An Introduction to Scheme Predicates

Summary: The textbook discusses the following common Scheme predicates:

Predicate	Example that returns True (#t)	Comment
number?	(number? 3.1415)	Is argument a number?
symbol?	(symbol? 'pi)	Is argument a symbol?
boolean?	(boolean? #t)	Is argument a boolean value?
pair?	(pair? '(a b))	Is argument a [cons] pair?
null?	(null? '())	Is argument the empty list?
procedure?	(procedure? car)	Is argument a procedure?
eq?	(eq? 'a 'a)	Do arguments represent identical symbols?
eqv?	(eqv? (car '(a a)) 'a)	Similar to eq?
equal?	(equal? '(b c) (cdr '(a b c)))	Are arguments the same symbols, numbers, booleans, or lists?

Steps for this Lab:

1. Check each of the above examples to confirm that the predicates return true (#t) as expected. In each case, explain why this result is obtained.

2. For each of the above predicates, find two rather different examples where the result is false (#f). Again, explain why this result is obtained.

3. eq? and eqv? produce similar results, although the internal details of how comparisons are performed within Scheme may be somewhat different. Both of these standard functions (and all other elements of Scheme) are defined formally in the Revised (5) Report on the Algorithmic Language Scheme.

   This document is available through Netscape:

   • If you are viewing this lab directly through Netscape, click here to obtain the document from a server at M.I.T. (If the M.I.T. server is down, the earlier Revised (4) Report is available from the University of Indiana by clicking here.)
   • If you are reading a printout of this lab, proceed as follows: First, within Netscape, click on the file option and then Open URL.
     Next, in the box that comes up, type the address http://www-swiss.ai.mit.edu:80/~jaffer/r5rs_toc.html
     (The alternative address from Indiana is http://www.cs.indiana.edu/scheme-repository/R4RS/r4rs_toc.html .) Finally, click on the button Open in Browser.

   To find information on eq?, scroll down the table of contents to the main heading Standard procedures, and click on the subheader Equivalence predicates. When the Standard procedures ... Equivalence predicates page appears, scroll down to find descriptions for both eq? and eqv?. While the discussion may contain more detail that you really want or need, note that this document gives a precise statement about how Scheme works. Also, the description contains several examples.

   Within Netscape, click the Back button once to return to the Table of Contents for the Scheme report. Since this document can serve as a valuable reference, you may want to record the URL address for future use. The easiest way to do this is to move the mouse to the Bookmarks menu and select the Add Bookmark option.

   Now, move to the Bookmarks menu again and note that this Revised (5) Report ... is listed. In the future, you can return to this page just by selecting this bookmark.

4. Use the Revised (5) Report ... to answer each of the following:

   1. Find a procedure which gives the integer quotient of two integer values; that is, the procedure should return the quotient of two integers, ignoring any remaineder. Thus, the integer quotient of 7 and 2 should be 3.

   2. Find a procedure which returns the denominator of a fraction (after reducing to lowest terms). Thus, this procedure should return 2 when applied to both 17/2 and 34/4.
   3. Find a procedure which returns the largest number on a list of numbers.

5. Review pp. 26-29 in the textbook, and run the following comparisons:

   
   (eq? 'a 'a)        (eqv? 'a 'a)        (equal? 'a 'a)
   (eq? 'a 'b)        (eqv? 'a 'b)        (equal? 'a 'b)
   (eq?  3  3)        (eqv?  3  3)        (equal?  3  3)
   (eq?  3  4)        (eqv?  3  4)        (equal?  3  4)
   (eq? '(a) '(a))    (eqv? '(a) '(a))    (equal? '(a) '(a))
   (eq? "abc" "abc")  (eqv? "abc" "abc")  (equal? "abc" "abc")
   

   In each case, be sure you can explain the results obtained.

6. Run the example that starts on the bottom of page 26 in the textbook, to be sure you understand the results obtained.

7. Textbook Exercises: Do exercises 1.10 and 1.11 from the textbook. In each case, the exercise asks you to state a conclusion. After you have a tentative answer, run several examples to test whether your conclusion is true, and revise your answer if necessary. Once you are confident in your answer, argue why your conclusion is correct.

8. Textbook Exercises: Work on exercises 1.12-1.15 from the textbook. In each case, check your answer by entering the expression into Chez Scheme. Be sure you can justify each result.

This document is available on the World Wide Web as

http://www.math.grin.edu/~walker/courses/151.fa98/lab-predicates.html