**Continuation Passing Style** (CPS for short) is a style of programming in which functions do not return values; rather, they pass control onto a *continuation*, which specifies what happens next. In this chapter, we are going to consider how that plays out in Haskell and, in particular, how CPS can be expressed with a monad.

## Contents

## What are continuations?Edit

To dispel puzzlement, we will have a second look at an example from way back in the book, when we introduced the `($)`

operator:

> map ($ 2) [(2*), (4*), (8*)] [4,8,16]

There is nothing out of ordinary about the expression above, except that it is a little quaint to write that instead of `map (*2) [2, 4, 8]`

. The `($)`

section makes the code appear backwards, as if we are applying a value to the functions rather than the other way around. And now, the catch: such an innocent-looking reversal is at heart of continuation passing style!

From a CPS perspective, `($ 2)`

is a *suspended computation*: a function with general type `(a -> r) -> r`

which, given another function as argument, produces a final result. The `a -> r`

argument is the *continuation*; it specifies how the computation will be brought to a conclusion. In the example, the functions in the list are supplied as continuations via `map`

, producing three distinct results. Note that suspended computations are largely interchangeable with plain values: `flip ($)`

^{[1]} converts any value into a suspended computation, and passing `id`

as its continuation gives back the original value.

### What are they good for?Edit

There is more to continuations than just a parlour trick to impress Haskell newbies. They make it possible to explicitly manipulate, and dramatically alter, the control flow of a program. For instance, returning early from a procedure can be implemented with continuations. Exceptions and failure can also be handled with continuations - pass in a continuation for success, another continuation for fail, and invoke the appropriate continuation. Other possibilities include "suspending" a computation and returning to it at another time, and implementing simple forms of concurrency (notably, one Haskell implementation, Hugs, uses continuations to implement cooperative concurrency).

In Haskell, continuations can be used in a similar fashion, for implementing interesting control flow in monads. Note that there usually are alternative techniques for such use cases, especially in tandem with laziness. In some circumstances, CPS can be used to improve performance by eliminating certain construction-pattern matching sequences (i.e. a function returns a complex structure which the caller will at some point deconstruct), though a sufficiently smart compiler *should* be able to do the elimination ^{[2]}.

## Passing continuationsEdit

An elementary way to take advantage of continuations is to modify our functions so that they return suspended computations rather than ordinary values. We will illustrate how that is done with two simple examples.

`pythagoras`

Edit

**Example:** A simple module, no continuations

```
-- We assume some primitives add and square for the example:
add :: Int -> Int -> Int
add x y = x + y
square :: Int -> Int
square x = x * x
pythagoras :: Int -> Int -> Int
pythagoras x y = add (square x) (square y)
```

Modified to return a suspended computation, `pythagoras`

looks like this:

**Example:** A simple module, using continuations

```
-- We assume CPS versions of the add and square primitives,
-- (note: the actual definitions of add_cps and square_cps are not
-- in CPS form, they just have the correct type)
add_cps :: Int -> Int -> ((Int -> r) -> r)
add_cps x y = \k -> k (add x y)
square_cps :: Int -> ((Int -> r) -> r)
square_cps x = \k -> k (square x)
pythagoras_cps :: Int -> Int -> ((Int -> r) -> r)
pythagoras_cps x y = \k ->
square_cps x $ \x_squared ->
square_cps y $ \y_squared ->
add_cps x_squared y_squared $ k
```

How the `pythagoras_cps`

example works:

- square x and throw the result into the (\x_squared -> ...) continuation
- square y and throw the result into the (\y_squared -> ...) continuation
- add x_squared and y_squared and throw the result into the top level/program continuation
`k`

.

We can try it out in GHCi by passing `print`

as the program continuation:

*Main> pythagoras_cps 3 4 print 25

If we look at the type of `pythagoras_cps`

without the optional parentheses around `(Int -> r) -> r`

and compare it with the original type of `pythagoras`

, we note that the continuation was in effect added as an extra argument, thus justifying the "continuation passing style" moniker.

