Revised^5 Report on the Algorithmic Language Scheme
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Chapter 6
Standard procedures
This chapter describes Scheme's built-in procedures. The initial (or
``top level'') Scheme environment starts out with a number of variables
bound to locations containing useful values, most of which are primitive
procedures that manipulate data. For example, the variable
abs
is
bound to (a location initially containing) a procedure of one argument
that computes the absolute value of a number, and the variable
is bound to a procedure that computes sums. Built-in procedures that
can easily be written in terms of other built-in procedures are identified as
``library procedures''.
A program may use a top-level definition to bind any variable. It may
subsequently alter any such binding by an assignment (see
4.1.6
).
These operations do not modify the behavior of Scheme's built-in
procedures. Altering any top-level binding that has not been introduced by a
definition has an unspecified effect on the behavior of the built-in procedures.
6.1 Equivalence predicates
predicate
is a procedure that always returns a boolean
value (
#t
or
#f
). An
equivalence predicate
is
the computational analogue of a mathematical equivalence relation (it is
symmetric, reflexive, and transitive). Of the equivalence predicates
described in this section,
eq?
is the finest or most
discriminating, and
equal?
is the coarsest.
Eqv?
is
slightly less discriminating than
eq?
procedure:
eqv?
obj
obj
The
eqv?
procedure defines a useful equivalence relation on objects.
Briefly, it returns
#t
if
obj
and
obj
should
normally be regarded as the same object. This relation is left slightly
open to interpretation, but the following partial specification of
eqv?
holds for all implementations of Scheme.
The
eqv?
procedure returns
#t
if:
obj
and
obj
are both
#t
or both
#f
obj
and
obj
are both symbols and
(string=? (symbol->string obj1)
(symbol->string obj2))
===>
#t
Note:
This assumes that neither
obj
nor
obj
is an ``uninterned
symbol'' as alluded to in section
6.3.3
. This report does
not presume to specify the behavior of
eqv?
on implementation-dependent
extensions.
obj
and
obj
are both numbers, are numerically
equal (see
, section
6.2
), and are either both
exact
or both inexact
obj
and
obj
are both characters and are the same
character according to the
char=?
procedure
(section
6.3.4
).
both
obj
and
obj
are the empty list.
obj
and
obj
are pairs, vectors, or strings that denote the
same locations in the store (section
3.4
).
obj
and
obj
are procedures whose location tags are
equal (section
4.1.4
).
The
eqv?
procedure returns
#f
if:
obj
and
obj
are of different types
(section
3.2
).
one of
obj
and
obj
is
#t
but the other is
#f
obj
and
obj
are symbols but
(string=? (symbol->string
obj
(symbol->string
obj
))
===>
#f
one of
obj
and
obj
is an exact number but the other
is an inexact number.
obj
and
obj
are numbers for which the
procedure returns
#f
obj
and
obj
are characters for which the
char=?
procedure returns
#f
one of
obj
and
obj
is the empty list but the other
is not.
obj
and
obj
are pairs, vectors, or strings that denote
distinct locations.
obj
and
obj
are procedures that would behave differently
(return different value(s) or have different side effects) for some arguments.
(eqv? 'a 'a) ===>
#t
(eqv? 'a 'b) ===>
#f
(eqv? 2 2) ===>
#t
(eqv? '() '()) ===>
#t
(eqv? 100000000 100000000) ===>
#t
(eqv? (cons 1 2) (cons 1 2)) ===>
#f
(eqv? (lambda () 1)
(lambda () 2)) ===>
#f
(eqv? #f 'nil) ===>
#f
(let ((p (lambda (x) x)))
(eqv? p p)) ===>
#t
The following examples illustrate cases in which the above rules do
not fully specify the behavior of
eqv?
. All that can be said
about such cases is that the value returned by
eqv?
must be a
boolean.
(eqv? "" "") ===>
unspecified
(eqv? '#() '#()) ===>
unspecified
(eqv? (lambda (x) x)
(lambda (x) x)) ===>
unspecified
(eqv? (lambda (x) x)
(lambda (y) y)) ===>
unspecified
The next set of examples shows the use of
eqv?
with procedures
that have local state.
Gen-counter
must return a distinct
procedure every time, since each procedure has its own internal counter.
Gen-loser
, however, returns equivalent procedures each time, since
the local state does not affect the value or side effects of the
procedures.
(define gen-counter
(lambda ()
(let ((n 0))
(lambda () (set! n (+ n 1)) n))))
(let ((g (gen-counter)))
(eqv? g g)) ===>
#t
(eqv? (gen-counter) (gen-counter))
===>
#f
(define gen-loser
(lambda ()
(let ((n 0))
(lambda () (set! n (+ n 1)) 27))))
(let ((g (gen-loser)))
(eqv? g g)) ===>
#t
(eqv? (gen-loser) (gen-loser))
===>
unspecified
(letrec ((f (lambda () (if (eqv? f g) 'both 'f)))
(g (lambda () (if (eqv? f g) 'both 'g))))
(eqv? f g))
===>
unspecified
(letrec ((f (lambda () (if (eqv? f g) 'f 'both)))
(g (lambda () (if (eqv? f g) 'g 'both))))
(eqv? f g))
===>
#f
Since it is an error to modify constant objects (those returned by
literal expressions), implementations are permitted, though not
required, to share structure between constants where appropriate. Thus
the value of
eqv?
on constants is sometimes
implementation-dependent.
(eqv? '(a) '(a)) ===>
unspecified
(eqv? "a" "a") ===>
unspecified
(eqv? '(b) (cdr '(a b))) ===>
unspecified
(let ((x '(a)))
(eqv? x x)) ===>
#t
Rationale:
The above definition of
eqv?
allows implementations latitude in
their treatment of procedures and literals: implementations are free
either to detect or to fail to detect that two procedures or two literals
are equivalent to each other, and can decide whether or not to
merge representations of equivalent objects by using the same pointer or
bit pattern to represent both.
procedure:
eq?
obj
obj
Eq?
is similar to
eqv?
except that in some cases it is
capable of discerning distinctions finer than those detectable by
eqv?
Eq?
and
eqv?
are guaranteed to have the same
behavior on symbols, booleans, the empty list, pairs, procedures,
and non-empty
strings and vectors.
Eq?
's behavior on numbers and characters is
implementation-dependent, but it will always return either true or
false, and will return true only when
eqv?
would also return
true.
Eq?
may also behave differently from
eqv?
on empty
vectors and empty strings.
(eq? 'a 'a) ===>
#t
(eq? '(a) '(a)) ===>
unspecified
(eq? (list 'a) (list 'a)) ===>
#f
(eq? "a" "a") ===>
unspecified
(eq? "" "") ===>
unspecified
(eq? '() '()) ===>
#t
(eq? 2 2) ===>
unspecified
(eq? #
A #
A) ===>
unspecified
(eq? car car) ===>
#t
(let ((n (+ 2 3)))
(eq? n n)) ===>
unspecified
(let ((x '(a)))
(eq? x x)) ===>
#t
(let ((x '#()))
(eq? x x)) ===>
#t
(let ((p (lambda (x) x)))
(eq? p p)) ===>
#t
Rationale:
It will usually be possible to implement
eq?
much
more efficiently than
eqv?
, for example, as a simple pointer
comparison instead of as some more complicated operation. One reason is
that it may not be possible to compute
eqv?
of two numbers in
constant time, whereas
eq?
implemented as pointer comparison will
always finish in constant time.
Eq?
may be used like
eqv?
in applications using procedures to implement objects with state since
it obeys the same constraints as
eqv?
library procedure:
equal?
obj
obj
Equal?
recursively compares the contents of pairs, vectors, and
strings, applying
eqv?
on other objects such as numbers and symbols.
A rule of thumb is that objects are generally
equal?
if they print
the same.
Equal?
may fail to terminate if its arguments are
circular data structures.
(equal? 'a 'a) ===>
#t
(equal? '(a) '(a)) ===>
#t
(equal? '(a (b) c)
'(a (b) c)) ===>
#t
(equal? "abc" "abc") ===>
#t
(equal? 2 2) ===>
#t
(equal? (make-vector 5 'a)
(make-vector 5 'a)) ===>
#t
(equal? (lambda (x) x)
(lambda (y) y)) ===>
unspecified
6.2 Numbers
Numerical computation has traditionally been neglected by the Lisp
community. Until Common Lisp there was no carefully thought out
strategy for organizing numerical computation, and with the exception of
the MacLisp system [
20
] little effort was made to
execute numerical code efficiently. This report recognizes the excellent work
of the Common Lisp committee and accepts many of their recommendations.
In some ways this report simplifies and generalizes their proposals in a manner
consistent with the purposes of Scheme.
It is important to distinguish between the mathematical numbers, the
Scheme numbers that attempt to model them, the machine representations
used to implement the Scheme numbers, and notations used to write numbers.
