[转]tspl3: Continuation Passing Style

Section 3.3. Continuations

During the evaluation of a Scheme expression, the implementation must keep track of two things: (1) what to evaluate and (2) what to do with the value. Consider the evaluation of (null? x) within the expression below.

(if (null? x) (quote ()) (cdr x))

The implementation must first evaluate (null? x) and, based on its value, evaluate either (quote ()) or (cdr x). "What to evaluate" is (null? x), and "what to do with the value" is to make the decision which of (quote ()) and (cdr x) to evaluate and to do so. We call "what to do with the value" the continuation of a computation.

Thus, at any point during the evaluation of any expression, there is a continuation ready to complete, or at least continue, the computation from that point. Let's assume that x has the value (a b c). Then we can isolate six continuations during the evaluation of (if (null? x) (quote ()) (cdr x)), the continuations waiting for:

1. the value of (if (null? x) (quote ()) (cdr x)),
2. the value of (null? x),
3. the value of null?,
4. the value of x,
5. the value of cdr, and
6. the value of x (again).

The continuation of (cdr x) is not listed because it is the same as the one waiting for (if (null? x) (quote ()) (cdr x)).

Scheme allows the continuation of any expression to be obtained with the procedure call-with-current-continuation, which may be abbreviated call/cc in most implementations.

We use the shorter name here. If the implementation you are using does not recognize call/cc, simply define it as follows

(define call/cc call-with-current-continuation)

or use the longer name in your code.

call/cc must be passed a procedure p of one argument. call/cc constructs a concrete representation of the the current continuation and passes it to p. The continuation itself is represented by a procedure k. Each time k is applied to a value, it returns the value to the continuation of the call/cc application. This value becomes, in essence, the value of the application of call/cc.

If p returns without invoking k, the value returned by the procedure becomes the value of the application of call/cc.

Consider the simple examples below.

(call/cc
(lambda (k)
(* 5 4))) 20

(call/cc
(lambda (k)
(* 5 (k 4)))) 4

(+ 2
(call/cc
(lambda (k)
(* 5 (k 4))))) 6

In the first example, the continuation is obtained and bound to k, but k is never used, so the value is simply the product of 5 and 4. In the second, the continuation is invoked before the multiplication, so the value is the value passed to the continuation, 4. In the third, the continuation includes the addition by 2; thus, the value is the value passed to the continuation, 4, plus 2.

Here is a less trivial example, showing the use of call/cc to provide a nonlocal exit from a recursion.

(define product
(lambda (ls)
(call/cc
(lambda (break)
(let f ((ls ls))
(cond
((null? ls) 1)
((= (car ls) 0) (break 0))
(else (* (car ls) (f (cdr ls))))))))))

(product '(1 2 3 4 5)) 120
(product '(7 3 8 0 1 9 5)) 0

The nonlocal exit allows product to return immediately, without performing the pending multiplications, when a zero value is detected.

Each of the continuation invocations above returns to the continuation while control remains within the procedure passed to call/cc. The following example uses the continuation after this procedure has already returned.

(let ((x (call/cc (lambda (k) k))))
(x (lambda (ignore) "hi"))) "hi"

The continuation obtained by this invocation of call/cc may be described as "Take the value, bind it to x, and apply the value of x to the value of (lambda (ignore) "hi")." Since (lambda (k) k) returns its argument, x is bound to the continuation itself; this continuation is applied to the procedure resulting from the evaluation of (lambda (ignore) "hi"). This has the effect of binding x (again!) to this procedure and applying the procedure to itself. The procedure ignores its argument and returns "hi".

The following variation of the example above is probably the most confusing Scheme program of its size; it may be easy to guess what it returns, but it takes some work to verify that guess.

(((call/cc (lambda (k) k)) (lambda (x) x)) "HEY!") "HEY!"

The value of the call/cc is its own continuation, as in the preceding example. This is applied to the identity procedure (lambda (x) x), so the call/cc returns a second time with this value. Then, the identity procedure is applied to itself, yielding the identity procedure. This is finally applied to "HEY!", yielding "HEY!".

