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Summary: In this laboratory, you will begin your experimnets with recursion by (1) considering some pre-defined recursive procedures and (2) designing your own recursive procedures.
a. Reflect on the key aspects of recursion.
b. Start DrScheme.
Here are the two versions of
sum as given in
the reading on recursion.
;;; Procedures: ;;; sum ;;; new-sum ;;; Parameters: ;;; numbers, a list of numbers. ;;; Purpose: ;;; Find the sum of the elements of a given list of numbers ;;; Produces: ;;; total, a number. ;;; Preconditions: ;;; All the elements of numbers must be numbers. ;;; Postcondition: ;;; total is the result of adding together all of the elements of ls. ;;; If all the values in numbers are exact, total is exact. ;;; If any values in numbers are inexact, total is inexact. (define sum (lambda (numbers) (if (null? numbers) 0 (+ (car numbers) (sum (cdr numbers)))))) (define new-sum (lambda (numbers) (new-sum-helper 0 numbers))) ;;; Procedure: ;;; new-sum-helper ;;; Parameters: ;;; sum-so-far, a number. ;;; remaining, a list of numbers. ;;; Purpose: ;;; Add sum-so-far to the sum of the elements of a given list of numbers ;;; Produces: ;;; total, a number. ;;; Preconditions: ;;; All the elements of remaining must be numbers. ;;; sum-so-far must be a number. ;;; Postcondition: ;;; total is the result of adding together sum-so-far and all of the ;;; elements of remaining. ;;; If both sum-so-far and all the values in remaining are exact, ;;; total is exact. ;;; If either sum-so-far or any values in remaining are inexact, ;;; total is inexact. (define new-sum-helper (lambda (sum-so-far remaining) (if (null? remaining) sum-so-far (new-sum-helper (+ sum-so-far (car remaining)) (cdr remaining)))))
a. Verify experimentally that they work.
b. Which version do you prefer? Why?
c. Here's an alternative definition of
(define newer-sum (lambda (numbers) (new-sum-helper (car numbers) (cdr numbers))))
Is it superior or inferior to the previous definition? Why?
Here is a variant of the
largest-of-list procedure given in
the reading on recursion.
;;; Procedures: ;;; largest-of-list ;;; Parameters: ;;; numbers, a list of real numbers. ;;; Purpose: ;;; Find the largest element of a given list of real numbers ;;; Produces: ;;; largest, a real number. ;;; Preconditions: ;;; numbers is not empty. ;;; All the values in numbers are real numbers. That is, numbers ;;; contains only numbers, and none of those numbers are complex. ;;; Postconditions: ;;; largest is an element of numbers (and, by implication, is real). ;;; largest is greater than or equal to every element of numbers. (define largest-of-list (lambda (numbers) ; If the list has only one element (if (null? (cdr numbers)) ; Use that one element (car numbers) ; Otherwise, take the greater of ; (a) the first element of the list ; (b) the largest remaining element (max (car numbers) (largest-of-list (cdr numbers))))))
a. Experiment with this procedure to verify that it works as advertised.
b. There's at least one kind of list for which this procedure fails to work. Can you tell what kind? Why might we have made that decision?
Here's another version of
largest-of-list, one that
uses a technique like
new-sum (that is, it has a helper
that adds an extra parameter to keep track of the largest value so far).
(define new-largest-of-list (lambda (numbers) (new-largest-of-list-helper (car numbers) (cdr numbers)))) (define new-largest-of-list-helper (lambda (largest-so-far remaining-numbers) ; If no elements remain ... (if (null? remaining-numbers) ; Use the largest value seen so far largest-so-far ; Otherwise, update the guess using the next element ; and continue (new-largest-of-list-helper (max largest-so-far (car remaining-numbers)) (cdr remaining-numbers)))))
a. Verify experimentally that this procedure works correctly.
b. You may find it helpful to
Add the following two lines to
new-largest-of-list-helper (right before the outermost
(display (list 'new-largest-of-list-helper remaining-numbers largest-so-far)) (newline)
What effect does this change have?
c. How is
new-largest-of-list procedure similar to
largest-of-list? How is it different?
d. Which of the two versions do you prefer? Why?
As some students note, there is no need for us to define any of the
largest-of-list, since Scheme gives us
an equivalent procedure in
max. While that complaint has
a glimmer of validity, it is, in fact, useful to figure out how to write
many of the built-in Scheme procedures. One reason is that learning
the structure of those procedures makes it easier to write other, similar,
Consider the problem of finding one of the longest strings in a list
of strings, a string whose length is at least as large as the length
of each of the other strings. (We say
one of the longest strings
because more than one string may have the same length.) For example,
> (longest-string-in-list (list "zebras" "have" "stripes") "stripes" > (longest-string-in-list (list "stripes" "often" "appear" "on" "zebras") "stripes" > (longest-string-in-list (list "this" "is" "a" "silly" "list")) "silly" > (longest-string-in-list (list "this" "is" "a" "long" "list")) "this" ; or "list", or although it seems unlikely, "long"
We might document this procedure as follows:
;;; Procedure: ;;; longest-string-in-list ;;; Parameters: ;;; los, a nonempty list of strings ;;; Purpose: ;;; Finds a longest string in los. ;;; Produces: ;;; longest, a string ;;; Preconditions: ;;; (none) ;;; Postconditions: ;;; For any string in los, (string-length longest) >= (string-length str) . ;;; longest is an element of los. ;;; Problems: ;;; los may contain many strings that meet the postconditions. We do ;;; not specify which of those strings is returned.
Using either of the patterns of
Note that you might find it helpful to
(longer-of-two str1 str2) procedure that finds the
larger of two strings.
Define and test a Scheme procedure,
that takes a list of numbers as its argument and returns the result of
multiplying them all together. For example,
> (product (list 3 5 8)) 120 > (product (list 1 2 3 4 5 0)) 0
should not be 0. It should be the identity for multiplication,
(sum null) is the identity for addition.
Write a Scheme procedure,
that, given a list of numbers (including both negative and positive
numbers), returns the value closest to zero in the list.
Hint: Think about how, given two numbers, you determine which is closer to zero.
Hint: Think about how this problem is similar to a problem or problems we've solved before.
Define and test a Scheme procedure,
values), that takes a list of numbers as its argument and
returns a list of their squares.
> (square-each-element (list -7 3 12 0 4/5)) (49 9 144 0 16/25)
Hint: For the base case, consider what the procedure should return when given a null list; for the other case, separate the car and the cdr of the given list and consider how to operate on them so as to construct the desired result.
If you find that you not only finish this laboratory early, but also finish the two extra exercises early, you can start your next homework assignment.
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