I would like to generate infinite stream of numbers with Rand monad from System.Random.MWC.Monad. If only there would be a MonadFix instance for this monad, or instance like this:
instance (PrimMonad m) => MonadFix m where
...
then one could write:
runWithSystemRandom (mfix (\ xs -> uniform >>= \x -> return (x:xs)))
There isn't one though.
I was going through MonadFix docs but I don't see an obvious way of implementing this instance.
You can write a MonadFix instance. However, the code will not generate an infinite stream of distinct random numbers. The argument to mfix is a function that calls uniform exactly once. When the code is run, it will call uniform exactly once, and create an infinite list containing the result.
You can try the equivalent IO code to see what happens:
import System.Random
import Control.Monad.Fix
main = print . take 10 =<< mfix (\xs -> randomIO >>= (\x -> return (x : xs :: [Int])))
It seems that you want to use a stateful random number generator, and you want to run the generator and collect its results lazily. That isn't possible without careful use of unsafePerformIO. Unless you need to produce many random numbers quickly, you can use a pure RNG function such as randomRs instead.
A question: how do you wish to generate your initial seed?
The problem is that MWS is built on the "primitive" package which abstracts only IO and strict (Control.Monad.ST.ST s). It does not also abstract lazy (Control.Monad.ST.Lazy.ST s).
Perhaps one could make instances for "primitive" to cover lazy ST and then MWS could be lazy.
UPDATE: I can make this work using Control.Monad.ST.Lazy by using strictToLazyST:
module Main where
import Control.Monad(replicateM)
import qualified Control.Monad.ST as S
import qualified Control.Monad.ST.Lazy as L
import qualified System.Random.MWC as A
foo :: Int -> L.ST s [Int]
foo i = do rest <- foo $! succ i
return (i:rest)
splam :: A.Gen s -> S.ST s Int
splam = A.uniformR (0,100)
getS :: Int -> S.ST s [Int]
getS n = do gen <- A.create
replicateM n (splam gen)
getL :: Int -> L.ST s [Int]
getL n = do gen <- createLazy
replicateM n (L.strictToLazyST (splam gen))
createLazy :: L.ST s (A.Gen s)
createLazy = L.strictToLazyST A.create
makeLots :: A.Gen s -> L.ST s [Int]
makeLots gen = do x <- L.strictToLazyST (A.uniformR (0,100) gen)
rest <- makeLots gen
return (x:rest)
main = do
print (S.runST (getS 8))
print (L.runST (getL 8))
let inf = L.runST (foo 0) :: [Int]
print (take 10 inf)
let inf3 = L.runST (createLazy >>= makeLots) :: [Int]
print (take 10 inf3)
(This would be better suited as a comment to Heatsink's answer, but it's a bit too long.)
MonadFix instances must adhere to several laws. One of them is left shrinking/thightening:
mfix (\x -> a >>= \y -> f x y) = a >>= \y -> mfix (\x -> f x y)
This law allows to rewrite your expression as
mfix (\xs -> uniform >>= \x -> return (x:xs))
= uniform >>= \x -> mfix (\xs -> return (x:xs))
= uniform >>= \x -> mfix (return . (x :))
Using another law, purity mfix (return . h) = return (fix h), we can further simplify to
= uniform >>= \x -> return (fix (x :))
and using the standard monad laws and rewriting fix (x :) as repeat x
= liftM (\x -> fix (x :)) uniform
= liftM repeat uniform
Therefore, the result is indeed one invocation of uniform and then just repeating the single value indefinitely.
Related
How can I get a maximum element of an effectful container where computing attribute to compare against also triggers an effect?
There has to be more readable way of doing things like:
latest dir = Turtle.fold (z (ls dir)) Fold.maximum
z :: MonadIO m => m Turtle.FilePath -> m (UTCTime, Turtle.FilePath)
z mx = do
x <- mx
d <- datefile x
return (d, x)
I used overloaded version rather than non-overloaded maximumBy but the latter seems better suite for ad-hoc attribute selection.
How can I be more methodic in solving similar problems?
