ListT done right
Contents
Introduction
The Haskell hierarchical libraries implement a ListT monad transformer. There are, however, some problems with that implementation.

ListT
imposes unnecessary strictness. 
ListT
isn't really a monad transformer, ie.ListT m
isn't always a monad for a monadm
.
See the #Examples below for demonstrations of these problems.
Implementation
The following implementation tries to provide a replacement for the ListT transformer using the following technique. Instead of associating a monadic side effect with a list of values (m [a]
), it lets each element of the list have its own side effects, which only get `excecuted' if this element of the list is really inspected.
There is also a ListT done right alternative.
import Data.Maybe
import Control.Monad.State
import Control.Monad.Reader
import Control.Monad.Error
import Control.Monad.Cont
 The monadic list type
data MList' m a = MNil  a `MCons` MList m a
type MList m a = m (MList' m a)
 This can be directly used as a monad transformer
newtype ListT m a = ListT { runListT :: MList m a }
 A "lazy" run function, which only calculates the first solution.
runListT' :: Functor m => ListT m a > m (Maybe (a, ListT m a))
runListT' (ListT m) = fmap g m where
g MNil = Nothing
g (x `MCons` xs) = Just (x, ListT xs)
 In ListT from Control.Monad this one is the data constructor ListT, so sadly, this code can't be a dropin replacement.
liftList :: Monad m => [a] > ListT m a
liftList [] = ListT $ return MNil
liftList (x:xs) = ListT . return $ x `MCons` (runListT $ liftList xs)
instance Functor m => Functor (ListT m) where
fmap f (ListT m) = ListT $ fmap (fmap f) m
instance Functor m => Functor (MList' m) where
fmap _ MNil = MNil
fmap f (x `MCons` xs) = f x `MCons` fmap (fmap f) xs
 Why on earth isn't Monad declared `class Functor m => Monad m'?
 I assume that a monad is always a functor, so the contexts
 get a little larger than actually necessary
instance (Functor m, Monad m) => Monad (ListT m) where
return x = ListT . return $ x `MCons` return MNil
m >>= f = joinListT $ fmap f m
instance MonadTrans ListT where
lift = ListT . liftM (`MCons` return MNil)
instance (Functor m, Monad m) => MonadPlus (ListT m) where
mzero = liftList []
(ListT xs) `mplus` (ListT ys) = ListT $ xs `mAppend` ys
 Implemenation of join
joinListT :: (Functor m, Monad m) => ListT m (ListT m a) > ListT m a
joinListT (ListT xss) = ListT . joinMList $ fmap (fmap runListT) xss
joinMList :: (Functor m, Monad m) => MList m (MList m a) > MList m a
joinMList = (=<<) joinMList'
joinMList' :: (Functor m, Monad m) => MList' m (MList m a) > MList m a
joinMList' MNil = return MNil
joinMList' (x `MCons` xs) = x `mAppend` joinMList xs
mAppend :: (Functor m, Monad m) => MList m a > MList m a > MList m a
mAppend xs ys = (`mAppend'` ys) =<< xs
mAppend' :: (Functor m, Monad m) => MList' m a > MList m a > MList m a
mAppend' MNil ys = ys
mAppend' (x `MCons` xs) ys = return $ x `MCons` mAppend xs ys
 These things typecheck, but I haven't made sure what they do is sensible.
 (callCC almost certainly has to be changed in the same way as throwError)
instance (MonadIO m, Functor m) => MonadIO (ListT m) where
liftIO = lift . liftIO
instance (MonadReader s m, Functor m) => MonadReader s (ListT m) where
ask = lift ask
local f = ListT . local f . runListT
instance (MonadState s m, Functor m) => MonadState s (ListT m) where
get = lift get
put = lift . put
instance (MonadCont m, Functor m) => MonadCont (ListT m) where
callCC f = ListT $
callCC $ \c >
runListT . f $ \a >
ListT . c $ a `MCons` return MNil
instance (MonadError e m, Functor m) => MonadError e (ListT m) where
throwError = lift . throwError
{ This (perhaps more straightforward) implementation has the disadvantage
that it only catches errors that occur at the first position of the
list.
m `catchError` h = ListT $ runListT m `catchError` \e > runListT (h e)
}
 This is better because errors are caught everywhere in the list.
(m :: ListT m a) `catchError` h = ListT . deepCatch . runListT $ m
where
deepCatch :: MList m a > MList m a
deepCatch ml = fmap deepCatch' ml `catchError` \e > runListT (h e)
deepCatch' :: MList' m a > MList' m a
deepCatch' MNil = MNil
deepCatch' (x `MCons` xs) = x `MCons` deepCatch xs
Examples
Here are some examples that show why the old ListT is not right, and how to use the new ListT instead.
