This package provides adjunctions for Haskell.
Contributions and bug reports are welcome!
Please feel free to contact me through github or on the #haskell IRC channel on irc.freenode.net.
-Edward Kmett
Simple adjunctions
Home Page: http://hackage.haskell.org/package/adjunctions
License: Other
travis/cabal-apt-install packdeps packunused
doesn't seem to work, at least packdeps
and packunused
are not available to the script afterwards.
We now have
collect1 :: Functor1 w => (forall x. g x -> f x) -> w g -> f (w Identity)
We can generalize this immediately:
coll2 :: (Applicative h, Functor1 w) => (forall x. g x -> f x) -> w g -> f (w h)
coll2 f = cotraverse1 (map1 (pure . runIdentity)) . map1 f
I don't know if it's possible to weaken the Applicative
constraint.
There is an isomorphism between Compose f g
and Day f g
of representable functors
c2d :: (Representable f, Representable g) => Compose f g a -> Day f g a
c2d = tabulate . index
d2c :: (Representable f, Representable g) => Day f g a -> Compose f g a
d2c = tabulate . index
This means every instance of Compose f g
is a potential instance for Day
, would these instances be useful and worth adding?
instance (Alternative f, Representable f, Alternative g, Representable g) => Alternative (Day f g) where
empty :: Day f g a
empty = c2d empty
(<|>) :: Day f g a -> Day f g a -> Day f g a
(d2c -> a) <|> (d2c -> b) = c2d (a <|> b)
This allows for an Alternative
definition as well as an inordinate number of other instances.
We also get a handful of classes the other way: some instance for Compose f g
that work for functors like Coyoneda f
, (_ ->)
, Cofree
, Day
and recursively Compose
(it would be an orphan instance I suppose)
instance (Comonad f, Representable f, Comonad g, Representable g) => Comonad (Compose f g) where
extract :: Compose f g a -> a
extract = extract . c2d
duplicate :: Compose f g a -> Compose f g (Compose f g a)
duplicate = fmap d2c . d2c . duplicate . c2d
instance (ComonadApply f, ComonadApply g, Representable g, Representable f) => ComonadApply (Compose f g) where
(<@>) :: Compose f g (a -> b) -> (Compose f g a -> Compose f g b)
(c2d -> f) <@> (c2d -> x) = d2c (f <@> x)
Is there a home for this class + associated data family?
A common example of data families
class Key key where
data TotalMap key :: * -> *
lookup :: key -> TotalMap key val -> val
with some modification
class Functor (TotalMap key) => Key key where
data TotalMap key :: Type -> Type
(¡) :: TotalMap key val -> (key -> val)
tab :: (key -> val) -> TotalMap key val
can form a Representable
functor for any instance of Key
instance Key key => Distributive (TotalMap key) where
distribute :: Functor f => f (TotalMap key val) -> TotalMap key (f val)
distribute = distributeRep
instance Key key => Representable (TotalMap key) where
type Rep (TotalMap key) = key
index :: TotalMap key val -> (key -> val)
index = (¡)
tabulate :: (key -> val) -> TotalMap key val
tabulate = tab
Since TotalMap key
can be made an instance of all these nice classes
instance Key key => Applicative (TotalMap key) where
pure :: a -> TotalMap key a
pure = pureRep
(<*>) :: TotalMap key (a -> b) -> (TotalMap key a -> TotalMap key b)
(<*>) = apRep
instance Key key => Monad (TotalMap key) where
return :: a -> TotalMap key a
return = pureRep
(>>=) :: TotalMap key a -> (a -> TotalMap key b) -> TotalMap key b
(>>=) = bindRep
instance Key key => MonadFix (TotalMap key) where
mfix :: (a -> TotalMap key a) -> TotalMap key a
mfix = mfixRep
instance (Key key, Monoid key) => Comonad (TotalMap key) where
extract :: TotalMap key a -> a
extract = extractRep
extend :: (TotalMap key a -> b) -> (TotalMap key a -> TotalMap key b)
extend = extendRep
instance Key key => MonadReader key (TotalMap key) where
ask :: TotalMap key key
ask = askRep
local :: (key -> key) -> (TotalMap key a -> TotalMap key a)
local = localRep
And we can get any of these instances of Key
automatically gets us all these nice things
Key key :- MonadFix (TotalMap key)
Key key :- Applicative (TotalMap key)
Key key :- Monad (TotalMap key)
Key key :- MonadReader key (TotalMap key)
(Key key, Monoid key) :- Comonad (TotalMap key)
instance Key Bool where
data TotalMap Bool val = BoolMap val val deriving Functor
(¡) :: TotalMap Bool val -> (Bool -> val)
BoolMap f _ ¡ False = f
BoolMap _ t ¡ True = t
tab :: (Bool -> val) -> TotalMap Bool val
tab f = BoolMap (f False) (f True)
instance (Key key1, Key key2) => Key (key1, key2) where
data TotalMap (key1, key2) val = PairMap (TotalMap key1 (TotalMap key2 val))
(¡) :: TotalMap (key1, key2) val -> ((key1, key2) -> val)
PairMap keys ¡ (k1, k2) = keys ¡ k1 ¡ k2
tab :: ((key1, key2) -> val) -> TotalMap (key1, key2) val
tab f = PairMap $
tab @key1 $ \key1 ->
tab @key2 $ \key2 ->
f (key1, key2)
deriving instance (Key key1, Key key2) => Functor (TotalMap (key1, key2))
Two of the new methods are incompatible with newtype deriving.
