MonadCont under the hood
This tutorial is a response to the following Stack Overflow question. There's a short but useful description of Cont and MonadCont operations in the Control.Monad.Cont documentation, but it doesn't really describe how the continuation monad does its thing. This is an attempt at much more detailed explanation of what Cont and MonadCont are doing under the hood.
This tutorial assumes a working knowledge of Haskell, though of course it doesn't assume that you understood the implementation of
Control.Monad.Cont the first time you read it!
Introducing Continuations and the Cont type
Continuations are functions that represent "the remaining computation to do." Their representation here is
a -> r, which is simply a function that takes some value produced by the current computation, of some type
a, and returns the final result of type
r from it.
Cont r a (instances of which I will, in this tutorial, refer to as Cont objects) represents a continuation-passing-style function that takes a single continuation as its only input. In other words, its guts are a function that:
- takes a continuation as an argument
- does whatever it needs to do
- produces a value of type
rat the end, presumably by invoking the continuation.
Note that whatever it needs to do, i.e. whatever values it needs to be able to use to do its thing, must already be bound up into the
Cont object. So, generally, we won't be dealing with
Cont objects directly, but with functions that can ultimately produce one.
Sequencing Continuation-Style Computations
Cont objects can be chained together, so that the continuation you pass in threads through the guts of all the
Cont objects in the chain before it's finally invoked. The way they chain is the way
Cont works: each object in the chain invokes a continuation that has the next object's computation prepended to the final continuation. Let's say we have a chain of
f1 -> f2 -> f3, and let's say you had a continuation
c3 that you want to pass to the chain. Then:
f3needs to invoke
c3when it's done.
f2needs to invoke a continuation
c2that will invoke
f3, which will invoke
f1needs to invoke a continuation
which will invoke
which will invoke
which will invoke
which will finally invoke
To chain the
Cont objects together, then, we need to create the appropriate continuation functions
c2 and make sure they get passed as the continuation argument to
Extending to Monad
Extending this idea to the
Monad class in general, there's an extra wrinkle: we allow for the value of one computation to affect which
Cont object gets invoked next. In this world:
returntakes a value and produces a
Contobject that just passes that value to its continuation.
- The bind operator
Contobject, and a function that produces another
Contobject given a value from the first, and chains them together into one
Contobject. That object, when invoked, is going to:
- take a single continuation object
- produce an intermediate value,
- use that intermediate value to select/create the next
Contobject to invoke,
- invoke that
- take a single continuation object
Understanding the Monad
return a = Cont ($ a)
is equivalent to the following code:
return a = Cont $ \c -> c a
Why? The code
($ a) is a slice of the operator
$, which represents application. In other words,
($ a) can be equivalently written
\f -> f a, or "take a function as input and apply it to a."
return in the
Cont monad passes the result of its computation directly to the continuation it's given.
A mathematical analogon that may help is the "insert a" functional: it takes a function f, and the result is the value f(a) of that function at x=a. Call it "insert(a)", then
insert(a) <---> return a insert(a)(f(x) = x^2) <---> runCont (return a) (\x -> x^2) == f(a) == (\x -> x^2) a == a^2 == a^2
m >>= k = Cont $ \c -> runCont m $ \a -> runCont (k a) c
is a terse way of saying the following:
m >>= k = let s c = runCont m c t c = \a -> runCont (k a) c in Cont $ \c -> s (t c)
Do you see what's happening?
(k a) has become part of the continuation that
m is given, and
m passes its value to
k by simply passing its value to its continuation. The
Cont objects are being created "just in time" to be used, based on the computation so far.
Exploring the Monad
Here's a simple example that should help illustrate the monad in action:
f :: Int -> Cont r Int f x = Cont $ \c -> c (x * 3) g :: Int -> Cont r Int g x = Cont $ \c -> c (x - 2)
These are simple functions that produce
Cont objects, given an intermediate value
x. You can see that the value being passed to the continuation is an Int, though we don't put any restrictions on what that continuation can ultimately produce.