Sum of squares
Here's a silly example how to use ListT. It checks if an Int
n
is a sum of two squares. Each inspected possibility is printed, and if the number is indeed a sum of squares, another message is printed. Note that with our ListT, runMyTest only evaluates the side effects needed to find the first representation of n
as a sum of squares, which would be impossible with the ListT implementation of Control.Monad.List.ListT
.
myTest :: Int > ListT IO (Int, Int)
myTest n = do
let squares = liftList . takeWhile (<=n) $ map (^(2::Int)) [0..]
x < squares
y < squares
lift $ print (x,y)
guard $ x + y == n
lift $ putStrLn "Sum of squares."
return (x,y)
runMyTest :: Int > IO (Int, Int)
runMyTest = fmap (fst . fromJust) . runListT' . myTest
A little example session (runMyTest'
is implemented in exactly the same way as runMyTest
, but uses Control.Monad.List.ListT
):
*Main> runMyTest 5 (0,0) (0,1) (0,4) (1,0) (1,1) (1,4) Sum of squares. *Main> runMyTest' 5 (0,0) (0,1) (0,4) (1,0) (1,1) (1,4) Sum of squares. (4,0) (4,1) Sum of squares. (4,4)
Grouping effects
I didn't understand the statement "ListT m
isn't always a monad", even after I understood why it is too strict. I found the answer in Composing Monads. It's in fact a direct consequence of the unnecessary strictness. ListT m
is not associative (which is one of the monad laws), because grouping affects when side effects are run (which may in turn affect the answers). Consider
import Control.Monad.List
import Data.IORef
test1 :: ListT IO Int
test1 = do
r < liftIO (newIORef 0)
(next r `mplus` next r >> next r `mplus` next r) >> next r `mplus` next r
test2 :: ListT IO Int
test2 = do
r < liftIO (newIORef 0)
next r `mplus` next r >> (next r `mplus` next r >> next r `mplus` next r)
next :: IORef Int > ListT IO Int
next r = liftIO $ do x < readIORef r
writeIORef r (x+1)
return x
Under Control.Monad.List.ListT, test1 returns the answers
[6,7,8,9,10,11,12,13]
while test2 returns the answers
[4,5,6,7,10,11,12,13]
. Under the above ListT (if all answers are forced), both return
[2,3,5,6,9,10,12,13]
.
Order of printing
Here is another (simpler?) example showing why "ListT m
isn't always a monad".
a,b,c :: ListT IO ()
[a,b,c] = map (liftIO . putChar) ['a','b','c']
t1 :: ListT IO ()
t1 = ((a `mplus` a) >> b) >> c
t2 :: ListT IO ()
t2 = (a `mplus` a) >> (b >> c)
Under Control.Monad.List.ListT
, running runListT t1
prints "aabbcc", while runListT t2
instead prints "aabcbc". Under the above ListT, they both print "abc" (if all answers were forced, they would print "abcabc").
Relation to Nondet
NonDeterminism describes another monad transformer that can also be used to model nondeterminism. In fact, ListT
and NondetT
are quite similar with the following two functions translating between them
toListT :: (Monad m) => NondetT m a > ListT m a
toListT (NondetT fold) = ListT $ fold ((return.) . MCons) (return MNil)
toNondetT :: (Monad m) => ListT m a > NondetT m a
toNondetT (ListT ml) = NondetT (\c n > fold c n ml) where
fold :: Monad m => (a > m b > m b) > m b > MList m a > m b
fold c n xs = fold' c n =<< xs
fold' :: Monad m => (a > m b > m b) > m b > MList' m a > m b
fold' _ n MNil = n
fold' c n (x `MCons` xs) = c x (fold c n xs)
ListT
is smaller than NondetT
in the sense that toListT . toNondetT
is the identity (is it ok to call ListT
`retract'?). However, these functions don't define an isomorphism (check for example NondetT (\_ n > liftM2 const n n)
).
I propose to replace every occurence of `fmap` in the above code with `liftM`, thereby moving `class Functor` and the complaint about it not being a superclass of `Monad` completely out of the picture. I'd simply do it, if there wasn't this feeling that I have overlooked something obvious. What is it?  Udo Stenzel
There's no particular reason why I used fmap, except that the page has the (unfortunate!) title "ListT Done Right", and having Functor superclass of Monad certainly is the right thing. But I agree, that mistake has long been done and I feel my halfhearted cure is worse than the disease. You can find an alternative, more concise definition of a ListT transformer based on evenstyle lists here: ListT done right alternative
amb has AmbT, which could be considered as 'ListT done right' (since Amb is identical to the list monad).