CS 257 (Spring 2004) Homework 4*

• Problem 1 (Bacterial Chemotaxis)

• Problem 2 (Marroquin Pattern)

  The Marroquin pattern of order n is formed by superimposing three square arrays of points of size 2n + 1. Consequently, each array has an odd number of rows and columns. The arrays are rotated about the center (i.e., n + 1) point. The orientations of the arrays (relative to the x-axis) are 0, 60, and 120 degrees. You will write a function called marroquin which takes a single integer argument, n, and returns the Marroquin pattern of order n. Example:

  >(marroquin 9)
  #<struct:graphic>
  >
  

  

• Problem 3 (Pig Latin)

  The following description of Pig Latin is taken directly from Wikipedia:
  • For words that begin with consonants, move the leading consonants to the end of the word and add "ay." Thus, "ball" becomes "all-bay"; "button" becomes "utton-bay"; "star" becomes "ar-stay"; "three" becomes "ee-thray";
  • For words beginning with "qu," move the "qu" to the end of the word and add ay. Thus "question" becomes "estion-quay"; and
  • For words that begin with vowels, simply add "way" to the end of the word. Thus, "a" becomes "a-way"; "at" becomes "at-way"; "ermine" becomes "ermine-way."
  Write a Scheme function which will take an English word and returns the translation of the word into Pig Latin (both represented as strings).

• Problem 4 (Recognizing Tail Recursive Definitions)

  The definition of a function, foo, is a <tail definition> if it has the following form:

  (define foo (lambda (<symbol>*) <tail wrt foo>))

  where <symbol>* is zero or more symbols and an expression is <tail wrt foo> if it is not an application of foo:

  <contains no foo applications>

  or

  if it is an application of foo to zero or more expressions which contain no applications of foo:

  (foo <contains no foo applications>*)

  or

  if all applications of foo are contained in expressions which are (themselves) tail recursive with respect to foo and which occur in any of the following contexts:
  • (if <contains no foo applications> <tail wrt foo> <tail wrt foo>)
  • (and <contains no foo applications>* <tail wrt foo>)
  • (or <contains no foo applications>* <tail wrt foo>)

  where <contains no foo applications> is an expression which contains no applications of foo, and <contains no foo applications>* is zero or more expressions which contain no applications of foo.

  Write a function, tail-definition?, which takes an expression, expr, as an argument, and returns #t if the expression is a <tail definition> and #f otherwise. For example:

  > (tail-definition? '(define fact (lambda (x acc) (if (= x 0) acc (fact (- x 1) (* x acc))))))
  #t
  > (tail-definition? '(define fact (lambda (x) (if (= x 0) 1 (* (fact (- x 1)))))))
  #f
  > (tail-definition? '(define member? (lambda (item ls) (and (not (null? ls)) (or (eq? item (car ls)) (member? item (cdr ls)))))))
  #t
  >
  

  Hint: Divide and conquer. Write helper-functions, tail-wrt? and no-applications-of?. Both take a function name, fname, and an expression, expr, as arguments. tail-wrt? returns #t if expr is tail with respect to fname and #f otherwise. no-applications-of? returns #t if there are no applications of fname in expr, and #f otherwise.

  Note: You may assume that there are no let's, let*'s, letrec's, or cond's.

• Problem 5 (Polynomial Arithmetic)

  Get the code for polynomial arithmetic from the class homepage. Re-write the function, p*, which multiplies two polynomials, so that it is completely tail-recursive. Hint: This should be done in two parts. First, re-write the subroutine, t*, which multiplies a polynomial by a term so that it is tail-recursive. After you have tested p* and verified that it works correctly with the tail-recursive t*, then re-write p* itself, so that the entire function is tail-recursive. You will need to add tail-recursive subroutines to the letrec (e.g., t*-it and p*-it) to do this.

What You Should Hand In

• Listings of all your code.
• Representative graphic output (where applicable).

When You Should Hand It In

* This webpage is located at http://cs.unm.edu/~williams/cs257/homework4s04.html