Is Haskell's mapM not lazy?

问题:

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

回答1:

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.

回答2:

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.

回答3:

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]

回答4:

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

回答5:

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