`thrice`

Edit

**Example:** A simple higher order function, no continuations

```
thrice :: (a -> a) -> a -> a
thrice f x = f (f (f x))
```

*Main> thrice tail "foobar" "bar"

A higher order function such as `thrice`

, when converted to CPS, takes as arguments functions in CPS form as well. Therefore, `f :: a -> a`

will become `f_cps :: a -> ((a -> r) -> r)`

, and the final type will be `thrice_cps :: (a -> ((a -> r) -> r)) -> a -> ((a -> r) -> r)`

. The rest of the definition follows quite naturally from the types - we replace `f`

by the CPS version, passing along the continuation at hand.

**Example:** A simple higher order function, with continuations

```
thrice_cps :: (a -> ((a -> r) -> r)) -> a -> ((a -> r) -> r)
thrice_cps f_cps x = \k ->
f_cps x $ \fx ->
f_cps fx $ \ffx ->
f_cps ffx $ k
```

## The `Cont`

monadEdit

Having continuation-passing functions, the next step is providing a neat way of composing them, preferably one which does not require the long chains of nested lambdas we have seen just above. A good start would be a combinator for applying a CPS function to a suspended computation. A possible type for it would be:

```
chainCPS :: ((a -> r) -> r) -> (a -> ((b -> r) -> r)) -> ((b -> r) -> r)
```

(You may want to try implementing it before reading on. Hint: start by stating that the result is a function which takes a `b -> r`

continuation; then, let the types guide you.)

And here is the implementation:

```
chainCPS s f = \k -> s $ \x -> f x $ k
```

We supply the original suspended computation `s`

with a continuation which makes a new suspended computation (produced by `f`

) and passes the final continuation `k`

to it. Unsurprisingly, it mirrors closely the nested lambda pattern of the previous examples.

Doesn't the type of `chainCPS`

look familiar? If we replace `(a -> r) -> r`

with `(Monad m) => m a`

and `(b -> r) -> r`

with `(Monad m) => m b`

we get the `(>>=)`

signature. Furthermore, our old friend `flip ($)`

plays a `return`

-like role, in that it makes a suspended computation out of a value in a trivial way. Lo and behold, we have a monad! All we need now ^{[3]} is a `Cont r a`

type to wrap suspended computations, with the usual wrapper and unwrapper functions.

```
cont :: ((a -> r) -> r) -> Cont r a
runCont :: Cont r a -> (a -> r) -> r
```

The monad instance for `Cont`

follows directly from our presentation, the only difference being the wrapping and unwrapping cruft:

```
instance Monad (Cont r) where
return x = cont ($ x)
s >>= f = cont $ \c -> runCont s $ \x -> runCont (f x) c
```

The end result is that the monad instance makes the continuation passing (and thus the lambda chains) implicit. The monadic bind applies a CPS function to a suspended computation, and `runCont`

is used to provide the final continuation. For a simple example, the Pythagoras example becomes:

**Example:** The `pythagoras`

example, using the Cont monad

```
-- Using the Cont monad from the transformers package.
import Control.Monad.Trans.Cont
add_cont :: Int -> Int -> Cont r Int
add_cont x y = return (add x y)
square_cont :: Int -> Cont r Int
square_cont x = return (square x)
pythagoras_cont :: Int -> Int -> Cont r Int
pythagoras_cont x y = do
x_squared <- square_cont x
y_squared <- square_cont y
add_cont x_squared y_squared
```

`callCC`

Edit

While it is always pleasant to see a monad coming forth naturally, a hint of disappointment might linger at this point. One of the promises of CPS was precise control flow manipulation through continuations. And yet, after converting our functions to CPS we promptly hid the continuations behind a monad. To rectify that, we shall introduce `callCC`

, a function which gives us back explicit control of continuations - but only where we want it.

