On Sun, Jun 27, 2010 at 6:25 AM, Max Bolingbroke <batterseapower@hotmail.com> 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