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.
The operator call-with-current-continuationcalls 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 (k3)))))
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 (k3)))
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
(+13)
which returns 4. In sum,
(+1 (call/cc
(lambda (k)
(+2 (k3)))))
=>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:
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:
(r5)
=>6
Note that r will abandon its own continuation,
which is better illustrated by embedding the call to
r inside some context:
(+3 (r5))
=>6
The continuations provided by call/cc are thus
abortive 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:
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:
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,
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.
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:
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.
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.
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.
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:
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:
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
(definecall/cccall-with-current-continuation) in
your initialization code and protect yourself from
RSI.