This report uses the types
number
complex
real
rational
, and
integer
to refer to both mathematical numbers
and Scheme numbers. Machine representations such as fixed point and
floating point are referred to by names such as
fixnum
and
flonum
6.2.1 Numerical types
Mathematically, numbers may be arranged into a tower of subtypes
in which each level is a subset of the level above it:
number
complex
real
rational
integer
For example, 3 is an integer. Therefore 3 is also a rational,
a real, and a complex. The same is true of the Scheme numbers
that model 3. For Scheme numbers, these types are defined by the
predicates
number?
complex?
real?
rational?
and
integer?
There is no simple relationship between a number's type and its
representation inside a computer. Although most implementations of
Scheme will offer at least two different representations of 3, these
different representations denote the same integer.
Scheme's numerical operations treat numbers as abstract data, as
independent of their representation as possible. Although an implementation
of Scheme may use fixnum, flonum, and perhaps other representations for
numbers, this should not be apparent to a casual programmer writing
simple programs.
It is necessary, however, to distinguish between numbers that are
represented exactly and those that may not be. For example, indexes
into data structures must be known exactly, as must some polynomial
coefficients in a symbolic algebra system. On the other hand, the
results of measurements are inherently inexact, and irrational numbers
may be approximated by rational and therefore inexact approximations.
In order to catch uses of inexact numbers where exact numbers are
required, Scheme explicitly distinguishes exact from inexact numbers.
This distinction is orthogonal to the dimension of type.
6.2.2 Exactness
Scheme numbers are either
exact
or
inexact
. A number is
exact if it was written as an exact constant or was derived from
exact numbers using only exact operations. A number is
inexact if it was written as an inexact constant,
if it was
derived using inexact ingredients, or if it was derived using
inexact operations. Thus inexactness is a contagious
property of a number.
If two implementations produce exact results for a
computation that did not involve inexact intermediate results,
the two ultimate results will be mathematically equivalent. This is
generally not true of computations involving inexact numbers
since approximate methods such as floating point arithmetic may be used,
but it is the duty of each implementation to make the result as close as
practical to the mathematically ideal result.
Rational operations such as
should always produce
exact results when given exact arguments.
If the operation is unable to produce an exact result,
then it may either report the violation of an implementation restriction
or it may silently coerce its
result to an inexact value.
See section
6.2.3
With the exception of
inexact->exact
, the operations described in
this section must generally return inexact results when given any inexact
arguments. An operation may, however, return an exact result if it can
prove that the value of the result is unaffected by the inexactness of its
arguments. For example, multiplication of any number by an exact zero
may produce an exact zero result, even if the other argument is
inexact.
6.2.3 Implementation restrictions
Implementations of Scheme are not required to implement the whole
tower of subtypes given in section
6.2.1
but they must implement a coherent subset consistent with both the
purposes of the implementation and the spirit of the Scheme language.
For example, an implementation in which all numbers are real
may still be quite useful.
Implementations may also support only a limited range of numbers of
any type, subject to the requirements of this section. The supported
range for exact numbers of any type may be different from the
supported range for inexact numbers of that type. For example,
an implementation that uses flonums to represent all its
inexact real numbers may
support a practically unbounded range of exact integers
and rationals
while limiting the range of inexact reals (and therefore
the range of inexact integers and rationals)
to the dynamic range of the flonum format.
Furthermore
the gaps between the representable inexact integers and
rationals are
likely to be very large in such an implementation as the limits of this
range are approached.
An implementation of Scheme must support exact integers
throughout the range of numbers that may be used for indexes of
lists, vectors, and strings or that may result from computing the length of a
list, vector, or string. The
length
vector-length
and
string-length
procedures must return an exact
integer, and it is an error to use anything but an exact integer as an
index. Furthermore any integer constant within the index range, if
expressed by an exact integer syntax, will indeed be read as an exact
integer, regardless of any implementation restrictions that may apply
outside this range. Finally, the procedures listed below will always
return an exact integer result provided all their arguments are exact integers
and the mathematically expected result is representable as an exact integer
within the implementation:
+ - *
quotient remainder modulo
max min abs
numerator denominator gcd
lcm floor ceiling
truncate round rationalize
expt
Implementations are encouraged, but not required, to support
exact integers and exact rationals of
practically unlimited size and precision, and to implement the
above procedures and the
procedure in
such a way that they always return exact results when given exact
arguments. If one of these procedures is unable to deliver an exact
result when given exact arguments, then it may either report a
violation of an
implementation restriction or it may silently coerce its result to an
inexact number. Such a coercion may cause an error later.
An implementation may use floating point and other approximate
representation strategies for inexact numbers.
This report recommends, but does not require, that the IEEE 32-bit
and 64-bit floating point standards be followed by implementations that use
flonum representations, and that implementations using
other representations should match or exceed the precision achievable
using these floating point standards [
12
].
In particular, implementations that use flonum representations
must follow these rules: A flonum result
must be represented with at least as much precision as is used to express any of
the inexact arguments to that operation. It is desirable (but not required) for
potentially inexact operations such as
sqrt
, when applied to exact
arguments, to produce exact answers whenever possible (for example the
square root of an exact 4 ought to be an exact 2).
If, however, an
exact number is operated upon so as to produce an inexact result
(as by
sqrt
), and if the result is represented as a flonum, then
the most precise flonum format available must be used; but if the result
is represented in some other way then the representation must have at least as
much precision as the most precise flonum format available.
Although Scheme allows a variety of written
notations for
numbers, any particular implementation may support only some of them.
For example, an implementation in which all numbers are real
need not support the rectangular and polar notations for complex
numbers. If an implementation encounters an exact numerical constant that
it cannot represent as an exact number, then it may either report a
violation of an implementation restriction or it may silently represent the
constant by an inexact number.
6.2.4 Syntax of numerical constants
The syntax of the written representations for numbers is described formally in
section
7.1.1
. Note that case is not significant in numerical
constants.
A number may be written in binary, octal, decimal, or
hexadecimal by the use of a radix prefix. The radix prefixes are
#b
(binary),
#o
(octal),
#d
(decimal), and
#x
(hexadecimal). With
no radix prefix, a number is assumed to be expressed in decimal.
numerical constant may be specified to be either exact or
inexact by a prefix. The prefixes are
#e
for exact, and
#i
for inexact. An exactness
prefix may appear before or after any radix prefix that is used. If
the written representation of a number has no exactness prefix, the
constant may be either inexact or exact. It is
inexact if it contains a decimal point, an
exponent, or a ``
'' character in the place of a digit,
otherwise it is exact.
In systems with inexact numbers
of varying precisions it may be useful to specify
the precision of a constant. For this purpose, numerical constants
may be written with an exponent marker that indicates the
desired precision of the inexact
representation. The letters
, and
specify the use of
short
single
double
, and
long
precision, respectively. (When fewer
than four internal
inexact
representations exist, the four size
specifications are mapped onto those available. For example, an
implementation with two internal representations may map short and
single together and long and double together.) In addition, the
exponent marker
specifies the default precision for the
implementation. The default precision has at least as much precision
as
double
, but
implementations may wish to allow this default to be set by the user.
3.14159265358979F0
Round to single --- 3.141593
0.6L0
Extend to long --- .600000000000000
6.2.5 Numerical operations
The reader is referred to section
1.3.3
for a summary
of the naming conventions used to specify restrictions on the types of
arguments to numerical routines.
The examples used in this section assume that any numerical constant written
using an exact notation is indeed represented as an exact
number. Some examples also assume that certain numerical constants written
using an inexact notation can be represented without loss of
accuracy; the inexact constants were chosen so that this is
likely to be true in implementations that use flonums to represent
inexact numbers.
procedure:
number?
obj
procedure:
complex?
obj
procedure:
real?
obj
procedure:
rational?
obj
procedure:
integer?
obj
These numerical type predicates can be applied to any kind of
argument, including non-numbers. They return
#t
if the object is
of the named type, and otherwise they return
#f
In general, if a type predicate is true of a number then all higher
type predicates are also true of that number. Consequently, if a type
predicate is false of a number, then all lower type predicates are
also false of that number.
If
is an inexact complex number, then
(real?
is true if
and only if
(zero? (imag-part
))
is true. If
is an inexact
real number, then
(integer?
is true if and only if
(=
(round
))
(complex? 3+4i) ===>
#t
(complex? 3) ===>
#t
(real? 3) ===>
#t
(real? -2.5+0.0i) ===>
#t
(real? #e1e10) ===>
#t
(rational? 6/10) ===>
#t
(rational? 6/3) ===>
#t
(integer? 3+0i) ===>
#t
(integer? 3.0) ===>
#t
(integer? 8/4) ===>
#t
Note:
The behavior of these type predicates on inexact numbers
is unreliable, since any inaccuracy may affect the result.
