Summary: Many of Scheme's control structures, such as conditionals (which you'll learn about in a subsequent reading), need mechanisms for constructing tests that return the values true or false. These tests can also be useful for gathering information about a variety of kinds of values. In this reading, we consider the types, basic procedures, and mechanisms for combining results that support such tests.
When writing complex programs, we often need to ask questions about the values with which we are computing. Is this pixel a shade of red? Is this image at least 100x100? Are these two colors close enough to be indistinguishable? Is this a light or dark color? Most frequently, these questions (which we often phrase as tests) are used in control structures. For example, we might decide to do one thing for large images and another for small images or we might replace light colors by white and dark colors by black.
To express these kinds of questions, we need a variety of tools. First, we need a type in which to express the valid answers to questions. Second, we need a collection of procedures that can answer simple questions. Third, we need ways to combine questions. Finally, we need control structures that use these questions. In the subsequent sections of this reading, we consider each of these issues. We return to more complex control structures in a subsequent reading.
A Boolean value is a datum that reflects the
outcome of a single yes-or-no test. For instance, if one were to
ask Scheme to compute whether pure red has a high blue component,
it would be able to determine that it does not, and it would signal
this result by displaying the Boolean value for “no”
or “false”, which is
#f. There is only
one other Boolean value, the one meaning “yes” or
“true”, which is
#t. These are called
“Boolean values” in honor of the logician George Boole
who was the first to develop a satisfactory formal theory of them.
(Some folks now talk about “fuzzy logic” that includes
values other than “true” and “false”, but
that's beyond the scope of this course.)
A predicate is a procedure that always returns a
Boolean value. A procedure call in which the procedure is a predicate
performs some yes-or-no test on its arguments. For instance, the
number? (the question mark is part of the name
of the procedure) takes one argument and returns
that argument is a number,
#f if it does not. Similarly,
even? takes one argument, which must be
an integer, and returns
#t if the integer is even and
#f if it is odd. The names of most Scheme predicates
end with question marks, and Grinnell's computer scientists recommend
this useful convention, even though it is not required by the rules
of the programming language. (If you ever notice that we've failed to
include a question mark in a predicate and you're the first to tell us,
we'll give you some extra credit.)
Scheme provides a wide variety of basic predicates and MediaScheme adds a few more. We will consider a few right now, but learn more as the course progresses.
The simplest predicates let you test the “type” of a value. Scheme provides a number of such predicates.
number?tests whether its argument is a number.
integer?tests whether its argument is an integer.
real?tests whether its argument is a real number.
string?tests whether its argument is a string.
procedure?tests whether its argument is a procedure.
boolean?tests whether its argument is a Boolean value.
MediaScheme adds a few special predicates that are tailored to working with colors and images. Because these types lack the specificity of the internal representation of the built-in types, these predicates give answers that typically represent whether we can interpret the value as the type, not whether it was actually built using one of the constructors for that type.
image?tests whether its argument can be interpreted as an image.
rgb?tests whether its argument can be interpreted as an RGB color.
color-name?tests whether its argument can be interpreted as a color name.
color?tests whether its argument can be interpreted as a color. (That is, it checks whether it's an RGB color, a color-name, or one of a few other representations of colors that MediaScheme supports.)
drawing?tests whether its argument can be interpreted as a drawing.
turtle?tests whether its argument seems to be a turtle.
Scheme provides a variety of predicates for testing whether two values can be understood to be the same.
eq?tests whether its two arguments are identical, in the very narrow sense of occupying the same storage location in the computer's memory. In practice, this is useful information only if at least one argument is known to be a symbol, a Boolean value, or an integer.
eqv?tests whether its two arguments “should normally be regarded as the same object” (as the language standard declares). Note, however, that two collections of values can have the same elements without being “regarded as the same object”. Also note that in Scheme's view the number 5, which is “exact”, is not necessarily the same object as the number 5.0, which might be an approximation.
equal?tests whether its two arguments are the same or, in the case of lists, whether they have the same contents.
=tests whether its arguments, which must all be numbers, are numerically equal; 5 and 5.0 are numerically equal for this purpose.
