CSCI 6610		  Automata and Formal Languages		     Spring 2004

			        Homework set #7

		  Due in class on *** Thursday, April 29 ***

    These exercises/problems are all from the text by Sipser except for 1.

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.
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	1*.   (25 pts.)  Prove that CYCLIC is NL-complete.
	8.13* (25 pts.)
	9.7(b) (5 pts.)
	9.7(d) (5 pts.)
	9.7(h) (5 pts.)
	9.7(i) (5 pts.)
	9.14  (15 pts.)
	9.15  (15 pts.)

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

* 1: a directed cycle in a digraph is a directed walk 
   (following the directions of the edges) containing at least one edge,
   which returns to the starting node (also called  a closed walk) and 
   which repeats no nodes except for returning to the start.
   A digraph is called cyclic iff it contains a directed cycle.
	CYCLIC = { <G> : G is a cyclic digraph }
   You may use the obvious fact that G is cyclic iff G contains a closed
   directed walk with at least one edge (so you don't have to check for
   repeated nodes).  

* 8.13: let TQCNF denote the subset of TQBF in which the matrix of the
   totally quantified boolean formula is in conjunctive normal form (CNF).
   The problem is to prove that TQCNF is PSPACE-complete.  You are to prove
   the PSPACE-hardness by showing that TQBF is polynomial-time reducible
   to TQCNF.  This will require some extra boolean variables, suitably
   quantified.