`callCC`

is a very peculiar function; one that is best introduced with examples. Let us start with a trivial one:

**Example:** `square`

using `callCC`

```
-- Without callCC
square :: Int -> Cont r Int
square n = return (n ^ 2)
-- With callCC
squareCCC :: Int -> Cont r Int
squareCCC n = callCC $ \k -> k (n ^ 2)
```

The argument passed to `callCC`

is a function, whose result is a suspended computation (general type `Cont r a`

) which we will refer to as "the `callCC`

computation". *In principle*, the `callCC`

computation is what the whole `callCC`

expression evaluates to. The caveat, and what makes `callCC`

so special, is due to `k`

, the argument to the argument. It is a function which acts as an *eject button*: calling it anywhere will lead to the value passed to it being made into a suspended computation, which then is inserted into control flow at the point of the `callCC`

invocation. That happens unconditionally; in particular, whatever follows a `k`

invocation in the `callCC`

computation is summarily discarded. From another perspective, `k`

captures *the rest of the computation* following the `callCC`

; calling it throws a value into the continuation at that particular point ("callCC" stands for "call with current continuation"). While in this simple example the effect is merely that of a plain `return`

, `callCC`

opens up a number of possibilities, which we are now going to explore.

### Deciding when to use `k`

Edit

`callCC`

gives us extra power over what is thrown into a continuation, and when that is done. The following example begins to show how we can use this extra power.

**Example:** Our first proper `callCC`

function

```
foo :: Int -> Cont r String
foo x = callCC $ \k -> do
let y = x ^ 2 + 3
when (y > 20) $ k "over twenty"
return (show $ y - 4)
```

`foo`

is a slightly pathological function that computes the square of its input and adds three; if the result of this computation is greater than 20, then we return from the `callCC`

computation (and, in this case, from the whole function) immediately, throwing the string `"over twenty"`

into the continuation that will be passed to `foo`

. If not, then we subtract four from our previous computation, `show`

it, and throw it into the continuation. Remarkably, `k`

here is used just like the 'return' *statement* from an imperative language, that immediately exits the function. And yet, this being Haskell, `k`

is just an ordinary first-class function, so you can pass it to other functions like `when`

, store it in a `Reader`

, etc.

Naturally, you can embed calls to `callCC`

within do-blocks:

**Example:** More developed `callCC`

example involving a do-block

```
bar :: Char -> String -> Cont r Int
bar c s = do
msg <- callCC $ \k -> do
let s0 = c : s
when (s0 == "hello") $ k "They say hello."
let s1 = show s0
return ("They appear to be saying " ++ s1)
return (length msg)
```

When you call `k`

with a value, the entire `callCC`

call takes that value. In effect, that makes `k`

a lot like an 'goto' statement in other languages: when we call `k`

in our example, it pops the execution out to where you first called `callCC`

, the `msg <- callCC $ ...`

line. No more of the argument to `callCC`

(the inner do-block) is executed. Hence the following example contains a useless line:

**Example:** Popping out a function, introducing a useless line

```
quux :: Cont r Int
quux = callCC $ \k -> do
let n = 5
k n
return 25
```

`quux`

will return `5`

, and not `25`

, because we pop out of `quux`

before getting to the `return 25`

line.

### Behind the scenesEdit

We have deliberately broken a trend here: normally when we introduce a function we give its type straight away, but in this case we chose not to. The reason is simple: the type is pretty complex, and it does not immediately give insight into what the function does, or how it works. After the initial presentation of `callCC`

, however, we are in a better position to tackle it. Take a deep breath...

```
callCC :: ((a -> Cont r b) -> Cont r a) -> Cont r a
```

We can make sense of that based on what we already know about `callCC`

. The overall result type and the result type of the argument have to be the same (i.e. `Cont r a`

), as in the absence of an invocation of `k`

the corresponding result values are one and the same. Now, what about the type of `k`

? As mentioned above, `k`

's argument is made into a suspended computation inserted at the point of the `callCC`

invocation; therefore, if the latter has type `Cont r a`

`k`

's argument must have type `a`

. As for `k`

's result type, interestingly enough it doesn't matter as long as it is wrapped in the same `Cont r`

monad; in other words, the `b`

stands for an arbitrary type. That happens because the suspended computation made out of the `a`

argument will receive whatever continuation follows the `callCC`

, and so the continuation taken by `k`

's result is irrelevant.

