a '[b]) = map $(cofmap 'a 'b) $(cofmap 'a '(T b1 b2)) = fmap $(cofmap 'a 'b2) -- when a only occurs in the last parameter, b2 $(cofmap 'a '(b -> c)) = \x b -> $(cofmap 'a' 'c) (x ($(fmap 'a 'c) b))
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
:
Deriving Foldable instances works the same way as Functor instances, only Foldable instances are not possible for function types at all. Here the derived instance for the type T above is: instance Foldable T where foldr f z (T1 x1 x2 x3) = $(foldr 'a 'b1) x1 ( $(foldr 'a 'a) x2 ( $(foldr 'a 'b2) x3 z ) ) The cases are: $(foldr 'a 'b) = \x z -> z -- when b does not contain a $(foldr 'a 'a) = f $(foldr 'a '(b1,b2)) = \x z -> case x of (x1,x2) -> $(foldr 'a 'b1) x1 ( $(foldr 'a 'b2) x2 z ) $(foldr 'a '(T b1 b2)) = \x z -> foldr $(foldr 'a 'b2) z x -- when a only occurs in the last parameter, b2 Note that the arguments to the real foldr function are the wrong way around, since (f :: a -> b -> b), while (foldr f :: b -> t a -> b).
In addition to foldr
, DeriveFoldable
also generates a definition for foldMap
as of GHC 7.8.1 (addressing #7436). The pseudo-definition for $(foldMap)
would look something like this:
$(foldMap 'a 'b) = \x -> mempty -- when b does not contain a $(foldMap 'a 'a) = f $(foldMap 'a '(b1,b2)) = \x -> case x of (x1, x2) -> mappend ($(foldMap 'a 'b1) x1) ($(foldMap 'a 'b2) x2) $(foldMap 'a '(T b1 b2)) = \x -> foldMap $(foldMap 'a 'b2) x -- when a only occurs in the last parameter, b2
DeriveTraversable
From TcGenDeriv.hs:
Again, Traversable is much like Functor and Foldable. The cases are: $(traverse 'a 'b) = pure -- when b does not contain a $(traverse 'a 'a) = f $(traverse 'a '(b1,b2)) = \x -> case x of (x1,x2) -> (,) <$> $(traverse 'a 'b1) x1 <*> $(traverse 'a 'b2) x2 $(traverse 'a '(T b1 b2)) = traverse $(traverse 'a 'b2) -- when a only occurs in the last parameter, b2 Note that the generated code is not as efficient as it could be. For instance: data T a = T Int a deriving Traversable gives the function: traverse f (T x y) = T <$> pure x <*> f y instead of: traverse f (T x y) = T x <$> f y
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:
data CovFun1 a = CovFun1 (Int -> a) data CovFun2 a = CovFun2 ((a -> Int) -> a) data CovFun3 a = CovFun3 (((Int -> a) -> Int) -> a)
but none of these can:
data ContraFun1 a = ContraFun1 (a -> Int) data ContraFun2 a = ContraFun2 ((Int -> a) -> Int) data ContraFun3 a = ContraFun3 (((a -> Int) -> a) -> Int)
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:
CovFun1/ContraFun1 :: n -> p CovFun2/ContraFun2 :: (p -> n) -> p CovFun3/ContraFun3 :: ((n -> p) -> n) -> p
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:
- The data type has at least one type parameter. (For example,
data NoArg = NoArg
cannot have aFunctor
instance.) - The data type's last type parameter cannot be used contravariantly. (see the section on covariant and contravariant positions.)
- 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 parametera
is only ever used as the last type argument in[]
andEither
, so both[a]
andEither Int a
values can befmap
ped. However, indata Wrong a = Wrong (Either a a)
, the type variablea
appears in a position other than the last, so trying tofmap
anEither a a
value would not typecheck.
Note that there are two exceptions to this rule: tuple and function types.