There is an isomorphism between f (Rep r)
and natural transformations r ~> f
:
one :: RepresentableOf rep r => Functor f => f rep -> r ~> f
one fins as = index as <$> fins
two :: RepresentableOf rep r => (r ~> f) -> f rep
two make = make positions
positions :: RepresentableOf rep r => r rep
positions = tabulate id
I don't know that it's useful, but there could be some Iso' (f rep) (r ~> f)
.
I got the idea of this isomorphism from Cofree Traversable Functors, where it is instantiated at RepresentableOf (Fin n) (Vec n)
. I'm curious if there is deeper history.
forall x. x -> .. (n-times) .. x -> f x
= Vec n ~> f
= f (Fin n)
In general,
ixSureDef :: (Representable f, Eq (Rep f)) => Rep f -> Lens' (f a) a
ixSureDef i f fa = (\x' -> Rep.tabulate (\k -> if k == i then x' else Rep.index fa k)) <$> f (Rep.index fa i)
Of course, this is horrifically inefficient. Is there a good place here for a subclass supporting such an operation? Alternatively (or additionally), is there room in lens for an IxSure
class similar to Ixed
but providing a Lens
instead of a Traversal
?
The index for Cofree
is currently Seq
from Data.Sequence:
adjunctions/src/Data/Functor/Rep.hs
Lines 505 to 510 in 3040e61
But it looks like the sequence is only ever accessed with consing and unconsing. If so, would it be possible to change the implementation to use lists instead? Maybe something like:
instance Representable f => Representable (Cofree f) where
type Rep (Cofree f) = [Rep f]
index = flip (foldr f extract) where
f x xs ys = xs (index (unwrap ys) x)
tabulate = coiterW (\f -> tabulate (\x -> f . (:) x))
tabulate
could alternatively be:
tabulate = unfold (\f -> (f [], tabulate (\x -> f . (:) x)))
-- Or
tabulate = coiterW (collect (flip (.)) (tabulate (:)))
The extra flexibility and lower overhead of lists would be handy, and I haven't seen any of the extra features of sequences being used with this instance.
So we don't forget to settle on #49 (comment)
Based on todays comments by Edward, Functor1
is not a good name!
21:48 edwardk Functor1, etc. doesn't work because I want FFunctor, FFoldable, FTraversable..
21:49 edwardk and the latter become Foldable1 and Traversable1 under that convention!
Looks like the distributive upload is just not on Hackage yet.
Hey there; this is a nicely thought out and very elegant lib!
I've found the following functions useful:
withIndex :: (Eq (Rep f), Representable f) => Rep f -> (a -> a) -> f a -> f a
setIndex :: (Eq (Rep f), Representable f) => Rep f -> a -> f a -> f a
With the relatively inefficient implementations:
withIndex :: (Eq (Rep f), Representable f) => Rep f -> (a -> a) -> f a -> f a
withIndex ind g fa = tabulate $ \rep ->
let existing = index fa rep
in if rep == ind then g existing
else existing
setIndex :: (Eq (Rep f), Representable f) => Rep f -> a -> f a -> f a
setIndex ind = withIndex ind . const
If these would be of interest to others I can write up a PR. Let me know what you think;
Is there a valid Representable
instance for Reverse
?
instance Representable f => Representable (Reverse f) where
type Rep (Reverse f) = Rep f
tabulate = Reverse . tabulate
index (Reverse f) i = index f i
gives the same result for these two expressions
>>> (!) = index; infixl 6 !
