io-transformer

An `IO` transformer

It has been claimed that there is no IO transformer in Haskell. This seems overly pessimistic. The obvious approach is to use

newtype IOT m a = IOT (m (IO a))

However, we get stuck when we try to define

join :: Monad m => m (IO (m (IO a))) -> m (IO a)

The typical approach would be to try to commute the IO (m ...) to m (IO ...) but there’s just no way to do that. We can’t even do it for some simple choices of m like Maybe let alone uniformly for all Monads m.

But there’s a another way. If we can turn all the ms into IO then we can just join them all together into one IO action. This suggests an IO transformer like

newtype IOT m a = IOT ((forall a. m a -> IO a) -> IO a)

In fact, it’s just a use of the Reader transformer.

newtype Morph f g = Morph { runMorph :: forall a. (f a -> g a) }
newtype IOT m a = IOT { runIOT :: ReaderT (Morph m IO) IO a }

Using ReaderT here is just a convenience that means we automatically get a Monad instance cheaply. It doesn’t play a role of any substance.

instance Monad (IOT m) where
  return = IOT . return
  m >>= f = IOT (runIOT m >>= (runIOT . f))

The MonadTrans instance is

instance MonadTrans IOT where
  lift act = IOT $ do
    f <- ask
    lift (runMorph f act)

That is, we just convert all the m actions into IO actions. (Note that the latter lift is in ReaderT). Does it satisfy the MonadTrans laws? Well (eliding runMorph and IOT wrapping/unwrapping for notational convenience),

lift (return x) = do
    f <- ask
    lift (f (return x))

which is

lift (return x) = do
    f <- ask
    return x

in the case where f (return x) = return x, i.e. lift . return = return. Additionally

lift (m >>= g) = do
    f <- ask
    lift (f (m >>= g))

which is

lift (m >>= g) = do
    f <- ask
    x <- lift (f m)
    f' <- ask
    lift (f' (g x))

i.e.

lift m >>= \x -> lift (g x)

that is

lift m >>= (lift . g)

in the case where f (m >>= g) = f m >>= (f . g). So lift (m >>= g) = lift m >>= (lift . g).

Thus we see that IOT is a monad transformer if the reader environment is a monad morphism f satisfying

f (return x) = return x
f (m >>= g) = f m >>= (f . g)

How to use `IOT`

You can run an IOT m action by providing a way of interpreting an m action in IO. (Remember the interpretation has to be a monad morphism for the monad laws to hold).

runIOT' :: (forall a. m a -> IO a) -> IOT m a -> IO a
runIOT' f m = runReaderT (runIOT m) (Morph f)

It seems that there are several uses of this transformer.

`Maybe`

For m = Maybe we can throw an error for Nothing.

handleMaybe :: Maybe a -> IO a
handleMaybe Nothing = error "Nothing"
handleMaybe (Just a) = return a

maybeExample :: IO ()
maybeExample = runIOT' handleMaybe $ do
  liftIO (putStrLn "Hello")
  lift Nothing
  liftIO (putStrLn "Goodbye")

> maybeExample 
Hello
*** Exception: Nothing

`Either`

For m = Either e we can throw an error which depends on the Left value.

handleEither :: Show e => Either e a -> IO a
handleEither (Left e) = error (show e)
handleEither (Right a) = return a

eitherExample :: IO ()
eitherExample = runIOT' handleEither $ do
  liftIO (putStrLn "Hello")
  lift (Left 1234)
  liftIO (putStrLn "Goodbye")

> eitherExample 
Hello
*** Exception: 1234

`Writer`

For m = Writer [w] we can print a log message.

handleWriter :: Show w => Writer [w] a -> IO a
handleWriter m = mapM_ print ws >> return a
  where (a, ws) = runWriter m

writerExample :: IO ()
writerExample = runIOT' handleWriter $ do
  liftIO (putStrLn "Hello")
  lift (tell ["log1", "log2"])
  liftIO (putStrLn "Goodbye")
  lift (tell ["log3"])

> writerExample 
Hello
"log1"
"log2"
Goodbye
"log3"

`State s`

For m = State s we can use an IORef to hold the state.

stateExample :: IO ()
stateExample = do
  ref <- newIORef 0
  let handleState :: State Integer a -> IO a
      handleState m = do
        s <- readIORef ref
        let (a, s') = runState m s
        writeIORef ref s'
        return a

  runIOT' handleState $ do
    x <- lift get
    liftIO (print x)
    lift (modify (+1))
    y <- lift get
    liftIO (print y)

> stateExample 
0
1

Is this really an `IO` transformer?

Firstly note that IOT Maybe, IOT (Either e) and IOT (State s) seem to give strictly less power than MaybeT IO, EitherT e IO and StateT s IO respectively. IOT (Writer [w]) does seem to be different to WriterT [w] IO in that it preserves the ordering of IO and Writer actions.

But is there some definition for “M transformer”, for an arbitrary monad M? And what is the justification for saying that Haskell has no IO transformer? Even though it may not be especially useful, this IOT seems like a reasonable candidate for one.

The code

{-# LANGUAGE Rank2Types #-}

import Control.Monad.Trans.Reader
import Control.Monad.Trans.Writer
import Control.Monad.Trans.State
import Control.Monad.Trans.Class
import Data.IORef

newtype Morph f g = Morph { runMorph :: forall a. (f a -> g a) }

newtype IOT m a = IOT { runIOT :: ReaderT (Morph m IO) IO a }

instance Monad (IOT m) where
  return = IOT . return
  m >>= f = IOT (runIOT m >>= (runIOT . f))

instance MonadTrans IOT where
  lift act = IOT $ do
    f <- ask
    lift (runMorph f act)

liftIO :: IO a -> IOT m a
liftIO = IOT . lift

handleMaybe :: Maybe a -> IO a
handleMaybe Nothing = error "Nothing"
handleMaybe (Just a) = return a

runIOT' :: (forall a. m a -> IO a) -> IOT m a -> IO a
runIOT' f m = runReaderT (runIOT m) (Morph f)

maybeExample :: IO ()
maybeExample = runIOT' handleMaybe $ do
  liftIO (putStrLn "Hello")
  lift Nothing
  liftIO (putStrLn "Goodbye")

handleEither :: Show e => Either e a -> IO a
handleEither (Left e) = error (show e)
handleEither (Right a) = return a

eitherExample :: IO ()
eitherExample = runIOT' handleEither $ do
  liftIO (putStrLn "Hello")
  lift (Left 1234)
  liftIO (putStrLn "Goodbye")

handleWriter :: Show w => Writer [w] a -> IO a
handleWriter m = mapM_ print ws >> return a
  where (a, ws) = runWriter m

writerExample :: IO ()
writerExample = runIOT' handleWriter $ do
  liftIO (putStrLn "Hello")
  lift (tell ["log1", "log2"])
  liftIO (putStrLn "Goodbye")
  lift (tell ["log3"])

stateExample :: IO ()
stateExample = do
  ref <- newIORef 0
  let handleState :: State Integer a -> IO a
      handleState m = do
        s <- readIORef ref
        let (a, s') = runState m s
        writeIORef ref s'
        return a

  runIOT' handleState $ do
    x <- lift get
    liftIO (print x)
    lift (modify (+1))
    y <- lift get

The H2 Wiki

io-transformer

An IO transformer

How to use IOT

Maybe

Either

Writer

State s

Is this really an IO transformer?