How to define an instance of Data.Foldable.Constrained? - haskell

Question

I've successfully defined Category, Functor, Semigroup, Monoid constrained. Now I'm stuck with Data.Foldable.Constrained. More precisely, I seem to have correctly defined the unconstrained functions fldl and fldMp, but I can't get them to be accepted as Foldable.Constrained instances.
My definition attempt is inserted as a comment.
{-# LANGUAGE OverloadedLists, GADTs, TypeFamilies, ConstraintKinds,
FlexibleInstances, MultiParamTypeClasses, StandaloneDeriving, TypeApplications #-}
import Prelude ()
import Control.Category.Constrained.Prelude
import qualified Control.Category.Hask as Hask
-- import Data.Constraint.Trivial
import Data.Foldable.Constrained
import Data.Map as M
import Data.Set as S
import qualified Data.Foldable as FL
main :: IO ()
main = print $ fmap (constrained #Ord (+1))
$ RMS ([(1,[11,21]),(2,[31,41])])
data RelationMS a b where
IdRMS :: RelationMS a a
RMS :: Map a (Set b) -> RelationMS a b
deriving instance (Show a, Show b) => Show (RelationMS a b)
instance Category RelationMS where
type Object RelationMS o = Ord o
id = IdRMS
RMS mp2 . RMS mp1
| M.null mp2 || M.null mp1 = RMS M.empty
| otherwise = RMS $ M.foldrWithKey
(\k s acc -> M.insert k (S.foldr (\x acc2 -> case M.lookup x mp2 of
Nothing -> acc2
Just s2 -> S.union s2 acc2
) S.empty s
) acc
) M.empty mp1
(°) :: (Object k a, Object k b, Object k c, Category k) => k a b -> k b c -> k a c
r1 ° r2 = r2 . r1
instance (Ord a, Ord b) => Semigroup (RelationMS a b) where
RMS r1 <> RMS r2 = RMS $ M.foldrWithKey (\k s acc -> M.insertWith S.union k s acc) r1 r2
instance (Ord a, Ord b) => Monoid (RelationMS a b) where
mempty = RMS M.empty
mappend = (<>)
instance Functor (RelationMS a) (ConstrainedCategory (->) Ord) Hask where
fmap (ConstrainedMorphism f) = ConstrainedMorphism $
\(RMS r) -> RMS $ M.map (S.map f) r
fldl :: (a -> Set b -> a) -> a -> RelationMS k b -> a
fldl f acc (RMS r) = M.foldl f acc r
fldMp :: Monoid b1 => (Set b2 -> b1) -> RelationMS k b2 -> b1
fldMp m (RMS r) = M.foldr (mappend . m) mempty r
-- instance Foldable (RelationMS a) (ConstrainedCategory (->) Ord) Hask where
-- foldMap f (RMS r)
-- | M.null r = mempty
-- | otherwise = FL.foldMap f r
-- ffoldl f = uncurry $ M.foldl (curry f)

Answer 1

You need FL.foldMap (FL.foldMap f) r in your definition so that you fold over the Map and the Set.
However, there's a critical error in your Functor instance; your fmap is partial. It's not defined on IdRMS.
I suggest using -Wall to have the compiler warn you about such issues.
The problem comes down to you need to be able to represent relations with finite and infinite domains. IdRMS :: RelationRMS a a can already be used to represent some relations of infinite domain, it isn't powerful enough to represent a relation like fmap (\x -> [x]) IdRMS.
One approach is to use Map a (Set b) for finite relations and a -> Set b for infinite relations.
data Relation a b where
Fin :: Map a (Set b) -> Relation a b
Inf :: (a -> Set b) -> Relation a b
image :: Relation a b -> a -> Set b
image (Fin f) a = M.findWithDefault (S.empty) a f
image (Inf f) a = f a
This changes the category instance accordingly:
instance Category Relation where
type Object Relation a = Ord a
id = Inf S.singleton
f . Fin g = Fin $ M.mapMaybe (nonEmptySet . concatMapSet (image f)) g
f . Inf g = Inf $ concatMapSet (image f) . g
nonEmptySet :: Set a -> Maybe (Set a)
nonEmptySet | S.null s = Nothing
| otherwise = Just s
concatMapSet :: Ord b => (a -> Set b) -> Set a -> Set b
concatMapSet f = S.unions . fmap f . S.toList
And now you can define a total Functor instance:
instance Functor (Relation a) (Ord ⊢ (->)) Hask where
fmap (ConstrainedMorphism f) = ConstrainedMorphism $ \case -- using {-# LANGUAGE LambdaCase #-}
Fin g -> Fin $ fmap (S.map f) g
Inf g -> Inf $ fmap (S.map f) g
But a new issue raises its head when defining the Foldable instance:
instance Foldable (Relation a) (Ord ⊢ (->)) Hask where
foldMap (ConstrainedMorphism f) = ConstrainedMorphism $ \case
Fin g -> Prelude.foldMap (Prelude.foldMap f) g
Inf g -> -- uh oh...problem!
We have f :: b -> m and g :: a -> Set b. Monoid m gives us append :: m -> m -> m, and we know Ord a, but in order to generate all the b values in the image of the relation, we need all the possible a values!
One way you could try to salvage this is to use Bounded and Enum as additional constraints on the relation's domain. Then you could try to enumerate all the possible a values with [minBound..maxBound] (this may not be list every value for all types; I'm not sure if that's a law for Bounded and Enum).
instance (Enum a, Bounded a) => Foldable (Relation a) (Ord ⊢ (->)) Hask where
foldMap (ConstrainedMorphism f) = ConstrainedMorphism $ \case
Fin g -> Prelude.foldMap (Prelude.foldMap f) g
Inf g -> Prelude.foldMap (Prelude.foldMap f . g) [minBound .. maxBound]