Continuations used in this manner are not always so puzzling. Consider the following definition of factorial that saves the continuation at the base of the recursion before returning 1, by assigning the top-level variable retry.

(define retry #f)

(define factorial
(lambda (x)
(if (= x 0)
(call/cc (lambda (k) (set! retry k) 1))
(* x (factorial (- x 1))))))

With this definition, factorial works as we expect factorial to work, except it has the side effect of assigning retry.

(factorial 4) 24
(retry 1) 24
(retry 2) 48

The continuation bound to retry might be described as "Multiply the value by 1, then multiply this result by 2, then multiply this result by 3, then multiply this result by 4." If we pass the continuation a different value, i.e., not 1, we will cause the base value to be something other than 1 and hence change the end result.

(retry 2) 48
(retry 5) 120

This mechanism could be the basis for a breakpoint package implemented with call/cc; each time a breakpoint is encountered, the continuation of the breakpoint is saved so that the computation may be restarted from the breakpoint (more than once, if desired).

Continuations may be used to implement various forms of multitasking. The simple "light-weight process" mechanism defined below allows multiple computations to be interleaved. Since it is nonpreemptive, it requires that each process voluntarily "pause" from time to time in order to allow the others to run.

(define lwp-list '())
(define lwp
(lambda (thunk)
(set! lwp-list (append lwp-list (list thunk)))))

(define start
(lambda ()
(let ((p (car lwp-list)))
(set! lwp-list (cdr lwp-list))
(p))))

(define pause
(lambda ()
(call/cc
(lambda (k)
(lwp (lambda () (k #f)))
(start)))))

The following light-weight processes cooperate to print an infinite sequence of lines containing "hey!".

(lwp (lambda () (let f () (pause) (display "h") (f))))
(lwp (lambda () (let f () (pause) (display "e") (f))))
(lwp (lambda () (let f () (pause) (display "y") (f))))
(lwp (lambda () (let f () (pause) (display "!") (f))))
(lwp (lambda () (let f () (pause) (newline) (f))))
(start)
hey!
hey!
hey!
hey! See Section 9.11 for an implementation of engines, which support preemptive multitasking, with call/cc.

Exercise 3.3.1

Use call/cc to write a program that loops indefinitely, printing a sequence of numbers beginning at zero. Do not use any recursive procedures, and do not use any assignments.

Exercise 3.3.2

Rewrite product without call/cc, retaining the feature that no multiplications are performed if any of the list elements are zero.

Exercise 3.3.3

What would happen if a process created by lwp as defined above were to terminate, i.e., simply return without calling pause? Define a quit procedure that allows a process to terminate without otherwise affecting the lwp system. Be sure to handle the case in which the only remaining process terminates.

Exercise 3.3.4

Each time lwp is called, the list of processes is copied because lwp uses append to add its argument to the end of the process list. Modify the original lwp code to use the queue datatype developed in Section 2.9 to avoid this problem.

Exercise 3.3.5

The light-weight process mechanism allows new processes to be created dynamically, although the example given in this section does not do so. Design an application that requires new processes to be created dynamically and implement it using the light-weight process mechanism.

Section 3.4. Continuation Passing Style

As we discussed in the preceding section, a continuation waits for the value of each expression. In particular, a continuation is associated with each procedure call. When one procedure invokes another via a nontail call, the called procedure receives an implicit continuation that is responsible for completing what is left of the calling procedure's body plus returning to the calling procedure's continuation. If the call is a tail call, the called procedure simply receives the continuation of the calling procedure.

We can make the continuations explicit by encapsulating "what to do" in an explicit procedural argument passed along on each call. For example, the continuation of the call to f in

(letrec ((f (lambda (x) (cons 'a x)))
(g (lambda (x) (cons 'b (f x))))
(h (lambda (x) (g (cons 'c x)))))
(cons 'd (h '()))) (d b a c)

conses the symbol b onto the value returned to it, then returns the result of this cons to the continuation of the call to g. This continuation is the same as the continuation of the call to h, which conses the symbol d onto the value returned to it. We can rewrite this in continuation-passing style, or CPS, by replacing these implicit continuations with explicit procedures.

