On Friday 20 May 2011 14:31:54, Kees Bleijenberg wrote:
I was playing with winghci and I tried:
mapM id [Just 1, Just 2, Just 3] result: Just [1,2,3]
I don't understand this answer.
From http://members.chello.nl/hjgtuyl/tourdemonad.html#mapM http://members.chello.nl/hjgtuyl/tourdemonad.html#mapM mapM mf xs takes a monadic function mf (having type Monad m => (a -> m b)) and applies it to each element in list xs; the result is a list inside a monad.
A few things I've found: mapM :: (a -> m b) -> [a] -> m [b] So in this case: a = Maybe Int (second arg in mapM id [Just1, Just 2, Just 3] and b = Int and m = Maybe. So id is :: Maybe Int -> Maybe Int
In more detail, [Just 1, Just 2, Just 3] :: [Maybe Int] That means a = Maybe Int, in mapM id [Just 1, Just 2, Just 3] id :: t -> t (for all t) in mapM id list, id must have type (a -> m b) for some Monad m, we have to unify the type (t -> t) with (a -> m b); that gives t = a (the argument types must be the same) and t = m b (the result types must be the same) Now a = Maybe Int = t = m b, hence m = Maybe, b = Int, thus mapM id [Just 1, Just 2, Just 3] :: m [b] = Maybe [Int]
mapM id [Just 1, Nothing, Just 3] result: Nothing. My first guess for the result: Just [Just 1, Nothing, Just 3]
No, that would have type Maybe [Maybe Int], but we've seen that it must have type Maybe [Int], so it's either Just (list of some Ints) or Nothing. Now, since mapM is polymorphic (works with all Monads and all parameter types), there's not much choice for doing something sensible. mapM can only use the function it gets as first argument and the Monad methods, so mapM foo [] = return [] mapM foo (x:xs) = ? well, it could ignore x (not sensible) or it can apply foo to x and then do something with that and xs; what can it do with xs? Well, mapM foo xs. So we have to somehow sensible combine foo x :: m b mapM foo xs :: m [b] The most natural is do y <- foo x ys <- mapM foo xs return (y:ys) or, without do-notation, foo x >>= \y -> mapM foo xs >>= \ys -> return (y:ys) or, with a combinator, liftM2 (:) (foo x) (mapM foo xs) Now, in the above, mapM id [Just 1, Nothing, Just 3], the Monad instance of Maybe says Nothing >>= _ = Nothing, so we get Just 1 >>= \y -> (Nothing >>= \z -> whatever) ~> Just 1 >>= \y -> Nothing ~> (\y -> Nothing) 1 ~> Nothing
when I do: mapM id [1,2,3] I get an error (id has wrong type, which makes sense)
Can somebody explain what is going on here?
Kees