P3

Scheme Programming

Objectives

• learn the basics of functional programming
• write a procedure that creates another procedure
• write recursive procedures to process lists

Specification

You should write the procedures described below. You should name each procedures as specified. (I need to know how to call them in order to test them.) All the procedures should return the specified result. Create a file called p3.scm which either contains all your code or loads files containing that code. I should be able to load p3.scm and then call all your functions. The last line of the file should be a display statement like

(display "p3 loaded\n")

• (polarToCartesian distance degrees) Define a function that takes polar coordinates (a distance from the origin and an angle in degrees) for a point and returns a list containing the x and y coordinates for the point. The built-in functions sin and cos need the angle to be converted to radians.

  (polarToCartesian 1 45)

  should return (.707 .707).
  
• (filter predicate list) Define a function that takes a predicate function and a simple list (all elements of a simple list are atoms) and returns a list containing only those elements for which (predicate element) is true.

  (filter even? '(1 2 3 4 5 6 7 8 9))

  should return (2 4 6 8).

  Would it ever make sense to filter nested lists in a similar way?
  

• (count-occurrences symb list) Define a function which takes a symbol followed by a list as parameters and returns the number of times that symbol occurs in the list. The list parameter could have nested lists inside it, you need to count the occurrences in those as well.

   (count-occurrences 'x '(x y x z))
  

  should return 2 and

   (count-occurrences 'x '((f x) y ((x z) x)))
  

  should return 3
  
• (minmax lon) Define a function that takes a list of numbers (you can assume that the list has only numbers in it) and returns a list containing the largest and smallest values in the list. You will probably find this is easier if you use let. It will be worth more points with a let than without.

  (minmax '(2 4 6 8 9 7 5 3 1))

  should return (1 9)
  
• (horner-eval coeff x) Define a function which has a list containing the coefficients of a polynomial (with the coefficient of x⁰ first and the coefficient of the highest power of x last)

  a₀ + a₁x + ... + a_n-1x^n-1 + a_nxⁿ

  and a value of x, and calculates the value of the polynomial at x using Horner's rule which restructures the calculation in the following way:
  

  a₀ + x(a₁ + x( a₂ + ... + x(a_n-1 + x a_n)...))

  
  For example, you would calculate
  

  1 + 3x + 5x³ + x⁵

  
  at x=2 by typing
  

      (horner-eval '(1 3 0 5 0 1) 2)    
  

  which should return 79.
• In order to handle large amounts of data, Google uses a programming model that has a map function that generates an intermediate set of data and a reduce function that somehow combines those results into a final result. This programming model is relatively easy to spread across multiple processors. (For more information about map-reduce, see the Google paper and Joel-on-Software.) In this exercise, you are going to implement a simplified programming model which applies a function to each element of a list and then combines these elements into a final value.
  • (mymap fun lis) The map function in Scheme takes a function and a list as arguments and applies the function to each element of the list. For example, (map square '(1 2 3 4)) produces (1 4 9 16) if square is a function that squares its argument. Write your own version of the map function and call it mymap.

    (mymap sqrt '(1 4 9 16))

    should produce (1 2 3 4)
  • (map-reduce fn1 fn2) Write a function called map-reduce that takes two functions as arguments and returns a function of a single argument which is a list. If map-reduce is called using (map-reduce f1 f2), the returned function maps f1 onto the list and then applies f2 to the resulting list to reduce it to a single value. Scheme has a function called apply which you can use to do the reduction part of map-reduce. apply takes a function and a list as arguments.

    ((map-reduce sqrt +) '(1 4 9 16))

    should produce 10
  • (sum-squares lis) Use map-reduce to create a function that sums the squares of the list that is its argument. You'll need to provide a function that squares its argument. (This one is basically trivial if you got the previous two parts working.)

Testing

I will put file called test3.scm which performs minimal testing of your implementation and show.scm which test3.scm uses, into the directory on onyx. Put both files in with your assignment files and load test3 into Scheme. Your procedures should at least load properly before running this test. I will do more extensive testing when I grade the assignment so it is to your advantage to think of as many ways to test your procedures as possible.

The functions you are defining are all independent of each other so you can get partial credit even if you don't get them all working. To make testing easier, any function you don't get working should have a stub with the same name that returns a message. Put your non-working code in a separate file and list it's name in your README file so I can look at what you tried.