Note:
In many implementations the
rational?
procedure will be the same
as
real?
, and the
complex?
procedure will be the same as
number?
, but unusual implementations may be able to represent
some irrational numbers exactly or may extend the number system to
support some kind of non-complex numbers.
procedure:
exact?
procedure:
inexact?
These numerical predicates provide tests for the exactness of a
quantity. For any Scheme number, precisely one of these predicates
is true.
procedure:
...
procedure:
...
procedure:
...
procedure:
<=
...
procedure:
>=
...
These procedures return
#t
if their arguments are (respectively):
equal, monotonically increasing, monotonically decreasing,
monotonically nondecreasing, or monotonically nonincreasing.
These predicates are required to be transitive.
Note:
The traditional implementations of these predicates in Lisp-like
languages are not transitive.
Note:
While it is not an error to compare inexact numbers using these
predicates, the results may be unreliable because a small inaccuracy
may affect the result; this is especially true of
and
zero?
When in doubt, consult a numerical analyst.
library procedure:
zero?
library procedure:
positive?
library procedure:
negative?
library procedure:
odd?
library procedure:
even?
These numerical predicates test a number for a particular property,
returning
#t
or
#f
. See note above.
library procedure:
max
...
library procedure:
min
...
These procedures return the maximum or minimum of their arguments.
(max 3 4) ===> 4 ; exact
(max 3.9 4) ===> 4.0 ; inexact
Note:
If any argument is inexact, then the result will also be inexact (unless
the procedure can prove that the inaccuracy is not large enough to affect the
result, which is possible only in unusual implementations). If
min
or
max
is used to compare numbers of mixed exactness, and the numerical
value of the result cannot be represented as an inexact number without loss of
accuracy, then the procedure may report a violation of an implementation
restriction.
procedure:
...
procedure:
...
These procedures return the sum or product of their arguments.
(+ 3 4) ===> 7
(+ 3) ===> 3
(+) ===> 0
(* 4) ===> 4
(*) ===> 1
procedure:
procedure:
(-
optional procedure:
(-
...
procedure:
procedure:
(/
optional procedure:
(/
...
With two or more arguments, these procedures return the difference or
quotient of their arguments, associating to the left. With one argument,
however, they return the additive or multiplicative inverse of their argument.
(- 3 4) ===> -1
(- 3 4 5) ===> -6
(- 3) ===> -3
(/ 3 4 5) ===> 3/20
(/ 3) ===> 1/3
library procedure:
abs
Abs
returns the absolute value of its argument.
(abs -7) ===> 7
procedure:
quotient
procedure:
remainder
procedure:
modulo
These procedures implement number-theoretic (integer)
division.
should be non-zero. All three procedures
return integers. If
is an integer:
(quotient
) ===>
(remainder
) ===> 0
(modulo
) ===> 0
If
is not an integer:
(quotient
) ===>
(remainder
) ===>
(modulo
) ===>
where
is
rounded towards zero,
0 < |
| < |
|, 0 < |
| < |
|,
and
differ from
by a multiple of
has the same sign as
, and
has the same sign as
From this we can conclude that for integers
and
with
not equal to 0,
(=
(+ (*
(quotient
))
(remainder
)))
===>
#t
provided all numbers 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 ; inexact
library procedure:
gcd
...
library procedure:
lcm
...
These procedures return the greatest common divisor or least common
multiple of their arguments. The result is always non-negative.
(gcd 32 -36) ===> 4
(gcd) ===> 0
(lcm 32 -36) ===> 288
(lcm 32.0 -36) ===> 288.0 ; inexact
(lcm) ===> 1
procedure:
numerator
procedure:
denominator
These procedures return the numerator or denominator of their
argument; the result is computed as if the argument was represented as
a fraction in lowest terms. The denominator is always positive. The
denominator of 0 is defined to be 1.
(numerator (/ 6 4)) ===> 3
(denominator (/ 6 4)) ===> 2
(denominator
(exact->inexact (/ 6 4))) ===> 2.0
procedure:
floor
procedure:
ceiling
procedure:
truncate
procedure:
round
These procedures return integers.
Floor
returns the largest integer not larger than
Ceiling
returns the smallest integer not smaller than
Truncate
returns the integer closest to
whose absolute
value is not larger than the absolute value of
Round
returns the
closest integer to
, rounding to even when
is halfway between two
integers.
Rationale:
Round
rounds to even for consistency with the default rounding
mode specified by the IEEE floating point standard.
Note:
If the argument to one of these procedures is inexact, then the result
will also be inexact. If an exact value is needed, the
result should be passed to the
inexact->exact
procedure.
(floor -4.3) ===> -5.0
(ceiling -4.3) ===> -4.0
(truncate -4.3) ===> -4.0
(round -4.3) ===> -4.0
(floor 3.5) ===> 3.0
(ceiling 3.5) ===> 4.0
(truncate 3.5) ===> 3.0
(round 3.5) ===> 4.0 ; inexact
(round 7/2) ===> 4 ; exact
(round 7) ===> 7
library procedure:
rationalize
x y
Rationalize
returns the
simplest
rational number
differing from
by no more than
. A rational number
is
simpler
than another rational number
if
and
(in lowest terms) and |
| and |
|. Thus 3/5 is simpler than 4/7.
Although not all rationals are comparable in this ordering (consider 2/7
and 3/5) any interval contains a rational number that is simpler than
every other rational number in that interval (the simpler 2/5 lies
between 2/7 and 3/5). Note that 0 = 0/1 is the simplest rational of
all.
(rationalize
(inexact->exact .3) 1/10) ===> 1/3 ; exact
(rationalize .3 1/10) ===> #i1/3 ; inexact
procedure:
exp
procedure:
log
procedure:
sin
procedure:
cos
procedure:
tan
procedure:
asin
procedure:
acos
procedure:
atan
procedure:
(atan
These procedures are part of every implementation that supports
general
real numbers; they compute the usual transcendental functions.
Log
computes the natural logarithm of
(not the base ten logarithm).
Asin
acos
, and
atan
compute arcsine (
sin
-1
),
arccosine (
cos
-1
), and arctangent (
tan
-1
), respectively.
The two-argument variant of
atan
computes
(angle
(make-rectangular
))
(see below), even in implementations
that don't support general complex numbers.
In general, the mathematical functions log, arcsine, arccosine, and
arctangent are multiply defined.
The value of
log
is defined to be the one whose imaginary
part lies in the range from -
(exclusive) to
(inclusive).
log
0 is undefined.
With
log
defined this way, the values of
sin
-1
cos
-1
and
tan
-1
are according to the following formulæ:
sin
-1
= -
log
+ (1 -
1/2
cos
-1
/ 2 -
sin
-1
tan
-1
= (
log
(1 +
) -
log
(1 -
)) / (2
The above specification follows [
27
], which in turn
cites [
19
]; refer to these sources for more detailed
discussion of branch cuts, boundary conditions, and implementation of
these functions. When it is possible these procedures produce a real
result from a real argument.
procedure:
sqrt
Returns the principal square root of
. The result will have
either positive real part, or zero real part and non-negative imaginary
part.
procedure:
expt
Returns
raised to the power
. For
log
is 1 if
= 0 and 0 otherwise.
procedure:
make-rectangular
procedure:
make-polar
procedure:
real-part
procedure:
imag-part
procedure:
magnitude
procedure:
angle
These procedures are part of every implementation that supports
general
complex numbers. Suppose
, and
are
real numbers and
is a complex number such that
Then
(make-rectangular
) ===>
(make-polar
) ===>
(real-part
) ===>
(imag-part
) ===>
(magnitude
) ===> |
(angle
) ===>
where -
with
+ 2
for some integer
Rationale:
Magnitude
is the same as
abs
for a real argument,
but
abs
must be present in all implementations, whereas
magnitude
need only be present in implementations that support
general complex numbers.
procedure:
exact->inexact
procedure:
inexact->exact
Exact->inexact
returns an inexact representation of
The value returned is the
inexact number that is numerically closest to the argument.
If an exact argument has no reasonably close inexact equivalent,
then a violation of an implementation restriction may be reported.
Inexact->exact
returns an exact representation of
. The value returned is the exact number that is numerically
closest to the argument.
If an inexact argument has no reasonably close exact equivalent,
then a violation of an implementation restriction may be reported.
These procedures implement the natural one-to-one correspondence between
exact and inexact integers throughout an
implementation-dependent range. See section
6.2.3
6.2.6 Numerical input and output
procedure:
number->string
procedure:
(number->string
z radix
must be an exact integer, either 2, 8, 10, or 16. If omitted,
defaults to 10.
The procedure
number->string
takes a
number and a radix and returns as a string an external representation of
the given number in the given radix such that
(let ((number
(radix
))
(eqv? number
(string->number (number->string number
radix)
radix)))
is true. It is an error if no possible result makes this expression true.
