Starting with a concrete instance of my question, we all know (and love) the Monad type class:
class ... => Monad m where
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> mb
...
Consider the following would-be instance, where we modify the standard list/"nondeterminism" instance using nub to retain only one copy of each "outcome":
type DistinctList a = DL { dL :: [a] }
instance Monad DistinctList where
return = DL . return
x >>= f = DL . nub $ (dL x) >>= (dL . f)
...Do you spot the error? The problem is that nub :: Eq a => [a] -> [a] and so x >>= f is only defined under the condition f :: Eq b => a -> DistinctList b, whereas the compiler demands f :: a -> DistinctList b. Is there some way I can proceed anyway?
Stepping back, suppose I have a would-be instance that is only defined under some condition on the parametric type's variable. I understand that this is generally not allowed because other code written with the type class cannot be guaranteed to supply parameter values that obey the condition. But are there circumstances where this still can be carried out? If so, how?
Here is an adaptation of the technique applied in set-monad to your case.
Note there is, as there must be, some "cheating". The structure includes extra value constructors to represent "return" and "bind". These act as suspended computations that need to be run. The Eq instance is there part of the run function, while the constructors that create the "suspension" are Eq free.
{-# LANGUAGE GADTs #-}
import qualified Data.List as L
import qualified Data.Functor as F
import qualified Control.Applicative as A
import Control.Monad
-- for reference, the bind operation to be implemented
-- bind operation requires Eq
dlbind :: Eq b => [a] -> (a -> [b]) -> [b]
dlbind xs f = L.nub $ xs >>= f
-- data structure comes with incorporated return and bind
-- `Prim xs` wraps a list into a DL
data DL a where
Prim :: [a] -> DL a
Return :: a -> DL a
Bind :: DL a -> (a -> DL b) -> DL b
-- converts a DL to a list
run :: Eq a => DL a -> [a]
run (Prim xs) = xs
run (Return x) = [x]
run (Bind (Prim xs) f) = L.nub $ concatMap (run . f) xs
run (Bind (Return x) f) = run (f x)
run (Bind (Bind ma f) g) = run (Bind ma (\a -> Bind (f a) g))
-- lifting of Eq and Show instance
-- Note: you probably should provide a different instance
-- one where eq doesn't depend on the position of the elements
-- otherwise you break functor laws (and everything else)
instance (Eq a) => Eq (DL a) where
dxs == dys = run dxs == run dys
-- this "cheats", i.e. it will convert to lists in order to show.
-- executing returns and binds in the process
instance (Show a, Eq a) => Show (DL a) where
show = show . run
-- uses the monad instance
instance F.Functor DL where
fmap = liftM
-- uses the monad instance
instance A.Applicative DL where
pure = return
(<*>) = ap
-- builds the DL using Return and Bind constructors
instance Monad DL where
return = Return
(>>=) = Bind
-- examples with bind for a "normal list" and a "distinct list"
list = [1,2,3,4] >>= (\x -> [x `mod` 2, x `mod` 3])
dlist = (Prim [1,2,3,4]) >>= (\x -> Prim [x `mod` 2, x `mod` 3])
And here is a dirty hack to make it more efficient, addressing the points raised below about evaluation of bind.
{-# LANGUAGE GADTs #-}
import qualified Data.List as L
import qualified Data.Set as S
import qualified Data.Functor as F
import qualified Control.Applicative as A
import Control.Monad
dlbind xs f = L.nub $ xs >>= f
data DL a where
Prim :: Eq a => [a] -> DL a
Return :: a -> DL a
Bind :: DL b -> (b -> DL a) -> DL a
-- Fail :: DL a -- could be add to clear failure chains
run :: Eq a => DL a -> [a]
run (Prim xs) = xs
run (Return x) = [x]
run b@(Bind _ _) =
case foldChain b of
(Bind (Prim xs) f) -> L.nub $ concatMap (run . f) xs
(Bind (Return a) f) -> run (f a)
(Bind (Bind ma f) g) -> run (Bind ma (\a -> Bind (f a) g))
-- fold a chain ((( ... >>= f) >>= g) >>= h
foldChain :: DL u -> DL u
foldChain (Bind b2 g) = stepChain $ Bind (foldChain b2) g
foldChain dxs = dxs
-- simplify (Prim _ >>= f) >>= g
-- if (f x = Prim _)
-- then reduce to (Prim _ >>= g)
-- else preserve (Prim _ >>= f) >>= g
stepChain :: DL u -> DL u
stepChain b@(Bind (Bind (Prim xs) f) g) =
let dys = map f xs
pms = [Prim ys | Prim ys <- dys]
ret = [Return ys | Return ys <- dys]
bnd = [Bind ys f | Bind ys f <- dys]
in case (pms, ret, bnd) of
-- ([],[],[]) -> Fail -- could clear failure
(dxs@(Prim ys:_),[],[]) -> let Prim xs = joinPrims dxs (Prim $ mkEmpty ys)
in Bind (Prim $ L.nub xs) g
_ -> b
stepChain dxs = dxs
-- empty list with type via proxy
mkEmpty :: proxy a -> [a]
mkEmpty proxy = []
-- concatenate Prims in on Prim
joinPrims [] dys = dys
joinPrims (Prim zs : dzs) dys = let Prim xs = joinPrims dzs dys in Prim (zs ++ xs)
instance (Ord a) => Eq (DL a) where
dxs == dys = run dxs == run dys
instance (Ord a) => Ord (DL a) where
compare dxs dys = compare (run dxs) (run dys)
instance (Show a, Eq a) => Show (DL a) where
show = show . run
instance F.Functor DL where
fmap = liftM
instance A.Applicative DL where
pure = return
(<*>) = ap
instance Monad DL where
return = Return
(>>=) = Bind
-- cheating here, Prim is needed for efficiency
return' x = Prim [x]
s = [1,2,3,4] >>= (\x -> [x `mod` 2, x `mod` 3])
t = (Prim [1,2,3,4]) >>= (\x -> Prim [x `mod` 2, x `mod` 3])
r' = ((Prim [1..1000]) >>= (\x -> return' 1)) >>= (\x -> Prim [1..1000])