Chapter 19

R⁵RS compatibility

The features described in this chapter are exported from the (rnrs r5rs (6))library and provide some functionality of the preceding revision of this report [8] that was omitted from the main part of the current report.

These are the same as the inexact and exact procedures; see report section on “Generic conversions”.

These procedures implement number-theoretic (integer) division. N₂ must be non-zero. All three procedures return integer objects. If n₁/n₂ is an integer object:

    (quotient n₁ n₂)   ‌⇒ n₁/n₂

    (remainder n₁ n₂)  ‌⇒ 0

    (modulo n₁ n₂)     ‌⇒ 0

If n₁/n₂ is not an integer object:

    (quotient n₁ n₂)   ‌⇒ n_q

    (remainder n₁ n₂)  ‌⇒ n_r

    (modulo n₁ n₂)     ‌⇒ n_m

where n_q is n₁/n₂ rounded towards zero, 0 < |n_r| < |n₂|, 0 < |n_m| < |n₂|, n_r and n_m differ from n₁ by a multiple of n₂, n_r has the same sign as n₁, and n_m has the same sign as n₂.

Consequently, for integer objects n₁ and n₂ with n₂ not equal to 0,

     (= n₁ (+ (* n₂ (quotient n₁ n₂))

           (remainder n₁ n₂)))

                                 ‌⇒  #t

provided all number object involved in that computation are exact.

(modulo 13 4)           ‌⇒  1

(remainder 13 4)        ‌⇒  1

(modulo -13 4)          ‌⇒  3

(remainder -13 4)       ‌⇒  -1

(modulo 13 -4)          ‌⇒  -3

(remainder 13 -4)       ‌⇒  1

(modulo -13 -4)         ‌⇒  -1

(remainder -13 -4)      ‌⇒  -1

(remainder -13 -4.0)    ‌⇒  -1.0

Note:‌ These procedures could be defined in terms of div and mod (see report section on “Arithmetic operations”) as follows (without checking of the argument types): (define (sign n)

  (cond

    ((negative? n) -1)

    ((positive? n) 1)

    (else 0)))

(define (quotient n1 n2)

  (* (sign n1) (sign n2) (div (abs n1) (abs n2))))

(define (remainder n1 n2)

  (* (sign n1) (mod (abs n1) (abs n2))))

(define (modulo n1 n2)

  (* (sign n2) (mod (* (sign n2) n1) (abs n2))))

The delay construct is used together with the procedure force to implement lazy evaluation or call by need. (delay <expression>) returns an object called a promise which at some point in the future may be asked (by the force procedure) to evaluate <expression>, and deliver the resulting value. The effect of <expression> returning multiple values is unspecified.

Promise must be a promise. The force procedure forces the value of promise. If no value has been computed for the promise, then a value is computed and returned. The value of the promise is cached (or “memoized”) so that if it is forced a second time, the previously computed value is returned.

(force (delay (+ 1 2)))   ‌⇒  3

(let ((p (delay (+ 1 2))))

  (list (force p) (force p)))  

                               ‌⇒  (3 3)

(define a-stream

  (letrec ((next

            (lambda (n)

              (cons n (delay (next (+ n 1)))))))

    (next 0)))

(define head car)

(define tail

  (lambda (stream) (force (cdr stream))))

(head (tail (tail a-stream)))  

                               ‌⇒  2

Promises are mainly intended for programs written in functional style. The following examples should not be considered to illustrate good programming style, but they illustrate the property that only one value is computed for a promise, no matter how many times it is forced.

(define count 0)

(define p

  (delay (begin (set! count (+ count 1))

                (if (> count x)

                    count

                    (force p)))))

(define x 5)

p                     ‌⇒  a promise

(force p)             ‌⇒  6

p                     ‌⇒  a promise, still

(begin (set! x 10)

       (force p))     ‌⇒  6

Here is a possible implementation of delay and force. Promises are implemented here as procedures of no arguments, and force simply calls its argument:

(define force

  (lambda (object)

    (object)))

The expression

(delay <expression>)

has the same meaning as the procedure call

(make-promise (lambda () <expression>))

as follows

(define-syntax delay

  (syntax-rules ()

    ((delay expression)

     (make-promise (lambda () expression))))),

where make-promise is defined as follows:

(define make-promise

  (lambda (proc)

    (let ((result-ready? #f)

          (result #f))

      (lambda ()

        (if result-ready?

            result

            (let ((x (proc)))

              (if result-ready?

                  result

                  (begin (set! result-ready? #t)

                         (set! result x)

                         result))))))))

N must be the exact integer object 5. The null-environment procedure returns an environment specifier suitable for use with eval (see chapter 16) representing an environment that is empty except for the (syntactic) bindings for all keywords described in the previous revision of this report [8], including bindings for =>, ..., else, and _ that are the same as those in the (rnrs base (6)) library.

N must be the exact integer object 5. The scheme-report-environment procedure returns an environment specifier for an environment that is empty except for the bindings for the identifiers described in the previous revision of this report [8], omitting load, interaction-environment, transcript-on, transcript-off, and char-ready?. The variable bindings have as values the procedures of the same names described in this report, and the keyword bindings, including =>, ..., else, and _ are the same as those described in this report.