Exam 1: Scheme Fundamentals

Distributed: Friday, 22 September 2006
Due: 9:00 a.m., Friday, 29 September 2006
No extensions.

This page may be found online at http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2006F/Exams/exam.01.html.

This exam is also available in PDF format.

Contents

• Preliminaries
• Problems
  • Problem 1: Determining Types
  • Problem 2: Simulating if and cond with or and and
  • Problem 3: Analyzing Append
  • Problem 4: Other List Operations
• Some Questions and Answers
• Errors
• Other Extra Credit

Preliminaries

There are four problems on the exam. Some problems have subproblems. Each problem is worth twenty-five (25) points. The point value associated with a problem does not necessarily correspond to the complexity of the problem or the time required to solve the problem.

This examination is open book, open notes, open mind, open computer, open Web. However, it is closed person. That means you should not talk to other people about the exam. Other than as restricted by that limitation, you should feel free to use all reasonable resources available to you. As always, you are expected to turn in your own work. If you find ideas in a book or on the Web, be sure to cite them appropriately.

Although you may use the Web for this exam, you may not post your answers to this examination on the Web (at least not until after I return exams to you). And, in case it's not clear, you may not ask others (in person, via email, via IM, by posting a please help message, or in any other way) to put answers on the Web.

This is a take-home examination. You may use any time or times you deem appropriate to complete the exam, provided you return it to me by the due date.

I expect that someone who has mastered the material and works at a moderate rate should have little trouble completing the exam in a reasonable amount of time. In particular, this exam is likely to take you about four to six hours, depending on how well you've learned topics and how fast you work. You should not work more than eight hours on this exam. Stop at eight hours and write There's more to life than CS and you will earn at least 80 points on this exam.

I would also appreciate it if you would write down the amount of time each problem takes. Each person who does so will earn two points of extra credit. Since I worry about the amount of time my exams take, I will give two points of extra credit to the first two people who honestly report that they've spent at least five hours on the exam or completed the exam. (At that point, I may then change the exam.)

You must include both of the following statements on the cover sheet of the examination. Please sign and date each statement. Note that the statements must be true; if you are unable to sign either statement, please talk to me at your earliest convenience. You need not reveal the particulars of the dishonesty, simply that it happened. Note also that inappropriate assistance is assistance from (or to) anyone other than Professor Rebelsky (that's me) or Professor Davis.

1. I have neither received nor given inappropriate assistance on this examination.
2. I am not aware of any other students who have given or received inappropriate assistance on this examination.

Because different students may be taking the exam at different times, you are not permitted to discuss the exam with anyone until after I have returned it. If you must say something about the exam, you are allowed to say This is among the hardest exams I have ever taken. If you don't start it early, you will have no chance of finishing the exam. You may also summarize these policies. You may not tell other students which problems you've finished. You may not tell other students how long you've spent on the exam.

You must present your exam to me in two forms: both physically and electronically. That is, you must write all of your answers using the computer, print them out, number the pages, put your name on the top of every page, and hand me the printed copy. You must also email me a copy of your exam. You should create the emailed version by copying the various parts of your exam and pasting them into an email message. In both cases, you should put your answers in the same order as the problems. Failure to name and number the printed pages will lead to a penalty of two points. Failure to turn in both versions may lead to a much worse penalty.

In many problems, I ask you to write code. Unless I specify otherwise in a problem, you should write working code and include examples that show that you've tested the code.

Just as you should be careful and precise when you write code and documentation, so should you be careful and precise when you write prose. Please check your spelling and grammar. Since I should be equally careful, the whole class will receive one point of extra credit for each error in spelling or grammar you identify on this exam. I will limit that form of extra credit to five points.

I will give partial credit for partially correct answers. You ensure the best possible grade for yourself by emphasizing your answer and including a clear set of work that you used to derive the answer.

I may not be available at the time you take the exam. If you feel that a question is badly worded or impossible to answer, note the problem you have observed and attempt to reword the question in such a way that it is answerable. If it's a reasonable hour (before 10 p.m. and after 8 a.m.), feel free to try to call me in the office (269-4410) or at home (236-7445).