For this class, you are not required to understand
the difference between the
eqv? procedures. In particular, you need not
plan to use the
eqv? procedure. At least for the
first half of the semester, you also need not understand the difference
procedures. Feel free to use
exclusively, except when dealing with numbers, in which case you should
Scheme also provides many numeric predicates, some of which you may have already explored.
<tests whether its arguments, which must all be numbers, are in strictly ascending numerical order. (The
<operation is one of the few built-in predicates that does not have an accompanying question mark.)
>tests whether its arguments, which must all be numbers, are in strictly descending numerical order.
<=tests whether its arguments, which must all be numbers, are in ascending numerical order, allowing equality.
>=tests whether its arguments, which must all be numbers, are in descending numerical order, allowing equality.
even?tests whether its argument, which must be an integer, is even.
odd?tests whether its argument, which must be an integer, is odd.
zero?tests whether its argument, which must be a number, is equal to zero.
positive?tests whether its argument, which must be a real number, is positive.
negative?tests whether its argument, which must be a real number, is negative.
exact?tests whether its argument, which must be a number, is represented exactly.
inexact?tests whether its argument, which must be a number, is not represented exactly.
Not all the procedures we use to work with Boolean values are strictly
Another useful Boolean procedure is
takes one argument and returns
#t if the argument is
#f if the argument is anything else.
For example, one can test whether
picture is not an
(not (image? picture))
If Scheme says that the value of this expression is
picture is not an image.
or keywords have
simple logical meanings. In particular, the and
of a collection of Boolean values is true if all are true and false
if any value is false, the or of a collection of
Boolean values is true if any of the values is true and false if all
the values are false. For example,
(and #t #t #t)
(and (< 1 2) (< 2 3))
(and (odd? 1) (odd? 3) (odd? 5) (odd? 6))
(or (odd? 1) (odd? 3) (odd? 5) (odd? 6))
(or (even? 1) (even? 3) (even? 4) (even? 5))
You may note that we were careful to describe
or as “keywords” rather than
as “procedures”. The distinction is an important one.
Although keywords look remarkably like procedures, Scheme distinguishes
keywords from procedures by the order of evaluation of the parameters.
For procedures, all the parameters are evaluated and then the procedure
is applied. For keywords, not all parameters need be evaluated,
and custom orders of evaluation are possible.
or were procedures, we could not
guarantee their control behavior. We'd also get some ugly errors. For
example, consider the extended version of the
(define new-even? (lambda (val) (and (integer? val) (even? val))))
new-even? is called with 2.3 as a
parameter. In the keyword implementation of
the first test,
new-even? returns false. If
and were a procedure, we would still evaluate
(, and that test would
generate an error, since
even? can only be called
Although many computer scientists, philosophers, and mathematicians prefer the purity of dividing the world into “true” and “false”, Scheme supports a somewhat more general separation. In Scheme, anything that is not false is considered “truish”. Hence, you can use expressions that return values other than truth values wherever a truth value is expected. For example,
(and #t 1)
(or 3 #t #t)
(not (not 1))
MediaScheme provides one additional keyword predicate,
This procedure takes one parameter, a name (as we would use in
define), and determines whether it has been
defined by a top-level
(define x 2)
(let ((y 2)) (defined? y))
defined? looks like a procedure, it is instead
a keyword. Why? Because
defined? treats its
parameter differently (not evaluating it, as most procedures do,
but taking it directly as a name), it can't be implemented as a pure
procedure. Hence, if you try to use
as a value (e.g., as a parameter to another procedure), you will get
defined?Interactions:1:0: defined?: bad syntax in: defined?
