Continuation

General or introductory materials

Powerful 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“ (Continuation monad 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))

Links

Wikipedia's Continuation is a surprisingly good introductory material on this topic. See also Continuation-passing style.
Yet Another Haskell Tutorial written by Hal Daume III contains a section on continuation passing style (4.6 Continuation Passing Style, pp 53-56)
HaWiki has a page on ContinuationPassingStyle, and some related pages linked from there, too.
David Madore's A page about call/cc describes the concept, and his The Unlambda Programming Language page shows how he implemented this construct in an esoteric functional programming language.

Examples

Citing haskellized Scheme examples from Wikipedia

Quoting Wikipedia's Continuation#Examples, but Scheme examples are translated to Haskell, and some straightforward modifications are made.

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.

Continuation monad

Jeff Newbern's All About Monads contains a section on it.
Control.Monad.Cont is contained by Haskell Hierarchical Libraries.