So I know nothing about Turtle; no idea whether this fits well with the rest of the Turtle ecosystem. But since you convinced me in the comments that maximumByM is worth writing by hand, here's how I would do it:
maximumOnM :: (Monad m, Ord b) => (a -> m b) -> [a] -> m a
maximumOnM cmp [x] = return x -- skip the effects if there's no need for comparison
maximumOnM cmp (x:xs) = cmp x >>= \b -> go x b xs where
go x b [] = return x
go x b (x':xs) = do
b' <- cmp x'
if b < b' then go x' b' xs else go x b xs
I generally prefer the *On versions of things -- which take a function that maps to an Orderable element -- to the *By versions -- which take a function that does the comparison directly. A maximumByM would be similar but have a type like Monad m => (a -> a -> m Ordering) -> [a] -> m a, but this would likely force you to redo effects for each a, and I'm guessing it's not what you want. I find *On more often matches with the thing I want to do and the performance characteristics I want.
Since you're already familiar with Fold, you might want to get to know FoldM, which is similar.
data FoldM m a b =
-- FoldM step initial extract
forall x . FoldM (x -> a -> m x) (m x) (x -> m b)
You can write:
maximumOnM ::
(Ord b, Monad m)
=> (a -> m b) -> FoldM m a (Maybe a)
maximumOnM f = FoldM combine (pure Nothing) (fmap snd)
where
combine Nothing a = do
f_a <- f a
pure (Just (f_a, a))
combine o#(Just (f_old, old)) new = do
f_new <- f new
if f_new > f_old
then pure $ Just (f_new, new)
else pure o
Now you can use Foldl.foldM to run the fold on a list (or other Foldable container). Like Fold, FoldM has an Applicative instance, so you can combine multiple effectful folds into one that interleaves the effects of each of them and combines their results.
It's possible to run effects on foldables using reducers package.
I'm not sure if it's correct, but it leverages existing combinators and instances (except for Bounded (Maybe a)).
import Data.Semigroup.Applicative (Ap(..))
import Data.Semigroup.Reducer (foldReduce)
import Data.Semigroup (Max(..))
import System.IO (withFile, hFileSize, IOMode(..))
-- | maxLength
--
-- >>> getMax $ maxLength ["abc","a","hello",""]
-- 5
maxLength :: [String] -> (Max Int)
maxLength = foldReduce . map (length)
-- | maxLengthIO
--
-- Note, this runs IO...
--
-- >>> (getAp $ maxLengthIO ["package.yaml", "src/Lib.hs"]) >>= return . getMax
-- Just 1212
--
-- >>> (getAp $ maxLengthIO []) >>= return . getMax
-- Nothing
maxLengthIO :: [String] -> Ap IO (Max (Maybe Integer))
maxLengthIO xs = foldReduce (map (fmap Just . f) xs) where
f :: String -> IO Integer
f s = withFile s ReadMode hFileSize
instance Ord a => Bounded (Maybe a) where
maxBound = Nothing
minBound = Nothing
According to the Control.Arrow documentation, for many monads (those for which the >>= operation is strict) the instance MonadFix m => ArrowLoop (Kleisli m) does not satisfy the right-tightening law (loop (f >>> first h) = loop f >>> h) required by the ArrowLoop class. Why is that so?
This is multi-faceted question with several different angles, and it goes back to the value-recursion (mfix/mdo) in Haskell. See here for background information. I'll try to address the right-tightening issue here in some detail.
Right-tightening for mfix
Here's the right-tightening property for mfix:
mfix (λ(x, y). f x >>= λz. g z >>= λw. return (z, w))
= mfix f >>= λz. g z >>= λw. return (z, w)
Here it is in picture form:
The dotted-lines show where the "knot-tying" is happening. This is essentially the same law as mentioned in the question, except it is expressed in terms of mfix and value-recursion. As shown in Section 3.1 of this work, for a monad with a strict bind operator, you can always write an expression that distinguishes the left-hand side of this equation from the right hand-side, thus failing this property. (See below for an actual example in Haskell.)
When an arrow is created via the Kleisli construction from a monad with mfix, the corresponding loop operator fails the corresponding property in the same way.
Domain-theory and approximations
In domain theoretic terms, the mismatch will always be an approximation.
