Continuationpassing style
In functional programming, continuationpassing style (CPS) is a style of programming in which control is passed explicitly in the form of a continuation. This is contrasted with direct style, which is the usual style of programming. Gerald Jay Sussman and Guy L. Steele, Jr. coined the phrase in AI Memo 349 (1975), which sets out the first version of the Scheme programming language.[1][2] John C. Reynolds gives a detailed account of the numerous discoveries of continuations.[3]
A function written in continuationpassing style takes an extra argument: an explicit "continuation", i.e. a function of one argument. When the CPS function has computed its result value, it "returns" it by calling the continuation function with this value as the argument. That means that when invoking a CPS function, the calling function is required to supply a procedure to be invoked with the subroutine's "return" value. Expressing code in this form makes a number of things explicit which are implicit in direct style. These include: procedure returns, which become apparent as calls to a continuation; intermediate values, which are all given names; order of argument evaluation, which is made explicit; and tail calls, which simply call a procedure with the same continuation, unmodified, that was passed to the caller.
Programs can be automatically transformed from direct style to CPS. Functional and logic compilers often use CPS as an intermediate representation where a compiler for an imperative or procedural programming language would use static single assignment form (SSA).[4] SSA is formally equivalent to a subset of CPS (excluding nonlocal control flow, which does not occur when CPS is used as intermediate representation).[5] Functional compilers can also use Anormal form (ANF) (but only for languages requiring eager evaluation), rather than with 'thunks' (described in the examples below) in CPS. CPS is used more frequently by compilers than by programmers as a local or global style.
Examples
In CPS, each procedure takes an extra argument representing what should be done with the result the function is calculating. This, along with a restrictive style prohibiting a variety of constructs usually available, is used to expose the semantics of programs, making them easier to analyze. This style also makes it easy to express unusual control structures, like catch/throw or other nonlocal transfers of control.
The key to CPS is to remember that (a) every function takes an extra argument, its continuation, and (b) every argument in a function call must be either a variable or a lambda expression (not a more complex expression). This has the effect of turning expressions "insideout" because the innermost parts of the expression must be evaluated first, so CPS explicitizes the order of evaluation as well as the control flow. Some examples of code in direct style and the corresponding CPS appear below. These examples are written in the Scheme programming language; by convention the continuation function is represented as a parameter named "k
":
(define (pyth x y)
(sqrt (+ (* x x) (* y y))))

(define (pyth& x y k)
(*& x x (lambda (x2)
(*& y y (lambda (y2)
(+& x2 y2 (lambda (x2py2)
(sqrt& x2py2 k))))))))

(define (factorial n)
(if (= n 0)
1 ; NOT tailrecursive
(* n (factorial ( n 1)))))

(define (factorial& n k)
(=& n 0 (lambda (b)
(if b ; growing continuation
(k 1) ; in the recursive call
(& n 1 (lambda (nm1)
(factorial& nm1 (lambda (f)
(*& n f k)))))))))

(define (factorial n)
(faux n 1))
(define (faux n a)
(if (= n 0)
a ; tailrecursive
(faux ( n 1) (* n a))))

(define (factorial& n k) (faux& n 1 k))
(define (faux& n a k)
(=& n 0 (lambda (b)
(if b ; unmodified continuation
(k a) ; in the recursive call
(& n 1 (lambda (nm1)
(*& n a (lambda (nta)
(faux& nm1 nta k)))))))))

