Undeepening Continuations

Posted on March 13, 2015
Tags: tutorial Categories: haskell

Continuation was always a mystery to me since I started to learn monad.

My friend Javran once sent me a link to ‘the mother of all monads’, from where I start to rethink the significance of the continuation monad. That article, to me as a novice, wasn’t enough explanative for me to comprehend. Therefore, to demystify the significance of the continuation monad, I spend several hours to read through articles about it last night. In the final, I eventually carried my understanding to continuation out into code and achieved enlightenment. (not really)

So let’s get started. First of all, we need to know, at least have a primitive concept of, what continuation is used for. So here is my understanding,

A continuation is an intermediate state that allows computation to run at that point.

You don’t need to try hardly on comprehending the statement above. By going through this tutorial you will make your own understanding out.

Computation with Holes

As we said above a continuation is a intermediate state that we can delay the computation to the future. Let’s see how these functions look like:

transformOne = ??? 1

doSomethingOnOne = do
  putStrLn "before execution"
  ??? 1
  putStrLn "after execution"

So we’re defining two functions with this feature. The first one is rather easy to see, on the blank (???) there should be a function, which takes an integer and yields whatever things. In the doSomethingOnOne there should also be a function like that in the first, expect being constrained to return an IO a.

Here’s the modified finished functions and their corresponding types:

transformOne :: (Int -> r) -> r
transformOne f = f 1

doSomethingOnOne :: (Int -> IO a) -> IO ()
doSomethingOnOne f = do
  putStrLn "before execution"
  val <- f 1
  putStrLn ("we get " ++ show val)

If you have other experiences about Control.Monad.Cont, you might be able to recognize the pattern in the types. Strictly, the latter one isn’t what continuation monad looks like because IO a is not the same as IO (). Well, this is normal in our real life programming, but we can’t call them continuation because they’re not composable. And one of the main purposes we want to generalize continuation is to make them composable so they could be an instance of Monad. Why? Well, a monad is just a MONOID in the category endofunctors, monoid implies composablity, what’s the problem? Why? Well, a monoid is a just a SEMIGROUP in combine with an identity object, what’s the problem? But why? Well, that’s how semigroup is defined, what’s the problem?

Anyway, we’ll see if we can make a continuation stronger, that is, in type of (a -> r) -> r, we make our life easier. For now, just remember it, that (a -> r) -> r is the type of continuation. Which means we can’t have the second piece of code above unless we add a return val at the end of the do notation. It also means you can’t have continuation like foo f = show (f 1), which modifies the result.

These functions are what we called continuations. It’s intuitive to think them as functions with a hole to be filled. To fill the hole in a continuation we need to feed it with a function that takes the passed in value and return something we needed from outside of the hole. I call these functions fed to a continuation hole-filler functions. For the arguments these hole-fillers take, I call them seed and the returning thing was barely named yield. I made these names just for referring the components more intuitively.

How to Fill in the Holes

We now see how (a -> r) -> r tastes like. You might think, well, continuation is weak that we are even restricted from modifying with the hole-filler’s yield. Well, yes, we cannot modify the yield. But continuation is not as weak as you might think, because modifying the yield is not the correct way to use a continuation.

If you just want to modify the result of a passed in function and carry on the computation, you want a Monad instead of a continuation. Yet without being capable to modify the yield, you’re allowed to modify the seed freely.

The significance of continuation is, in my understanding, just hole-filling. It’s like, if we’re playing with continuation, our aim will be giving out a seed to a incoming hole-filler, for which we don’t know what it is or what it will do. In the other words, we pass our result of computation IN instead of return it OUT.

If we’ve composed a bunch of continuation together, that is, we get a very deeply nested continuations, how can we take the result out? Since the computation always throw their results inwards, by the seeds. Can we acquire the seed out of the continuation? Just think of the simplest example: transformOne f = f 1. The answer is, id (id x = x). If we feed the continuation with id function as hole-filler, it will yield the seed without any modification and then we can acquire it.

As we take out the result, we can do whatever we want with it. Also we can take some actions directly onto the result. Think of feed a continuation with a print hole-filler, then we will see the seed printed out as expect.

Now let’s think, how do we generate a continuation from a single value such that we can take out result out by feeding the continuation with id? The solution isn’t very hard:

genCont :: a -> (a -> r) -> r
genCont val f = f val

or more point-lesslyfreely:

genCont = flip ($)

So far, we have gain a basic concept how continuations are created and composed together.

Functor Property of Continuations

We now look on how to transform the seed with a function. If we have a function f :: a -> b, we hope we can use it to transform a continuation with type (a -> r) -> r into (b -> r) -> r. What does it mean? If I have a continuation with seed of type Int, how can use feed it with a hole-filler that eats Strings? The answer is to have a transforming function of type Int -> String.

Here’s how we might these functions look like:

toStr :: Int -> String
toStr = show

foo :: (Int -> r) -> r
foo f = f 1

bar :: (String -> r) -> r
bar f = f "1"

If now we have toStr and foo, how should we combine them together to form bar?

