Functional Problem Solving (CSC 151 2014F) : Readings
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Summary: Many programs need to make choices. In this reading, we consider Scheme's conditional expressions, expressions that allow programs to behave differently in different situations. We also consider how conditionals help us draw some simple shapes.
When Scheme encounters a procedure call, it looks at all of the subexpressions within the parentheses and evaluates each one. Sometimes, however, the programmer wants Scheme to exercise more discretion. Specifically, the programmer wants to select just one subexpression for evaluation from two or more alternatives. In such cases, one uses a conditional expression, an expression that tests whether some condition is met and selects the subexpression to evaluate on the basis of the outcome of that test.
For instance, suppose we want to increase a value by 20% if it is greater than 127 and reduce it by 20% if it is less than 128. (You may recall a similar problem from a recent laboratory exercise.) While it is possible to write an interesting expression to do this computation, many programmers would prefer something a bit clearer.
To write a procedure that like this, we benefit from a mechanism that allows us to explicitly tell Scheme how to choose which expression to evaluate. Such mechanisms are the primary subject of this reading.
The simplest conditional expression in Scheme is an
if
expression. An if
expression typically has three components: a test, a consequent, and
an alternative. It selects one or the other of these expressions,
depending on the outcome of a test. The general form is
(iftest
consequent
alternative
)
We'll return to the particular details in a moment. For now, let's consider the conditional we might write for the procedure to make a component more extreme.
(if (> component 127) ; If the component is greater than 127 (* component 1.2) ; Increment it by 20% (* component 0.8)) ; Otherwise, decrement it by 20%
To turn this expression into a procedure, we need
to add the define
keyword, a name (such as
component-enhance
), a lambda expression, and such.
We also want to give appropriate documentation and a bit of cleanup
to the results.
Here, then, is the complete definition of the
component-enhance
procedure:
;;; Procedure: ;;; component-enhance ;;; Parameters: ;;; component, an integer ;;; Purpose: ;;; Compute an enhanced version of component. A large component gets ;;; larger. A small component gets smaller. ;;; Produces: ;;; new-component, an integer ;;; Preconditions: ;;; component is an integer between 0 and 255, inclusive ;;; Postconditions: ;;; If component > 127, then new-component is approximately 20% larger ;;; than component. ;;; If component < 128, then new-component is approximately 20% smaller ;;; than component. (define component-enhance (lambda (component) (if (> component 127) (min 255 (round (* 1.2 component))) (round (* 0.8 component)))))
In an if
-expression
of the form (
, the
if
test
consequent
alternative
)test
is always evaluated first. If its value is
#t
(which means “yes” or “true”),
then the consequent
is evaluated, and the
alternative
(the expression following the
consequent) is ignored. On the other hand, if the value of the test
is #f
, then the consequent is ignored and the alternative
is evaluated.
Scheme accepts if
-expressions in which the value of
the test is non-Boolean. However, all non-Boolean values are classified
as “truish” and cause the evaluation of the consequent.
It is also possible to write an if expression without the alternative.
Such an expression has the form (
.
In this case, the test is still evaluated first. If the test holds
(that is, has a value of if
test
consequent
)#t
or anything other than
#f
), the consequent is evaluated and its value is returned.
If the test fails to hold (that is, has value #f
), the
if expression has no value.
Scheme programmers tend to use the alternative-free if expression much less frequently than they use the traditional form. In general, your first inclination should be to provide both a consequent and an alternative when you write a conditional. Some Scheme programmers object to the alternative-free if expression enough that they discourage its use. In fact, the primary version of PLT Scheme used in MediaScheme will not permit you to write an if expression without an alternative.
Newer versions of Scheme include another kind of conditional, the
when
expression, which provides an alternative
to the alternative-free if expression.
