function exp(base, power: integer): integer;
    var ans:integer
    begin
        ans:= 1;
        while power > 0 do
            if odd(power) then
                begin
                    ans := ans*base;
                    power:= power- 1
                end
            else
                begin
                    base := base * base;
                    power := power div 2
                end;
        exp := ans
    end;

a. Your assignment is to write a high-level function to support list abstractions. The language Miranda allows the user to write list abstractions of the form:

   [f(x) | x <- startlist; cond(x)]

where startlist is a list of type 'a, f : 'a -> 'b (for some type 'b), and cond : 'a -> bool. This expression results in a list containing all elements of the form f(x), where x is an element in the list, "startlist", and "cond(x)" is true. For example, if sqr(x) = x*x and odd(x) is true iff x is an odd integer, then

    [sqr(x) | x <- [1,2,5,4,3]; odd(x)]

returns the list [1,25,9] (that is, the squares of the odd elements of the list - 1,5,3). Note that the list returned preserves the order of startlist.

You are to write a function

    listcomp : ('a -> 'b) -> ('a list) -> ('a -> bool) -> ('b list)

    listcomp f startlist cond = [f(x) | x <- startlist; cond(x)].  

(Hint: One way to do this is to divide the function up into two pieces, the first of which calculates the list, [x | x <- startlist; cond(x)], and then think how the "map" function can be used to compute the final answer. It's also pretty straightforward to do it all at once.)

b. Test your function by writing a function which extracts the list of all names of managerial employees over the age of 60 from a list of employee records, each of which has the following fields: "name" which is a string, "age" which is an integer, and "status" which has a value of managerial, clerical, or manual. (Be sure to define this function correctly. I'm always amazed at the number of people who miss this problem by carelessness!)

c. Generalize your function in part a to

    listcomp2 g slist1 slist2 cond = 
                            [g x y | x <- list1; y <- list2; cond x y].

   - parenBalance "(a*[b-(c/d)]-r)"; 
   val it = true : boolean 
   - parenBalance "(a*[b-(c/d]] -r)"; 
   val it = false : boolean 
   - parenBalance "())(()"; 
   val it = false : boolean

Use the definition:

    datatype 'a stack = Empty | Push of 'a * ('a stack )

and be sure to write a pop routine which returns the stack after the top element is removed. (Hint: It is trivial if you use pattern matching.) You can define pop Empty = Empty, or, if you are willing to learn about exceptions on your own, raise an exception.

You may find it useful to use the built-in function, explode, which takes a string to a list of the characters in the list. E.g., explode "hello" = [#"h",#"e",#"l",#"l",#"o"].

        e ::= n | id | ~e | e + e | e - e | e * e | e / e | (e) 

Rather than working directly with the syntax above, we will presume that we have a parser which parses input into an abstract syntax tree, which your interpreter should use. Abstract syntax trees are just a convenient method of expressing parsed expressions; the definition of the ML datatype is

    datatype arithExp = 
                AST_NUM of int | AST_NEG of arithExp | 
                AST_PLUS of  (arithExp * arithExp) |
                AST_MINUS of  (arithExp * arithExp) |
                AST_PRODUCT of  (arithExp * arithExp) |
                AST_QUOTIENT of  (arithExp * arithExp) |
                AST_ID of string | AST_ERROR of string;

Note that we no longer need the parentheses to group terms, as the example given above would simply be represented by:

        AST_PRODUCT (AST_NUM 3, AST_MINUS(AST_NUM 5,AST_NUM 1))

You are to write an ML function evaluate that takes an abstract syntax tree representing an arithExp and returns the result of evaluating it. Most of the time when you evaluate the tree, you will get an integer, but if evaluating the expression results in a division by zero, it should return an error message. To allow both of these to be returned, we define

        datatype arithValue = NUM of int | ERROR;

To get you started, here is the beginning of the definition of evaluate:

        fun evaluate (AST_NUM n) = NUM n
            evaluate (AST_ERROR str) = ERROR
          | evaluate ...

Finally, to make your life easier, I have written a parser in ML that will take a correctly written arithmetic expression and return the equivalent element of type arithExp. Thus you can use this to create examples to try out your evaluate function.

If you wish to use the parser, copy it from the link above and save it to a file named "parseArith.ml" in the same directory as your homework. Now the file containing your evaluate function should start with:

    use "parseArith.ml";

    fun evaluate (AST_NUM n) = NUM n
      | evaluate ...  

To use the parser, you apply the function parsestr to a string representing an arithmetic expression and you get out the corresponding element of type arithExp:

- parsestr "34-7*(5-2)";
val it =
  AST_MINUS
    (AST_NUM 34,AST_PRODUCT (AST_NUM 7,AST_MINUS (AST_NUM 5,AST_NUM 2)))
  : arithExp
- evaluate it;

- evaluate (parsestr "34-7*(5-2)");
val it = NUM 13;