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General or introductory materials

Helpful metaphors, images

Here is a collection of short descriptions, analogies or metaphors, that illustrate this difficult concept, or an aspect of it.

Imperative metaphors

  • In computing, a continuation is a representation of the execution state of a program (for example, the call stack) at a certain point in time (Wikipedia's Continuation).
  • At its heart, call/cc is something like the goto instruction (or rather, like a label for a goto instruction); but a Grand High Exalted goto instruction... The point about call/cc is that it is not a static (lexical) goto instruction but a dynamic one (David Madore's A page about call/cc)

Functional metaphors

  • Continuations represent the future of a computation, as a function from an intermediate result to the final result ([1] section in Jeff Newbern's All About Monads)
  • The idea behind CPS is to pass around as a function argument what to do next (Yet Another Haskell Tutorial written by Hal Daume III, 4.6 Continuation Passing Style, pp 53-56. It can also be read in wikified format).
  • Rather than return the result of a function, pass one or more Higher Order Functions to determine what to do with the result. Yes, direct sum like things (or in generally, case analysis, managing cases, alternatives) can be implemented in CPS by passing more continuations.

External links

More introductions and guides about continuations can be found here.


Citing haskellized Scheme examples from Wikipedia

Quoting the Scheme examples (with their explanatory texts) from Wikipedia's Continuation-passing style article, but Scheme examples are translated to Haskell, and some straightforward modifications are made to the explanations (e.g. replacing word Scheme with Haskell, or using abbreviated name fac instead of factorial).

In the Haskell programming language, the simplest of direct-style functions is the identity function:

 id :: a -> a
 id a = a

which in CPS becomes:

 idCPS :: a -> (a -> r) -> r
 idCPS a ret = ret a

where ret is the continuation argument (often also called k). A further comparison of direct and CPS style is below.

Direct style
Continuation-passing style
 mysqrt :: Floating a => a -> a
 mysqrt a = sqrt a
 print (mysqrt 4) :: IO ()
 mysqrtCPS :: a -> (a -> r) -> r
 mysqrtCPS a k = k (sqrt a)
 mysqrtCPS 4 print :: IO ()
 mysqrt 4 + 2 :: Floating a => a
 mysqrtCPS 4 (+ 2) :: Floating a => a
 fac :: Integral a => a -> a
 fac 0 = 1
 fac n'@(n + 1) = n' * fac n
 fac 4 + 2 :: Integral a => a
 facCPS :: a -> (a -> r) -> r
 facCPS 0 k = k 1
 facCPS n'@(n + 1) k = facCPS n $ \ret -> k (n' * ret)
 facCPS 4 (+ 2) :: Integral a => a

The translations shown above show that CPS is a global transformation; the direct-style factorial, fac takes, as might be expected, a single argument. The CPS factorial, facCPS takes two: the argument and a continuation. Any function calling a CPS-ed function must either provide a new continuation or pass its own; any calls from a CPS-ed function to a non-CPS function will use implicit continuations. Thus, to ensure the total absence of a function stack, the entire program must be in CPS.

As an exception, mysqrt calls sqrt without a continuation — here sqrt is considered a primitive operator; that is, it is assumed that sqrt will compute its result in finite time and without abusing the stack. Operations considered primitive for CPS tend to be arithmetic, constructors, accessors, or mutators; any O(1) operation will be considered primitive.

The quotation ends here.

Intermediate structures

The function Foreign.C.String.withCString converts a Haskell string to a C string. But it does not provide it for external use but restricts the use of the C string to a sub-procedure, because it will cleanup the C string after its use. It has signature withCString :: String -> (CString -> IO a) -> IO a. This looks like continuation and the functions from continuation monad can be used, e.g. for allocation of a whole array of pointers:

multiCont :: [(r -> a) -> a] -> ([r] -> a) -> a
multiCont xs = runCont (mapM Cont xs)

withCStringArray0 :: [String] -> (Ptr CString -> IO a) -> IO a
withCStringArray0 strings act =
      (map withCString strings)
      (\rs -> withArray0 nullPtr rs act)

However, the right associativity of mapM leads to inefficiencies here.


More general examples

Maybe it is confusing, that

  • the type of the (non-continuation) argument of the discussed functions (idCPS, mysqrtCPS, facCPS)
  • and the type of the argument of the continuations

coincide in the above examples. It is not a necessity (it does not belong to the essence of the continuation concept), so I try to figure out an example which avoids this confusing coincidence:

 newSentence :: Char -> Bool
 newSentence = flip elem ".?!"

 newSentenceCPS :: Char -> (Bool -> r) -> r
 newSentenceCPS c k = k (elem c ".?!")

but this is a rather uninteresting example. Let us see another one that uses at least recursion:

 mylength :: [a] -> Integer
 mylength [] = 0
 mylength (_ : as) = succ (mylength as)

 mylengthCPS :: [a] -> (Integer -> r) -> r
 mylengthCPS [] k = k 0
 mylengthCPS (_ : as) k = mylengthCPS as (k . succ)

 test8 :: Integer
 test8 = mylengthCPS [1..2006] id

 test9 :: IO ()
 test9 = mylengthCPS [1..2006] print

You can download the Haskell source code (the original examples plus the new ones): Continuation.hs.

Monads as stylised continuation-passing

After class today, a few of us were discussing the market for functional programmers. Talk turned to Clojure and Scala. A student who claims to understand monads said:

To understand monad tutorials, you really have to understand monads first.

Priceless. The topic of today's class was mutual recursion. I think we are missing a base case here.

Knowing and Doing: Student Wisdom on Monad Tutorials, Eugene Wallingford.

The partial application of (>>=) to monadic values means they can be used in the traditional continuation-passing style:

Monadic style
Continuation-passing style
m     :: M T

h     :: U -> M V

(>>=) :: M a -> (a -> M b) -> M b
m'    :: (T -> M b) -> M b
m'    =  (>>=) m

h'    :: U -> (V -> M b) -> M b
h' x  =  (>>=) (h x)


Continuation monad

Delimited continuation


Chris Barker: Continuations in Natural Language


Oleg Kiselyov's zipper-based file server/OS where threading and exceptions are all realized via delimited continuations.

Blog Posts