If
is inexact, the radix is 10, and the above expression
can be satisfied by a result that contains a decimal point,
then the result contains a decimal point and is expressed using the
minimum number of digits (exclusive of exponent and trailing
zeroes) needed to make the above expression
true [
];
otherwise the format of the result is unspecified.
The result returned by
number->string
never contains an explicit radix prefix.
Note:
The error case can occur only when
is not a complex number
or is a complex number with a non-rational real or imaginary part.
Rationale:
If
is an inexact number represented using flonums, and
the radix is 10, then the above expression is normally satisfied by
a result containing a decimal point. The unspecified case
allows for infinities, NaNs, and non-flonum representations.
procedure:
string->number
string
procedure:
(string->number
string radix
Returns a number of the maximally precise representation expressed by the
given
must be an exact integer, either 2, 8, 10,
or 16. If supplied,
is a default radix that may be overridden
by an explicit radix prefix in
(e.g.
"#o177"
). If
is not supplied, then the default radix is 10. If
is not
a syntactically valid notation for a number, then
string->number
returns
#f
(string->number "100") ===> 100
(string->number "100" 16) ===> 256
(string->number "1e2") ===> 100.0
(string->number "15##") ===> 1500.0
Note:
The domain of
string->number
may be restricted by implementations
in the following ways.
String->number
is permitted to return
#f
whenever
contains an explicit radix prefix.
If all numbers supported by an implementation are real, then
string->number
is permitted to return
#f
whenever
uses the polar or rectangular notations for complex
numbers. If all numbers are integers, then
string->number
may return
#f
whenever
the fractional notation is used. If all numbers are exact, then
string->number
may return
#f
whenever
an exponent marker or explicit exactness prefix is used, or if
appears in place of a digit. If all inexact
numbers are integers, then
string->number
may return
#f
whenever
a decimal point is used.
6.3 Other data types
This section describes operations on some of Scheme's non-numeric data types:
booleans, pairs, lists, symbols, characters, strings and vectors.
6.3.1 Booleans
The standard boolean objects for true and false are written as
#t
and
#f
What really
matters, though, are the objects that the Scheme conditional expressions
if
cond
and
or
do
) treat as
true
or false
. The phrase ``a true value''
(or sometimes just ``true'') means any object treated as true by the
conditional expressions, and the phrase ``a false value''
(or
``false'') means any object treated as false by the conditional expressions.
Of all the standard Scheme values, only
#f
counts as false in conditional expressions.
Except for
#f
all standard Scheme values, including
#t
pairs, the empty list, symbols, numbers, strings, vectors, and procedures,
count as true.
Note:
Programmers accustomed to other dialects of Lisp should be aware that
Scheme distinguishes both
#f
and the empty list
from the symbol
nil
Boolean constants evaluate to themselves, so they do not need to be quoted
in programs.
#t
===>
#t
#f
===>
#f
#f
===>
#f
library procedure:
not
obj
Not
returns
#t
if
obj
is false, and returns
#f
otherwise.
(not
#t
) ===>
#f
(not 3) ===>
#f
(not (list 3)) ===>
#f
(not
#f
) ===>
#t
(not '()) ===>
#f
(not (list)) ===>
#f
(not 'nil) ===>
#f
library procedure:
boolean?
obj
Boolean?
returns
#t
if
obj
is either
#t
or
#f
and returns
#f
otherwise.
(boolean?
#f
) ===>
#t
(boolean? 0) ===>
#f
(boolean? '()) ===>
#f
6.3.2 Pairs and lists
pair
(sometimes called a
dotted pair
) is a
record structure with two fields called the car and cdr fields (for
historical reasons). Pairs are created by the procedure
cons
The car and cdr fields are accessed by the procedures
car
and
cdr
. The car and cdr fields are assigned by the procedures
set-car!
and
set-cdr!
Pairs are used primarily to represent lists. A list can
be defined recursively as either the empty list
or a pair whose
cdr is a list. More precisely, the set of lists is defined as the smallest
set
such that
The empty list is in
If
list
is in
, then any pair whose cdr field contains
list
is also in
The objects in the car fields of successive pairs of a list are the
elements of the list. For example, a two-element list is a pair whose car
is the first element and whose cdr is a pair whose car is the second element
and whose cdr is the empty list. The length of a list is the number of
elements, which is the same as the number of pairs.
The empty list
is a special object of its own type
(it is not a pair); it has no elements and its length is zero.
Note:
The above definitions imply that all lists have finite length and are
terminated by the empty list.
The most general notation (external representation) for Scheme pairs is
the ``dotted'' notation
where
is the value of the car field and
is the value of the
cdr field. For example
(4 . 5)
is a pair whose car is 4 and whose
cdr is 5. Note that
(4 . 5)
is the external representation of a
pair, not an expression that evaluates to a pair.
A more streamlined notation can be used for lists: the elements of the
list are simply enclosed in parentheses and separated by spaces. The
empty list
is written
()
. For example,
(a b c d e)
and
(a . (b . (c . (d . (e . ())))))
are equivalent notations for a list of symbols.
A chain of pairs not ending in the empty list is called an
improper list
. Note that an improper list is not a list.
The list and dotted notations can be combined to represent
improper lists:
(a b c . d)
is equivalent to
(a . (b . (c . d)))
Whether a given pair is a list depends upon what is stored in the cdr
field. When the
set-cdr!
procedure is used, an object can be a
list one moment and not the next:
(define x (list 'a 'b 'c))
(define y x)
y ===> (a b c)
(list? y) ===>
#t
(set-cdr! x 4) ===>
unspecified
x ===> (a . 4)
(eqv? x y) ===>
#t
y ===> (a . 4)
(list? y) ===>
#f
(set-cdr! x x) ===>
unspecified
(list? x) ===>
#f
Within literal expressions and representations of objects read by the
read
procedure, the forms
, and
,@
the symbols
quote
quasiquote
unquote
, and
unquote-splicing
, respectively. The second element in each case
is
programs may be represented as lists. That is, according to Scheme's grammar, every
7.1.2
).
Among other things, this permits the use of the
read
procedure to
parse Scheme programs. See section
3.3
procedure:
pair?
obj
Pair?
returns
#t
if
obj
is a pair, and otherwise
returns
#f
(pair? '(a . b)) ===>
#t
(pair? '(a b c)) ===>
#t
(pair? '()) ===>
#f
(pair? '#(a b)) ===>
#f
procedure:
cons
obj
obj
Returns a newly allocated pair whose car is
obj
and whose cdr is
obj
. The pair is guaranteed to be different (in the sense of
eqv?
) from every existing object.
(cons 'a '()) ===> (a)
(cons '(a) '(b c d)) ===> ((a) b c d)
(cons "a" '(b c)) ===> ("a" b c)
(cons 'a 3) ===> (a . 3)
(cons '(a b) 'c) ===> ((a b) . c)
procedure:
car
pair
Returns the contents of the car field of
pair
. Note that it is an
error to take the car of the empty list
(car '(a b c)) ===> a
(car '((a) b c d)) ===> (a)
(car '(1 . 2)) ===> 1
(car '()) ===>
error
procedure:
cdr
pair
Returns the contents of the cdr field of
pair
Note that it is an error to take the cdr of the empty list.
(cdr '((a) b c d)) ===> (b c d)
(cdr '(1 . 2)) ===> 2
(cdr '()) ===>
error
procedure:
set-car!
pair obj
Stores
obj
in the car field of
pair
The value returned by
set-car!
is unspecified.
(define (f) (list 'not-a-constant-list))
(define (g) '(constant-list))
(set-car! (f) 3) ===>
unspecified
(set-car! (g) 3) ===>
error
procedure:
set-cdr!
pair obj
Stores
obj
in the cdr field of
pair
The value returned by
set-cdr!
is unspecified.
library procedure:
caar
pair
library procedure:
cadr
pair
library procedure:
cdddar
pair
library procedure:
cddddr
pair
These procedures are compositions of
car
and
cdr
, where
for example
caddr
could be defined by
(define caddr (lambda (x) (car (cdr (cdr x))))).
Arbitrary compositions, up to four deep, are provided. There are
twenty-eight of these procedures in all.
library procedure:
null?
obj
Returns
#t
if
obj
is the empty list
otherwise returns
#f
library procedure:
list?
obj
Returns
#t
if
obj
is a list, otherwise returns
#f
By definition, all lists have finite length and are terminated by
the empty list.
(list? '(a b c)) ===>
#t
(list? '()) ===>
#t
(list? '(a . b)) ===>
#f
(let ((x (list 'a)))
(set-cdr! x x)
(list? x)) ===>
#f
library procedure:
list
obj
...
Returns a newly allocated list of its arguments.