(when
test
body1
body2
...bodyn
)
When evaluating a when
expression, the Scheme
interpreter first evaluates the test
. If the
test holds, the interpreter evaluates each body
expression in turn.
cond
When there are more than two choices, it is often more convenient
to set them out using a cond
expression. Like if
,
cond
is a keyword. (Recall that keywords differ
from procedures in that the order of evaluation of the parameters may
differ.) The cond
keyword is followed by zero or
more lists-like expressions called cond
clauses.
(cond
[test-0
consequent-0
] ... [test-n
consequent-1
] [elsealternate
])
The first expression within a cond
clause is a test,
similar to the test in an if
expression. When the value
of such a test is found to be #f
, the subexpression
that follows the test is ignored and Scheme proceeds to the test
at the beginning of the next cond
clause. But when
a test is evaluated and the value turns out to be true, or even
“truish” (that is, anything other than #f
),
the consequent for that test is evaluated and its value is the value
of the whole cond expression.. Subsequent cond
clauses are completely ignored.
In other words, when Scheme encounters a cond
expression, it works its way through the cond
clauses, evaluating the test at the beginning of each one, until it
reaches a test that succeeds (one that does not
have #f
as its value). It then makes a ninety-degree turn
and evaluates the consequent in the selected cond
clause, retaining the value of the consequent.
If all of the tests in a cond
expression are found
to be false, the value of the cond
expression is
unspecified (that is, it might be anything!). To prevent the surprising
results that can ensue when one computes with unspecified values,
good programmers customarily end every cond
expression with a cond
clause in which the
keyword else
appears in place of a test. Scheme treats
such a cond
clause as if it had a test that
always succeeded. If it is reached, the subexpressions following
else
are evaluated, and the value of the last one is the
value of the whole cond
expression.
For example, here is a cond
expression that
produces black, white, or grey based only on the red component of a
color, c
.
(cond [(< (rgb-red c) 96) color-black] [(> (rgb-red c) 160) color-white] [else color-grey])
The expression has three cond
clauses. In the first, the test
is (< (rgb-red c) 96)
. If the red component of c
happens to be the small,
the value of this first test is #t
, so we evaluate
whatever comes after the test to find the value of the entire expression,
in this case, the color black.
If the red component of c
is not small, then we proceed
instead to the second cond
clause. Its test
is (> (rgb-red c) 160)
, which determines if the red
component is large. If it is, then we return the color white.
However, if c
has a red component that is neither small nor
large, then we proceed instead to the third cond
clause. Since the keyword else
appears in this
cond
clause in place of a test, we take that as
an automatic success and evaluate color-grey
, so that
that value of the whole cond
expression in this
case is the color grey.
Although we have written our conditionals with one consequent per test (and we encourage you to do the same), it is, in fact, possible to have multiple consequents per test.
(cond [test-0
consequent-0-0
consequent-0-1
...consequent-0-m
] ... [elsealternate-0
alternate-1
...alternate-a
])
In this case, when a test succeeds, each of the remaining subexpressions
(that is, consequents) in the same cond
clause
is evaluated in turn, and the value of the last one becomes the value
of the entire cond
expression.
As you may have noted from our discussion of cond
,
cond
expressions can use square brackets rather
than parenthesis to indicate structure - that is, they do not surround
an expression to evaluate (a procedure followed by its parameters).
Instead, they serve only to group things. In this case, the parentheses
group the guard and consequents for each cond
clause. The square brackets are just a notational convenience; parenthesis
will work just as well, and you'll see a lot of Scheme code that uses
parentheses rather than square brackets. Racket, like most modern
Scheme implementations, allows both because the square brackets add
a bit of clarity.
When writing cond
clauses, you should take
the time to verify that you've used the right number of parentheses
and square brackets. Each clause has its own open and close square
brackets (or open and close parenthesis). Typically, the test has
parentheses, unless it's the else
clause. Make sure to
include both sets.
and
and or
As we saw in the
reading on Boolean values, both and
and or
provide a type of conditional behavior.
