Monad Transformers

Monad hides the plumbing involved with sequencing a bunch of actions. But they don't support alternation between different types of actions.

Motivating Example

Read three Ints from user and sum them.

First read one Int...

readInt0 :: IO (Maybe Int)
readInt0 = do
  s <- getLine
  if all isDigit s
    then pure $ Just (read s)
    else pure Nothing

Or:

readInt :: IO (Maybe Int)
readInt = do {- IO -}
  s <- getLine
  pure $ do {- Maybe -}
    guard (all isDigit s)
    Just (read s)

Then run it three times:

addThree :: IO (Maybe Int)
addThree = do
  mi <- readInt
  mj <- readInt
  mk <- readInt
  case (mi, mj, mk) of
    (Just i, Just j, Just k) -> pure $ Just (i+j+k)
    _                        -> pure Nothing

Could use a do-block for the final expression.

addThree :: IO (Maybe Int)
addThree = do {- IO -}
  mi <- readInt
  mj <- readInt
  mk <- readInt
  pure $ do {- Maybe -}
    i <- mi
    j <- mj
    k <- mk
    pure $ i + j + k

But this waits for all three lines before reporting error. Might rather fail as soon as one bad line is entered.

addThree :: IO (Maybe Int)
addThree = do {- IO -}
  mi <- readInt
  case mi of
    Nothing -> pure Nothing
    Just i  -> do {- IO -}
      mj <- readInt
      case mj of
        Nothing -> pure Nothing
        Just j -> do {- IO -}
          mk <- readInt
          case mk of
            Nothing -> pure Nothing
            Just k  -> pure $ Just (i+j+k)

Okay, let's reduce the ugliness here. Define new, specialized versions of bind and pure to combine Maybe and IO, so that we can write:

addThree =
  readInt `bindIOMaybe` \i ->
  readInt `bindIOMaybe` \j ->
  readInt `bindIOMaybe` \k ->
    pureIOMaybe $ i + j + k

So:

pureIOMaybe :: a -> IO (Maybe a)
pureIOMaybe a = pure $ Just a

bindIOMaybe :: IO (Maybe a) -> (a -> IO (Maybe b)) -> IO (Maybe b)
action `bindIOMaybe` f = do
  ma <- action
  case ma of
    Nothing -> pure Nothing
    Just a  -> f a

These are "mashups" of pure and (>>=) from the IO and Maybe instances. In fact, these definitions work, not just for IO, but for any Monad type m:

pureMonadPlusMaybe :: Monad m => a -> m (Maybe a)
pureMonadPlusMaybe a = pure $ Just a

bindMonadPlusMaybe :: Monad m => m (Maybe a) -> (a -> m (Maybe b)) -> m (Maybe b)
action `bindMonadPlusMaybe` f = do
  ma <- action
  case ma of
    Nothing -> pure Nothing
    Just a  -> f a

Factoring this common plumbing makes definitions like addThree much better than before. Now let's plug this plumbing into the Monad interface, to benefit from library and syntactic conveniences provided to Monad.

Monad Transformers

As demonstrated by the motivating example, the pattern we want is to "alternate" between actions of two Monad types m and m':

do
  x1 <- e0  -- e0 :: m (m' a1)  ~= MPrimeT m a1
  x2 <- e1  -- e1 :: m (m' a2)  ~= MPrimeT m a2
  x3 <- e2  -- e2 :: m (m' a3)  ~= MPrimeT m a3
  ...
  en        -- en :: m (m' an)  ~= MPrimeT m an

For each such m', we will define a combination of m and m' — called MPrimeT m — that forms a Monad.

Convention: The monad transformer FooT creates a new monad from an existing Monad type m (the "underlying" or "inner monad") by adding the functionality of Foo (the "base monad") to m.

