Benjamin Franksen wrote:
As has been already mentioned in this thread, in
http://www.soi.city.ac.uk/~ross/papers/Applicative.html Conor McBride and
Ross Paterson invent/explain a new type class that is now part of the base
package (Control.Applicative). They also use/propose syntactic sugar for
it, i.e.
pure f <*> u1 <*> ... <*> un
~~> (| f u1 ... un |)
(I just made up the symbols '(|' and '|)', the concrete syntax would have to
be fixed by people more knowledgeable than me.)
The problem with [| and |] lifted to monads that this only works for
fully applied arguments, i.e. that
handle :: IO Handle
string :: IO String
[| hPutStr handle string |] :: IO ()
works but
[| hPutStr handle |]
= join (return hPutStr `ap` handle)
^= join ((f :: m (a -> b -> m c)) `ap` (x :: m a))
= join ( y :: m (b -> m c))
is a type error.
I think this is also what makes the (<- action) proposal so non-local
and what is the source of this whole discussion. The core problem is:
Functions like a -> b -> m c can't be partially applied to
monadic actions like m a without specifying the number of
arguments in advance. In other words, such functions aren't
curried correctly.
Clearly, LiftMn specifies the number of arguments. But _both_ the (<-)
proposal and idiom brackets specify the number of arguments too! Namely
by requiring that all arguments are fully applied. So, neither one is
capable of partially applying the first argument without saturating the
call, you can only write
handle :: IO Handle
-- define putStr in terms of the above hPutStr
putStr :: String -> IO ()
putStr = \x -> [| hPutStr handle (return x) |]
putStr = \x -> do { hPutStr (<- handle) x }
One way to get currying for monads is to work with functions
m a -> m b -> m c
However, this type is larger than a -> b -> m c , i.e. the function
from :: Monad m => (a -> b -> m c) -> (m a -> m b -> m c)
from f ma mb = ma >>= \a -> mb >>= \b -> f a b
is not surjective since we could perform the actions in a different order
from2 f ma mb = mb >>= \b -> ma >>= \a -> f a b
In other words, if someone gives you a value of type m a -> m b -> m c
, then you can't convert it to a -> b -> m c and back without
risking that you end up with a different result.
But there is another type suitable for currying
m (a -> m (b -> m c))
which I believe is in some way equivalent to a -> b -> m c
from :: Monad m => (a -> b -> m c) -> m (a -> m (b -> m c))
from f = return $ \a -> return $ \b -> f a b
to :: Monad m => m (a -> m (b -> m c)) -> (a -> b -> m c)
to f a b = f >>= \g -> g a >>= \h -> h b
but I'm not sure. My assumption is that we have an equivalence
forall a,b . m (a -> m b) ~ (a -> m b)
because any side effect executed by the extra m on the outside can well
be delayed until we are supplied a value a. Well, at least when all
arguments are fully applied, for some notion of "fully applied"
Anyway, here's how to curry with that type:
(@) :: Monad m => m (a -> m b) -> (m a -> m b)
(@) f x = join (f `ap` x)
hPutStr :: IO (Handle -> IO (String -> IO ()))
handle :: IO Handle
putStr :: IO (String -> IO ())
putStr = hPutStr @ handle
With the infix type synonym
type (~>) a b = a -> IO b
we can also write
hPutStr :: IO (Handle ~> String ~> () )
putStr :: IO (String ~> () )
This is of course the Kleisli-Arrow which explains why currying works.
Regards,
apfelmus
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