- 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:
- For derived
Foldable
andTraversable
instances, a data type cannot use function types. This restriction does not apply to derivedFunctor
instances, however. - For derived
Functor
andTraversable
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:
data T a b where T1 :: a -> b -> T a b -- Fine! Vanilla H-98 T2 :: b -> c -> T a b -- Fine! Existential c, but we can still map over 'b' T3 :: b -> T Int b -- Fine! Constraint 'a', but 'b' is still polymorphic deriving instance Functor (T a) {- instance Functor (T a) where fmap f (T1 a b) = T1 a (f b) fmap f (T2 b c) = T2 (f b) c fmap f (T3 x) = T3 (f x) -}
but the following is not legal:
data T a b where T4 :: Ord b => b -> T a b -- No! 'b' is constrained T5 :: b -> T b b -- No! 'b' is constrained T6 :: T a (b,b) -- No! 'b' is constrained
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:
data T a where T1 :: Ord a => a -> T a
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:
class Foldable t where fold :: Monoid m => t m -> m foldMap :: Monoid m => (a -> m) -> t a -> m foldr :: (a -> b -> b) -> b -> t a -> b foldr' :: (a -> b -> b) -> b -> t a -> b foldl :: (b -> a -> b) -> b -> t a -> b foldl' :: (b -> a -> b) -> b -> t a -> b foldr1 :: (a -> a -> a) -> t a -> a foldl1 :: (a -> a -> a) -> t a -> a toList :: t a -> [a] null :: t a -> Bool length :: t a -> Int elem :: Eq a => a -> t a -> Bool maximum :: forall a. Ord a => t a -> a minimum :: forall a. Ord a => t a -> a sum :: Num a => t a -> a product :: Num a => t a -> a
Therefore, a derived Foldable
instance for T
typechecks:
instance Foldable T where foldr f z (T1 a) = f a z -- foldr :: Ord a => (a -> b -> b) -> b -> T a -> b foldMap f (T1 a) = f a -- foldMap :: (Monoid m, Ord a) => (a -> m) -> T a -> m
Deriving Foldable
instances for GADTs with equality constraints could become murky, however. Consider this GADT:
data E a where E1 :: (a ~ Int) => a -> E a E2 :: Int -> E Int E3 :: (a ~ Int) => a -> E Int E4 :: (a ~ Int) => Int -> E a
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:
instance Foldable E where foldr f z (E1 e) = f e z foldr f z (E2 e) = f e z foldr f z (E3 e) = f e z foldr f z (E4 e) = f e z foldMap f (E1 e) = f e foldMap f (E2 e) = f e foldMap f (E3 e) = f e foldMap f (E4 e) = f e
it is much harder to determine which arguments are equivalent to a
. Also consider this case:
data UnknownConstraints a where UC :: Mystery a => Int -> UnknownConstraints a
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:
instance Foldable E where foldr f z (E1 e) = f e z foldr f z (E2 e) = z foldr f z (E3 e) = z foldr f z (E4 e) = z foldMap f (E1 e) = f e foldMap f (E2 e) = mempty foldMap f (E3 e) = mempty foldMap f (E4 e) = mempty
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:
data HigherKinded f a where HigherKinded :: f a -> HigherKinded f (f a)
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.
Proposal: alternative strategy for deriving Foldable
and Traversable
We adapt the algorithms for -XDeriveFoldable
and -XDeriveTraversable
based on that of -XDeriveFunctor
. However, there an important difference between deriving the former two typeclasses and the latter one, which is best illustrated by the following scenario:
data WithInt a = WithInt a Int# deriving (Functor, Foldable, Traversable)
The generated code for the Functor
instance is straightforward:
instance Functor WithInt where fmap f (WithInt a i) = WithInt (f a) i
But if we use too similar of a strategy for deriving the Foldable
and Traversable
instances, we end up with this code:
instance Foldable WithInt where foldMap f (WithInt a i) = f a <> mempty instance Traversable WithInt where traverse f (WithInt a i) = fmap WithInt (f a) <*> pure i
This is unsatisfying for two reasons:
- The
Traversable
instance doesn't typecheck!Int#
is of kind#
, butpure
expects an argument whose type is of kind*
. This effectively preventsTraversable
from being derived for any datatype with an unlifted argument type (Trac #11174).