*Note*

The arbitrariness of `k`

's result type explains why the following variant of the useless line example leads to a type error:

```
quux :: Cont r Int
quux = callCC $ \k -> do
let n = 5
when True $ k n
k 25
```

`k`

's result type *could* be anything of form `Cont r a`

; however, the `when`

constrains it to `Cont r ()`

, and so the closing `k 25`

does not match the result type of `quux`

. The solution is very simple: replace the final `k`

by a plain old `return`

.

To conclude this section, here is the implementation of `callCC`

. Can you identify `k`

in it?

```
callCC f = cont $ \h -> runCont (f (\a -> cont $ \_ -> h a)) h
```

The code is far from obvious. However, the amazing fact is that the implementations of `callCC`

, `return`

and `(>>=)`

for `Cont`

can be produced automatically from their type signatures - Lennart Augustsson's Djinn [1] is a program that will do this for you. See Phil Gossett's Google tech talk: [2] for background on the theory behind Djinn; and Dan Piponi's article: [3] which uses Djinn in deriving continuation passing style.

## Example: a complicated control structureEdit

We will now look at some more realistic examples of control flow manipulation. The first one, presented below, was originally taken from the "The Continuation monad" section of the All about monads tutorial, used with permission.

**Example:** Using Cont for a complicated control structure

```
{- We use the continuation monad to perform "escapes" from code blocks.
This function implements a complicated control structure to process
numbers:
Input (n) Output List Shown
========= ====== ==========
0-9 n none
10-199 number of digits in (n/2) digits of (n/2)
200-19999 n digits of (n/2)
20000-1999999 (n/2) backwards none
>= 2000000 sum of digits of (n/2) digits of (n/2)
-}
fun :: Int -> String
fun n = (`runCont` id) $ do
str <- callCC $ \exit1 -> do -- define "exit1"
when (n < 10) (exit1 (show n))
let ns = map digitToInt (show (n `div` 2))
n' <- callCC $ \exit2 -> do -- define "exit2"
when ((length ns) < 3) (exit2 (length ns))
when ((length ns) < 5) (exit2 n)
when ((length ns) < 7) $ do
let ns' = map intToDigit (reverse ns)
exit1 (dropWhile (=='0') ns') --escape 2 levels
return $ sum ns
return $ "(ns = " ++ (show ns) ++ ") " ++ (show n')
return $ "Answer: " ++ str
```

`fun`

is a function that takes an integer `n`

. The implementation uses `Cont`

and `callCC`

to set up a control structure using `Cont`

and `callCC`

that does different things based on the range that `n`

falls in, as stated by the comment at the top. Let us dissect it:

- Firstly, the
`(`runCont` id)`

at the top just means that we run the`Cont`

block that follows with a final continuation of`id`

(or, in other words, we extract the value from the suspended computation unchanged). That is necessary as the result type of`fun`

doesn't mention`Cont`

. - We bind
`str`

to the result of the following`callCC`

do-block:- If
`n`

is less than 10, we exit straight away, just showing`n`

. - If not, we proceed. We construct a list,
`ns`

, of digits of`n `div` 2`

. `n'`

(an`Int`

) gets bound to the result of the following inner`callCC`

do-block.- If
`length ns < 3`

, i.e., if`n `div` 2`

has less than 3 digits, we pop out of this inner do-block with the number of digits as the result. - If
`n `div` 2`

has less than 5 digits, we pop out of the inner do-block returning the original`n`

. - If
`n `div` 2`

has less than 7 digits, we pop out of*both*the inner and outer do-blocks, with the result of the digits of`n `div` 2`

in reverse order (a`String`

). - Otherwise, we end the inner do-block, returning the sum of the digits of
`n `div` 2`

.