That is, the left hand-side will always be less defined than the right. (More precisely, the lhs will be lower than the rhs, in the domain of PCPOs, the typical domain we use for Haskell semantics.) In practice, this means that the right hand side will terminate more often, and is to be preferred when that is an issue. Again, see Section 3.1 of this for details.
In practice
This may sound all abstract, and in a certain sense it is. More intuitively, the left hand side gets a chance to act on the recursive value as it is being produced since g is inside the "loop," and thus is able to interfere with the fixed-point computation. Here's an actual Haskell program to illustrate:
import Control.Monad.Fix
f :: [Int] -> IO [Int]
f xs = return (1:xs)
g :: [Int] -> IO Int
g [x] = return x
g _ = return 1
lhs = mfix (\(x, y) -> f x >>= \z -> g z >>= \w -> return (z, w))
rhs = mfix f >>= \z -> g z >>= \w -> return (z, w)
If you evaluate the lhs it will never terminate, while rhs will give you the infinite list of 1's as expected:
*Main> :t lhs
lhs :: IO ([Int], Int)
*Main> lhs >>= \(xs, y) -> return (take 5 xs, y)
^CInterrupted.
*Main> rhs >>= \(xs, y) -> return (take 5 xs, y)
([1,1,1,1,1],1)
I interrupted the computation in the first case as it is non-terminating. While this is a contrived example, it is the simplest to illustrate the point. (See below for a rendering of this example using the mdo-notation, which might be easier to read.)
Example monads
Typical examples of monads that do not satisfy this law include Maybe, List, IO, or any other monad that's based on an algebraic type with multiple constructors. Typical examples of monads that do satisfy this law are State and Environment monads. See Section 4.10 for a table summarizing these results.
Pure-right shrinking
Note that a "weaker" form of right-tightening, where the function g in the above equation is pure, follows from value-recursion laws:
mfix (λ(x, y). f x >>= λz. return (z, h z))
= mfix f >>= λz. return (z, h z)
This is the same law as before with, g = return . h. That is g cannot perform any effects. In this case, there is no way to distinguish the left-hand side from the right as you might expect; and the result indeed follows from value-recursion axioms. (See Section 2.6.3 for a proof.) The picture in this case looks like this:
This property follows from the sliding property, which is a version of dinaturality for value-recursion, and is known to be satisfied by many monads of interest: Section 2.4.
Impact on the mdo-notation
The failure of this law has an impact on how the mdo notation was designed in GHC. The translation includes the so called "segmentation" step precisely to avoid the failure of the right-shrinking law. Some people consider that a bit controversial as GHC automatically picks the segments, essentially applying the right-tightening law. If explicit control is needed, GHC provides the rec keyword to leave the decision to the users.
Using the mdo-notation and explicit do rec, the above example renders
as follows:
{-# LANGUAGE RecursiveDo #-}
f :: [Int] -> IO [Int]
f xs = return (1:xs)
g :: [Int] -> IO Int
g [x] = return x
g _ = return 1
lhs :: IO ([Int], Int)
lhs = do rec x <- f x
w <- g x
return (x, w)
rhs :: IO ([Int], Int)
rhs = mdo x <- f x
w <- g x
return (x, w)
One might naively expect that lhs and rhs above should be the same, but due to the failure of the right-shrinking law, they are not. Just like before, lhs gets stuck, while rhs successfully produces the value:
*Main> lhs >>= \(x, y) -> return (take 5 x, y)
^CInterrupted.
*Main> rhs >>= \(x, y) -> return (take 5 x, y)
([1,1,1,1,1],1)
Visually inspecting the code, we see that the recursion is simply for the function f, which justifies the segmentation that is automatically performed by the mdo-notation.
If the rec notation is to be preferred, the programmer will need to put it in minimal blocks to ensure termination. For instance, the above expression for lhs should be written as follows:
lhs :: IO ([Int], Int)
lhs = do rec x <- f x
w <- g x
return (x, w)
The mdo-notation takes care of this and places the recursion over minimal blocks without user intervention.
