Chapter 13

Jumps

One of the signal features of Scheme is its support for jumps or nonlocal control. Specifically, Scheme allows program control to jump to arbitrary locations in the program, in contrast to the more restrained forms of program control flow allowed by conditionals and procedure calls. Scheme's nonlocal control operator is a procedure named call-with-current-continuation. We will see how this operator can be used to create a breathtaking variety of control idioms.

13.1  call-with-current-continuation

The operator call-with-current-continuation calls its argument, which must be a unary procedure, with a value called the ``current continuation''. If nothing else, this explains the name of the operator. But it is a long name, and is often abbreviated call/cc.6

The current continuation at any point in the execution of a program is an abstraction of the rest of the program. Thus in the program

(+ 1 (call/cc
       (lambda (k)
         (+ 2 (k 3)))))

the rest of the program, from the point of view of the call/cc-application, is the following program-with-a-hole (with [] representing the hole):

(+ 1 [])

In other words, this continuation is a program that will add 1 to whatever is used to fill its hole.

This is what the argument of call/cc is called with. Remember that the argument of call/cc is the procedure

(lambda (k)
  (+ 2 (k 3)))

This procedure's body applies the continuation (bound now to the parameter k) to the argument 3. This is when the unusual aspect of the continuation springs to the fore. The continuation call abruptly abandons its own computation and replaces it with the rest of the program saved in k! In other words, the part of the procedure involving the addition of 2 is jettisoned, and k's argument 3 is sent directly to the program-with-the-hole:

(+ 1 [])

The program now running is simply

(+ 1 3)

which returns 4. In sum,

(+ 1 (call/cc
       (lambda (k)
         (+ 2 (k 3)))))
=>  4

The above illustrates what is called an escaping continuation, one used to exit out of a computation (here: the (+ 2 []) computation). This is a useful property, but Scheme's continuations can also be used to return to previously abandoned contexts, and indeed to invoke them many times. The ``rest of the program'' enshrined in a continuation is available whenever and how many ever times we choose to recall it, and this is what contributes to the great and sometimes confusing versatility of call/cc. As a quick example, type the following at the listener:

(define r #f)

(+ 1 (call/cc
       (lambda (k)
         (set! r k)
         (+ 2 (k 3)))))
=>  4

The latter expression returns 4 as before. The difference between this use of call/cc and the previous example is that here we also store the continuation k in a global variable r.

Now we have a permanent record of the continuation in r. If we call it on a number, it will return that number incremented by 1:

(r 5)
=>  6

Note that r will abandon its own continuation, which is better illustrated by embedding the call to r inside some context:

(+ 3 (r 5))
=>  6

The continuations provided by call/cc are thus abortive continuations.

13.2  Escaping continuations

Escaping continuations are the simplest use of call/cc and are very useful for programming procedure or loop exits. Consider a procedure list-product that takes a list of numbers and multiplies them. A straightforward recursive definition for list-product is:

(define list-product
  (lambda (s)
    (let recur ((s s))
      (if (null? s) 1
          (* (car s) (recur (cdr s)))))))

There is a problem with this solution. If one of the elements in the list is 0, and if there are many elements after 0 in the list, then the answer is a foregone conclusion. Yet, the code will have us go through many fruitless recursive calls to recur before producing the answer. This is where an escape continuation comes in handy. Using call/cc, we can rewrite the procedure as:

(define list-product
  (lambda (s)
    (call/cc
      (lambda (exit)
        (let recur ((s s))
          (if (null? s) 1
              (if (= (car s) 0) (exit 0)
                  (* (car s) (recur (cdr s))))))))))

If a 0 element is encountered, the continuation exit is called with 0, thereby avoiding further calls to recur.

13.3  Tree matching

A more involved example of continuation usage is the problem of determining if two trees (arbitrarily nested dotted pairs) have the same fringe, ie, the same elements (or leaves) in the same sequence. Eg,

(same-fringe? '(1 (2 3)) '((1 2) 3))
=>  #t

(same-fringe? '(1 2 3) '(1 (3 2)))
=>  #f

The purely functional approach is to flatten both trees and check if the results match.

(define same-fringe?
  (lambda (tree1 tree2)
    (let loop ((ftree1 (flatten tree1))
               (ftree2 (flatten tree2)))
      (cond ((and (null? ftree1) (null? ftree2)) #t)
            ((or (null? ftree1) (null? ftree2)) #f)
            ((eqv? (car ftree1) (car ftree2))
             (loop (cdr ftree1) (cdr ftree2)))
            (else #f)))))

(define flatten
  (lambda (tree)
    (cond ((null? tree) '())
          ((pair? (car tree))
           (append (flatten (car tree))
                   (flatten (cdr tree))))
          (else
           (cons (car tree)
                 (flatten (cdr tree)))))))

However, this traverses the trees completely to flatten them, and then again till it finds non-matching elements. Furthermore, even the best flattening algorithms will require conses equal to the total number of leaves. (Destructively modifying the input trees is not an option.)

We can use call/cc to solve the problem without needless traversal and without any consing. Each tree is mapped to a generator, a procedure with internal state that successively produces the leaves of the tree in the left-to-right order that they occur in the tree.