BTW, they can be equivalently written as:
f x = return (x * 3) g x = return (x - 2)
where they look very similar to normal functions. I'm writing them longhand to show you explicitly what the functions are doing.
h :: Int -> Cont r Int h x | x == 5 = f x | otherwise = g x
This is a more complicated function that chooses between two other
Cont objects, based on the input it's given.
Now let's create a top-level
Cont object that does some chaining:
doC :: Cont r Int doC = return 5 >>= h
And we'll invoke it like this:
finalC :: Show a => a -> String finalC x = "Done: " ++ show(x) runCont doC finalC
runCont doC produces a function of type
(Int -> a) -> a, which is invoked on a continuation of type
Show a => a -> String, which reduces in this context to
Int -> String. The final value produced will be a
String. Can you guess what it will say? What if you changed
return 5 to
Let's see if you're right:
return 5 produces a
Cont object that basically looks like this:
Cont $ \c -> c 5. So that part is easy.
h is a function that takes a value and produces a
Cont object depending on the value it's given.
Lemma: The sequence of terms
runCont Cont $ effectively cancel out, i.e.
runCont (Cont $ \c -> ...) is simply the function
\c -> .... This is because
runCont is a field selector of
Cont objects, and
Cont objects only have that one field.
(return 5) >>= h expands and simplifies to:
doC = let s c = c 5 t c = \a -> runCont (h a) c in Cont $ \c -> s (t c)
runCont doC finalC evaluates to:
runCont doC finalC => runCont (Cont $ \c -> s (t c)) finalC -- unfold doC => s (t finalC) -- simplify with lemma and apply to finalC => (t finalC) 5 -- unfold s => (\a -> runCont (h a) finalC) 5 -- unfold t => runCont (h 5) finalC -- apply \a... to 5 => runCont (f 5) finalC -- unfold h => runCont (Cont $ \c -> c (5*3)) finalC -- unfold f => (\c -> c (5*3)) finalC -- simplify with lemma => finalC (5*3) -- apply \c... to finalC => "Done: 15" -- apply *; apply finalC to final value!
If you changed doC to
return 4 >>= h, the derivation would be almost identical to the above, except that 4 would pass through to h, which would unfold to g instead. "Done: 2" should be the result.
MonadCont and callCC
One final extension to this monad, which can be extremely useful in practice, is the
MonadCont class, which provides a
callCC creates a
Cont object that invokes a function to construct a
Cont object, and then runs it with the continuation it's given. However, it provides that function an alternate continuation that can be invoked to "break out" of the computation and simply pass a value to the continuation that was active when
callCC was invoked. This function's operation is definitely easier to understand by seeing it in action. Evaluate the following code, replacing the corresponding functions above:
h :: Int -> (Int -> Cont r Int) -> Cont r Int h x abort | x == 5 = f x | otherwise = abort (-1) doC n = return n >>= \x -> callCC (\abort -> h x abort) >>= \y -> g y
runCont (doC 5) finalC. h should invoke f, and g will be invoked afterward, so you should get 13 as the final answer.
(doC 5) to
(doC 4). In this case, h will call abort, which passes -1 to g. -3 should be the final answer.
doC to move g inside the callCC abort context:
doC n = return n >>= \x -> callCC (\abort -> h x abort >>= \y -> g y)
and run with
(doC 4). In this case, h invokes abort and g is never invoked! -1 is the final answer.
Once you've converted all your operations to continuation-passing style by putting them in the
Cont monad, and have a handle on how
>>= works in that monad, understanding how
callCC works is surprisingly simple:
callCC f = Cont $ \c -> runCont (f (\a -> Cont $ \_ -> c a )) c
can be written as
callCC f = let backtrack a = Cont $ \_ -> c a in Cont $ \c -> runCont (f backtrack) c
The key is
backtrack, which takes whatever "inner" continuation is active when backtrack is invoked, completely ignores it, and simply passes its value to the "outer" continuation
c. (Compare this to the definition of
return, which always uses the continuation it's given.)
f is the function passed to
callCC, whose extent provides the context under which
backtrack can be used.