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CSC 236               Tutorial Exercises for Week 10              Fall 2007
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A.  This is the example in section 2.5 of the textbook, rephrased to use
    the same notation I use in lecture.

    Show that the following algorithm terminates:

        # Pre:  x, y in N
        while x != 0 or y != 0:
            if x != 0:
                x -= 1
            else:
                x = 16
                y -= 1


B.  Show that the following algorithm terminates:

        # Pre:  x in N, x >= 1
        while x != 1:
            if x % 2 == 0:  x /= 2
            else:           x = 3 * x + 1

    WARNING:  This is not as easy as it looks!
    Trace it for some values of x (or code it up to display the sequence of
    values generated) to get a feeling for what's going on.


C.  (a) List the elements of {a,ab}*.  Find a way to describe strings in
        this language, i.e., find a predicate P(s) such that P(s) is true
        iff s in {a,ab}*, for all strings s over alphabet {a,b,c}.

    (b) Find three different examples of a language L over alphabet {a,b,c}
        such that L = L.L

    (c) Find three different examples of a language L over alphabet {a,b,c}
        such that L = L*

    (d) Find three different examples of languages L_1, L_2 over alphabet
        {a,b,c} such that L_1.L_2 = L_2.L_1