(define tree->generator
  (lambda (tree)
    (let ((caller '*))
      (letrec
          ((generate-leaves
            (lambda ()
              (let loop ((tree tree))
                (cond ((null? tree) 'skip)
                      ((pair? tree)
                       (loop (car tree))
                       (loop (cdr tree)))
                      (else
                       (call/cc
                        (lambda (rest-of-tree)
                          (set! generate-leaves
                            (lambda ()
                              (rest-of-tree 'resume)))
                          (caller tree))))))
              (caller '()))))
        (lambda ()
          (call/cc
           (lambda (k)
             (set! caller k)
             (generate-leaves))))))))

When a generator created by tree->generator is called, it will store the continuation of its call in caller, so that it can know who to send the leaf to when it finds it. It then calls an internal procedure called generate-leaves which runs a loop traversing the tree from left to right. When the loop encounters a leaf, it will use caller to return the leaf as the generator's result, but it will remember to store the rest of the loop (captured as a call/cc continuation) in the generate-leaves variable. The next time the generator is called, the loop is resumed where it left off so it can hunt for the next leaf.

Note that the last thing generate-leaves does, after the loop is done, is to return the empty list to the caller. Since the empty list is not a valid leaf value, we can use it to tell that the generator has no more leaves to generate.

The procedure same-fringe? maps each of its tree arguments to a generator, and then calls these two generators alternately. It announces failure as soon as two non-matching leaves are found:

(define same-fringe?
  (lambda (tree1 tree2)
    (let ((gen1 (tree->generator tree1))
          (gen2 (tree->generator tree2)))
      (let loop ()
        (let ((leaf1 (gen1))
              (leaf2 (gen2)))
          (if (eqv? leaf1 leaf2)
              (if (null? leaf1) #t (loop))
              #f))))))

It is easy to see that the trees are traversed at most once, and in case of mismatch, the traversals extend only upto the leftmost mismatch. cons is not used.

13.4  Coroutines

The generators used above are interesting generalizations of the procedure concept. Each time the generator is called, it resumes its computation, and when it has a result for its caller returns it, but only after storing its continuation in an internal variable so the generator can be resumed again. We can generalize generators further, so that they can mutually resume each other, sending results back and forth amongst themselves. Such procedures are called coroutines [18].

We will view a coroutine as a unary procedure, whose body can contain resume calls. resume is a two-argument procedure used by a coroutine to resume another coroutine with a transfer value. The macro coroutine defines such a coroutine procedure, given a variable name for the coroutine's initial argument, and the body of the coroutine.

(define-macro coroutine
  (lambda (x . body)
    `(letrec ((+local-control-state
               (lambda (,x) ,@body))
              (resume
               (lambda (c v)
                 (call/cc
                  (lambda (k)
                    (set! +local-control-state k)
                    (c v))))))
       (lambda (v)
         (+local-control-state v)))))

A call of this macro creates a coroutine procedure (let's call it A) that can be called with one argument. A has an internal variable called +local-control-state that stores, at any point, the remaining computation of the coroutine. Initially this is the entire coroutine computation. When resume is called -- ie, invoking another coroutine B -- the current coroutine will update its +local-control-state value to the rest of itself, stop itself, and then jump to the resumed coroutine B. When coroutine A is itself resumed at some later point, its computation will proceed from the continuation stored in its +local-control-state.

13.4.1  Tree-matching with coroutines

Tree-matching is further simplified using coroutines. The matching process is coded as a coroutine that depends on two other coroutines to supply the leaves of the respective trees:

(define make-matcher-coroutine
  (lambda (tree-cor-1 tree-cor-2)
    (coroutine dont-need-an-init-arg
      (let loop ()
        (let ((leaf1 (resume tree-cor-1 'get-a-leaf))
              (leaf2 (resume tree-cor-2 'get-a-leaf)))
          (if (eqv? leaf1 leaf2)
              (if (null? leaf1) #t (loop))
              #f))))))

The leaf-generator coroutines remember who to send their leaves to:

(define make-leaf-gen-coroutine
  (lambda (tree matcher-cor)
    (coroutine dont-need-an-init-arg
      (let loop ((tree tree))
        (cond ((null? tree) 'skip)
              ((pair? tree)
               (loop (car tree))
               (loop (cdr tree)))
              (else
               (resume matcher-cor tree))))
      (resume matcher-cor '()))))

The same-fringe? procedure can now almost be written as

(define same-fringe?
  (lambda (tree1 tree2)
    (letrec ((tree-cor-1
              (make-leaf-gen-coroutine
               tree1
               matcher-cor))
             (tree-cor-2
              (make-leaf-gen-coroutine
               tree2
               matcher-cor))
             (matcher-cor
              (make-matcher-coroutine
               tree-cor-1
               tree-cor-2)))
      (matcher-cor 'start-ball-rolling))))

Unfortunately, Scheme's letrec can resolve mutually recursive references amongst the lexical variables it introduces only if such variable references are wrapped inside a lambda. And so we write:

(define same-fringe?
  (lambda (tree1 tree2)
    (letrec ((tree-cor-1
              (make-leaf-gen-coroutine
               tree1
               (lambda (v) (matcher-cor v))))
             (tree-cor-2
              (make-leaf-gen-coroutine
               tree2
               (lambda (v) (matcher-cor v))))
             (matcher-cor
              (make-matcher-coroutine
               (lambda (v) (tree-cor-1 v))
               (lambda (v) (tree-cor-2 v)))))
      (matcher-cor 'start-ball-rolling))))

Note that call/cc is not called directly at all in this rewrite of same-fringe?. All the continuation manipulation is handled for us by the coroutine macro.


6 If your Scheme does not already have this abbreviation, include (define call/cc call-with-current-continuation) in your initialization code and protect yourself from RSI.