(list 'a (+ 3 4) 'c) ===> (a 7 c)
(list) ===> ()
library procedure:
length
list
Returns the length of
list
(length '(a b c)) ===> 3
(length '(a (b) (c d e))) ===> 3
(length '()) ===> 0
library procedure:
append
list
...
Returns a list consisting of the elements of the first
list
followed by the elements of the other
list
s.
(append '(x) '(y)) ===> (x y)
(append '(a) '(b c d)) ===> (a b c d)
(append '(a (b)) '((c))) ===> (a (b) (c))
The resulting list is always newly allocated, except that it shares
structure with the last
list
argument. The last argument may
actually be any object; an improper list results if the last argument is not a
proper list.
(append '(a b) '(c . d)) ===> (a b c . d)
(append '() 'a) ===> a
library procedure:
reverse
list
Returns a newly allocated list consisting of the elements of
list
in reverse order.
(reverse '(a b c)) ===> (c b a)
(reverse '(a (b c) d (e (f))))
===> ((e (f)) d (b c) a)
library procedure:
list-tail
list
Returns the sublist of
list
obtained by omitting the first
elements. It is an error if
list
has fewer than
elements.
List-tail
could be defined by
(define list-tail
(lambda (x k)
(if (zero? k)
(list-tail (cdr x) (- k 1)))))
library procedure:
list-ref
list
Returns the
th element of
list
. (This is the same
as the car of
(list-tail
list
.)
It is an error if
list
has fewer than
elements.
(list-ref '(a b c d) 2) ===> c
(list-ref '(a b c d)
(inexact->exact (round 1.8)))
===> c
library procedure:
memq
obj list
library procedure:
memv
obj list
library procedure:
member
obj list
These procedures return the first sublist of
list
whose car is
obj
, where the sublists of
list
are the non-empty lists
returned by
(list-tail
list
for
less
than the length of
list
. If
obj
does not occur in
list
, then
#f
(not the empty list) is
returned.
Memq
uses
eq?
to compare
obj
with the elements of
list
, while
memv
uses
eqv?
and
member
uses
equal?
(memq 'a '(a b c)) ===> (a b c)
(memq 'b '(a b c)) ===> (b c)
(memq 'a '(b c d)) ===>
#f
(memq (list 'a) '(b (a) c)) ===>
#f
(member (list 'a)
'(b (a) c)) ===> ((a) c)
(memq 101 '(100 101 102)) ===>
unspecified
(memv 101 '(100 101 102)) ===> (101 102)
library procedure:
assq
obj alist
library procedure:
assv
obj alist
library procedure:
assoc
obj alist
Alist
(for ``association list'') must be a list of
pairs. These procedures find the first pair in
alist
whose car field is
obj
and returns that pair. If no pair in
alist
has
obj
as its
car, then
#f
(not the empty list) is returned.
Assq
uses
eq?
to compare
obj
with the car fields of the pairs in
alist
while
assv
uses
eqv?
and
assoc
uses
equal?
(define e '((a 1) (b 2) (c 3)))
(assq 'a e) ===> (a 1)
(assq 'b e) ===> (b 2)
(assq 'd e) ===>
#f
(assq (list 'a) '(((a)) ((b)) ((c))))
===>
#f
(assoc (list 'a) '(((a)) ((b)) ((c))))
===> ((a))
(assq 5 '((2 3) (5 7) (11 13)))
===>
unspecified
(assv 5 '((2 3) (5 7) (11 13)))
===> (5 7)
Rationale:
Although they are ordinarily used as predicates,
memq
memv
member
assq
assv
, and
assoc
do not
have question marks in their names because they return useful values rather
than just
#t
or
#f
6.3.3 Symbols
Symbols are objects whose usefulness rests on the fact that two
symbols are identical (in the sense of
eqv?
) if and only if their
names are spelled the same way. This is exactly the property needed to
represent identifiers
in programs, and so most
implementations of Scheme use them internally for that purpose. Symbols
are useful for many other applications; for instance, they may be used
the way enumerated values are used in Pascal.
The rules for writing a symbol are exactly the same as the rules for
writing an identifier; see sections
2.1
and
7.1.1
It is guaranteed that any symbol that has been returned as part of
a literal expression, or read using the
read
procedure, and
subsequently written out using the
write
procedure, will read back
in as the identical symbol (in the sense of
eqv?
). The
string->symbol
procedure, however, can create symbols for
which this write/read invariance may not hold because their names
contain special characters or letters in the non-standard case.
Note:
Some implementations of Scheme have a feature known as ``slashification''
in order to guarantee write/read invariance for all symbols, but
historically the most important use of this feature has been to
compensate for the lack of a string data type.
Some implementations also have ``uninterned symbols'', which
defeat write/read invariance even in implementations with slashification,
and also generate exceptions to the rule that two symbols are the same
if and only if their names are spelled the same.
procedure:
symbol?
obj
Returns
#t
if
obj
is a symbol, otherwise returns
#f
(symbol? 'foo) ===>
#t
(symbol? (car '(a b))) ===>
#t
(symbol? "bar") ===>
#f
(symbol? 'nil) ===>
#t
(symbol? '()) ===>
#f
(symbol?
#f
) ===>
#f
procedure:
symbol->string
symbol
Returns the name of
symbol
as a string. If the symbol was part of
an object returned as the value of a literal expression
(section
4.1.2
) or by a call to the
read
procedure,
and its name contains alphabetic characters, then the string returned
will contain characters in the implementation's preferred standard
case -- some implementations will prefer upper case, others lower case.
If the symbol was returned by
string->symbol
, the case of
characters in the string returned will be the same as the case in the
string that was passed to
string->symbol
. It is an error
to apply mutation procedures like
string-set!
to strings returned
by this procedure.
The following examples assume that the implementation's standard case is
lower case:
(symbol->string 'flying-fish)
===> "flying-fish"
(symbol->string 'Martin) ===> "martin"
(symbol->string
(string->symbol "Malvina"))
===> "Malvina"
procedure:
string->symbol
string
Returns the symbol whose name is
string
. This procedure can
create symbols with names containing special characters or letters in
the non-standard case, but it is usually a bad idea to create such
symbols because in some implementations of Scheme they cannot be read as
themselves. See
symbol->string
The following examples assume that the implementation's standard case is
lower case:
(eq? 'mISSISSIppi 'mississippi)
===>
#t
(string->symbol "mISSISSIppi")
===> the symbol with name "mISSISSIppi"
(eq? 'bitBlt (string->symbol "bitBlt"))
===>
#f
(eq? 'JollyWog
(string->symbol
(symbol->string 'JollyWog)))
===>
#t
(string=? "K. Harper, M.D."
(symbol->string
(string->symbol "K. Harper, M.D.")))
===>
#t
6.3.4 Characters
Characters are objects that represent printed characters such as
letters and digits.
Characters are written using the notation
or
For example:
; lower case letter
; upper case letter
; left parenthesis
; the space character
space
; the preferred way to write a space
newline
; the newline character
Case is significant in
following
space or parenthesis. This rule resolves the ambiguous case where, for
example, the sequence of characters ``
space
''
could be taken to be either a representation of the space character or a
representation of the character ``
'' followed
by a representation of the symbol ``
pace
.''
Characters written in the
notation are self-evaluating.
That is, they do not have to be quoted in programs.
Some of the procedures that operate on characters ignore the
difference between upper case and lower case. The procedures that
ignore case have ``
-ci
'' (for ``case
insensitive'') embedded in their names.
procedure:
char?
obj
Returns
#t
if
obj
is a character, otherwise returns
#f
procedure:
char=?
char
char
procedure:
charchar
char
procedure:
char>?
char
char
procedure:
char<=?
char
char
procedure:
char>=?
char
char
These procedures impose a total ordering on the set of characters. It
is guaranteed that under this ordering:
The upper case characters are in order. For example,
(charA #
B)
returns
#t
The lower case characters are in order. For example,
(chara #
b)
returns
#t
The digits are in order. For example,
(char0 #
9)
returns
#t
Either all the digits precede all the upper case letters, or vice versa.
Either all the digits precede all the lower case letters, or vice versa.
Some implementations may generalize these procedures to take more than
two arguments, as with the corresponding numerical predicates.
library procedure:
char-ci=?
char
char
library procedure:
char-cichar
char
library procedure:
char-ci>?
char
char
library procedure:
char-ci<=?
char
char
library procedure:
char-ci>=?
char
char
These procedures are similar to
char=?
et cetera, but they treat
upper case and lower case letters as the same. For example,
(char-ci=? #
A #
a)
returns
#t
. Some
implementations may generalize these procedures to take more than two
arguments, as with the corresponding numerical predicates.
library procedure:
char-alphabetic?
char
library procedure:
char-numeric?
char
library procedure:
char-whitespace?
char
library procedure:
char-upper-case?
letter
library procedure:
char-lower-case?
letter
These procedures return
#t
if their arguments are alphabetic,
numeric, whitespace, upper case, or lower case characters, respectively,
otherwise they return
#f
. The following remarks, which are specific to
the ASCII character set, are intended only as a guide: The alphabetic characters
are the 52 upper and lower case letters. The numeric characters are the
ten decimal digits. The whitespace characters are space, tab, line
feed, form feed, and carriage return.
procedure:
char->integer
char
procedure:
integer->char
Given a character,
char->integer
returns an exact integer
representation of the character. Given an exact integer that is the image of
a character under
char->integer
integer->char
returns that character. These procedures implement order-preserving isomorphisms
between the set of characters under the
char<=?
ordering and some
subset of the integers under the
<=
ordering. That is, if
(char<=?
