User:Benmachine/Cont

I found a bug with the <hask> tag. I put it on its own page so it doesn't ruin my user page.

It seems to me like Cont and ContT is way simpler than people make it. Essentially what it seems to be is the ability to give a name to the tail of a do-block. Consider this:

contstuff :: Magic a
contstuff = do
  thing1
  thing2
  -- Here I want to manipulate the rest of the computation.
  -- So I want a magic function that will give me the rest of it to
  -- play with.
  magic $ \rest ->
    -- Now I can just do it (tm), or do it twice, or discard it, or
    -- do it and then use the result to do it again... it's easy to
    -- imagine why this might be useful.
  thing3

The question is, what type should magic have? Well, let's say the whole do-block results in a thing of type r (without thinking too hard about what this means). Then certainly the function we give magic should result in type r as well, since it can run that do-block. The function should also accept a single parameter, referring to the tail of the computation. That's the rest of the do-block, which has type r, right? Well, more or less, with one caveat: we might bind the result of magic:

  x <- magic $ \rest -> -- ...
  thingInvolving x

so the rest of the do-block has an x in it that we need to supply (as well as other variables, but magic already has access to those). So the rest of the do-block can be thought of as a bit like a -> r. Given access to the rest of that do-block, we need to produce something of type r. So our lambda has type (a -> r) -> r and hence magic :: (a -> r) -> r -> Magic a

Magic a = Cont r a
magic = Cont

Tada!

The thing with Cont is I could implement it way before I understood it, because the types have really only one implementation, but here's a way of using the intuition above to implement Functor without thinking about the types too much:

instance Functor (Cont r) where
  fmap f (Cont g) = -- ...

Well, we've got to build a Cont value, and those always start the same way:

  fmap f (Cont g) = Cont $ \rest -> -- ...

Now what? Well, remember what g is. It looks like \rest -> stuffWith (rest val), where val is the 'value' of the computation (what would be bound with <-). So we want to give it a rest, but we don't want it to be called with the 'value' of the computation - we want f to be applied to it first. Well, that's easy:

  fmap f (Cont g) = Cont $ \rest -> g (\val -> rest (f val))

Load it in `ghci` and the types check. Amazing! Emboldened, let's try Applicative

instance Applicative (Cont r) where
  pure x = Cont $ \rest -> -- ...

We don't want to do anything special here. The rest of the computation wants a value, let's just give it one:

  pure x = Cont $ \rest -> rest x

What about <*>?

  Cont f <*> Cont x = Cont $ \rest -> -- ...

This is a little trickier, but if we look at how we did fmap we can guess at how we get the function and the value out to apply one to the other:

  Cont f <*> Cont x = Cont $ \rest -> f (\fn -> x (\val -> rest (fn val)))

Monad is a harder challenge, but the same basic tactic applies. Hint: remember to unwrap the newtype with runCont, case, or let. The latter two might be easier.

What about ContT?

The thing with ContT is that it's literally exactly the same trick. In fact I think the following definition works fine:

newtype ContT r m a = ContT (Cont (m r) a)
  deriving (Functor, Applicative, Monad)

runContT :: ContT r m a -> (a -> m r) -> m r
runContT (ContT m) = runCont m

The only reason the newtype exists at all is to make the kind compatible with things like MonadTrans.