Computer Science 115-03, Midterm. Friday, November 1, 1996.

1. [20 points]. What return values have been substituted for the BLANKS in the following code?

   : (define x 4)
   : (define y (cons 3 (cons x '())))
   : y
   
   ???  BLANK 1a. ???
   
   : (let ((x 3)
           (y (* x 2)))
       (+ x y))
   
   ???  BLANK 1b. ???
   
   : (cond 
        ((not (= x 4))  (error ``oops''))
        ((> x 4)        (+ x (car y)))
        ((= x 4)        (cdr y))
        (else           (* 2 x)))
   
   ???  BLANK 1c. ???
   
   : (define (sum term a next b)
       (if (> a b)
           5
           (+ (term a) (sum term (next a) next b))))
   
   : (sum (lambda (z) (* 2 z))
          10
          (lambda (z) (* 10 z))
          1000)
   
   ???  BLANK 1d. ???
   

2. [15 points] Write a scheme procedure that takes 3 numbers as arguments and returns \#t if there are exactly 2 distinct values among them. Your first line should look like this:

   (define (twovals? a b c) ...
   

3. [15 points] Draw Box and Pointer diagrams for the following list structures: (Note, the ``.'' in (3c) is the result of a dotted pair, not a period.)

   3a.  (3 4 9)             
   3b.  ((4 5) (3 9))       
   3c.  (17  ((9)) . 5)     
   

4. [15 points] Suppose that a, b, and c are bound to the lists of Question 3a, 3b, and 3c repectively.
   For Question 4a, give an expression using only car, cdr, and a that returns the value 9.
   For Question 4b, give an expression using only car, cdr, and b that returns the value 9.
   For Question 4c, give an expression using only car, cdr, and c that returns the value 9.

5. [15 points] Define a procedure reverse that takes a list as an argument and returns a list of the same elements in reverse order. For example:

   : (reverse (list 1 4 9 16 25))
   (25 16 9 4 1)
   

6. [20 points] Write a function repeated that takes a function f, a positive integer n and a list of numbers lis, and returns the list of numbers resulting from applying f to each element of lis n times. The first line of your definition and a sample of how it should work is as follows:

   : (define (repeated f n lis) ...
   
   : (repeated (lambda (x) (+ x 1)) 3 '(4 5 6))
   
   (7 8 9)
   
   : (repeated square 2 '(0 1 2 3))
   
   (0 1 16 81)
   

   Write 2 versions of repeated: one that generates an iterative process and one that generates a recursive process. For each version, describe the time and space requirements (using Theta(...) notation). Your answer should be in terms of n ; you may assume that the lis has constant size.

7. [20 points] Consider the following code:

   (define (test n)
     (define (pair x) (cons x x))
    (pair (random n)))
   

   Recall that (random n) returns a random number between 0 and n-1.
   • 7a. [6 points] Briefly describe normal-order evaluation and applicative-order evaluation.
   • 7b. [10 points] Write a procedure (normal?) that takes no arguments and returns #t if the interpreter uses normal order evaluation, and #f if it uses applicative order evaluation. Note that it must call the procedure test above, and it may occasionally make a mistake.
   • 7c. [4 points] What is the probability that your procedure (normal?) makes an error if it returns #t? If it returns #f?

: (define (f g) (g 2))

: (f square)
4

: (f (lambda (z) (* z (+ z 1))))
6

: (f f)

??? WHAT DOES THE INTERPRETER DO HERE?  EXPLAIN.  ???