Continuation

1 General or introductory materials

1.1 Powerful metaphors, images

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

1.1.1 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`)

1.1.2 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). It can be read also in wikified format.
• 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.

2 Examples

2.1 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.

2.2 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```