Computer Science 3675
Fall 2000
Answers to practice questions for quiz 2

  1. Given the definition 
          f([])   = []
      f(h::t) = (h*h)::f(t)  when h > 10
      f(h::t) = f(t)         when h <= 10
    show an inside-out evaluation of expression f([4,12,15,6,11]). Assume that arithmetic is done as soon as possible. 
        f([4,12,15,6,11]) = f([12,15,6,11])      (since 4 <= 10)
                      = 144::f([15,6,11])    (since 12 > 10 and 12*12=144)
                      = 144::225::f([6,11])  (since 15 > 10 and 15*15=225)
                      = 144::225::f([11])    (since 6 <= 10)
                      = 144::225::121::f([]) (since 11 > 10 and 11*11=121)
                      = 144::225::121::[]
                      = 144::225::[121]
                      = 144::[225,121]
                      = [144,225,121]
    Remarks. Be sure to follow the rules exactly. Do not forget about the [] at the end.

  2. Write an equational definition of a function called smallest so that smallest(n,x) is the smallest member of list n::x. For example, smallest(3, [6,4,7]) = 3 and smallest(8, [2,5]) = 2. You may presume that you have a function called min that takes the minimum of two numbers. For example, min(7,4) = 4. Here are two solutions.

    Definition 1: does not use min. 

        smallest(n,[])   = n
    smallest(n,h::t) = smallest(n,t) when n <= h
    smallest(n,h::t) = smallest(h,t) when n > h

    Definition 2: uses min. 

        smallest(n,[])   = n
    smallest(n,h::t) = smallest(min(n,h),t)

  3. By choosing among the operations lfold, rfold, map and select, write a definition for each of the following that does not use recursion or loops.

    1. Write a function called doubleAll that takes a list x of numbers as its parameter and produces a list of the doubles numbers in list x as its result. For example, doubleAll([5,2,19,3]) = [10,4,38,6]. 
             double(x) = x + x
       doubleAll = map double

      Remark. You can always use helper functions. This definition still does not have any recursion, either direct or mutual.

    2. Write a function called firstPos that returns the first positive number from a list of numbers. For example, firstPos([-4, -2, 0, 6, 12, -9]) = 6. Presume that the list has at least one positive number in it. 
             positive(x) = x > 0
       firstPos = select positive

  4. In a purely functional language, is it ever possible to compute the same expression twice in the same context and get different values? For example, if E is an expression, and you compute E two times, one right after another, could the first computation yield a different result from the second computation? Why or why not?

    Two evaluations of the same expression in the same context must always yield the same value. This is because there are no variables that can be changed.

  5. Consider the following definition written in Astarte. 
        Define f(?n) = (:n*n :: f(n+1):).
    What value does expression f(2) compute? Give the full value, as it would be if it were examined. (It suffices to give a clear description of the full value.)

    f(2) = [4,9,16,25,...]

    (an infinite list of the squares of the numbers 2,3,4,...)