>>> V3 'x' 'y' 'z' ! E _x
'x'
>>> Reverse (V3 'x' 'y' 'z') ! E _x
'x'
For future reference, once we have Bicomonad
class Bifunctor bi => Bicomonad bi where
fst :: bi a b -> a
snd :: bi a b -> b
bidup :: bi a b -> (bi a b `bi` bi a b)
I think we can define a Representable
instance for (Join f)
instance (Biapplicative f, Bicomonad f) => Representable (Join f) where
type Rep (Join f) = Bool
index :: Join f a -> (Bool -> a)
index (Join faa) = \case
False -> fst faa
True -> snd faa
tabulate :: (Bool -> a) -> Join f a
tabulate gen = Join (False `bipure` True)
unless it's unlawful. It captures the pattern of Representable Pair
for newtype Pair a = Pair (a, a)
.
I append this to cabal.project
:
source-repository-package
type: git
location: [email protected]:haskell/mtl.git
tag: v2.3.1-rc1
but compilation fails because of a conflict regarding liftCallCC.
I expected compilation to succeed.
Hi Edward,
I notice that the Representable
class define an isomorphism (between the functor and its representation), but that you avoid defining it in terms if lens
's Iso
. Is there a reason for this?
I have been using the following in my own code but I would rather it was in an official package.
import Control.Lens
import Data.Functor.Rep
tabulated :: Representable f => Iso (Rep f -> a) (Rep f -> b) (f a) (f b)
tabulated = iso tabulate index
There is an explicit isomorphism for this:
fromAdj :: Adjunction f u => f a -> (a, Rep u)
fromAdj = rightAdjunct (tabulate . (,))
toAdj :: Adjunction f u => a -> Rep u -> f a
toAdj = index . unit
And there should be a newtype wrapper, for instance named Adj
, that works like Co
in Data.Functor.Rep, and provides instances for Comonad, instances for Applicative and Monad when Rep u
is a Monoid, etc. Also, there would be an Adjunction between Adj f
and Co u
, so that Co u
has an Adjunction whenever u does.
The latest version of adjunctions (4.3) fails to build for me with the following error:
Preprocessing library adjunctions-4.3...
[ 1 of 11] Compiling Data.Functor.Contravariant.Rep ( src/Data/Functor/Contravariant/Rep.hs, dist/dist-sandbox-fffe542f/build/Data/Functor/Contravariant/Rep.o )
[ 2 of 11] Compiling Control.Monad.Trans.Conts ( src/Control/Monad/Trans/Conts.hs, dist/dist-sandbox-fffe542f/build/Control/Monad/Trans/Conts.o )
[ 3 of 11] Compiling Data.Functor.Contravariant.Adjunction ( src/Data/Functor/Contravariant/Adjunction.hs, dist/dist-sandbox-fffe542f/build/Data/Functor/Contravariant/Adjunction.o )
[ 4 of 11] Compiling Control.Monad.Trans.Contravariant.Adjoint ( src/Control/Monad/Trans/Contravariant/Adjoint.hs, dist/dist-sandbox-fffe542f/build/Control/Monad/Trans/Contravariant/Adjoint.o )
[ 5 of 11] Compiling Data.Functor.Rep ( src/Data/Functor/Rep.hs, dist/dist-sandbox-fffe542f/build/Data/Functor/Rep.o )
src/Data/Functor/Rep.hs:234:10:
No instance for (Distributive Dual)
arising from the superclasses of an instance declaration
In the instance declaration for ‘Representable Dual’
src/Data/Functor/Rep.hs:239:10:
No instance for (Distributive Monoid.Product)
arising from the superclasses of an instance declaration
In the instance declaration for ‘Representable Monoid.Product’
src/Data/Functor/Rep.hs:244:10:
No instance for (Distributive Sum)
arising from the superclasses of an instance declaration
In the instance declaration for ‘Representable Sum’
src/Data/Functor/Rep.hs:250:10:
No instance for (Distributive Complex)
arising from the superclasses of an instance declaration
In the instance declaration for ‘Representable Complex’
The previous version 4.2.2 doesn't have this issue, locking to 4.2.2 in the cabal file works. Using 7.8.4 with base-4.7.0.2.