Failure for Kleisli Arrows
After this lengthy detour, let us now go back to the original question about the corresponding law for arrows. Similar to the mfix case, we can construct a failing example for Kleisli arrows as well. In fact, the above example translates more or less directly:
{-# LANGUAGE Arrows #-}
import Control.Arrow
f :: Kleisli IO ([Int], [Int]) ([Int], [Int])
f = proc (_, ys) -> returnA -< (ys, 1:ys)
g :: Kleisli IO [Int] Int
g = proc xs -> case xs of
[x] -> returnA -< x
_ -> returnA -< 1
lhs, rhs :: Kleisli IO [Int] Int
lhs = loop (f >>> first g)
rhs = loop f >>> g
Just like in the case of mfix, we have:
*Main> runKleisli rhs []
1
*Main> runKleisli lhs []
^CInterrupted.
The failure of right-tightening for mfix of the IO-monad also prevents the Kleisli IO arrow from satisfying the right-tightening law in the ArrowLoop instance.
At first glance I thought these two functions would work the same:
firstM _ [] = return Nothing
firstM p (x:xs) = p x >>= \r -> if r then return (Just x) else firstM p xs
firstM' p xs = fmap listToMaybe (mapM p xs)
But they don't. In particular, firstM stops as soon as the first p x is true. But firstM', because of mapM, needs the evaluate the whole list.
Is there a "lazy mapM" that enables the second definition, or at least one that doesn't require explicit recursion?
There isn't (can't be) a safe, Monad-polymorphic lazy mapM. But the monad-loops package contains many lazy monadic variants of various pure functions, and includes firstM.
One solution is to use ListT, the list monad transformer. This type interleaves side effects and results, so you can peek at the initial element without running the whole computation first.
Here's an example using ListT:
import Control.Monad
import qualified ListT
firstM :: Monad m => (a -> Bool) -> [a] -> m (Maybe a)
firstM p = ListT.head . mfilter p . ListT.fromFoldable
(Note that the ListT defined in transformers and mtl is buggy and should not be used. The version I linked above should be okay, though.)
If there is, I doubt it's called mapM.
As I recall, mapM is defined in terms of sequence:
mapM :: Monad m => (a -> b) -> [a] -> m [b]
mapM f = sequence . map f
and the whole point of sequence is to guarantee that all the side-effects are done before giving you anything.
As opposed to using some alternate mapM, you could get away with just using map and sequence, so you could change the container from [a] to Just a:
firstM p xs = sequence $ listToMaybe (map p xs)
or even:
firstM p xs = mapM f $ listToMaybe xs
Now that mapM and sequence can operate on generic Traversables, not just lists.
Suppose the code
f :: IO [Int]
f = f >>= return . (0 :)
g :: IO [Int]
g = f >>= return . take 3
When I run g in ghci, it cause stackoverflow. But I was thinking maybe it could be evaluated lazily and produce [0, 0, 0] wrapped in IO. I suspect IO is to blame here, but I really have no idea. Obviously the following works:
f' :: [Int]
f' = 0 : f'
g' :: [Int]
g' = take 3 f'
Edit: In fact I am not interested in having such a simple function f, original code looked more along the lines:
h :: a -> IO [Either b c]
h a = do
(r, a') <- h' a
case r of
x#(Left _) -> h a' >>= return . (x :)
y#(Right _) -> return [y]
h' :: IO (Either b c, a)
-- something non trivial
main :: IO ()
main = mapM_ print . take 3 =<< h a
h does some IO computations and stores invalid (Left) responses in a list until a valid response (Right) is produced. The attempt is to construct the list lazily even though we are in the IO monad. So that someone reading the result of h can start consuming the list even before it is complete (because it may even be infinite). And if the one reading the results cares only for the first 3 entries no matter what, the rest of the list does not even have to be constructed. And I am getting the feeling that this will not be possible :/.
Yes, IO is to blame here. >>= for IO is strict in the "state of the world". If you write m >>= h, you'll get an action that first performs the action m, then applies h to the result, and finally performs the action h yields. It doesn't matter that your f action doesn't "do anything"; it has to be performed anyway. Thus you end up in an infinite loop starting the f action over and over.