- The generated code contains superfluous expressions. By the
Monoid
laws, we can reducef a <> mempty
tof a
, and by theApplicative
laws, we can reducefmap WithInt (f a) <*> pure i
tofmap (\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:
- In the generated expression, we only fold over arguments whose types mention the last type parameter. Any other argument types will simply produce useless
mempty
s orpure
s, so they can be safely ignored.
- In the case of
-XDeriveTraversable
, instead of applyingConName
, we apply\b_i ... b_k -> ConName a_1 ... a_n
, where
ConName
hasn
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 ofConName
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
data Foo a = Foo Int a Int a
would generate the following
Traversable
instance:
instance Traversable Foo where traverse f (Foo a1 a2 a3 a4) = fmap (\b2 b4 -> Foo a1 b2 a3 b4) (f a2) <*> f a4
Technically, this approach would also work for -XDeriveFunctor
as well, but we decide not to do so because:
- There's not much benefit to generating, e.g.,
(\b -> WithInt b i) (f a)
instead ofWithInt (f a) i
.
- There would be certain datatypes for which the above strategy would generate
Functor
code that would fail to typecheck. For example:
data Bar f a = Bar (forall f. Functor f => f a) deriving Functor
With the conventional algorithm, it would generate something like:
fmap f (Bar a) = Bar (fmap f a)
which typechecks. But with the strategy mentioned above, it would generate:
fmap f (Bar a) = (\b -> Bar b) (fmap f a)
which does not typecheck, since GHC cannot unify the rank-2 type variables in the types of
b
andfmap f a
.
Future work
The Bifunctor
class (born from the bifunctors library) was added to base
in GHC 7.10, and there are plans to add Bifoldable
and Bitraversable
to base
in the future. 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.
Classes
The classes are defined as follows:
class Bifunctor p where bimap :: (a -> b) -> (c -> d) -> p a c -> p b d class Bifoldable p where bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> p a b -> m bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> p a b -> c class (Bifunctor t, Bifoldable t) => Bitraversable t where bitraverse :: Applicative f => (a -> f c) -> (b -> f d) -> t a b -> f (t c d)
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
.
Algorithms
A pseudo-code algorithm for generating a bimap
implementation is:
We write the derivation of bimap code over the last two type variables 'a and 'b, for the given type 'c, as ($bimap 'a 'b 'c). We refer bimap's first and second map functions as f and g, respectively. $(bimap 'a 'b 'c) = \x -> x -- when c does not contain a or b $(bimap 'a 'b 'a) = f $(bimap 'a 'b 'b) = g $(bimap 'a 'b '(c1,c2)) = \x -> case x of (x1,x2) -> ($(bimap 'a 'b 'c1) x1, $(bimap 'a 'b 'c2) x2) $(bimap 'a 'b '(T c1 c2)) = bimap $(bimap 'a 'b 'c1) $(bimap 'a 'b 'c2) -- when a and b only occur in the last two parameters, c1 and c2 $(bimap 'a 'b '(c -> d)) = \x e -> $(bimap 'a 'b 'd) (x ($(cobimap 'a 'b 'c) e)) For functions, the type parameters, 'a and 'b, can occur in contravariant positions, which means we need to derive a function like: cobimap :: (a -> b) -> (c -> d) -> (f b d -> f a c) This is pretty much the same as $bimap, only without the $(cobimap 'a 'b 'a) and $(cobimap 'a 'b 'b) cases: $(cobimap 'a 'b 'c) = \x -> x -- when c does not contain a or b $(cobimap 'a 'b 'a) = error "type variable in contravariant position" $(cobimap 'a 'b 'b) = error "type variable in contravariant position" $(cobimap 'a 'b '(c1,c2)) = \x -> case x of (x1,x2) -> ($(cobimap 'a 'b 'c1) x1, $(cobimap 'a 'b 'c2) x2) $(cobimap 'a 'b '(T c1 c2)) = bimap $(cobimap 'a 'b 'c1) $(cobimap 'a 'b 'c2) -- when a and b only occur in the last two parameters, c1 and c2 $(cobimap 'a 'b '(c -> d)) = \x e -> $(cobimap 'a 'b 'd) (x ($(bimap 'a 'b 'c) e))