It appears that the lines it's complaining about were recently added. I'm not very familiar with this package but perhaps it's expecting newer versions of these classes than what I've got.
Diff between v4.2.2 and v4.3
I'm confused on why there is a Representable class in both representable-functors and adjunctions. Does the version here supersede the version in representable-functors? If yes, can the documentation in both packages be updated to reflect that?
Edward says here:
For any Functor that is representable, .. you build a 'zipping' applicative that is isomorphic to reader. For that you can show that (
m *> _ = m
) and (_ <* m = m
). So (<*
) and (*>
) are O(1) while the (<*>
) pays for every point used.
It seems the order he drops things is flipped, in any case I propose we make use of this and define
instance Representable f => Applicative (Co f) where
..
(*>) :: Co f a -> Co f b -> Co f b
_ *> bs = bs
(<*) :: Co f a -> Co f b -> Co f a
as <* _ = as
defaultimap ::
Representable f => (Rep f -> a -> b) -> f a -> f b
defaultimap f fa = tabulate (f <*> index fa)
proves that we can (modulo package issues) use
class (FunctorWithIndex (Rep f) f, Distributive f) => Representable f where ...
The only potential downside (aside from package dep and compatibility issues) is that someone could theoretically use something other than Rep f
as the index. This seems a bit unlikely, but not entirely impossible.
As for the packaging, I think it's weird that XxxWithIndex
live in lens
. Are the lensy defaults really worth tying these simple classes up with the behemoth?
related commit: ekmett/comonad@f607067
problematic:
instance (Adjunction f g, Adjunction f' g') =>
Adjunction (Coproduct f f') (Product g g') where
unit a = Pair (leftAdjunct left a) (leftAdjunct right a)
counit = coproduct (rightAdjunct fstP) (rightAdjunct sndP)
where
fstP (Pair x _) = x
sndP (Pair _ x) = x
should be somehow replaced with Sum, not sure how
* Variable not in scope: left :: f a -> Sum f f' a
* Perhaps you meant one of these:
data constructor `Left' (imported from Prelude),
left' (imported from Data.Profunctor)
Perhaps you want to add `left' to the import list in the import of
`Control.Arrow' (src\Data\Functor\Adjunction.hs:37:1-35).
The generic Rep
definition is too robotic, if I derive Representable Count
I most likely don't want to index by Rep Count = Either () (Either () ())
but by something like data Move = Rock | Paper | Scissors
!
data Count a = Count
{ rock :: a
, paper :: a
, scissors :: a
}
deriving stock (Functor, Generic1)
deriving anyclass Distributive -- dummy
deriving anyclass Representable -- Rep Count = () `Either` () `Either` ()
I have implemented a via type that allows us to derive Representable
with a specified Rep
:
-- >> index (Count 1 2 3) Rock
-- 1
-- >> index (Count 1 2 3) Paper
-- 2
-- >> index (Count 1 2 3) Scissors
-- 3
data Count a = Count ..
