ListT done right

(Difference between revisions)

1 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
m
.

See the #Examples below for demonstrations of these problems.

2 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, and the Pipes package provides its own version as well.

```import Data.Maybe

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 drop-in 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
return x = ListT . return \$ x `MCons` return MNil
m >>= f = joinListT \$ fmap f m

lift = ListT . liftM (`MCons` return MNil)

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)
liftIO = lift . liftIO

local f = ListT . local f . runListT

instance (MonadState s m, Functor m) => MonadState s (ListT m) where
get = lift get
put = lift . put

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```

3 Examples

Here are some examples that show why the old ListT is not right, and how to use the new ListT instead.

3.1 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
.
```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
):
```*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)
```

3.2 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```

[6,7,8,9,10,11,12,13]
[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]
.

3.3 Order of printing

Here is another (simpler?) example showing why "
ListT m
```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
, 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").

3.4 Order of ListT []

This is a simple example that doesn't use
IO
, only pure
ListT []
.
```v :: Int -> ListT [] Int
v 0 = ListT [[0, 1]]
v 1 = ListT [[0], [1]]

main = do
print \$ runListT \$ ((v >=> v) >=> v) 0
-- = [[0,1,0,0,1],[0,1,1,0,1],[0,1,0,0],[0,1,0,1],[0,1,1,0],[0,1,1,1]]
print \$ runListT \$ (v >=> (v >=> v)) 0
-- = [[0,1,0,0,1],[0,1,0,0],[0,1,0,1],[0,1,1,0,1],[0,1,1,0],[0,1,1,1]]```
Clearly,
ListT []
fails to preserve the associativity monad law. This example violates the requirement given in the documentation that the inner monad has to be commutative. However, all the preceding examples use
IO
which is neither commutative, so I suppose this example is valid at the end. Most likely, a proper implementation of
ListT
should not have such a requirement.

--PetrP 19:15, 27 September 2012 (UTC)

4 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 half-hearted cure is worse than the disease. You can find an alternative, more concise definition of a ListT transformer based on even-style 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).