I will also reserve time at the start of classes next week to discuss any general questions you have on the exam.

Problems

Problem 1: Determining Types

Topics: Types, Predicates

Your colleagues, Tyra and Tyler Typer, are thrilled that we've encountered a wide variety of basic types in our exploration of Scheme, including integers (exact and inexact), complex numbers, characters, strings, input-ports, and even procedures. However, they are concerned that Scheme provides no obvious mechanism for determining the type of a value.

They've turned to me to write such a procedure, and I've decided to delegate the work to you. Write a procedure, (type-of val) that returns the type of value val as a symbol. If the value has a type that you don't know, report a type of unknown. For example,

> (type-of 1)
exact-integer
> (type-of 3.4)
inexact-real
> (type-of null)
list
> (type-of list?)
procedure
> (type-of (cons 1 2))
unknown

As you might expect, one of the goals of this question is for you to figure out what types you know about.

Problem 2: Simulating if and cond with or and and

Topics: Booleans, Conditionals

In the reading on conditionals, we noted that it is possible to simulate the behavior of and and or using if and cond. Bob and Bonnie Boole (no relation to George Boole) think the converse should be possible. That is, they think that you should be able to rewrite if and cond clauses using just or and and. Unfortunately, they are better philosophers than coders. Hence, they have turned to you to work out the details.

a. Consider the following typical form of if

(if test 
    consequent 
    alternate)

Write an equivalent piece of code using and, not, and or.

b. Consider the following prototypical cond expression.

(cond
  (test0 consequent0)
  (test1 consequent1)
  (test2 consequent2)
  (else alternate))

Write an equivalent piece of code using and, not, and or.

Problem 3: Analyzing Append

Topics: Lists, Recursion, Efficiency

My computer science teachers, I. H. Ate and A. P. Pend, regularly discouraged me from using append in my list-building procedures, particularly my recursive list-building procedures. They never told me why, but I finally figured it out through experimentation. Following the philosophy that you learn better by doing than by being told, I challenge you to replicate my experiments (with some guidance).

We begin by defining our own version of cons that lets us figure out how often it gets called in a typical procedure.

(define my-cons
  (lambda (head tail)
    (display 'cons)
    (newline)
    (cons head tail)))

This definition by itself is not very interesting. Hence, we will use it in a prototypical definition of append.

(define my-append
  (lambda (first second)
    (if (null? first)
        second
        (my-cons (car first) (my-append (cdr first) second)))))

We can now see their interaction.

> (my-append (list 'a 'b 'c) (list 'd 'e 'f))
cons
cons
cons
(a b c d e f)

a. If you use my-append to join a list of length n to a list of length m, how many times does my-cons get called?

Now, let's write a reverse procedure that uses my-append (the thing Professors Ate and Pend so disliked). Since we need to build a singleton list for appending, we also write singleton.

(define singleton
  (lambda (sym)
    (my-cons sym null)))

(define my-reverse
  (lambda (lst)
    (if (null? lst)
        null
        (my-append (my-reverse (cdr lst)) (singleton (car lst))))))

Now, let's try it

> (my-reverse (list 'a 'b 'c))
cons
cons
cons
cons
cons
cons
(c b a)

Yes, the result is correct. We also call cons six times to make the result. Does that make sense? Well, there are three calls to create the singleton lists, and then we have to join them together, so I guess that's okay. But let's be safe and check a little more. What happens if we try to reverse a list of length four?

> (my-reverse (list 'a 'b 'c 'd))
cons
cons
cons
cons
cons
cons
cons
cons
cons
cons
(d c b a)

Hmmm ... that's a bit strange. The result is correct and has four elements, but it took ten calls to cons. I guess we should explore a bit more.

> (my-reverse (list 'a 'b 'c 'd 'e))
cons
... thirteen outputs elided
cons
(e d c b a)

Wow! Fifteen calls to cons for a list of size five. What is going on?

b. In your own words, explain why it is taking so many more calls to cons in the larger lists. You may find it helpful to try larger lists.

After observing these problems, I rewrote reverse to use a helper that accumulates the reverse as we go.

