Editing Continuation-passing style (section)

==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 <code>catch/throw</code> or other non-local transfers of control.

The key to CPS is to remember that (a) ''every'' function takes an extra argument known as its continuation, and (b) every argument in a function call must be either a variable or a [[Lambda (programming)|lambda expression]] (not a more complex expression). This has the effect of turning expressions "inside-out" because the innermost parts of the expression must be evaluated first, thus CPS makes explicit 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 programming language [[Scheme (programming language)|Scheme]]; by convention the continuation function is represented as a parameter named "<code>k</code>":
{|
!{{center|Direct style}}!!{{center|Continuation passing style}}
|-valign="top"
|<syntaxhighlight lang=scheme>
(define (pyth x y)
 (sqrt (+ (* x x) (* y y))))</syntaxhighlight>
||
<syntaxhighlight lang=scheme>
(define (pyth& x y k)
 (*& x x (lambda (x2)
          (*& y y (lambda (y2)
                   (+& x2 y2 (lambda (x2py2)
                              (sqrt& x2py2 k))))))))
</syntaxhighlight>
|-valign="top"
|
<syntaxhighlight lang=scheme>
(define (factorial n)
 (if (= n 0)
     1     ; NOT tail-recursive
     (* n (factorial (- n 1)))))
</syntaxhighlight>
||
<syntaxhighlight lang=scheme>

(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)))))))))
</syntaxhighlight>
|-valign="top"
|
<syntaxhighlight lang=scheme>
(define (factorial n)
 (f-aux n 1))
(define (f-aux n a)
 (if (= n 0)
     a        ; tail-recursive
     (f-aux (- n 1) (* n a))))
</syntaxhighlight>
||
<syntaxhighlight lang=scheme>
(define (factorial& n k) (f-aux& n 1 k))
(define (f-aux& 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)
                                (f-aux& nm1 nta k)))))))))
</syntaxhighlight>
|}

In the CPS versions, the primitives used, like <code>+&</code> and <code>*&</code> are themselves CPS, not direct style, so to make the above examples work in a Scheme system requires writing these CPS versions of primitives, with for instance <code>*&</code> defined by:
<syntaxhighlight lang=scheme>
(define (*& x y k)
 (k (* x y)))
</syntaxhighlight>  To do this in general, we might write a conversion routine:
<syntaxhighlight lang=scheme>
(define (cps-prim f)
 (lambda args
  (let ((r (reverse args)))
   ((car r) (apply f
             (reverse (cdr r)))))))
(define *& (cps-prim *))
(define +& (cps-prim +))</syntaxhighlight>

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 CPS primitives have been provided), we might call <code>(factorial& 10 (lambda (x) (display x) (newline)))</code>.

There is some variety between compilers in the way primitive functions are provided in CPS. Above is used the simplest convention, however sometimes Boolean primitives are provided that take two [[thunk]]s to be called in the two possible cases, so the <code>(=& n 0 (lambda (b) (if b ...)))</code> call inside <code>f-aux&</code> definition above would be written instead as <code>(=& n 0 (lambda () (k a)) (lambda () (-& n 1 ...)))</code>. Similarly, sometimes the <code>if</code> primitive is not included in CPS, and instead a function <code>if&</code> 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 direct-style ''factorial'' takes, as might be expected, a single argument;  the CPS ''factorial&'' 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.

===CPS in Haskell===
A function <code>pyth</code> to calculate a hypotenuse using the [[Pythagorean theorem]] can be written in [[Haskell]]. A traditional implementation of the <code>pyth</code> function looks like this:
<syntaxhighlight lang="haskell">
pow2 :: Float -> Float
pow2 x = x ** 2

add :: Float -> Float -> Float
add x y = x + y

pyth :: Float -> Float -> Float
pyth x y = sqrt (add (pow2 x) (pow2 y))
</syntaxhighlight>

To transform the traditional function to CPS, its signature must be changed. The function will get another argument of function type, and its return type depends on that function:
<syntaxhighlight lang="haskell">
pow2' :: Float -> (Float -> a) -> a
pow2' x cont = cont (x ** 2)

add' :: Float -> Float -> (Float -> a) -> a
add' x y cont = cont (x + y)

-- 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' x = \cont -> cont (sqrt x)

pyth' :: Float -> Float -> (Float -> a) -> a
pyth' x y cont = pow2' x (\x2 -> pow2' y (\y2 -> add' x2 y2 (\anb -> sqrt' anb cont)))
</syntaxhighlight>

First we calculate the square of ''a'' in <code>pyth'</code> function and pass a lambda function as a continuation which will accept a square of ''a'' as a first argument. And so on until the result of the calculations are reached. To get the result of this function we can pass <code>id</code> function as a final argument which returns the value that was passed to it unchanged: <code>pyth' 3 4 id == 5.0</code>.

The mtl [[Library (computing)|library]], which is shipped with [[Glasgow Haskell Compiler]] (GHC), has the module <code>Control.Monad.Cont</code>. This module provides the Cont type, which implements Monad and some other useful functions. The following snippet shows the <code>pyth'</code> function using Cont:
<syntaxhighlight lang="haskell">
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
</syntaxhighlight>

Not only has the syntax become cleaner, but this type allows us to use a function <code>callCC</code> with type <code>MonadCont m => ((a -> m b) -> m a) -> m a</code>. 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 <code>pyth</code> function if at least one of its arguments is negative returning zero:
<syntaxhighlight lang="haskell">
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
</syntaxhighlight>

===Continuations as objects===
{{See also|Callback (computer programming)}}

Programming with continuations can also be useful when a caller does not want to wait until the callee completes. For example, in [[user interface]] (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.{{Clarify|date=June 2019}}

<syntaxhighlight lang=javascript>
function confirmName() {
    fields.name = name;
    framework.Show_dialog_box(fields, confirmNameContinuation);
}

function confirmNameContinuation(fields) {
    name = fields.name;
}
</syntaxhighlight>
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 (programming language)|Java]] 8 using the [[Swing (Java)|Swing]] UI framework:

<syntaxhighlight lang=java>
void buttonHandler() {
    // This is executing in the Swing UI thread.
    // We can access UI widgets here to get query parameters.
    int parameter = getField();

    new Thread(() -> {
        // 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.
        int result = lookup(parameter);

        javax.swing.SwingUtilities.invokeLater(() -> {
            // This code runs in the UI thread and can use
            // the fetched data to fill in UI widgets.
            setField(result);
        });
    }).start();
}
</syntaxhighlight>