The libraries

import Control.Monad.Trans.Maybe
import Control.Monad.Trans.List
import Control.Monad.Trans.State
import Control.Monad.Trans.Reader
import Control.Monad.Trans.Writer

define the following:

newtype MaybeT m a    = MaybeT  { runMaybeT  :: m (Maybe a)   }
newtype ListT m a     = ListT   { runListT   :: m [a]         }
newtype StateT s m a  = StateT  { runStateT  :: s -> m (a, s) }
newtype ReaderT r m a = ReaderT { runReaderT :: r -> m a      }
newtype WriterT w m a = WriterT { runWriterT :: m (a, w)      }

Each FooT, partially applied to all but its last arguments, is a Monad (when the underlying type m is).

MaybeT

Let's work through MaybeT, which is the combination we need for our example:

newtype MaybeT m a = MaybeT { runMaybeT :: m (Maybe a) }

We'll implement Monad first and take the free instances of Functor and Applicative:

instance Monad m => Monad (MaybeT m) where
 -- return :: a -> MaybeT m a
    return = MaybeT . return . Just

 -- (>>=) :: MaybeT m a -> (a -> MaybeT m b) -> MaybeT m b
    MaybeT mma >>= f = MaybeT $ do
      ma <- mma
      case ma of
        Nothing -> return Nothing
        Just a  -> runMaybeT $ f a

instance Monad m => Functor (MaybeT m) where {fmap f x = pure f <*> x}
instance Monad m => Applicative (MaybeT m) where {pure = return; (<*>) = ap}

The implementations of return and (>>=) are just like pureMonadPlusMaybe and bindMonadPlusMaybe implementations, but with MaybeT wrappers and runMaybeT unwrappers sprinkled in.

Example

Let's try to re-implement readInt and addThree to be MaybeT IO Int actions rather than "raw" IO (Maybe Int) actions.

We'll need to "lift" m actions to MaybeT m actions:

liftMaybeT :: Monad m => m a -> MaybeT m a
liftMaybeT ma = MaybeT (fmap Just ma)

And we'll define a version of guard for MaybeT m:

guardMaybeT :: Monad m => Bool -> MaybeT m ()
guardMaybeT True  = MaybeT $ pure $ Just ()
guardMaybeT False = MaybeT $ pure Nothing

Now we can define the following (notice the difference compared to the nested do blocks in the final version of readInt; now there is just one, with the "stacking" taking place inside)...

maybeReadInt :: MaybeT IO Int
maybeReadInt = do {- MaybeT IO -}
  s <- liftMaybeT getLine
  guardMaybeT $ all isDigit s
  pure $ read s

... and run it thrice:

maybeAddThree :: MaybeT IO Int
maybeAddThree = do {- MaybeT IO -}
  i <- maybeReadInt
  j <- maybeReadInt
  k <- maybeReadInt
  pure $ i+j+k

> runMaybeT maybeReadInt
> runMaybeT maybeAddThree

This is like the final version of addThree, achieved via the MaybeT plumbing — which allows Maybe to be composed with any Monad, not just IO.

There are two minor improvements to make, which will clean up maybeReadInt a bit more.

MonadTrans

First, liftMaybeT is specific to the MaybeT transformer. But, as we will see, we will want to lift functions to different transformers. Type class to describe a bunch of monad transformers:

class MonadTrans t where
    lift :: Monad m => m a -> t m a

instance MonadTrans MaybeT where
 -- lift :: Monad m => m a -> MaybeT m a
 -- lift = liftMaybeT
 -- lift ma = MaybeT (fmap Just ma)
    lift = MaybeT . fmap Just

This will be just as easy for all other monad transformers.

Alternative

Second, let's make MaybeT m an Alternative, so that we can simply use guard rather than guardMaybeT.

instance (Monad m, Alternative m) => Alternative (MaybeT m) where
 -- empty :: MaybeT m a
    empty = MaybeT empty

 -- (<|>) :: MaybeT m a -> MaybeT m a -> MaybeT m a
    MaybeT mma <|> MaybeT mmb = MaybeT $ mma <|> mmb

(Note: This instance is defined with different class constraints in the library.)

Now our final version of maybeReadInt:

maybeReadInt :: MaybeT IO Int
maybeReadInt = do
  s <- lift getLine
  guard $ all isDigit s
  pure $ read s

Source Files

• MaybeT.hs