CS 3540 Quiz 2

Quiz 2

Instructions

The quiz consists of five questions.
Read all the questions thoroughly before you begin. It is worth your time to plan ahead!
The quiz is worth 60 points. Each question is worth twelve points.
Partial credit will be given where possible, so show your work.
The quiz lasts forty (40) minutes. It is due at 1:45 PM.

Because the primary goal of this portion of the course is to demonstrate the ability to write recursive programs, you may not use any built-in function that does the recursion for you, such as map, filter, reverse, or string->list.

Problem 1

Write a structurally recursive function (drop-last los) that takes one argument, a list of symbols los:

<list-of-symbols> ::= ()
                    | (<symbol> . <list-of-symbols>)

drop-last returns a list of all the symbols in los except the last one. For example:

> (drop-last '(a b c d))
'(a b c)
> (drop-last '(x))
'()

If los is the empty list, drop-last returns the empty list.

Problem 2

Write a structurally recursive function (tree-count test? tree) that takes two arguments: a boolean function test? and a binary tree tree of this form:.

<binary-tree> ::= <number>
                |  (<number> <binary-tree> <binary-tree>)

tree-count returns the number of values in the tree that satisfy test?, a one-argument function. For example:

> (tree-count positive? '(1 (2 3 4) 5))    ; all five are positive
5
> (tree-count (lambda (n) (> n 10))        ; nodes greater than 10
              '(8 (13 11 (5 24 6)) (15 (12 10 14) 20)))
7

Note: tree-count does not need an extra parameter to solve this problem! Use regular structural recursion.

Problem 3

Write a mutually recursive function (slist-contains? s slst) that takes two arguments, a symbol s and an s-list slst:

           <s-list> ::= ()
                      | (<symbol-expression> . <s-list>)

<symbol-expression> ::= <symbol> 
                      | <s-list>

slist-contains? returns true if s appears anywhere in slst, and false otherwise. For example:

> (slist-contains? 'eugene '((a b) (((b g r) (f r)) c (d e)) b))
#f

> (slist-contains? '/ '(if (zero? n) zero (/ total n)))
#t

slist-contains? must be mutually recursive with the function symexp-contains?, which determines if a symbol appears anywhere in a symbol expression.

Problem 4

Write brief answers for each of the following items. One phrase, sentence, or Racket expression is sufficient for each.

How does program derivation improve the efficiency of a mutually recursive solution?
This version of the factorial function is tail recursive:
```
(define (factorial n acc)
  (if (zero? n)
      acc
      (factorial (sub1 n) (* n acc))))
```
What does that statement mean, in terms of the way the function executes at run time?
In what situations do we need to define an interface procedure to our structurally recursive function?
The expression (occurs-free? x exp) is not always equal to the expression (not (occurs-bound? x exp)) for a given x and exp. Give an expression in the little language that illustrates why.

Extra Credit. There are two ways in which this is true. Give distinct examples that illustrate both ways.

Problem 5

Write a structurally recursive function (contains-varref? v exp) that takes two arguments: v, a symbol, and exp, an expression in our little language:

<exp> ::= <varref>
        | (lambda (<var>) <exp>)
        | (<exp> <exp>)

contains-varref? returns true if v is used as a variable reference anywhere in exp, and false otherwise. For example:

> (contains-varref? 'y 'y)
#t
> (contains-varref? 'x '(lambda (x) x))
#t
> (contains-varref? 'y '(lambda (y) x))
#f
> (contains-varref? 'x '(f (lambda (y) x)))
#t
> (contains-varref? 'x '( (lambda (x) y)      ; x is not used here
                          (lambda (y) x) ))   ; but x is used here
#t

You must use these syntax procedures for the little language, as provided in class.

exp?    varref?     lambda?            app?
                    lambda->param      app->proc
                    lambda->body       app->arg