Continuation-passing Style - Examples

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 non-local 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 "inside-out" because the innermost parts of the expression must be evaluated first, so CPS explicates the order of evaluation as well as the control flow. Some examples of code in direct style and the corresponding CPS style appear below. These examples are written in the Scheme programming language; by convention the continuation function is represented as a parameter named "k":

Direct style	Continuation passing style
(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 tail-recursive (* 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) (f-aux n 1)) (define (f-aux n a) (if (= n 0) a ; tail-recursive (f-aux (- n 1) (* n a))))	(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)))))))))

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 (cps-prim f) (lambda args (let ((r (reverse args))) ((car r) (apply f (reverse (cdr r))))))) (define *& (cps-prim *)) (define +& (cps-prim +))

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 CPS-style 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 f-aux& 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 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.