In particular, and
evaluates each argument
in turn until it hits a value that is #f
and then
returns #f
(or returns the last value if none return
#f
). Similarly, or
evaluates each
argument in turn until it finds one that is not #f
,
in which case it returns that value, or until it runs out of values,
in which case it returns #f
.
That is, (
behaves much like the following or
exp0
exp1
... expn
)cond
expression, except
that the or
version evaluates each expression once,
rather than twice.
(cond
[exp0
exp0
] [exp1
exp1
] ... [expn
expn
] [else #f])
Similarly, (
behaves much like the following and
exp0
exp1
... expn
)cond
expression.
(cond [(notexp0
) #f] [(notexp1
) #f] ... [(notexpn
) #f] [elseexpn
])
Most beginning programmers find the cond
versions
much more understandable, but some advanced Scheme programmers use the
and
and or
forms because they
find them clearer. Certainly, the cond
equivalents
for both or
and and
are
quite repetitious.
So, what does any of this have to do with images? Well, we've already seen one thing: We can use conditionals in writing color transformations. We also need conditionals to let us write procedures for rendering drawing values on the screen. Why? Different types of drawing values need to be rendered differently. For an important example, ellipses should be rendered as ellipses, and rectangles should be rendered as rectangles. How? Well, we'll generally want to figure out what kind of drawing it is and then use GIMP commands to draw it there. We might describe the algorithm in English as something like
If the drawing is a rectangle select the appropriate region set the foreground color fill the rectangle Otherwise, if the drawing is an ellipse select the appropriate region set the foreground color fill the ellipse Otherwise, we don't know what kind of drawing it is Use some default behavior
First, we need to be able to find out the type of a drawing. We use
the drawing-type
procedure to figure that out.
The two most important types are ellipse
and
rectangle
. (There are other types, but those are the
only ones we'll deal with right now.)
We also need to know how to get all of the components. These are
given by
drawing-left
,
drawing-top
,
drawing-width
,
drawing-height
, and
drawing-color
. You can get a bit more information
on all of them on
the reference page.
Now, we're ready to put it all together. This procedure is among the longest and most complex you've seen so far, in part because it accounts for three different conditions. (Can you think of any ways to make it simpler or more concise?)
;;; Procedure: ;;; render! ;;; Parameters: ;;; drawing, a drawing value ;;; image, an image id ;;; Purpose: ;;; Render drawing on the image. ;;; Produces: ;;; [Nothing; called for side effect] ;;; Preconditions: ;;; drawing can be rendered on image. ;;; drawing is a simple drawing, that is one of ;;; (drawing-rectangle? drawing) or ;; (drawing-ellipse? drawing) holds. ;;; Postconditions: ;;; drawing has been rendered on image. (define render! (lambda (drawing image) (context-set-fgcolor! (drawing-color drawing)) (cond [(equal? (drawing-type drawing) 'rectangle) (image-select-rectangle! image REPLACE (drawing-left drawing) (drawing-top drawing) (drawing-width drawing) (drawing-height drawing))] [(equal? (drawing-type drawing) 'ellipse) (image-select-ellipse! image REPLACE (drawing-left drawing) (drawing-top drawing) (drawing-width drawing) (drawing-height drawing))]) (image-fill-selection! image)))
(if
test
consequent
alternative
)
test
. If its value is truish (that is,
anything but false), evaluate consequent
and return its value. If the value of test is false (#f), evaluate
and return alternative
.
(cond
(test1
consequents1
)
(test2
consequents2
)
...
(testn
consequentsn
)
(else
alternative
))
(when
test
exp1
exp2
...
expn
)
test
. If it holds, evaluate
each expression in turn. Otherwise, do nothing.
(and
exp1
exp2
...
expn
)
(or
exp1
exp2
...
expn
)
a. Assuming num
is defined as an integer, write an
if
expression that produces double the value of
num
if it is odd, and half the value otherwise.
b. Write a cond
expression that produces the symbol
positive
if num
is greater than zero, the
symbol negative
if num
is less than zero,
and the symbol neither
otherwise.