(define my-reverse
  (lambda (lst)
    (my-reverse-helper null lst)))
(define my-reverse-helper
  (lambda (reversed-so-far remaining)
    (if (null? remaining)
        reversed-so-far
        (my-reverse-helper 
         (my-append (singleton (car remaining)) reversed-so-far)
         (cdr remaining)))))

Is this any better? Let's see ... For a list of size three, it does six calls to cons. About the same. For a list of size four, it does eight calls to cons. Better. For a list of size five, it does ten calls to cons. Still better.

c. In your own words, explain why this new version is so much better.

Of course, it seems a bit silly to make a one-element list if we're going to immediately append it to another list. We should instead use cons (or, for testing, my-cons).

(define my-reverse
  (lambda (lst)
    (my-reverse-helper null lst)))
(define my-reverse-helper
  (lambda (reversed-so-far remaining)
    (if (null? remaining)
        reversed-so-far
        (my-reverse-helper 
         (my-cons (car remaining) reversed-so-far)
         (cdr remaining)))))

d. Explain the benefits of this improvement.

Problem 4: Other List Operations

Topics: List operations, Recursion

Lisa and Lisle Lister note that in the problem above, and in homework 6, we were able to define our own versions of built-in list operations. When they asked me about this, I told them that most of the built-in list operations can be defined in terms of only a few, more primitive operations (cons, car, cdr, null, and null?). That comment inspired this problem.

a. Define (my-cadr lst), which extracts the second element of a list, using only the five primitive operations above. (You need not use all five.)

b. Define (my-cddr lst), which extracts all but the first two elements of a list, using only the five primitive operations above. (You need not use all five.)

c. Define (my-length lst), which determines the length of lst, using the five primitives above, conditionals, and any numeric operations you deem appropriate. Note that my-length probably needs to be defined recursively.

d. Define (my-list-ref index lst), which extracts the element of lst that is preceded by index elements. Again, you may use the five primitives above, conditionals, and any numeric operations you deem appropriate. Note that my-list-ref probably needs to be defined recursively.

If you'd prefer to define this as (my-list-ref lst index), thereby matching the order of list-ref, that's fine, too.

Some Questions and Answers

These are some of the questions students have asked about the exam and my answers to those questions.

Problem 1

Can we use integer? and real? and other predicates, or do we have to define them ourselves?

It is quite difficult to define these predicates yourself, so you should use the built-in ones.

Do we have to check for odd, even, and negative numbers?

No.

Will we get extra credit if we do?

If you consider negative values extra, then yes.

While doing problem 1 on the exam, I recalled that Scheme uses real and rational almost interchangeably. Would you prefer us using one or the other when writing type-of, or is it just a matter of personal preference?

I prefer rational for exact numbers and real for inexact, but it is up to you.

Problem 2

Do we have to use all three of and, or, and not, or only those that we deem necessary?

Only those you consider necessary.

Do you have to write working code, or just something equivalent?

Just something equivalent.

Can I use the eq? procedure?

No.

Problem 3

No questions received yet.

Problem 4

You specify for parts 3 and 4 that conditionals are ok. Does that preclude us from using if on parts 1 and 2, then?

You may use if on parts 1 and 2, but you should be able to do without it. (Note that you do not have to check preconditions for either procedure.)

Errors

Here you will find errors of spelling, grammar, and design that students have noted. Remember, each error found corresponds to a point of extra credit for everyone. I usually limit such extra credit to five points. However, if I make an astoundingly large number of errors, then I will provide more extra credit.

• Sam wrote and if or cond clause when he probably meant an .[MP, 1 point]
• You must present you exam to me should be your exam. [TP, 1 point]
• In transcribing a question, Sam forgot the symbol that indicates that the question is, indeed, a question. (That is, he forgot the question mark.) [PR, 1 point]
• Sam reversed the parameters to index-of. [AB, 1 point]

Other Extra Credit

I challenged you to find a reasonable test that tri.2211.ss failed to pass. A few people tried, but as one wrote To be honest, the tests that I didn't get to work were limitations of Scheme rather than the program not working.

For those people's valiant efforts, everyone gets two points of extra credit.