deriving stock (Functor, Generically1)
deriving anyclass Distributive -- dummy
deriving Representable via Count `ShapedBy` Move -- Rep Count = Move
data Move = Rock | Paper | Scissors deriving stock Generic
-- >> (pi :# 10) `index` False
-- 3.141592653589793
-- >> (pi :# 10) `index` True
-- 10.0
--
-- >> tabulate @Pair id
-- False :# True
data Pair a = a :# a
deriving stock (Show, Functor, Generic1)
deriving anyclass Distributive -- dummy
deriving Representable via Pair `ShapedBy` Bool -- Rep Pair = Bool
{-# Language BlockArguments #-}
{-# Language FlexibleContexts #-}
{-# Language ImportQualifiedPost #-}
{-# Language InstanceSigs #-}
{-# Language PolyKinds #-}
{-# Language RankNTypes #-}
{-# Language ScopedTypeVariables #-}
{-# Language StandaloneKindSignatures #-}
{-# Language TypeApplications #-}
{-# Language TypeFamilies #-}
{-# Language TypeOperators #-}
{-# Language UndecidableInstances #-}
import Data.Coerce
import Data.Distributive
import Data.Functor.Rep hiding (gtabulate, gindex)
import Data.Kind
import GHC.Generics hiding (Rep)
import GHC.Generics qualified as GHC
type ShapedBy :: (k -> Type) -> argument -> (k -> Type)
newtype ShapedBy f arg a = ShapedBy (f a)
instance (Coercible (GHC.Rep rep ()) (RepToRep f), Generic1 f, Generic rep, GTabulate (Rep1 f), GIndex (Rep1 f)) => Functor (ShapedBy f rep) where
fmap = fmapRep
instance (Coercible (GHC.Rep rep ()) (RepToRep f), Generic1 f, Generic rep, GTabulate (Rep1 f), GIndex (Rep1 f)) => Distributive (ShapedBy f rep) where
distribute = distributeRep
collect = collectRep
instance (Coercible (GHC.Rep rep ()) (RepToRep f), Generic1 f, Generic rep, GTabulate (Rep1 f), GIndex (Rep1 f)) => Representable (ShapedBy f rep) where
type Rep (ShapedBy f rep) = rep
index :: ShapedBy f rep a -> (rep -> a)
index (ShapedBy as) = gindex as . roundtrip where
roundtrip :: rep -> RepToRep f
roundtrip = coerce . GHC.from @rep @()
tabulate :: forall a. (rep -> a) -> ShapedBy f rep a
tabulate make = ShapedBy $ gtabulate (make . roundtrip) where
roundtrip :: RepToRep f -> rep
roundtrip = GHC.to @rep @() . coerce
this uses the GRep'
machinary from adjunctions except RepToRep'
takes a generic representation and returns another generic representation.
type RepToRep :: (Type -> Type) -> Type
type RepToRep f = RepToRep' (Rep1 f) ()
gtabulate :: Generic1 f => GTabulate (Rep1 f) => (RepToRep f -> a) -> f a
gtabulate = to1 . gtabulate'
gindex :: Generic1 f => GIndex (Rep1 f) => f a -> RepToRep f -> a
gindex = gindex' . from1
type
RepToRep' :: (Type -> Type) -> (Type -> Type)
type family
RepToRep' rep
class GTabulate rep where
gtabulate' :: (RepToRep' rep () -> a) -> rep a
class GIndex rep where
gindex' :: rep a -> (RepToRep' rep () -> a)
type instance
RepToRep' Par1 = U1
instance GTabulate Par1 where
gtabulate' :: (U1 () -> a) -> Par1 a
gtabulate' f = Par1 (f U1)
instance GIndex Par1 where
gindex' :: Par1 a -> (U1 () -> a)
gindex' (Par1 a) U1 = a
type instance
RepToRep' (rep1 :*: rep2) = RepToRep' rep1 :+: RepToRep' rep2
instance (GTabulate rep1, GTabulate rep2) => GTabulate (rep1 :*: rep2) where
gtabulate' :: ((RepToRep' rep1 :+: RepToRep' rep2) () -> a) -> (rep1 :*: rep2) a
gtabulate' f = gtabulate' (f . L1) :*: gtabulate' (f . R1)
instance (GIndex rep1, GIndex rep2) => GIndex (rep1 :*: rep2) where
gindex' :: (rep1 :*: rep2) a -> ((RepToRep' rep1 :+: RepToRep' rep2) () -> a)
gindex' (a :*: _) (L1 i) = gindex' a i
gindex' (_ :*: b) (R1 j) = gindex' b j
type instance
RepToRep' (Rec1 f) = Rec0 (WrappedRep f)
instance Representable f => GTabulate (Rec1 f) where
gtabulate' :: forall a. (Rec0 (WrappedRep f) () -> a) -> Rec1 f a
gtabulate' = coerce do
tabulate @f @a
instance Representable f => GIndex (Rec1 f) where
gindex' :: forall a. Rec1 f a -> (Rec0 (WrappedRep f) () -> a)
gindex' = coerce do
index @f @a
type instance
RepToRep' (M1 i c rep) = RepToRep' rep
instance GTabulate rep => GTabulate (M1 i c rep) where