Note that in the CPS versions, the primitives used, like +&
and *&
are themselves CPS, not direct style, so to make the above examples work in a Scheme system we would need to write these CPS versions of primitives, with for instance *&
defined by:
(define (*& x y k)
(k (* x y)))
To do this in general, we might write a conversion routine:
(define (cpsprim f)
(lambda args
(let ((r (reverse args)))
((car r) (apply f
(reverse (cdr r)))))))
(define *& (cpsprim *))
(define +& (cpsprim +))
In order to call a procedure written in CPS from a procedure written in direct style, it is necessary to provide a continuation that will receive the result computed by the CPS procedure. In the example above (assuming that CPSstyle primitives have been provided), we might call (factorial& 10 (lambda (x) (display x) (newline)))
.
There is some variety between compilers in the way primitive functions are provided in CPS. Above we have used the simplest convention, however sometimes boolean primitives are provided that take two thunks to be called in the two possible cases, so the (=& n 0 (lambda (b) (if b ...)))
call inside faux&
definition above would be written instead as (=& n 0 (lambda () (k a)) (lambda () (& n 1 ...)))
. Similarly, sometimes the if
primitive itself is not included in CPS, and instead a function if&
is provided which takes three arguments: a boolean condition and the two thunks corresponding to the two arms of the conditional.
The translations shown above show that CPS is a global transformation. The directstyle factorial takes, as might be expected, a single argument; the CPS factorial& takes two: the argument and a continuation. Any function calling a CPSed function must either provide a new continuation or pass its own; any calls from a CPSed function to a nonCPS function will use implicit continuations. Thus, to ensure the total absence of a function stack, the entire program must be in CPS.
CPS in Haskell
In this section we will write a function pyth
that calculates the hypotenuse using the Pythagorean theorem. A traditional implementation of the pyth
function looks like this:
pow2 :: Float > Float
pow2 a = a ** 2
add :: Float > Float > Float
add a b = a + b
pyth :: Float > Float > Float
pyth a b = sqrt (add (pow2 a) (pow2 b))
To transform the traditional function to a CPS one, we need to change its signature. The function will get another argument of function type, and its return type depends on that function:
pow2' :: Float > (Float > a) > a
pow2' a cont = cont (a ** 2)
add' :: Float > Float > (Float > a) > a
add' a b cont = cont (a + b)
 Types a > (b > c) and a > b > c are equivalent, so CPS function
 may be viewed as a higher order function
sqrt' :: Float > ((Float > a) > a)
sqrt' a = \cont > cont (sqrt a)
pyth' :: Float > Float > (Float > a) > a
pyth' a b cont = pow2' a (\a2 > pow2' b (\b2 > add' a2 b2 (\anb > sqrt' anb cont)))
First we calculate the square of a in pyth'
function and pass a lambda function as a continuation which will accept a square of a as a first argument. And so on until we reach the result of our calculations. To get the result of this function we can pass id
function as a final argument which returns the value that was passed to it unchanged: pyth' 3 4 id == 5.0
.
mtl library which is shipped with Ghc compiler has Control.Monad.Cont module. This module provides Cont type, which implements Monad and some other useful functions. The following snippet shows the pyth'
function using Cont type:
pow2_m :: Float > Cont a Float
pow2_m a = return (a ** 2)
pyth_m :: Float > Float > Cont a Float
pyth_m a b = do
a2 < pow2_m a
b2 < pow2_m b
anb < cont (add' a2 b2)
r < cont (sqrt' anb)
return r
Not only has the syntax become cleaner, but this type allows us to use a function callCC
with type MonadCont m => ((a > m b) > m a) > m a
. This function has one argument of a function type; that function argument accepts the function too, which discards all computations going after its call. For example let's break the execution of the pyth
function if at least one of its arguments is negative returning zero:
pyth_m :: Float > Float > Cont a Float
pyth_m a b = callCC $ \exitF > do  $ sign helps to avoid parentheses: a $ b + c == a (b + c)
when (b < 0  a < 0) (exitF 0.0)  when :: Applicative f => Bool > f () > f ()
a2 < pow2_m a
b2 < pow2_m b
anb < cont (add' a2 b2)
r < cont (sqrt' anb)
return r
Continuations as objects
Programming with continuations can also be useful when a caller does not want to wait until the callee completes. For example, in userinterface (UI) programming, a routine can set up dialog box fields and pass these, along with a continuation function, to the UI framework. This call returns right away, allowing the application code to continue while the user interacts with the dialog box. Once the user presses the "OK" button, the framework calls the continuation function with the updated fields. Although this style of coding uses continuations, it is not full CPS.
function confirmName() {
fields.name = name;
framework.Show_dialog_box(fields, confirmNameContinuation);
}
function confirmNameContinuation(fields) {
