Graham Klyne writes:
Is there a function that switches monadic layering?
f :: (Monad m1,Monad m2) => m1 (m2 a) -> m2 (m1 a)
Does this even make sense in the general case? I'm thinking along the lines of a generalization of sequence to non-list monads.
In order to do that, the two monads have to be composable. Mark Jones's paper "Functional Programming with Overloading and Higher-Order Polymorphism"[1] brings up a similar function while discussing monad composition and transformers: swap :: m (n a) -> n (m a) He ends up defining two classes of monads with appropriate swap functions: class Monad m => Into m where into :: Monad n => m (n a) -> n (m a) class Monad m => OutOf m where outof :: Monad n => n (m a) -> m (n a) List, Maybe, Error, and Writer are instances of Into, and Reader is an instance of OutOf. For Either String, the instance would be something like this: instance Into (Either String) where into (Left e) = return (Left e) into (Right m) = fmap Right m The only wrinkle is that Jones declares Monad as a subclass of Functor, but the Prelude doesn't for some reason. In that case, you could still use: into (Right m) = m >>= return . Right [1] http://www.cse.ogi.edu/~mpj/pubs/springschool.html
... map looks like a version of liftM specialized to lists: ... Similarly, concat appears to be a monadic join on lists.
They are. It's easier to see with a slightly different definition of
Monad:
class Functor f where
fmap :: (a -> b) -> f a -> f b
-- when f is a Monad and (>>=) is defined, fmap may be defined as
-- fmap f m = m >>= return . f
class Functor m => Monad m where
return :: a -> m a
join :: m (m a) -> m a
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
fail :: String -> m a
-- minimal definition: return and (>>=) or return, join, and fmap
join m = m >>= id
m >>= k = join (fmap k m)
m >> n = m >>= (\_ -> n)
fail s = error s
Then you can define the list monad like so:
instance Functor [] where
fmap = map
instance Monad [] where
return x = [x]
join = concat
fail _ = []
liftM plays the same role as fmap.
--
David Menendez