CSCI 6610		  Automata and Formal Languages		     Spring 2004

			        Homework set #4

		        Due by 2:30 PM on Friday, March 5

            These exercises/problems are all from the text by Sipser.

NOTES: "Show" always means PROVE in the exercise/problem descriptions.
       Be sure to consult the note below for any problem followed by a * in the 
       listing, as this indicates a hint and/or directions.  Hints are optional,
       but all other directions such as "modify", "let", "do",... are mandatory.
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	7.12  (15 pts.)
	7.15* (25 pts.)
	7.22* (25 pts.)
	7.29* (15 pts.)
	7.30* (20 pts.)

************   Directions and/or Hints   ****************

* 7.15: you may assume that the input is of the form
		#1^(x_1)01^(x_2)0...01^(x_k)01^t
	where {x_1, x_2, ... , x_k} is the sequence of non-negative integers
	and t is the target value for a subset sum.  This is a form of "unary"
	representation for the numbers.  It is possible to convert these to 
	binary, but that is not necessary, nor even helpful.  What IS helpful
	is the dynamic programming approach to keeping track of which numbers
	can be obtained as sums of subsets of {x_1, ... , x_i} for 
	i = 0, 1, 2, ... , k.

* 7.22: you will need the result of part (a) when you verify part (b);
	in proving that the existence of a != assignment for phi' implies that
	phi is satisfiable, you may need to take the negation of the
	!= assignment in order to find an assignment which satisfies phi.
	(Here phi is in 3-CNF, and we map from <phi> to <phi'> in reducing
	3-SAT to !=SAT>)

* 7.29: you can develop a satisfying assignment one variable at a time, 
	based on asking a series of questions about membership in SAT.

* 7.30: you can determine the maximum size of a clique in G by asking a 
	series of questions about membership in CLIQUE, then develop a 
	clique of that size one node at at a time, based on asking another 
	series of questions about membership in clique.