Computer Science 3675
Section 001
Fall 2015
Programming Assignment 2

Overview of assignment

For this assignment, you will write the inc, sum, and product functions in an equational style instead of an imperative style. You will add to that functions to decrement, subtract and compare two numbers. You should find the function definitions fairly short and simple.

Language

The language that you will use is Cinnameg. There is material in the book on Cinnameg, and you can find documentation here. It is installed on xlogin.cs.ecu.edu as /usr/local/bin/cmgc [a compiler] and /usr/local/bin/cmgr [an interpreter].

Cinnameg is not a purely functional language, and it allows you to use other styles. For this assignment, stay within the functional subset for writing the arithmetic and comparison functions. Do not use loops, or anything that changes something once you have already created it. Write in a declarative, equational style.

Representing nonnegative integers

A nonnegative integer can be represented in binary as a list of zeros and ones. The most convenient way to do so is to put the least significant bit first. For example, the binary number 1101₂ (13 decimal) would be represented by the list [1,0,1,1].

Of course, list [1,0,1,1,0] also represents 13. Leading zeros in the standard representation (as in 01101) are ignored, and they become trailing zeros in the list representation. Say that a list is normalized if it does not end on 0. So [1,0,1,1] is normalized, but [1,0,1,1,0] is not. Note that list [0] is not normalized, but [ ] is. The empty list is a representation of 0.

The assignment

Your assignment is to write the following functions that work on lists as representations of binary numbers.

1. inc(x) produces x + 1 (as a normalized list);

2. dec(x) produces max(0, x − 1) (as a normalized list);

3. sum(x,y) produces x + y (as a normalized list);

4. diff(x,y) produces max(0, x − y) (as a normalized list);

5. product(x,y) produces the product x * y (as a normalized list)

6. compareInts(x,y) produces a character: '<' if x < y, '>' if x > y and '=' if x = y.

For example inc([1,1,0,1,0,1]) = [0,0,1,1,0,1], and sum([1,1], [1,1]) = [0,1,1]. The result list of each function should be normalized even if the parameter lists are not. For example, inc([1,0,0,0]) = [0,1].

Normalizing

You can handle normalization as follows. An example for inc should suffice. First, write an implementation of inc that does not worry about normalizing its result. You might call it incn, for inc-nonnormalizing. Now define inc as follows.

   Define inc x = normalize(incn x).

Requirement

Cinnameg allows large integers. But for this assigment, you are expected to handle the algorithms for adding and multiplying directly using lists of bits. It is not acceptable to convert from a list to an integer, do the arithmetic on the integers, and then convert back to a list. You are allowed to do arithmetic on small integers.

Hints

Think about the number that each list represents. Here are some correspondences between lists and numbers. In the left-hand column, x stands for a list of bits. In the right-hand column, x stands for the corresponding number.

For example, [0,1,1] = 0::[1,1]. List [1,1] stands for 3. So [0,1,1] stands for 2(3) = 6. Similarly, [1,0,0,1] = 1::[0,0,1]. List [0,0,1] stands for 4. So [1,0,0,1] stands for 2(4) + 1 = 9.

Think about your functions first in terms of arithmetic, then convert to lists. For example, you know that 2x + (2y + 1) = 2(x+y) + 1. Converting that to list notation, and writing sum(a,b) for a + b, yields

sumn(0::x, 1:y) = 1::sumn(x,y)

Writing and running the program

Write one package that defines and exports your definitions. Do not export the helper functions that do not normalize their results. See arithmetic.cmg for a skeleton of this module. Use the skeleton as a starting point.

Please include at least one Example for each function. I have written inc for you. You will need to provide dec, sum, diff, product and compareInts.

To test your functions, use a separate module. See here for a skeleton of a testing module. Add tests.

To compile arithmetic.cmg, use command

  cmgc arithmetic

  cmgc -l arithmetic

The interpreter is called cmgr. You will want to compile both arithmetic.cmg and arithtest.cmg and then run arithtest. To do that use commands

  cmgc arithmetic
  cmgc arithtest
  cmgr -t arithtest

Submitting your work

Be sure to include your name in the program. Also, near the top of the program, write the amount of time spent (in hours). When you are ready to turn in your work, use command

   ~abrahamsonk/3675/bin/submit 2 arithmetic.cmg arithtest.cmg