Can we write predicates that work with colors? Certainly. One simple question is whether we might consider two colors near to each other. What are criteria for making that decision? One possibility is that we will consider two colors similar if all of their components are within 8 of each other. We can define that predicate as follows:
;;; Procedure: ;;; colors-similar? ;;; Parameters: ;;; color1, an RGB color ;;; color2, an RGB color ;;; Purpose: ;;; Determines if color1 and color2 are similar. ;;; Produces: ;;; similar?, a Boolean value ;;; Preconditions: ;;; [No additional] ;;; Postconditions: ;;; If color1 and color2 are close enough to be considered similar, ;;; then similar? is #t. ;;; Otherwise, similar? is #f. ;;; We use a proprietary technique to decide what "close enough" means. (define colors-similar? (lambda (color1 color2) (and (>= 8 (abs (- (rgb-red color1) (rgb-red color2)))) (>= 8 (abs (- (rgb-green color1) (rgb-green color2)))) (>= 8 (abs (- (rgb-blue color1) (rgb-bluecolor2)))))))
Here's a pair of useful predicates: One computes whether a color might reasonably be considered light; another computes whether a color might reasonably consider dark.
;;; Procedure: ;;; rgb-light? ;;; Parameters: ;;; color, an RGB color ;;; Purpose: ;;; Determine if the color seems light. ;;; Produces: ;;; light?, a Boolean value ;;; Preconditions: ;;; [None] ;;; Postconditions: ;;; light? is true (#t) if color's intensity is relatively high. ;;; light? is false (#f) otherwise. (define rgb-light? (lambda (color) (<= 192 (+ (* 0.30 (rgb-red color)) (* 0.59 (rgb-green color)) (* 0.11 (rgb-blue color)))))) ;;; Procedure: ;;; rgb-dark? ;;; Parameters: ;;; color, an RGB color ;;; Purpose: ;;; Determine if the color seems dark. ;;; Produces: ;;; dark?, a Boolean value ;;; Preconditions: ;;; [None] ;;; Postconditions: ;;; dark? is true (#t) if color's intensity is relatively low. ;;; dark? is false (#f) otherwise. (define rgb-dark? (lambda (color) (>= 64 (+ (* 0.30 (rgb-red color)) (* 0.59 (rgb-green color)) (* 0.11 (rgb-blue color))))))
We've seen how
can be used to combine tests. But
or can be used for so much more. In fact, they
can be used as control structures.
and-expression, the expressions that follow
and are evaluated in succession until
one is found to have the value
#f (in which case the rest
of the expressions are skipped and the
#f becomes the
value of the entire
and-expression). If, after
evaluating all of the expressions, none is found to be
then the value of the last expression becomes the value of the entire
and expression. This evaluation strategy gives
the programmer a way to combine several tests into one that will
succeed only if all of its parts succeed.
This strategy also gives the programmer a way to avoid
meaningless tests. For example, we should not make the comparison
( unless we are sure that
< a b)
b are numbers.
or expression, the expressions that follow
or are evaluated in succession until
one is found to have a value other than
#f, in which case
the rest of the expressions are skipped and this value becomes the
value of the entire
or-expression. If all of the
expressions have been evaluated and all have the value
then the value of the
#f. This gives the programmer a way to combine several
tests into one that will succeed if any of its
In these cases,
and returns the last parameter it encounters
(or false, if it encounters a false value) while
the first non-false value it encounters. For example,
(and 1 2 3)
(define x 'two)
(define y 3)
(+ x y)+: expects type <number> as 1st argument, given: two; other arguments were: 3
(and (number? x) (number? y) (+ x y))
(define x 2)
(and (number? x) (number? y) (+ x y))
(or 1 2 3)
(or 1 #f 3)
(or #f 2 3)
(or #f #f 3)
We can use the ideas above to make an addition procedure that returns
#f if either parameter is not a number. We might say that
such a procedure is a bit safer than the normal addition procedure.