- If
- We end this do-block, returning the String
`"(ns = X) Y"`

, where X is`ns`

, the digits of`n `div` 2`

, and Y is the result from the inner do-block,`n'`

.

- If
- Finally, we return out of the entire function, with our result being the string "Answer: Z", where Z is the string we got from the
`callCC`

do-block.

## Example: exceptionsEdit

One use of continuations is to model exceptions. To do this, we hold on to *two* continuations: one that takes us out to the handler in case of an exception, and one that takes us to the post-handler code in case of a success. Here's a simple function that takes two numbers and does integer division on them, failing when the denominator is zero.

**Example:** An exception-throwing `div`

```
divExcpt :: Int -> Int -> (String -> Cont r Int) -> Cont r Int
divExcpt x y handler = callCC $ \ok -> do
err <- callCC $ \notOk -> do
when (y == 0) $ notOk "Denominator 0"
ok $ x `div` y
handler err
{- For example,
runCont (divExcpt 10 2 error) id --> 5
runCont (divExcpt 10 0 error) id --> *** Exception: Denominator 0
-}
```

How does it work? We use two nested calls to `callCC`

. The first labels a continuation that will be used when there's no problem. The second labels a continuation that will be used when we wish to throw an exception. If the denominator isn't 0, `x `div` y`

is thrown into the `ok`

continuation, so the execution pops right back out to the top level of `divExcpt`

. If, however, we were passed a zero denominator, we throw an error message into the `notOk`

continuation, which pops us out to the inner do-block, and that string gets assigned to `err`

and given to `handler`

.

A more general approach to handling exceptions can be seen with the following function. Pass a computation as the first parameter (more precisely, a function which takes an error-throwing function and results in the computation) and an error handler as the second parameter. This example takes advantage of the generic `MonadCont`

class ^{[4]} which covers both `Cont`

and the corresponding `ContT`

transformer by default, as well as any other continuation monad which instantiates it.

**Example:** General `try`

using continuations.

```
import Control.Monad.Cont
tryCont :: MonadCont m => ((err -> m a) -> m a) -> (err -> m a) -> m a
tryCont c h = callCC $ \ok -> do
err <- callCC $ \notOk -> do
x <- c notOk
ok x
h err
```

And here is our `try`

in action:

**Example:** Using `try`

```
data SqrtException = LessThanZero deriving (Show, Eq)
sqrtIO :: (SqrtException -> ContT r IO ()) -> ContT r IO ()
sqrtIO throw = do
ln <- lift (putStr "Enter a number to sqrt: " >> readLn)
when (ln < 0) (throw LessThanZero)
lift $ print (sqrt ln)
main = runContT (tryCont sqrtIO (lift . print)) return
```

In this example, error throwing means escaping from an enclosing `callCC`

. The `throw`

in `sqrtIO`

jumps out of `tryCont`

's inner `callCC`

.

## Example: coroutinesEdit

In this section we make a CoroutineT monad that provides a monad with `fork`

, which enqueues a new suspended coroutine, and `yield`

, that suspends the current thread.

```
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
-- We use GeneralizedNewtypeDeriving to avoid boilerplate. As of GHC 7.8, it is safe.
import Control.Applicative
import Control.Monad.Cont
import Control.Monad.State
-- The CoroutineT monad is just ContT stacked with a StateT containing the suspended coroutines.
newtype CoroutineT r m a = CoroutineT {runCoroutineT' :: ContT r (StateT [CoroutineT r m ()] m) a}
deriving (Functor,Applicative,Monad,MonadCont,MonadIO)
-- Used to manipulate the coroutine queue.
getCCs :: Monad m => CoroutineT r m [CoroutineT r m ()]
getCCs = CoroutineT $ lift get
putCCs :: Monad m => [CoroutineT r m ()] -> CoroutineT r m ()
putCCs = CoroutineT . lift . put
-- Pop and push coroutines to the queue.
dequeue :: Monad m => CoroutineT r m ()
dequeue = do
current_ccs <- getCCs
case current_ccs of
[] -> return ()
(p:ps) -> do
putCCs ps
p
queue :: Monad m => CoroutineT r m () -> CoroutineT r m ()
queue p = do
ccs <- getCCs
putCCs (ccs++[p])
-- The interface.
yield :: Monad m => CoroutineT r m ()
yield = callCC $ \k -> do
queue (k ())
dequeue
fork :: Monad m => CoroutineT r m () -> CoroutineT r m ()
fork p = callCC $ \k -> do
queue (k ())
p
dequeue
-- Exhaust passes control to suspended coroutines repeatedly until there isn't any left.
exhaust :: Monad m => CoroutineT r m ()
exhaust = do
exhausted <- null <$> getCCs
if not exhausted
then yield >> exhaust
else return ()
-- Runs the coroutines in the base monad.
runCoroutineT :: Monad m => CoroutineT r m r -> m r
runCoroutineT = flip evalStateT [] . flip runContT return . runCoroutineT' . (<* exhaust)
```

