CSCI 6610		  Automata and Formal Languages		     Spring 2004

			        Homework set #5

		      *** Due in class Monday, March 29 *** 

                 These problems are all from the text by Sipser.

NOTES: "Show" always means PROVE in the problem descriptions.
       Be sure to consult the note below for any problem followed by a * in the 
       listing, as this indicates a hint. 
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	7.16(a)* (10 pts.)
	7.16(b)  (10 pts.)
	7.21*    (20 pts.)
	7.23     (20 pts.)
	7.25     (20 pts.)
	7.34     (20 pts.)

*********************************   Hints   ************************************

* 7.16(a): you can make use of the fact that G contains a simple path from a to
	   b of length <= k iff G contains some path (not necessarily simple)
	   from a to b of length <= k.  This is because every a-b path, say P, 
	   contains a simple a-b path Q, and of course Q's length is <= P's 
	   length.  Then, use the "dynamic programming" idea yet again, to 
	   calculate the set R_i of vertices reachable from a by some path of 
	   length <= i, for i = 0, 1, ... , k.
* 7.21:	   reduce CLIQUE to HALF-CLIQUE to show NP-hardness.  To do this,
	   consider modifying the graph in a CLIQUE instance by adding disjoint
	   isolated vertices or else a complete graph joined to the original
	   with all possible edges (depending on how the specified clique size
	   compares to the number of nodes in the graph).