;;; Procedure: ;;; safe-add ;;; Parameters: ;;; x, a number [verified] ;;; y, a number [verified] ;;; Purpose: ;;; Add x and y. ;;; Produces: ;;; sum, a number. ;;; Preconditions: ;;; (No additional preconditions) ;;; Postconditions: ;;; sum = x + y ;;; Problems: ;;; If either x or y is not a number, sum is #f. (define safe-add (lambda (x y) (and (number? x) (number? y) (+ x y))))
Let's compare this version to the standard addition procedure,
(+ 2 3)
(safe-add 2 3)
(+ 2 'three)Error: +: argument 2 must be: number
(safe-add 2 'three)
If we'd prefer to return 0, rather than
#f, we could add an
;;; Procedure: ;;; safer-add ;;; Parameters: ;;; x, a number [verified] ;;; y, a number [verified] ;;; Purpose: ;;; Add x and y. ;;; Produces: ;;; sum, a number. ;;; Preconditions: ;;; [No additional preconditions] ;;; Postconditions: ;;; If both x and y are numbers, sum = x + y ;;; Problems: ;;; If either x or y is not a number, sum is 0. (define safer-add (lambda (x y) (or (and (number? x) (number? y) (+ x y)) 0)))
In most cases,
safer-add acts much like
safe-add. However, when we use the result of the
two procedures as an argument to another procedure, we get a little
bit further through the calculation.
(* 4 (+ 2 3))
(* 4 (safer-add 2 3))
(* 4 (+ 2 'three))Error: +: argument 2 must be: number
(* 4 (safe-add 2 'three))Error: *: argument 2 must be: number
(* 4 (safer-add 2 'three))
Different situations will call for different choices between those strategies.
Here's a simple application of the preceding strategies: We can write a procedure that, given a color, returns black if the color is dark, white if the color is light, and grey if the color is neither dark nor light.
How? Well, we can use
and to compute either black,
if the color is dark, or
#f, if the color is not dark.
(and (rgb-dark? color) black)
Similarly, we can use
and to compute either white,
if the color is light, or
#f if the color is not light.
(and (rgb-light? color) white)
Finally, we can use
or to put it all together.
;;; Procedure: ;;; rgb-bgw ;;; Parameters: ;;; color, an RGB color ;;; Purpose: ;;; Convert an RGB color to black, grey, or white, depending on ;;; the intensity of the color. ;;; Produces: ;;; bgw, an RGB color ;;; Preconditions: ;;; rgb-light? and rgb-dark? are defined. ;;; Postconditions: ;;; If (rgb-light? color) and not (rgb-dark? color), then bgw is white. ;;; If (rgb-dark? color) and not (rgb-light? color), then bgw is black. ;;; If neither (rgb-light? color) nor (rgb-dark? color), then bgw is ;;; grey. ;;; Problems: ;;; In the unexpected case that none of the above conditions holds, ;; bgw will be one of black, white, and grey. (define rgb-bgw (let ((black (rgb-new 0 0 0)) (grey (rgb-new 128 128 128)) (white (rgb-new 255 255 255))) (lambda (color) (or (and (rgb-light? color) white) (and (rgb-dark? color) black) grey))))
Here's a slightly stranger, but potentially useful, example. As you've
noted, we often want to associate the name
an image on which we experiment. Unfortunately, if you put something
like the following in your definitions pane, each time you click
, you get a new image.
(define canvas (image-show (image-new 200 200)))
Is there something we can do to ensure that we don't create a new
image each time we click run? We can generate the image by hand once,
observe its number, and then use
define to associate the
canvas with that number.
(define canvas 2) ; The number of the image we created by hand
However, if we subsequently
close the image,
canvas no longer refers to an actual
Using the Boolean operations, we can write an expression that will
check whether we've created an image (but not too many images) and,
if so, associate
canvas with that image. If there is
no available image, the expression will build and show a new image and
canvas with that new image.
(define canvas (or (and (image? 1) 1) (and (image? 2) 2) (and (image? 3) 3) (and (image? 4) 4) (image-show (image-new 200 200))))
Although it may feel like the
define needs to be
or, the nature of these various keywords
makes the particular order necessary.
Clearly, this will only work the first four times we create an image. It seems tempting to use one of our looping structures. However, we do not yet know a looping structure that will handle this type of situation.
Copyright (c) 2007-9 Janet Davis, Matthew Kluber, Samuel A. Rebelsky, and Jerod Weinman. (Selected materials copyright by John David Stone and Henry Walker and used by permission.)
This material is based upon work partially supported by the National Science Foundation under Grant No. CCLI-0633090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
This work is licensed under a Creative Commons
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