Some example usage:

```
printOne n = do
liftIO (print n)
yield
example = runCoroutineT $ do
fork $ replicateM_ 3 (printOne 3)
fork $ replicateM_ 4 (printOne 4)
replicateM_ 2 (printOne 2)
```

Outputting:

3 4 3 2 4 3 2 4 4

## Example: Implementing pattern matchingEdit

An interesting usage of CPS functions is to implement our own pattern matching. We will illustrate how this can be done by some examples.

**Example:** Built-in pattern matching

```
check :: Bool -> String
check b = case b of
True -> "It's True"
False -> "It's False"
```

Now we have learnt CPS, we can refactor the code like this.

**Example:** Pattern matching in CPS

```
type BoolCPS r = r -> r -> r
true :: BoolCPS r
true x _ = x
false :: BoolCPS r
false _ x = x
check :: BoolCPS String -> String
check b = b "It's True" "It's False"
```

*Main> check true "It's True" *Main> check false "It's False"

What happens here is that, instead of plain values, we represent `True`

and `False`

by functions that would choose either the first or second argument they are passed. Since `true`

and `false`

behave differently, we can achieve the same effect as pattern matching. Furthermore, `True`

, `False`

and `true`

, `false`

can be converted back and forth by `\b -> b True False`

and `\b -> if b then true else false`

.

We should see how this is related to CPS in this more complicated example.

**Example:** More complicated pattern matching and its CPS equivalence

```
data Foobar = Zero | One Int | Two Int Int
type FoobarCPS r = r -> (Int -> r) -> (Int -> Int -> r) -> r
zero :: FoobarCPS r
zero x _ _ = x
one :: Int -> FoobarCPS r
one x _ f _ = f x
two :: Int -> Int -> FoobarCPS r
two x y _ _ f = f x y
fun :: Foobar -> Int
fun x = case x of
Zero -> 0
One a -> a + 1
Two a b -> a + b + 2
funCPS :: FoobarCPS Int -> Int
funCPS x = x 0 (+1) (\a b -> a + b + 2)
```

*Main> fun zero 0 *Main> fun $ one 3 4 *Main> fun $ two 3 4 9

Similar to former example, we represent values by functions. These function-values pick the corresponding (i.e. match) continuations they are passed to and pass to the latter the values stored in the former. An interesting thing is that this process involves in no comparison. As we know, pattern matching can work on types that are not instances of `Eq`

: the function-values "know" what their patterns are and would automatically pick the right continuations. If this is done from outside, say, by an `pattern_match :: [(pattern, result)] -> value -> result`

function, it would have to inspect and compare the patterns and the values to see if they match -- and thus would need `Eq`

instances.

## Notes

- ↑ That is,
`\x -> ($ x)`

, fully spelled out as`\x -> \k -> k x`

- ↑ attoparsec is an example of performance-driven usage of CPS.
- ↑ Beyond verifying that the monad laws hold, which is left as an exercise to the reader.
- ↑ Found in the
`mtl`

package, module Control.Monad.Cont.