Thankfully, there is a way around this, because IO is an instance of MonadFix. You can "magically" access the result of an IO action from within that action. Critically, that access must be sufficiently lazy, or you'll throw yourself into an infinite loop.
import Control.Monad.Fix
import Data.Functor ((<$>))
f :: IO [Int]
f = mfix (\xs -> return (0 : xs))
-- This `g` is just like yours, but prettier IMO
g :: IO [Int]
g = take 3 <$> f
There's even a bit of syntactic sugar in GHC for this letting you use do notation with the rec keyword or mdo notation.
{-# LANGUAGE RecursiveDo #-}
f' :: IO [Int]
f' = do
rec res <- (0:) <$> (return res :: IO [Int])
return res
f'' :: IO [Int]
f'' = mdo
res <- f'
return (0 : res)
For more interesting examples of ways to use MonadFix, see the Haskell Wiki.
It sounds like you want a monad that mixes the capabilities of lists and IO. Luckily, that's just what ListT is for. Here's your example in that form, with an h' that computes the Collatz sequence and asks the user how they feel about each element in the sequence (I couldn't really think of anything convincing that fit the shape of your outline).
import Control.Monad.IO.Class
import qualified ListT as L
h :: Int -> L.ListT IO (Either String ())
h a = do
(r, a') <- liftIO (h' a)
case r of
x#(Left _) -> L.cons x (h a')
y#(Right _) -> return y
h' :: Int -> IO (Either String (), Int)
h' 1 = return (Right (), 1)
h' n = do
putStrLn $ "Say something about " ++ show n
s <- getLine
return (Left s, if even n then n `div` 2 else 3*n + 1)
main = readLn >>= L.traverse_ print . L.take 3 . h
Here's how it looks in ghci:
> main
2
Say something about 2
small
Left "small"
Right ()
> main
3
Say something about 3
prime
Left "prime"
Say something about 10
not prime
Left "not prime"
Say something about 5
fiver
Left "fiver"
I suppose modern approaches would use pipes or conduits or iteratees or something, but I don't know enough about them to talk about the tradeoffs compared to ListT.
I'm not sure if this is an appropriate usage, but unsafeInterleaveIO would get you the behavior you're asking for, by deferring the IO actions of f until the value inside of f is asked for:
module Tmp where
import System.IO.Unsafe (unsafeInterleaveIO)
f :: IO [Int]
f = unsafeInterleaveIO f >>= return . (0 :)
g :: IO [Int]
g = f >>= return . take 3
*Tmp> g
[0,0,0]
UPDATE: Okay this question becomes potentially very straightforward.
q <- mapM return [1..]
Why does this never return?
Does mapM not lazily deal with infinite lists?
The code below hangs. However, if I replace line A by line B, it doesn't hang anymore. Alternatively, if I preceed line A by a "splitRandom $", it also doesn't hang.
Q1 is: Is mapM not lazy? Otherwise, why does replacing line A with line B "fix this" code?
Q2 is: Why does preceeding line A with splitRandom "solve" the problem?
import Control.Monad.Random
import Control.Applicative
f :: (RandomGen g) => Rand g (Double, [Double])
f = do
b <- splitRandom $ sequence $ repeat $ getRandom
c <- mapM return b -- A
-- let c = map id b -- B
a <- getRandom
return (a, c)
splitRandom :: (RandomGen g) => Rand g a -> Rand g a
splitRandom code = evalRand code <$> getSplit
t0 = do
(a, b) <- evalRand f <$> newStdGen
print a
print (take 3 b)
The code generates an infinite list of random numbers lazily. Then it generates a single random number. By using splitRandom, I can evaluate this latter random number first before the infinite list. This can be demonstrated if I return b instead of c in the function.
However, if I apply the mapM to the list, the program now hangs. To prevent this hanging, I have to apply splitRandom again before the mapM. I was under the impression that mapM can lazily
Well, there's lazy, and then there's lazy. mapM is indeed lazy in that it doesn't do more work than it has to. However, look at the type signature:
mapM :: (Monad m) => (a -> m b) -> [a] -> m [b]
Think about what this means: You give it a function a -> m b and a bunch of as. A regular map can turn those into a bunch of m bs, but not an m [b]. The only way to combine the bs into a single [b] without the monad getting in the way is to use >>= to sequence the m bs together to construct the list.
