An 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 1101 (13 decimal) would be represented by the list [1,0,1,1]. Of course, list [1,0,1,1,0] also represents the same number. Leading zeros are ignored. The empty list is a representation of 0.
Your assignment is to write four functions: sum, difference product and shift. The first three compute the sum, difference and product of two binary numbers, each represented as a list. The shift function shifts a number to the left (equivalent to multiplying by a power of 2). Expression shift(x,k) should produce the result of adding k 0's to the end of binary number x, or, equivalently, the result of multiplying x by 2k.
For example, sum([1,0,1], [0,0,1,1]) = [1,0,0,0,1] (since 5+12=17), difference([0,0,1,1], [1,0,1]) = [1,1,1] (since 12 - 5 = 7) and product([1,0,1], [0,1]) = [0,1,0,1] (since (5)(2) = 10). Your answers might have additional 0 bits on the end -- don't worry about that. The difference function requires its first number to be no smaller than its second, so that no negative numbers result.
Write the functions in an equational style in Astarte. Check your equations to see whether they make sense. Then test your functions!.
Make your functions definitions simple and elegant. Include a comment for each explaining what it does (but not how it works). That is, give a contract for each function.
Do not start worrying about Astarte syntax until you have a set of equations that you believe are correct and complete, in the sense that every case is covered by at least one equation. When you think you have solid equational definitions, code them up.
Think of a list as representing a number. Let N(L) be the number that list L represents. For example, N([0,0,1]) = 4. Then N(h::t) = h + 2*N(t). When inspecting equations, see if they make any sense when the lists are thought of as numbers. If you write A = B, then you should expect that N(A) = N(B). For example, one equation that you might write is
sum(0::a,0::b) = 0::sum(a,b)
Check this out, keeping in mind that sum(x,y) is supposed to be x+y.
In terms of the numbers that the lists represent, the equation says that
(0 + 2*a) + (0 + 2*b) = (0 + 2*(a+b))
which is easily seen to be true. Check all of your equations this way.
You will find some two helper functions useful. Write a function inc(x) that adds one to x and dec(x) that subtracts 1 from x. That is, if x is a list that represents number n, then inc(x) is a list that represents number n+1 and dec(x) is the list that represents the number n-1. (Function dec requires that its input be a positive number.) Think about the cases to handle. For example,
A manual is available for the Astarte programming language. You can also get the manual from the 320 lab server. Please look at that before writing and running your program. The most relevant pages for you are the following. Read only as much of each that appears to be useful. When you get into more esoteric looking material, try writing your program instead of just reading on.
You might want to consult these.
To test your functions, the simplest approach is just to compute a few values and see what they are. For example, here is a very rudimentary (and not very complete) test of the sum function.
Execute
Let six = [0,1,1].
Let seven = [1,1,1].
Let eight = [0,0,0,1].
Writeln[$six, " + ", $six, " = ", $(sum(six,six))].
Writeln[$six, " + ", $seven, " = ", $(sum(six,seven))].
Writeln[$six, " + ", $eight, " = ", $(sum(six,eight))].
Writeln[$seven, " + ", $eight, " = ", $(sum(seven,eight))].
%Execute
You may use the built-in arithmetic operators (+,-,...) provided they are only used on small numbers (say, less than 10). It is not acceptable to convert your binary list to a large integer and then to rely on the built-in operations to do the arithmetic. Do the algorithms yourself.
Turn in the source code of your program using the handin program. Put it all in one file. If your file is called arithmetic.ast, you will hand it in as follows.
alias handin "/export/stu/classes/csci3675/bin/handin csci3675" handin 2 arithmetic.astDo not include your tests. Be sure to include your name in the program.