How to make callCC more dynamic?

Question

I thought the right type for ContT should be

newtype ContT m a = ContT {runContT :: forall r. (a -> m r) -> m r}

and other control op

Answer 1

Although there is no way to win the given game, if we can twist the game a little bit we could win!

newtype ContT' m a =
    ContT' { runContT' :: forall r. (Maybe a -> m (Maybe r)) -> m (Maybe r) }

As we have seen in the other answer, the problem we have is, to win we have to generate a value for an arbitrary type our opponent played, but we do not know how to do so.

By forcing all raw types (r and a) being decorated by Maybe, we can bypass this problem, and are able to generate a value of any Maybe t -- simply give Nothing to them!

We got to show this is a Monad.

instance Monad m => Monad (ContT' m) where
    return a = ContT' $ \k -> k (Just a)
    a >>= f = ContT' $ \c -> runContT' a (
        maybe (return Nothing) (\v -> runContT' (f v) c)) 

And then We can implement callCC:

class Monad m => MonadCont' m where
    callCC' :: ((a -> forall b.m b) -> m a) -> m a

instance Monad m => MonadCont' (ContT' m) where
    callCC' k = ContT' $ \c -> 
        runContT' (k (\a -> ContT' $ \_ -> c (Just a) >> return Nothing)) c  

I am still working out how to implemen reset and shift.