In fact, mapM is precisely equivalent to sequence . map.
In general, for any monadic expression, if the value is used at all, the entire chain of >>=s leading to the expression must be forced, so applying sequence to an infinite list can't ever finish.
If you want to work with an unbounded monadic sequence, you'll either need explicit flow control--e.g., a loop termination condition baked into the chain of binds somehow, which simple recursive functions like mapM and sequence don't provide--or a step-by-step sequence, something like this:
data Stream m a = Nil | Stream a (m (Stream m a))
...so that you only force as many monad layers as necessary.
Edit:: Regarding splitRandom, what's going on there is that you're passing it a Rand computation, evaluating that with the seed splitRandom gets, then returning the result. Without the splitRandom, the seed used by the single getRandom has to come from the final result of sequencing the infinite list, hence it hangs. With the extra splitRandom, the seed used only needs to thread though the two splitRandom calls, so it works. The final list of random numbers works because you've left the Rand monad at that point and nothing depends on its final state.
Okay this question becomes potentially very straightforward.
q <- mapM return [1..]
Why does this never return?
It's not necessarily true. It depends on the monad you're in.
For example, with the identity monad, you can use the result lazily and it terminates fine:
newtype Identity a = Identity a
instance Monad Identity where
Identity x >>= k = k x
return = Identity
-- "foo" is the infinite list of all the positive integers
foo :: [Integer]
Identity foo = do
q <- mapM return [1..]
return q
main :: IO ()
main = print $ take 20 foo -- [1 .. 20]
Here's an attempt at a proof that mapM return [1..] doesn't terminate. Let's assume for the moment that we're in the Identity monad (the argument will apply to any other monad just as well):
mapM return [1..] -- initial expression
sequence (map return [1 ..]) -- unfold mapM
let k m m' = m >>= \x ->
m' >>= \xs ->
return (x : xs)
in foldr k (return []) (map return [1..]) -- unfold sequence
So far so good...
-- unfold foldr
let k m m' = m >>= \x ->
m' >>= \xs ->
return (x : xs)
go [] = return []
go (y:ys) = k y (go ys)
in go (map return [1..])
-- unfold map so we have enough of a list to pattern-match go:
go (return 1 : map return [2..])
-- unfold go:
k (return 1) (go (map return [2..])
-- unfold k:
(return 1) >>= \x -> go (map return [2..]) >>= \xs -> return (x:xs)
Recall that return a = Identity a in the Identity monad, and (Identity a) >>= f = f a in the Identity monad. Continuing:
-- unfold >>= :
(\x -> go (map return [2..]) >>= \xs -> return (x:xs)) 1
-- apply 1 to \x -> ... :
go (map return [2..]) >>= \xs -> return (1:xs)
-- unfold >>= :
(\xs -> return (1:xs)) (go (map return [2..]))
Note that at this point we'd love to apply to \xs, but we can't yet! We have to instead continue unfolding until we have a value to apply:
-- unfold map for go:
(\xs -> return (1:xs)) (go (return 2 : map return [3..]))
-- unfold go:
(\xs -> return (1:xs)) (k (return 2) (go (map return [3..])))
-- unfold k:
(\xs -> return (1:xs)) ((return 2) >>= \x2 ->
(go (map return [3..])) >>= \xs2 ->
return (x2:xs2))
-- unfold >>= :
(\xs -> return (1:xs)) ((\x2 -> (go (map return [3...])) >>= \xs2 ->
return (x2:xs2)) 2)
At this point, we still can't apply to \xs, but we can apply to \x2. Continuing:
-- apply 2 to \x2 :
(\xs -> return (1:xs)) ((go (map return [3...])) >>= \xs2 ->
return (2:xs2))
-- unfold >>= :
(\xs -> return (1:xs)) (\xs2 -> return (2:xs2)) (go (map return [3..]))
Now we've gotten to a point where neither \xs nor \xs2 can be reduced yet! Our only choice is:
-- unfold map for go, and so on...
(\xs -> return (1:xs))
(\xs2 -> return (2:xs2))
(go ((return 3) : (map return [4..])))
