CS442, Spring 2006, HW #4 Answers

1.  It is not satisfiable.  There is no way we can set values for a,b
    and c to make the formula be true.  The easy (but tedious) way to
    check the answer is to try all 8 combinations.  A more complex
    proof is to notice that "a" must be false (there is a "not a"
    clause).  Since there is a "a or c" clause and "a" is false, "c"
    must be true.  But, there's also a "a or not c" clause, so "c"
    must be false.   Since it can't be both, the formula cannot
    satisfied.

2. (i) valid. (ii) not valid. (iii) valid

3. (i)   Show 2 consecutive numbers in the list that are not in order
         (if such 2 numbers exist, the list is out of order).
   (ii)  Show 2 repeated letters in the word (say the index of the
         letters in the word, for example, for the word "letter",
         indicate 3&4 (3rd and 4th letter)).
   (iii) Show the solution.
   (iv)  Indicate the 3 words: 2 words and the third one being their
         concatentation.  It is easy to check that the 3rd word is the
         concatentation of the 2 words and that all three are in the
         list.

4. Be creative.

5. (i) We have the words : spa, ran, shape, apt
	score = 4.
   (ii) If we switch "AN" and "AY" we get : ray, shape, spa, apt, yes,
	score = 5.  You could also say that swapping "SK" and "AY" we
        get: has, shape, spa, apt, pay, score = 5.  

6. while(condition) stays in the loop as long as condition is true.
   If condition is always true, the program will never halt.

	(i)   condition is (n == n+1).  This one is never true (no
	      matter how you pick an integer n, it will never be equal
	      to n+1) ANSWER: for any value of n, tuna1 will halt.
	(ii)  condition is (n>10) so the program will halt if n ever
	      gets < or = 10.  Now, we have to look if that's ever
	      possible.  If it's impossible that n ever gets <=10,
	      tuna2 will never halt.  We will look at what happens
	      inside the loop : n=n-1, so at each step, n decreases by
	      1.  If, at the beginning, n is bigger than 10, tuna2
	      will loop (at each step decreasing n by 1) until n will
	      eventually reach 10 and then it will stop looping,
	      because the condition will become false; if at the
	      beginning n <=10, tuna2 will stop because the condition
	      is false.  ANSWER: For any value of n, tuna2 will halt 
	(iii) condition is (n < 40).  What happens inside the loop: n
	      is decreased by 1 at each step; if n is bigger or equal
	      to 40 in the beginning, tuna3 will halt, because the
	      condition is not true; if n<40 in the beginning, tuna3
	      will never stop because at each step n becomes smaller
	      and smaller and there's no way it can get bigger or
	      equal to 40, so that to make tuna3 halt.  ANSWER : for
	      n>=40 tuna3 halts.

7.			L
			/\
		   L	       W
	          /\	       /\
	       W     W       L    L
	      /\     /\     /\    /\
             L  L   L  L   L  W  L  W
            /\  |   |  |   |  |  /\  \
           L  W W   W  W   W  L  L W  L

8. (i)   mystery1
   (ii)  mystery3
   (iii) mystery2