name = fields.name;
}
A similar idea can be used when the function must run in a different thread or on a different processor. The framework can execute the called function in a worker thread, then call the continuation function in the original thread with the worker's results. This is in Java using the Swing UI framework:
void buttonHandler() {
// This is executing in the Swing UI thread.
// We can access UI widgets here to get query parameters.
final int parameter = getField();
new Thread(new Runnable() {
public void run() {
// This code runs in a separate thread.
// We can do things like access a database or a
// blocking resource like the network to get data.
final int result = lookup(parameter);
javax.swing.SwingUtilities.invokeLater(new Runnable() {
public void run() {
// This code runs in the UI thread and can use
// the fetched data to fill in UI widgets.
setField(result);
}
});
}
}).start();
}
Using Java 8 lambda notation above code shortens to (final
keyword might be skipped):
void buttonHandler() {
int parameter = getField();
new Thread(() > {
int result = lookup(parameter);
javax.swing.SwingUtilities.invokeLater(() > setField(result));
}).start();
}
Tail calls
Every call in CPS is a tail call, and the continuation is explicitly passed. Using CPS without tail call optimization (TCO) will cause not only the constructed continuation to potentially grow during recursion, but also the call stack. This is usually undesirable, but has been used in interesting ways—see the Chicken Scheme compiler. As CPS and TCO eliminate the concept of an implicit function return, their combined use can eliminate the need for a runtime stack. Several compilers and interpreters for functional programming languages use this ability in novel ways.[6]
Use and implementation
Continuation passing style can be used to implement continuations and control flow operators in a functional language that does not feature firstclass continuations but does have firstclass functions and tailcall optimization. Without tailcall optimization, techniques such as trampolining, i.e. using a loop that iteratively invokes thunkreturning functions, can be used; without firstclass functions, it is even possible to convert tail calls into just gotos in such a loop.
Writing code in CPS, while not impossible, is often errorprone. There are various translations, usually defined as one or twopass conversions of pure lambda calculus, which convert direct style expressions into CPS expressions. Writing in trampolined style, however, is extremely difficult; when used, it is usually the target of some sort of transformation, such as compilation.
Functions using more than one continuation can be defined to capture various control flow paradigms, for example (in Scheme):
(define (/& x y ok err)
(=& y 0.0 (lambda (b)
(if b
(err (list "div by zero!" x y))
(ok (/ x y))))))
It is of note that CPS transform is conceptually a Yoneda embedding.[7] It is also similar to the embedding of lambda calculus in πcalculus.[8][9]
Use in other fields
Outside of computer science, CPS is of more general interest as an alternative to the conventional method of composing simple expressions into complex expressions. For example, within linguistic semantics, Chris Barker and his collaborators have suggested that specifying the denotations of sentences using CPS might explain certain phenomena in natural language.[10]
In mathematics, the Curry–Howard isomorphism between computer programs and mathematical proofs relates continuationpassing style translation to a variation of doublenegation embeddings of classical logic into intuitionistic (constructive) logic. Unlike the regular doublenegation translation, which maps atomic propositions p to ((p → ⊥) → ⊥), the continuation passing style replaces ⊥ by the type of the final expression. Accordingly, the result is obtained by passing the identity function as a continuation to the CPSstyle expression, as in the above example.
Classical logic itself relates to manipulating the continuation of programs directly, as in Scheme's callwithcurrentcontinuation control operator, an observation due to Tim Griffin (using the closely related C control operator).[11]
See also
Notes
 Sussman, Gerald Jay; Steele, Guy L., Jr. (December 1975). . AI Memo. 349: 19.
That is, in this continuationpassing programming style, a function always "returns" its result by "sending" it to another function. This is the key idea.
 Sussman, Gerald Jay; Steele, Guy L., Jr. (December 1998). "Scheme: A Interpreter for Extended Lambda Calculus" (reprint). HigherOrder and Symbolic Computation. 11 (4): 405–439. doi:10.1023/A:1010035624696.
We believe that this was the first occurrence of the term "continuationpassing style" in the literature. It has turned out to be an important concept in source code analysis and transformation for compilers and other metaprogramming tools. It has also inspired a set of other "styles" of program expression.
 Reynolds, John C. (1993). "The Discoveries of Continuations". Lisp and Symbolic Computation. 6 (3–4): 233–248. CiteSeerX 10.1.1.135.4705. doi:10.1007/bf01019459.