) ===>
#t
and (<=
) ===>
#t
and
and
are in the domain of
integer->char
, then
(<= (char->integer
(char->integer
)) ===>
#t
(char<=? (integer->char
(integer->char
)) ===>
#t
library procedure:
char-upcase
char
library procedure:
char-downcase
char
These procedures return a character
char
such that
(char-ci=?
char
char
. In addition, if
char
is
alphabetic, then the result of
char-upcase
is upper case and the
result of
char-downcase
is lower case.
6.3.5 Strings
Strings are sequences of characters.
Strings are written as sequences of characters enclosed within doublequotes
). A doublequote can be written inside a string only by escaping
it with a backslash (
), as in
"The word
"recursion
" has many meanings."
A backslash can be written inside a string only by escaping it with another
backslash. Scheme does not specify the effect of a backslash within a
string that is not followed by a doublequote or backslash.
A string constant may continue from one line to the next, but
the exact contents of such a string are unspecified.
The
length
of a string is the number of characters that it
contains. This number is an exact, non-negative integer that is fixed when the
string is created. The
valid indexes
of a string are the
exact non-negative integers less than the length of the string. The first
character of a string has index 0, the second has index 1, and so on.
In phrases such as ``the characters of
string
beginning with
index
start
and ending with index
end
,'' it is understood
that the index
start
is inclusive and the index
end
is
exclusive. Thus if
start
and
end
are the same index, a null
substring is referred to, and if
start
is zero and
end
is
the length of
string
, then the entire string is referred to.
Some of the procedures that operate on strings ignore the
difference between upper and lower case. The versions that ignore case
have ``
-ci
'' (for ``case insensitive'') embedded in their
names.
procedure:
string?
obj
Returns
#t
if
obj
is a string, otherwise returns
#f
procedure:
make-string
procedure:
(make-string
char
Make-string
returns a newly allocated string of
length
. If
char
is given, then all elements of the string
are initialized to
char
, otherwise the contents of the
string
are unspecified.
library procedure:
string
char
...
Returns a newly allocated string composed of the arguments.
procedure:
string-length
string
Returns the number of characters in the given
string
procedure:
string-ref
string
must be a valid index of
string
String-ref
returns character
of
string
using zero-origin indexing.
procedure:
string-set!
string k char
must be a valid index of
string
String-set!
stores
char
in element
of
string
and returns an unspecified value.
(define (f) (make-string 3
*))
(define (g) "***")
(string-set! (f) 0
?) ===>
unspecified
(string-set! (g) 0
?) ===>
error
(string-set! (symbol->string 'immutable)
?) ===>
error
library procedure:
string=?
string
string
library procedure:
string-ci=?
string
string
Returns
#t
if the two strings are the same length and contain the same
characters in the same positions, otherwise returns
#f
String-ci=?
treats
upper and lower case letters as though they were the same character, but
string=?
treats upper and lower case as distinct characters.
library procedure:
stringstring
string
library procedure:
string>?
string
string
library procedure:
string<=?
string
string
library procedure:
string>=?
string
string
library procedure:
string-cistring
string
library procedure:
string-ci>?
string
string
library procedure:
string-ci<=?
string
string
library procedure:
string-ci>=?
string
string
These procedures are the lexicographic extensions to strings of the
corresponding orderings on characters. For example,
stringis
the lexicographic ordering on strings induced by the ordering
charon characters. If two strings differ in length but
are the same up to the length of the shorter string, the shorter string
is considered to be lexicographically less than the longer string.
Implementations may generalize these and the
string=?
and
string-ci=?
procedures to take more than two arguments, as with
the corresponding numerical predicates.
library procedure:
substring
string start end
String
must be a string, and
start
and
end
must be exact integers satisfying
start
end
(string-length
string
).
Substring
returns a newly allocated string formed from the characters of
string
beginning with index
start
(inclusive) and ending with index
end
(exclusive).
library procedure:
string-append
string
...
Returns a newly allocated string whose characters form the concatenation of the
given strings.
library procedure:
string->list
string
library procedure:
list->string
list
String->list
returns a newly allocated list of the
characters that make up the given string.
List->string
returns a newly allocated string formed from the characters in the list
list
, which must be a list of characters.
String->list
and
list->string
are
inverses so far as
equal?
is concerned.
library procedure:
string-copy
string
Returns a newly allocated copy of the given
string
library procedure:
string-fill!
string char
Stores
char
in every element of the given
string
and returns an
unspecified value.
6.3.6 Vectors
Vectors are heterogenous structures whose elements are indexed
by integers. A vector typically occupies less space than a list
of the same length, and the average time required to access a randomly
chosen element is typically less for the vector than for the list.
The
length
of a vector is the number of elements that it
contains. This number is a non-negative integer that is fixed when the
vector is created. The
valid indexes
of a
vector are the exact non-negative integers less than the length of the
vector. The first element in a vector is indexed by zero, and the last
element is indexed by one less than the length of the vector.
Vectors are written using the notation
#(
obj
...
For example, a vector of length 3 containing the number zero in element
0, the list
(2 2 2 2)
in element 1, and the string
"Anna"
in
element 2 can be written as following:
#(0 (2 2 2 2) "Anna")
Note that this is the external representation of a vector, not an
expression evaluating to a vector. Like list constants, vector
constants must be quoted:
'#(0 (2 2 2 2) "Anna")
===> #(0 (2 2 2 2) "Anna")
procedure:
vector?
obj
Returns
#t
if
obj
is a vector, otherwise returns
#f
procedure:
make-vector
procedure:
(make-vector
k fill
Returns a newly allocated vector of
elements. If a second
argument is given, then each element is initialized to
fill
Otherwise the initial contents of each element is unspecified.
library procedure:
vector
obj
...
Returns a newly allocated vector whose elements contain the given
arguments. Analogous to
list
(vector 'a 'b 'c) ===> #(a b c)
procedure:
vector-length
vector
Returns the number of elements in
vector
as an exact integer.
procedure:
vector-ref
vector k
must be a valid index of
vector
Vector-ref
returns the contents of element
of
vector
(vector-ref '#(1 1 2 3 5 8 13 21)
5)
===> 8
(vector-ref '#(1 1 2 3 5 8 13 21)
(let ((i (round (* 2 (acos -1)))))
(if (inexact? i)
(inexact->exact i)
i)))
===> 13
procedure:
vector-set!
vector k obj
must be a valid index of
vector
Vector-set!
stores
obj
in element
of
vector
The value returned by
vector-set!
is unspecified.
(let ((vec (vector 0 '(2 2 2 2) "Anna")))
(vector-set! vec 1 '("Sue" "Sue"))
vec)
===> #(0 ("Sue" "Sue") "Anna")
(vector-set! '#(0 1 2) 1 "doe")
===>
error
; constant vector
library procedure:
vector->list
vector
library procedure:
list->vector
list
Vector->list
returns a newly allocated list of the objects contained
in the elements of
vector
List->vector
returns a newly
created vector initialized to the elements of the list
list
(vector->list '#(dah dah didah))
===> (dah dah didah)
(list->vector '(dididit dah))
===> #(dididit dah)
library procedure:
vector-fill!
vector fill
Stores
fill
in every element of
vector
The value returned by
vector-fill!
is unspecified.
6.4 Control features
This chapter describes various primitive procedures which control the
flow of program execution in special ways.
The
procedure?
predicate is also described here.
procedure:
procedure?
obj
Returns
#t
if
obj
is a procedure, otherwise returns
#f
(procedure? car) ===>
#t
(procedure? 'car) ===>
#f
(procedure? (lambda (x) (* x x)))
===>
#t
(procedure? '(lambda (x) (* x x)))
===>
#f
(call-with-current-continuation procedure?)
===>
#t
procedure:
apply
proc
arg
...
args
Proc
must be a procedure and
args
must be a list.
Calls
proc
with the elements of the list
(append (list
arg
...
args
as the actual
arguments.
(apply + (list 3 4)) ===> 7
(define compose
(lambda (f g)
(lambda args
(f (apply g args)))))
((compose sqrt *) 12 75) ===> 30
library procedure:
map
proc
list
list
...