So you can see that, because of foldr, we're building up a series of functions to apply, starting from the end of the list and working our way back up. Because at each step the input list is infinite, this unfolding will never terminate and we will never get an answer.
This makes sense if you look at this example (borrowed from another StackOverflow thread, I can't find which one at the moment). In the following list of monads:
mebs = [Just 3, Just 4, Nothing]
we would expect sequence to catch the Nothing and return a failure for the whole thing:
sequence mebs = Nothing
However, for this list:
mebs2 = [Just 3, Just 4]
we would expect sequence to give us:
sequence mebs = Just [3, 4]
In other words, sequence has to see the whole list of monadic computations, string them together, and run them all in order to come up with the right answer. There's no way sequence can give an answer without seeing the whole list.
Note: The previous version of this answer asserted that foldr computes starting from the back of the list, and wouldn't work at all on infinite lists, but that's incorrect! If the operator you pass to foldr is lazy on its second argument and produces output with a lazy data constructor like a list, foldr will happily work with an infinite list. See foldr (\x xs -> (replicate x x) ++ xs) [] [1...] for an example. But that's not the case with our operator k.
This question is showing very well the difference between the IO Monad and other Monads. In the background the mapM builds an expression with a bind operation (>>=) between all the list elements to turn the list of monadic expressions into a monadic expression of a list. Now, what is different in the IO monad is that the execution model of Haskell is executing expressions during the bind in the IO Monad. This is exactly what finally forces (in a purely lazy world) something to be executed at all.
So IO Monad is special in a way, it is using the sequence paradigm of bind to actually enforce execution of each step and this is what our program makes to execute anything at all in the end. Others Monads are different. They have other meanings of the bind operator, depending on the Monad. IO is actually the one Monad which execute things in the bind and this is the reason why IO types are "actions".
The following example show that other Monads do not enforce execution, the Maybe monad for example. Finally this leds to the result that a mapM in the IO Monad returns an expression, which - when executed - executes each single element before returning the final value.
There are nice papers about this, start here or search for denotational semantics and Monads:
Tackling the awkward squad: http://research.microsoft.com/en-us/um/people/simonpj/papers/marktoberdorf/mark.pdf
Example with Maybe Monad:
module Main where
fstMaybe :: [Int] -> Maybe [Int]
fstMaybe = mapM (\x -> if x == 3 then Nothing else Just x)
main = do
let r = fstMaybe [1..]
return r
Let's talk about this in a more generic context.
As the other answers said, the mapM is just a combination of sequence and map. So the problem is why sequence is strict in certain Monads. However, this is not restricted to Monads but also Applicatives since we have sequenceA which share the same implementation of sequence in most cases.
Now look at the (specialized for lists) type signature of sequenceA :
sequenceA :: Applicative f => [f a] -> f [a]
How would you do this? You were given a list, so you would like to use foldr on this list.
sequenceA = foldr f b where ...
--f :: f a -> f [a] -> f [a]
--b :: f [a]
Since f is an Applicative, you know what b coule be - pure []. But what is f?
Obviously it is a lifted version of (:):
(:) :: a -> [a] -> [a]
So now we know how sequenceA works:
sequenceA = foldr f b where
f a b = (:) <$> a <*> b
b = pure []
or
sequenceA = foldr ((<*>) . fmap (:)) (pure [])
Assume you were given a lazy list (x:_|_). The above definition of sequenceA gives
sequenceA (x:_|_) === (:) <$> x <*> foldr ((<*>) . fmap (:)) (pure []) _|_
=== (:) <$> x <*> _|_
So now we see the problem was reduced to consider weather f <*> _|_ is _|_ or not. Obviously if f is strict this is _|_, but if f is not strict, to allow a stop of evaluation we require <*> itself to be non-strict.
So the criteria for an applicative functor to have a sequenceA that stops on will be
the <*> operator to be non-strict. A simple test would be
const a <$> _|_ === _|_ ====> strict sequenceA
-- remember f <$> a === pure f <*> a
If we are talking about Moands, the criteria is
_|_ >> const a === _|_ ===> strict sequence