apfelmus wrote:
class DiMonad m where returnR :: a -> m e a bindR :: m e a -> (a -> m e b) -> m e b
returnL :: e -> m e a bindL :: m e a -> (e -> m e' a) -> m e' a
type TwoCont e a = (e -> R) -> (a -> R) -> R
A final question remains: does the dimonad abstraction cover the full power of TwoCont? I mean, it still seems like there's an operation missing that supplies new left and right continuations at once.
I think that this missing operation is bind2 :: m e a -> (e -> m e' a') -> (a -> m e' a') -> m e' a' It executes the second or the third argument depending on whether the first argument is a failure or a success. First, bind2 can be defined for TwoCont bind2 m fe fa = \e' a' -> m (\e -> (fe e) e' a') (\a -> (fa a) e' a') Apparently, bindL and bindR can be expressed with bind2 bindL m f = bind2 m f returnR bindR m f = bind2 m returnL f The question is whether bind2 can be expressed by bindL and bindR or whether bind2 offers more than both. It turns out that bind2 can be formulated from bindL and bindR alone fmapR f m = m `bindR` returnR . f bind2 m fe fa = ((Left `fmapR` m) `bindL` (\e -> Right `fmapR` fe e)) `bindR` (\aa' -> case aa' of Left a -> fa a Right a' -> returnR a') The definitions is rather cumbersome and we omit the proof that both definitions for bind2 are the same for TwoConts. For a general proof, we'd need an axiomatic characterization of dimonads and bind2. Recast in the light of MonadError, bind2 gives rise to a combinator bind2 :: MonadError e m => m a -> (e -> m b) -> (a -> m b) -> m b that executes either failure or success path. The important point is the inequality bind2 m fe fa ≠ (m >>= fa) `catchError` fe There's no equivalent to bind2 (with a better name, of course) in the libraries at the moment. The only function that does come near bind2 in the libraries is Control.Exception.try :: IO a -> IO (Either Exception a) which can be used rather easily to implement bind2 of course. Interestingly, the existence of try shows that there is a natural isomorphism DiMonad m => m e a ≅ m () (Either e a) so that dimonads do not add anything beyond what monads can already do. (More precisely, try gives one direction. The other direction is rather obvious.) Regards, apfelmus