Some might recognize what we’re trying to implement is just fmap, if we make continuation an instance of Functor. We can do that, let’s try. First we wrap the function in this pattern it into a newtype, call it ‘Cont’.

newtype Cont r a = Cont { runCont :: (a -> r) -> r }

The reason I write Cont r a instead of Cont a r is, the varying type for a continuation is its type of seed, rather than that of yield. So now we try:

instance Functor (Cont r) where
  fmap f (Cont cnt) = ???

The returning value of fmap should be a continuation with seed type b. So first of all, we should have a continuation that takes an argument with a hole-filler with type b -> r.

fmap :: (a -> b) -> (Cont r a) -> (Cont r b)
fmap f (Cont cnt) = Cont $ \(hf :: b -> r) -> (??? :: r)

We need a r as result, which should be generated by the passed in hf. And hf takes a value with type b. Obviously, this b should be transformed by f from a, i.e. f (??? ::: a). So now we would have:

fmap :: (a -> b) -> (Cont r a) -> (Cont r b)
fmap f (Cont cnt) = Cont $ \hf -> hf (f (??? :: a))

Now the problem has been converted to, how can we extract a from the passed in continuation cnt? The word extract implies that we have to get INTO it for what we want. Just as what we did above with id. (cnt id) will give us a, pretty cool. For some purpose, here I will expand the definition of id, which is \x -> x, you’ll see why I do it in that way soon:

fmap :: (a -> b) -> (Cont r a) -> (Cont r b)
fmap f (Cont cnt) = Cont $ \hf -> hf (f (cnt (\a -> a)))

Looks good, seems in this way we can extract a out of cnt. But wait, it doesn’t typecheck? Let’s see what happens. Without the type constrain above, the code compiles. When we query the type of fmap, it was in type (a -> b) -> (Cont a a) -> (Cont r b).

This is not what we want. The returning type of cnt shouldn’t be restricted to a as what it is, rather, it should share the same r with the one in Cont r b as fmap finally returns. So we start to see how we can achieve that.

fmap :: (a -> b) -> (Cont r a) -> (Cont r b)
fmap f (Cont cnt) = Cont $ \hf -> hf (f (cnt (\a -> (??? :: r))))

However, f expects to have an argument with type a isn’t it? How can we have the a in the context of f while keeping the returning type of cnt to be r?

Okay, we look for where r is needed in the context and we easily found it:

fmap :: (a -> b) -> (Cont r a) -> (Cont r b)
fmap f (Cont cnt) = Cont $ \hf -> ((hf (f a)) :: r)

That means, if we have the a to feed into f, then we get a b to feed into the hole-filler hf, and the hole-filler will yield an r, which is what we wanted. But we don’t have an a for f at this point. The solution is to wrap the extraction of a around hf (f a):

fmap :: (a -> b) -> (Cont r a) -> (Cont r b)
fmap f (Cont cnt) = Cont $ \hf -> cnt (\a -> (hf (f a)))

It typechecks, so it should be correct. Try it out:

(fmap (+1) $ genCont 3) id     -- => 4

Good.

Monadic Continuations

We now make continuation an instance of a monad:

instance Monad (Cont r) where
  return           = ???
  (Cont cnt) >>= f = ???

The type of return is a -> Cont r a. Unwrap the Cont we have a -> (a -> r) -> r. Looks familiar? Yup it is just genCont we have above. The aim of return is to create a continuation from a value, the same as what genCont does. So this part is easy.

return x = Cont $ \hf -> hf x

-- or pointlessly
return = Cont . flip ($)

Actually, after we deduced fmap, we’ll feel much easier to catch >>=. The significance is to take the value out from the continuation supplied as the first argument of >>=. The way to do this is similar to fmap.

-- cnt     :: (a -> r) -> r
-- f       :: a -> Cont r b
-- (>>=)   :: (Cont r a) -> (a -> Cont r b) -> Cont r b
-- hf      :: b -> r
-- runCont :: (Cont r b) -> ((b -> r) -> r)

(Cont cnt) >>= f = Cont $ \hf -> cnt (\x -> runCont (f x) hf)

This solution typechecks. Let’s look into it to see what it does in >>=. First of all, a holefiller

Why monad?

Disclaimer: DO NOT READ THIS SECTION. This section was purely my OWN understanding to continuation monad. I try to write correct things but my thought was specific and could be quite misleading to beginners. If you think you haven’t understood continuation yet, don’t read this section because it will muddle you up once again. Otherwise, you’ve understood continuation well, you don’t need to read on my premitive and partial and inaccurate opinion. Anyway, don’t read it.

Continuation, is just a kind of computations. A continuation could generate a dependent output that relies on a value, either a plain value or the output of another continuation, we call it composability property. (Although I used some general terms here, the way continuations take input is still very different from that of functions) On the other hand, we know we can produce a continuation that will generate a specified plain output. Therefore a continuation is a monad, whose ‘bind’ operation is the ‘compose’ operation of continuations.

composed. ‘Composing’ two computations means to collapse them into one. On the other hand, computations are definitely the arrows in the category of inputs and outputs. On the third hand, we know we can always create a computation that gives no matter what input, because it is a at the

We’re now entering the domain of monad, so we now need to make use of our knowledge of how to implement (>>=).

References

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