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.
- “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/ccis something like the
gotoinstruction (or rather, like a label for a
gotoinstruction); but a Grand High Exalted
gotoinstruction... The point about
call/ccis that it is not a static (lexical)
gotoinstruction but a dynamic one“ (David Madore's A page about
- “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))
- “Rather than return the result of a function, pass one or more HigherOrderFunctions to determine what to do with the result“ (HaWiki's ContinuationPassingStyle). Yes, direct sum like things (or in generally, case analysis, managing cases, alternatives) can be implemented in CPS by passing more continuations.
- 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/ccdescribes the concept, and his The Unlambda Programming Language page shows how he implemented this construct in an esoteric functional programming language.
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
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
ret is the continuation argument (often also called
k). A further comparison of direct and CPS style is below.
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,
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.
More general examples
Maybe it is confusing, that
- the type of the (non-continuation) argument of the discussed functions (
- 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.