(letrec ((f (lambda (x k) (k (cons 'a x))))
(g (lambda (x k)
(f x (lambda (v) (k (cons 'b v))))))
(h (lambda (x k) (g (cons 'c x) k))))
(h '() (lambda (v) (cons 'd v))))

Like the implicit continuation of h and g in the preceding example, the explicit continuation passed to h and on to g,

(lambda (v) (cons 'd v))

conses the symbol d onto the value passed to it. Similarly, the continuation passed to f,

(lambda (v) (k (cons 'b v)))

conses b onto the value passed to it, then passes this on to the continuation of g. Expressions written in CPS are more complicated, of course, but this style of programming has some useful applications. CPS allows a procedure to pass more than one result to its continuation, because the procedure that implements the continuation can take any number of arguments.

(define car&cdr
(lambda (p k)
(k (car p) (cdr p))))

(car&cdr '(a b c)
(lambda (x y)
(list y x))) ((b c) a)
(car&cdr '(a b c) cons) (a b c)
(car&cdr '(a b c a d) memv) (a d)

(This can be done with multiple values as well; see Section 5.7.) CPS also allows a procedure to take separate "success" and "failure" continuations, which may accept different numbers of arguments. An example is integer-divide below, which passes the quotient and remainder of its first two arguments to its third, unless the second argument (the divisor) is zero, in which case it passes an error message to its fourth argument.

(define integer-divide
(lambda (x y success failure)
(if (= y 0)
(failure "divide by zero")
(let ((q (quotient x y)))
(success q (- x (* q y)))))))

(integer-divide 10 3 list (lambda (x) x)) (3 1)
(integer-divide 10 0 list (lambda (x) x)) "divide by zero"

The procedure quotient, employed by integer-divide, returns the quotient of its two arguments, truncated toward zero.

Explicit success and failure continuations can sometimes help to avoid the extra communication necessary to separate successful execution of a procedure from unsuccessful execution. Furthermore, it is possible to have multiple success or failure continuations for different flavors of success or failure, each possibly taking different numbers and types of arguments. See Sections 9.10 and 9.11 for extended examples that employ continuation-passing style.

At this point you may be wondering about the relationship between CPS and the continuations obtained via call/cc. It turns out that any program that uses call/cc can be rewritten in CPS without call/cc, but a total rewrite of the program (sometimes including even system-defined primitives) may be necessary. Try to convert the product example on page 72 into CPS before looking at the version below.

(define product
(lambda (ls k)
(let ((break k))
(let f ((ls ls) (k k))
(cond
((null? ls) (k 1))
((= (car ls) 0) (break 0))
(else (f (cdr ls)
(lambda (x)
(k (* (car ls) x))))))))))

(product '(1 2 3 4 5) (lambda (x) x)) 120
(product '(7 3 8 0 1 9 5) (lambda (x) x)) 0

Exercise 3.4.1

Rewrite the reciprocal example first given in Section 2.1 to accept both success and failure continuations, like integer-divide above.

Exercise 3.4.2

Rewrite the retry example from page 73 to use CPS.

Exercise 3.4.3

Rewrite the following expression in CPS to avoid using call/cc.

(define reciprocals
(lambda (ls)
(call/cc
(lambda (k)
(map (lambda (x)
(if (= x 0)
(k "zero found")
(/ 1 x)))
ls)))))

(reciprocals '(2 1/3 5 1/4)) (1/2 3 1/5 4)
(reciprocals '(2 1/3 0 5 1/4)) "zero found"

[Hint: A single-list version of map is defined on page 44.]

posted @ 2014-05-23 16:25  Scan.  阅读(211)  评论(0编辑  收藏  举报