Laboratory: Verifying Preconditions
Summary: In this laboratory, you will consider mechanisms for verifying the preconditions of procedures. You will also consider some issues in the documentation of such procedures.
Preparation
Make a copy of preconditions-lab.rkt.
Exercises
Exercise 1: Are They All Drawings?
a. In the reading, we called drawings-leftmost on three erroneous inputs: the empty list, the value 1, and a list containing the value 2. Check to see whether you get the same error messages as were presented in the reading.
b. You may have noted that drawings-leftmost requires an all-drawings? procedure. Here’s a definition of that procedure.
(define all-drawings?
(lambda (lst)
(or (null? lst)
(and (drawing? (car lst))
(all-drawings? (cdr lst))))))
Add this definition to your definitions pane and check to see whether you now get the “appropriate” output.
c. What preconditions should all-drawings? have?
d. Is it necessary to test those preconditions? Why or why not?
e. Compare your answers to the those in notes at the end of this lab.
f. Document the all-drawings? procedure.
Exercise 2: Exploring Errors
In the corresponding reading, there is an extended version of drawings-leftmost that reports different errors. Replace your current drawings-leftmost with the new version and verify that it does, in fact, report different errors.
Exercise 3: Finding Values
Here is a procedure, index-of, that takes two arguments - a value, val, and a list, vals - and returns the index of val in vals.
(define index-of
(lambda (val vals)
(cond
; If the value appears first in the list, then its index is 0.
[(equal? val (car vals))
0]
; Otherwise, we find the index in the cdr. Since we've
; thrown away the car in finding that index, we need to add 1
; to get its index in the overall list.
[else
(+ 1 (index-of val (cdr vals)))])))
Here are some examples of index-of in use.
> (index-of "red" (list "red" "green" "blue" "yellow"))
0
> (index-of "blue" (list "red" "green" "blue" "yellow"))
2
a. What preconditions should index-of have?
b. What will happen if the preconditions are violated? Verify your answer experimentally.
c. Arrange for index-of to explicitly signal an error (by invoking the error procedure) if vals is not a list.
d. Arrange for index-of to explicitly signal an error (by invoking the error procedure) if val does not occur at all as an element of vals.
> (index-of "black" (list "red" "green" "blue" "yellow"))
Error: The value does not appear in the list.
e. Some programmers return special values to signal an error to the caller, rather than throw an error. If val does not occur as an element of vals, why might it be better to have index-of return a special value (such as -1 or #f) rather than throwing an error? Explain your answer.
f. If val does not occur as an element of vals, why might be better to have index-of throw an error?
g. Once you have explained your answers to e and f, you may want to check our notes on this problem.
h. Rewrite index-of using a husk-and-kernel strategy.
When you’re done thinking about these questions, add your definition of index-of to your notes, as it is a very useful procedure.
Exercise 4: Changing Colors
Consider the following procedure that increments the red component of color by 64.
;;; Procedure:
;;; irgb-much-redder
;;; Parameters:
;;; color, an integer-encoded RGB color
;;; Purpose:
;;; To produce a color that is much redder than color.
;;; Produces:
;;; newcolor, an integer-encoded RGB color
;;; Preconditions:
;;; FORTHCOMING
;;; Postconditions:
;;; (irgb-red new-color) = (+ 64 (irgb-red color))
;;; (irgb-green new-color) = (irgb-green color)
;;; (irgb-blue new-color) = (irgb-blue color)
(define irgb-much-redder
(lambda (color)
(irgb (+ 64 (irgb-red color)) (irgb-green color) (irgb-blue color))))
a. What preconditions must be met in order for irgb-much-redder to meet its postconditions?
b. Should we test those preconditions? Why or why not?
Exercise 5: Weighted Color Averages
In a number of exercises, we were required to blend two colors. For example, we blended colors in a variety of ways to make interesting images, and we made a color more grey by averaging it with grey. In blending two colors, we are, in essence, creating an average of the two colors, but an average in which each color contributes a different fraction.
For this problem, we might write a procedure, (irgb-weighted-average fraction color1 color2), that makes a new color, each of whose components is computed by multiplying the corresponding component of color1 by fraction and adding that to the result of multiplying the corresponding component of color2 by (1-*fraction). For example, we might compute the red component with
(+ (* fraction (irgb-red color1)) (* (- 1 fraction) (irgb-red color2)))
a. What preconditions should irgb-weighted-average have? (Think about restrictions on fraction, color1, and color2.)
b. How might you formally specify the postconditions for irgb-weighted-average?
c. Here is a simple definition of rgb-weighted-average.
(define irgb-weighted-average
(lambda (fraction color1 color2)
(let ([frac2 (- 1 fraction)])
(irgb (+ (* fraction (irgb-red color1)) (* frac2 (irgb-red color2)))
(+ (* fraction (irgb-green color1)) (* frac2 (irgb-green color2)))
(+ (* fraction (irgb-blue color1)) (* frac2 (irgb-blue color2)))))))
Rewrite irgb-weighted-average to use a husk-and-kernel strategy to test for preconditions.
For Those With Extra Time
Extra 1: Substitution
Consider a procedure, (list-substitute lst old new), that builds a new list by substituting new ... old whenever old appears in lst.
> (list-substitute (list "black" "red" "green" "blue" "black") "black" "white")
("white" "red" "green" "blue" "white")
> (list-substitute (list "black" "red" "green" "blue" "black") "yellow" "white")
("black" "red" "green" "blue" "black")
> (list-substitute null "yellow" "white")
()
a. Document this procedure, making sure to carefully consider the preconditions.
b. Implement this procedure, making sure to check the preconditions.
Extra 2: Substituting Colors, Revisited
Consider a procedure, (drawings-partially-recolor drawings old new), that, given a list of drawings and two colors, makes a new copy of drawings by using the color new whenever a drawing with color old appeared in the original list.
a. What preconditions does this procedure have?
b. Implement this procedure, using the husk-and-kernel structure to ensure that old ... new are rgb colors and that drawings is a list of drawings, before starting the recursion.
Extra 3: index-of, Revisited
Rewrite index-of using tail recursion and a husk and kernel.
Notes on the Problems
Notes on Problem 1: Are They All Drawings?
Here are some possible solutions.
a. The all-drawings? procedure needs a list as a parameter.
b. It depends on how we will use all-drawings?. If we are sure that it will only be called correctly (e.g., after we’ve already tested that the parameter is a list or in a context in which we can prove that the parameter is list), then it need not check its preconditions. Otherwise, it should check its preconditions.
Notes on Exercise 3: Finding Values
e. If index-of explicitly checks its precondition using member?, we end up duplicating work. That is, we scan the list once to see if the value is there, and once to see its index. Even if index-of does not explicitly check its precondition, the caller may be called upon to do so, which still duplicates the work. By having index-of return a special value, we permit the client to have index-of do both.
f. In some cases, programs should stop when there is no index for a specified value. For example, a program that tries to look up a grade for a student should not continue if the student does not appear in the list. There are also some instances in which careless programmers do not check the return value, which can lead to unpredictable behavior.