On Sun, Jun 27, 2010 at 6:25 AM, Max Bolingbroke wrote: By the way, you can use this stuff to solve the restricted monad
problem (e.g. make Set an instance of Monad). This is not that useful
until we find out what the mother of all MonadPlus is, though, because
we really need a MonadPlus Set instance. Code below. Cheers,
Max \begin{code}
{-# LANGUAGE RankNTypes #-}
import Control.Applicative import Data.Set (Set)
import qualified Data.Set as S newtype CodensityOrd m a = CodensityOrd { runCodensityOrd :: forall b.
Ord b => (a -> m b) -> m b } -- liftCodensityOrd :: Monad m => m a -> CodensityOrd m a
-- liftCodensityOrd m = CodensityOrd ((>>=) m) -- lowerCodensityOrd :: (Ord a, Monad m) => CodensityOrd m a -> m a
-- lowerCodensityOrd m = runCodensityOrd m return instance Functor (CodensityOrd f) where
fmap f m = CodensityOrd (\k -> runCodensityOrd m (k . f)) instance Applicative (CodensityOrd f) where
pure x = CodensityOrd (\k -> k x)
mf <*> mx = CodensityOrd (\k -> runCodensityOrd mf (\f ->
runCodensityOrd mx (\x -> k (f x)))) instance Monad (CodensityOrd f) where
return = pure
m >>= k = CodensityOrd (\c -> runCodensityOrd m (\a ->
runCodensityOrd (k a) c)) liftSet :: Ord a => Set a -> CodensityOrd Set a
liftSet m = CodensityOrd (bind m)
where bind :: (Ord a, Ord b) => Set a -> (a -> Set b) -> Set b
mx `bind` fxmy = S.fold (\x my -> fxmy x `S.union` my) S.empty mx lowerSet :: Ord a => CodensityOrd Set a -> Set a
lowerSet m = runCodensityOrd m S.singleton main = print $ lowerSet $ monadicPlus (liftSet $ S.fromList [1, 2, 3])
(liftSet $ S.fromList [1, 2, 3]) monadicPlus :: Monad m => m Int -> m Int -> m Int
monadicPlus mx my = do
x <- mx
y <- my
return (x + y) \end{code} I've pointed out the Codensity Set monad on the Haskell channel. It is an
interesting novelty, but it unfortunately has somewhat funny semantics in
that the intermediate sets that you obtain are based on what you would get
if you reparenthesized all of your binds and associating them to the right.
One way to think about how Codensity adjusts a monad is that it can take
something that is almost a monad (you need to get in and out of Codensity).
Another thing to note is that Codensity is slightly more powerful than the
original type you embedded.
An interesting example is that you can show that Codensity Reader ~ State.
Take a look at the code in monad-ran on hackage for how Ran (StateT s m) is
implemented for an example.
-Edward Kmett