The
list
s must be lists, and
proc
must be a
procedure taking as many arguments as there are
list
and returning a single value. If more
than one
list
is given, then they must all be the same length.
Map
applies
proc
element-wise to the elements of the
list
s and returns a list of the results, in order.
The dynamic order in which
proc
is applied to the elements of the
list
s is unspecified.
(map cadr '((a b) (d e) (g h)))
===> (b e h)
(map (lambda (n) (expt n n))
'(1 2 3 4 5))
===> (1 4 27 256 3125)
(map + '(1 2 3) '(4 5 6)) ===> (5 7 9)
(let ((count 0))
(map (lambda (ignored)
(set! count (+ count 1))
count)
'(a b))) ===> (1 2)
or
(2 1)
library procedure:
for-each
proc
list
list
...
The arguments to
for-each
are like the arguments to
map
, but
for-each
calls
proc
for its side effects rather than for its
values. Unlike
map
for-each
is guaranteed to call
proc
on
the elements of the
list
s in order from the first element(s) to the
last, and the value returned by
for-each
is unspecified.
(let ((v (make-vector 5)))
(for-each (lambda (i)
(vector-set! v i (* i i)))
'(0 1 2 3 4))
v) ===> #(0 1 4 9 16)
library procedure:
force
promise
Forces the value of
promise
(see
delay
section
4.2.5
).
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
Force
and
delay
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)))
We define the expression
(delay
to have the same meaning as the procedure call
(make-promise (lambda ()
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))))))))
Rationale:
A promise may refer to its own value, as in the last example above.
Forcing such a promise may cause the promise to be forced a second time
before the value of the first force has been computed.
This complicates the definition of
make-promise
Various extensions to this semantics of
delay
and
force
are supported in some implementations:
Calling
force
on an object that is not a promise may simply
return the object.
It may be the case that there is no means by which a promise can be
operationally distinguished from its forced value. That is, expressions
like the following may evaluate to either
#t
or to
#f
depending on the implementation:
(eqv? (delay 1) 1) ===>
unspecified
(pair? (delay (cons 1 2))) ===>
unspecified
Some implementations may implement ``implicit forcing,'' where
the value of a promise is forced by primitive procedures like
cdr
and
+:
(+ (delay (* 3 7)) 13) ===> 34
procedure:
call-with-current-continuation
proc
Proc
must be a procedure of one
argument. The procedure
call-with-current-continuation
packages
up the current continuation (see the rationale below) as an ``escape
procedure''
and passes it as an argument to
proc
. The escape procedure is a Scheme procedure that, if it is
later called, will abandon whatever continuation is in effect at that later
time and will instead use the continuation that was in effect
when the escape procedure was created. Calling the escape procedure
may cause the invocation of
before
and
after
thunks installed using
dynamic-wind
The escape procedure accepts the same number of arguments as the continuation to
the original call to
call-with-current-continuation
Except for continuations created by the
call-with-values
procedure, all continuations take exactly one value. The
effect of passing no value or more than one value to continuations
that were not created by
call-with-values
is unspecified.
The escape procedure that is passed to
proc
has
unlimited extent just like any other procedure in Scheme. It may be stored
in variables or data structures and may be called as many times as desired.
The following examples show only the most common ways in which
call-with-current-continuation
is used. If all real uses were as
simple as these examples, there would be no need for a procedure with
the power of
call-with-current-continuation
(call-with-current-continuation
(lambda (exit)
(for-each (lambda (x)
(if (negative? x)
(exit x)))
'(54 0 37 -3 245 19))
#t
)) ===> -3
(define list-length
(lambda (obj)
(call-with-current-continuation
(lambda (return)
(letrec ((r
(lambda (obj)
(cond ((null? obj) 0)
((pair? obj)
(+ (r (cdr obj)) 1))
(else (return
#f
))))))
(r obj))))))
(list-length '(1 2 3 4)) ===> 4
(list-length '(a b . c)) ===>
#f
Rationale:
A common use of
call-with-current-continuation
is for
structured, non-local exits from loops or procedure bodies, but in fact
call-with-current-continuation
is extremely useful for implementing a
wide variety of advanced control structures.
Whenever a Scheme expression is evaluated there is a
continuation
wanting the result of the expression. The continuation
represents an entire (default) future for the computation. If the expression is
evaluated at top level, for example, then the continuation might take the
result, print it on the screen, prompt for the next input, evaluate it, and
so on forever. Most of the time the continuation includes actions
specified by user code, as in a continuation that will take the result,
multiply it by the value stored in a local variable, add seven, and give
the answer to the top level continuation to be printed. Normally these
ubiquitous continuations are hidden behind the scenes and programmers do not
think much about them. On rare occasions, however, a programmer may
need to deal with continuations explicitly.
Call-with-current-continuation
allows Scheme programmers to do
that by creating a procedure that acts just like the current
continuation.
Most programming languages incorporate one or more special-purpose
escape constructs with names like
exit
return
, or
even
goto
. In 1965, however, Peter Landin [
16
invented a general purpose escape operator called the J-operator. John
Reynolds [
24
] described a simpler but equally powerful
construct in 1972. The
catch
special form described by Sussman
and Steele in the 1975 report on Scheme is exactly the same as
Reynolds's construct, though its name came from a less general construct
in MacLisp. Several Scheme implementors noticed that the full power of the
catch
construct could be provided by a procedure instead of by a
special syntactic construct, and the name
call-with-current-continuation
was coined in 1982. This name is
descriptive, but opinions differ on the merits of such a long name, and
some people use the name
call/cc
instead.
procedure:
values
obj
...
Delivers all of its arguments to its continuation.
Except for continuations created by the
call-with-values
procedure, all continuations take exactly one value.
Values
might be defined as follows:
(define (values . things)
(call-with-current-continuation
(lambda (cont) (apply cont things))))
procedure:
call-with-values
producer consumer
Calls its
producer
argument with no values and
a continuation that, when passed some values, calls the
consumer
procedure with those values as arguments.
The continuation for the call to
consumer
is the
continuation of the call to
call-with-values
(call-with-values (lambda () (values 4 5))
(lambda (a b) b))
===> 5
(call-with-values * -) ===> -1
procedure:
dynamic-wind
before thunk after
Calls
thunk
without arguments, returning the result(s) of this call.
Before
and
after
are called, also without arguments, as required
by the following rules (note that in the absence of calls to continuations
captured using
call-with-current-continuation
the three arguments are
called once each, in order).
Before
is called whenever execution
enters the dynamic extent of the call to
thunk
and
after
is called
whenever it exits that dynamic extent. The dynamic extent of a procedure
call is the period between when the call is initiated and when it
returns. In Scheme, because of
call-with-current-continuation
, the
dynamic extent of a call may not be a single, connected time period.
It is defined as follows:
The dynamic extent is entered when execution of the body of the
called procedure begins.
The dynamic extent is also entered when execution is not within
the dynamic extent and a continuation is invoked that was captured
(using
call-with-current-continuation
) during the dynamic extent.
It is exited when the called procedure returns.
It is also exited when execution is within the dynamic extent and
a continuation is invoked that was captured while not within the
dynamic extent.
If a second call to
dynamic-wind
occurs within the dynamic extent of the
call to
thunk
and then a continuation is invoked in such a way that the
after
s from these two invocations of
dynamic-wind
are both to be
called, then the
after
associated with the second (inner) call to
dynamic-wind
is called first.
If a second call to
dynamic-wind
occurs within the dynamic extent of the
call to
thunk
and then a continuation is invoked in such a way that the
before
s from these two invocations of
dynamic-wind
are both to be
called, then the
before
associated with the first (outer) call to
dynamic-wind
is called first.
If invoking a continuation requires calling the
before
from one call
to
dynamic-wind
and the
after
from another, then the
after
is called first.
The effect of using a captured continuation to enter or exit the dynamic
extent of a call to
before
or
after
is undefined.
(let ((path '())
(c #f))
(let ((add (lambda (s)
(set! path (cons s path)))))
(dynamic-wind
(lambda () (add 'connect))
(lambda ()
(add (call-with-current-continuation
(lambda (c0)
(set! c c0)
'talk1))))
(lambda () (add 'disconnect)))
(if (< (length path) 4)
(c 'talk2)
(reverse path))))
===> (connect talk1 disconnect
connect talk2 disconnect)
6.5
Eval
procedure:
eval
expression environment-specifier
Evaluates
expression
in the specified environment and returns its value.
Expression
must be a valid Scheme expression represented as data,
and
environment-specifier
must be a value returned by one of the
three procedures described below.
Implementations may extend
eval
to allow non-expression programs
(definitions) as the first argument and to allow other
values as environments, with the restriction that
eval
is not
allowed to create new bindings in the environments associated with
null-environment
or
scheme-report-environment
(eval '(* 7 3) (scheme-report-environment 5))
===> 21
(let ((f (eval '(lambda (f x) (f x x))
(null-environment 5))))
(f + 10))
===> 20
procedure:
scheme-report-environment
version
procedure:
null-environment
version
Version
must be the exact integer
corresponding to this revision of the Scheme report (the
Revised
Report on Scheme).
