Melding Monads

In a previous post, I demonstrated how to use the lazy state monad to turn Geraint Jones and Jeremy Gibbon’s breadth-first labeling algorithm into a reusable abstraction, and mentioned that I did not know how to combine the result with continuations. In this post, I will do exactly that.

A hat tip goes to Twan van Laarhoven for suggesting that I split my fresh operator into two, which I adopt in this post. In the comments, down is defined as:

type Fresh n a = State [n] a

down :: Fresh n a -> Fresh n a
down m = do
    (n:ns) <- get
    put ns
    a <- m
    ns' <- get
    put (n:ns')
    return a

Now, if we naïvely apply the continuation transformer in the Monad Transformer Library, the get and put operators are lifted as follows:

type FreshCC r n a = ContT r (State [n]) a

lift :: Monad m => m a -> ContT r m a
lift m = ContT (m >>=)

down :: FreshCC r n a -> FreshCC r n a
down m = do
    (n:ns) <- lift get
    lift (put ns)
    a <- m
    ns' <- lift get
    lift (put (n:ns'))
    return a

The problem is that this replaces the >>= of the lazy state monad with the >>= of the continuation transformer, and these two operators have different strictness properties. This, in turn, leads to non-termination. The trick is not to lift get and put, but rather conjugate down:

lower :: Monad m => ContT a m a -> m a
lower m = runContT m return

down :: FreshCC a n a -> FreshCC r n a
down m = lift $ do
    (n:ns) <- get
    put ns
    a <- lower m
    ns' <- get
    put (n:ns')
    return a

-- *or*

down = lift . original_down . lower

Now, the conjugation preserves the use of the lazy state monad’s >>= in the definition of down, however it changes the type of the argument from FreshCC r n a to FreshCC a n a! The other definitions contained in the previous post stay much the same.

Feel free to download freshcc.hs, a full working demonstration of this post. One can even use callCC, fresh, and down in the same computation and terminate! Sadly, I don’t have any demonstrations of this combination, nor do I have any applications in mind. My intuition about callCC is currently quite limited in this context.

I have implemented Dan Friedman’s angel-devil-milestone control operators in conjunction with fresh and down, and have used it to duplicate labels and change the shape of a spirited tree; but I’m definitely cheating at this point, as all I have done is observe that the definitions compute something. I have no control over what’s going on, nor do I know what the definitions are supposed to compute. (yet)