 Appel, Andrew W. (April 1998). "SSA is Functional Programming". ACM SIGPLAN Notices. 33 (4): 17–20. CiteSeerX 10.1.1.34.3282. doi:10.1145/278283.278285.

 Kelsey, Richard A. (March 1995). "A Correspondence between Continuation Passing Style and Static Single Assignment Form". ACM SIGPLAN Notices. 30 (3): 13–22. CiteSeerX 10.1.1.489.930. doi:10.1145/202530.202532.
 Appel, Andrew W. (1992). Compiling with Continuations. Cambridge University Press. ISBN 0521416957.
 Mike Stay, "The Continuation Passing Transform and the Yoneda Embedding"
 Mike Stay, "The Pi Calculus II"
 Boudol, Gérard (1997). "The πCalculus in Direct Style". CiteSeerX 10.1.1.52.6034. Cite journal requires
journal=
(help)  Barker, Chris (20020901). "Continuations and the Nature of Quantification" (PDF). Natural Language Semantics. 10 (3): 211–242. doi:10.1023/A:1022183511876. ISSN 1572865X.
 Griffin, Timothy (January 1990). A formulaeastype notion of control. Proceedings of the Conference on the Principles of Programming Languages. 17. pp. 47–58. doi:10.1145/96709.96714. ISBN 9780897913430.
References
 Continuation Passing C (CPC)  programming language for writing concurrent systems, designed and developed by Juliusz Chroboczek and Gabriel Kerneis. github repository
 The construction of a CPSbased compiler for ML is described in: Appel, Andrew W. (1992). Compiling with Continuations. Cambridge University Press. ISBN 9780521416955.
 Danvy, Olivier; Filinski, Andrzej (1992). "Representing Control, A Study of the CPS Transformation". Mathematical Structures in Computer Science. 2 (4): 361–391. CiteSeerX 10.1.1.46.84. doi:10.1017/S0960129500001535.
 Chicken Scheme compiler, a Scheme to C compiler that uses continuationpassing style for translating Scheme procedures into C functions while using the Cstack as the nursery for the generational garbage collector
 Kelsey, Richard A. (March 1995). "A Correspondence between Continuation Passing Style and Static Single Assignment Form". ACM SIGPLAN Notices. 30 (3): 13–22. CiteSeerX 10.1.1.3.6773. doi:10.1145/202530.202532.
 Appel, Andrew W. (April 1998). "SSA is Functional Programming". ACM SIGPLAN Notices. 33 (4): 17–20. CiteSeerX 10.1.1.34.3282. doi:10.1145/278283.278285.
 Danvy, Olivier; Millikin, Kevin; Nielsen, Lasse R. (2007). "On OnePass CPS Transformations". BRICS, Department of Computer Science, University of Aarhus. p. 24. ISSN 09090878. RS076. Retrieved 26 October 2007.
 Dybvig, R. Kent (2003). The Scheme Programming Language. Prentice Hall. p. 64. Direct link: "Section 3.4. Continuation Passing Style".