gtabulate' :: (RepToRep' rep () -> a) -> M1 i c rep a
gtabulate' = M1 . gtabulate'
instance GIndex rep => GIndex (M1 i c rep) where
gindex' :: M1 i c rep a -> (RepToRep' rep () -> a)
gindex' = gindex' . unM1
type instance
RepToRep' (f :.: rep) = Rec0 (WrappedRep f) :*: RepToRep' rep
instance (Representable f, GTabulate rep) => GTabulate (f :.: rep) where
gtabulate' :: forall a. ((Rec0 (WrappedRep f) :*: RepToRep' rep) () -> a) -> (f :.: rep) a
gtabulate' make = Comp1 do tabulate (gtabulate' <$> f) where
f :: Rep f -> RepToRep' rep () -> a
f a b = make (K1 (WrapRep a) :*: b)
instance (Representable f, GIndex rep) => GIndex (f :.: rep) where
gindex' :: (f :.: rep) a -> ((Rec0 (WrappedRep f) :*: RepToRep' rep) () -> a)
gindex' (Comp1 reps) (K1 (WrapRep a) :*: b) = gindex' (index reps a) b
There should be a function called something like indexM
in the Representable typeclass, with the type
indexM :: (Representable f, Monad m) => f a -> Rep f -> m a
Its default definition would simply be
indexM f k = return $ index f k
But for some Representable
s, it could produce a more defined result, for the same reasons as in Data.Vector
. An example for Product
:
indexM (Pair a _) (Left i) = indexM a i
indexM (Pair _ b) (Right j) = indexM b j
The result passed to return
contains no reference to the original Product
value. This could be a win for people using Representable
s for memoization.
Whenever ekmett/distributive#52 lands, we can and presumably should add
instance (Representable m, Monad m) => Representable (WrappedMonad m) where
type Rep (WrappedMonad m) = Rep m
index (WrapMonad m) k = index m k
tabulate f = WrapMonad (tabulate f)
Does this instance exist anywhere? F l r a b
is Kleisli r a b
with a phantom l
argument. It was inspired by the "In terms of representable functors" section.
type F :: (Type -> Type) -> (Type -> Type) -> Type -> (Type -> Type)
newtype F l r a b = F (a -> r b)
deriving Functor
via Kleisli r a
instance Adjunction l r => Distributive (F l r a) where
distribute :: Functor f => f (F l r a b) -> F l r a (f b)
distribute = distributeRep
instance Adjunction l r => Representable (F l r a) where
type Rep (F l r a) = l a
index :: F l r a b -> (l a -> b)
index (F f) = rightAdjunct f
tabulate :: (l a -> b) -> F l r a b
tabulate f = F (leftAdjunct f)
Hi (not using the bleeding edge), let's say I have
data Pair a = a :# a
instance Distributive Pair where
distribute :: Functor f => f (Pair a) -> Pair (f a)
distribute = distributeRep
instance Representable Pair where
type Rep Pair = Bool
index :: Pair a -> (Bool->a)
index (f:#_) False = f
index (_:#t) True = t
tabulate :: (Bool->a) -> Pair a
tabulate make = make False :# make True
Let's say I want to derive some instances via
Co Pair
data Pair a ..
deriving (Functor, Applicative, Monad, MonadReader Bool)
via Co Pair
Spot the mistake?
Because the Functor (Co f)
instance is stock derived it's the only instance that doesn't make use of Representable
and depends on f
being a functor already
-- | Data.Functor.Rep
newtype Co f a = Co { unCo :: f a } deriving Functor
This doesn't trigger an error? Maybe I'm doing it wrong. In any case this means that we trigger a cyclic definition fmap = fmap
.
I propose the following definition, which is in line with all other instances
instance Representable f => Functor (Co f) where
fmap :: (a->a') -> (Co f a->Co f a')
fmap = fmapRep
There doesn't seem to be a version of adjunctions available that advertises itself as working with the new comonad 4.0. Please could you release one? Thanks!
When playing with a bit of code I needed a combine function, which I think should be part of the adjunctions library.
combine :: Adjunction f u => u a -> f b -> (a, b)
combine ua = rightAdjunct (\b -> fmap (\a -> (a, b)) ua)
https://gist.github.com/1621224
greetings,
Sjoerd Visscher
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