Fundamentals of Computer Science I (CSC-151.02 2000F)

Procedures that return multiple values

When we invoke any of the procedures that we have discussed so far in the course, we get back a single result. However, there are many computations that we can describe most naturally as having two or more results. A typical example is the division of integers, as taught in elementary schools: When you divide 647 by 7, you get a quotient of 92 and a remainder of 3. It is not obvious that one of these results is more important or more significant than the other, and in fact Scheme provides a primitive for each one: quotient for the quotient, remainder for the remainder:

> (quotient 647 7)
> (remainder 647 7)

But since the same underlying computation is carried out in either case, it would make more sense to have a single procedure divide that would, when invoked, produce both results:

> (divide 647 7)

The procedure call (divide 647 7) is an example of a multiple-valued expression in Scheme. Evaluating a multiple-valued expression gives you several results, as the interaction displayed above shows. Note that this result is not the same as displaying multiple values nor is it the same as returning a list of values. We are, in fact, returning "many values" from one procedure.

Other kinds of expressions can also have multiple values:

However, identifiers, constants, lambda-expressions, and-expressions, and or-expressions are never multiple-valued.

Of course, we have to be careful about where we write multiple-valued expressions. We can't put one in the test position of an if-expression, for instance. If the test expression could be multiple-valued, it might have both #t and #f among its values! No, the test has to produce exactly one value, so that we know unambiguously whether to evaluate the consequent or the alternate.

Similarly, we can't write a multiple-valued expression as a subexpression of a procedure call, since the procedure to be called has to be some one particular procedure, and each of the arguments that we supply to it has to be some one particular value.

However, we can write a multiple-valued expression as the body of a lambda-expression, thus constructing our own multiple-result procedures, and this is the context in which you can expect to see them most frequently.

The values procedure

All multiple-valued expressions enter Scheme, directly or indirectly, through calls to the primitive procedure values. Values is a variable-arity procedure that returns its arguments without change -- all of them.

> (values 650 682 513 861)

When the values procedure is given only one argument, it returns that argument without change:

> (values #\a)

It is also possible to call values with no arguments, in which case, of course, it returns no values:

> (values)

This is not an error or a non-terminating computation, but simply a computation that produces nothing when it finishes, like a committee that decides not to issue a report (which is sometimes the most useful thing a committee can do). You've seen procedures that return nothing before; newline and display both return nothing.

As a quick example, let's define a procedure mixed-number-parts that takes a rational number as its argument and returns its integer part and its proper fractional part:

;;; Procedure:
;;;   mixed-number-parts
;;; Parameters:
;;;   A rational number
;;; Purpose:
;;;   Separate a rational number into whole and fractional parts.
;;; Produces:
;;;   The whole part
;;;   The fracional part
;;; Preconditions:
;;;   The parameters must be a rational number.
;;; Postconditions:
;;;   None
(define mixed-number-parts
  (lambda (num)
    (let ((integer-part (truncate num)))
      (values integer-part (- num integer-part)))))

The call-with-values procedure

Sometimes we want to use the values of a multiple-valued expression in some further computation instead of returning them directly. The fact that a multiple-valued expression cannot be used as an argument in a procedure call makes it difficult to do this.

Scheme's solution is another primitive procedure, call-with-values, that manages the flow of data from a multiple-valued expression into a larger computation. Call-with-values is a higher-order procedure that takes two arguments, a producer procedure that generates multiple values and a consumer procedure that accepts them.

The producer procedure takes no arguments and has a multiple-valued expression as its body. Call-with-values invokes the producer and collects all of the values that it returns. Call-with-values then invokes the consumer, providing the collected values as arguments. The arity of the consumer must therefore be compatible with the number of values delivered by the producer. Call-with-values returns whatever the consumer returns.

Suppose, for instance, that we want to recover the integer part and the proper fractional part of 1173/83 and then subtract the fractional part from the integer part. Here's how we could invoke call-with-values to perform the computation:

> (call-with-values (lambda () (mixed-number-parts 1173/83)) -)

The producer in this case is (lambda () (mixed-number-parts 1173/83), which returns two values when invoked. The consumer is -, which accepts two values and returns their difference.

As a more practical example, let's define a procedure that takes two arguments, a predicate pred and a list ls, and returns two lists, one consisting of all of the elements of ls that satisfy pred, the other consisting of all of the elements of ls that do not satisfy pred.

Our basic strategy is list recursion. In the base case, where ls is the empty list, we want to return two lists, both empty. That's easy -- we'll just write (values null null). In any other case, we divide ls into its car and its cdr and invoke the procedure recursively to deal with the cdr. The recursive call will return two lists: the list of elements of the cdr that satisfy pred, and the list of elements of the cdr that do not. We cons the car of the list onto one or the other of these recursive results, depending on whether it does or does not satisfy pred, and return both the result of the cons and the other recursive result.

Translating into Scheme:

(define partition
  (lambda (pred ls)
    (letrec ((recurrer
              (lambda (ls)
                (if (null? ls)
                    (values null null)
                      (lambda () (recurrer (cdr ls)))
                      (lambda (ins outs)
                        (if (pred (car ls))
                            (values (cons (car ls) ins) outs)
                            (values ins (cons (car ls) outs)))))))))
      (recurrer ls))))

Notice how one uses call-with-values to manage the transfer of two values from the multiple-valued expression (recurrer (cdr ls)) into the part of the computation that consumes those values.

Think of the body of the producer, the expression `(recurrer (cdr ls))', as ready to supply its results when asked. Think of the body of the consumer, the inner if-expression, as ready to receive those results and operate on them to construct the final values that recurrer will return. The role of call-with-values is to activate and mediate these two packaged components of the overall computation.

Disclaimer Often, these pages were created "on the fly" with little, if any, proofreading. Any or all of the information on the pages may be incorrect. Please contact me if you notice errors.

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