Scheme-report-environment
returns a specifier for an
environment that is empty except for all bindings defined in
this report that are either required or both optional and
supported by the implementation.
Null-environment
returns
a specifier for an environment that is empty except for the
(syntactic) bindings for all syntactic keywords defined in
this report that are either required or both optional and
supported by the implementation.
Other values of
version
can be used to specify environments
matching past revisions of this report, but their support is not
required. An implementation will signal an error if
version
is neither
nor another value supported by
the implementation.
The effect of assigning (through the use of
eval
) a variable
bound in a
scheme-report-environment
(for example
car
) is unspecified. Thus the environments specified
by
scheme-report-environment
may be immutable.
optional procedure:
interaction-environment
This procedure returns a specifier for the environment that
contains implementation-defined bindings, typically a superset of
those listed in the report. The intent is that this procedure
will return the environment in which the implementation would evaluate
expressions dynamically typed by the user.
6.6 Input and output
6.6.1 Ports
Ports represent input and output devices. To Scheme, an input port is a
Scheme object that can deliver characters upon command, while an output port
is a Scheme object that can accept characters.
library procedure:
call-with-input-file
string proc
library procedure:
call-with-output-file
string proc
String
should be a string naming a file, and
proc
should be a procedure that accepts one argument.
For
call-with-input-file
the file should already exist; for
call-with-output-file
the effect is unspecified if the file
already exists. These procedures call
proc
with one argument: the
port obtained by opening the named file for input or output. If the
file cannot be opened, an error is signalled. If
proc
returns,
then the port is closed automatically and the value(s) yielded by the
proc
is(are) returned. If
proc
does not return, then
the port will not be closed automatically unless it is possible to
prove that the port will never again be used for a read or write
operation.
Rationale:
Because Scheme's escape procedures have unlimited extent, it is
possible to escape from the current continuation but later to escape back in.
If implementations were permitted to close the port on any escape from the
current continuation, then it would be impossible to write portable code using
both
call-with-current-continuation
and
call-with-input-file
or
call-with-output-file
procedure:
input-port?
obj
procedure:
output-port?
obj
Returns
#t
if
obj
is an input port or output port
respectively, otherwise returns
#f
procedure:
current-input-port
procedure:
current-output-port
Returns the current default input or output port.
optional procedure:
with-input-from-file
string thunk
optional procedure:
with-output-to-file
string thunk
String
should be a string naming a file, and
proc
should be a procedure of no arguments.
For
with-input-from-file
the file should already exist; for
with-output-to-file
the effect is unspecified if the file
already exists.
The file is opened for input or output, an input or output port
connected to it is made the default value returned by
current-input-port
or
current-output-port
(and is used by
(read)
(write
obj
, and so forth),
and the
thunk
is called with no arguments. When the
thunk
returns,
the port is closed and the previous default is restored.
With-input-from-file
and
with-output-to-file
return(s) the
value(s) yielded by
thunk
If an escape procedure
is used to escape from the continuation of these procedures, their
behavior is implementation dependent.
procedure:
open-input-file
filename
Takes a string naming an existing file and returns an input port capable of
delivering characters from the file. If the file cannot be opened, an error is
signalled.
procedure:
open-output-file
filename
Takes a string naming an output file to be created and returns an output
port capable of writing characters to a new file by that name. If the file
cannot be opened, an error is signalled. If a file with the given name
already exists, the effect is unspecified.
procedure:
close-input-port
port
procedure:
close-output-port
port
Closes the file associated with
port
, rendering the
port
incapable of delivering or accepting characters.
These routines have no effect if the file has already been closed.
The value returned is unspecified.
6.6.2 Input
library procedure:
read
library procedure:
(read
port
Read
converts external representations of Scheme objects into the
objects themselves. That is, it is a parser for the nonterminal
7.1.2
and
6.3.2
).
Read
returns the next
object parsable from the given input
port
, updating
port
to point to
the first character past the end of the external representation of the object.
If an end of file is encountered in the input before any
characters are found that can begin an object, then an end of file
object is returned. The port remains open, and further attempts
to read will also return an end of file object. If an end of file is
encountered after the beginning of an object's external representation,
but the external representation is incomplete and therefore not parsable,
an error is signalled.
The
port
argument may be omitted, in which case it defaults to the
value returned by
current-input-port
. It is an error to read from
a closed port.
procedure:
read-char
procedure:
(read-char
port
Returns the next character available from the input
port
, updating
the
port
to point to the following character. If no more characters
are available, an end of file object is returned.
Port
may be
omitted, in which case it defaults to the value returned by
current-input-port
procedure:
peek-char
procedure:
(peek-char
port
Returns the next character available from the input
port
without
updating
the
port
to point to the following character. If no more characters
are available, an end of file object is returned.
Port
may be
omitted, in which case it defaults to the value returned by
current-input-port
Note:
The value returned by a call to
peek-char
is the same as the
value that would have been returned by a call to
read-char
with the
same
port
. The only difference is that the very next call to
read-char
or
peek-char
on that
port
will return the
value returned by the preceding call to
peek-char
. In particular, a call
to
peek-char
on an interactive port will hang waiting for input
whenever a call to
read-char
would have hung.
procedure:
eof-object?
obj
Returns
#t
if
obj
is an end of file object, otherwise returns
#f
. The precise set of end of file objects will vary among
implementations, but in any case no end of file object will ever be an object
that can be read in using
read
procedure:
char-ready?
procedure:
(char-ready?
port
Returns
#t
if a character is ready on the input
port
and
returns
#f
otherwise. If
char-ready
returns
#t
then
the next
read-char
operation on the given
port
is guaranteed
not to hang. If the
port
is at end of file then
char-ready?
returns
#t
Port
may be omitted, in which case it defaults to
the value returned by
current-input-port
Rationale:
Char-ready?
exists to make it possible for a program to
accept characters from interactive ports without getting stuck waiting for
input. Any input editors associated with such ports must ensure that
characters whose existence has been asserted by
char-ready?
cannot
be rubbed out. If
char-ready?
were to return
#f
at end of
file, a port at end of file would be indistinguishable from an interactive
port that has no ready characters.
6.6.3 Output
library procedure:
write
obj
library procedure:
(write
obj port
Writes a written representation of
obj
to the given
port
. Strings
that appear in the written representation are enclosed in doublequotes, and
within those strings backslash and doublequote characters are
escaped by backslashes.
Character objects are written using the
notation.
Write
returns an unspecified value. The
port
argument may be omitted, in which case it defaults to the value
returned by
current-output-port
library procedure:
display
obj
library procedure:
(display
obj port
Writes a representation of
obj
to the given
port
. Strings
that appear in the written representation are not enclosed in
doublequotes, and no characters are escaped within those strings. Character
objects appear in the representation as if written by
write-char
instead of by
write
Display
returns an unspecified value.
The
port
argument may be omitted, in which case it defaults to the
value returned by
current-output-port
Rationale:
Write
is intended
for producing machine-readable output and
display
is for producing
human-readable output. Implementations that allow ``slashification''
within symbols will probably want
write
but not
display
to
slashify funny characters in symbols.
library procedure:
newline
library procedure:
(newline
port
Writes an end of line to
port
. Exactly how this is done differs
from one operating system to another. Returns an unspecified value.
The
port
argument may be omitted, in which case it defaults to the
value returned by
current-output-port
procedure:
write-char
char
procedure:
(write-char
char port
Writes the character
char
(not an external representation of the
character) to the given
port
and returns an unspecified value. The
port
argument may be omitted, in which case it defaults to the value
returned by
current-output-port
6.6.4 System interface
Questions of system interface generally fall outside of the domain of this
report. However, the following operations are important enough to
deserve description here.
optional procedure:
load
filename
Filename
should be a string naming an existing file
containing Scheme source code. The
load
procedure reads
expressions and definitions from the file and evaluates them
sequentially. It is unspecified whether the results of the expressions
are printed. The
load
procedure does not affect the values
returned by
current-input-port
and
current-output-port
Load
returns an unspecified value.
Rationale:
For portability,
load
must operate on source files.
Its operation on other kinds of files necessarily varies among
implementations.
optional procedure:
transcript-on
filename
optional procedure:
transcript-off
Filename
must be a string naming an output file to be
created. The effect of
transcript-on
is to open the named file
for output, and to cause a transcript of subsequent interaction between
the user and the Scheme system to be written to the file. The
transcript is ended by a call to
transcript-off
, which closes the
transcript file. Only one transcript may be in progress at any time,
though some implementations may relax this restriction. The values
returned by these procedures are unspecified.
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