CSCI 6610		  Automata and Formal Languages		     Spring 2004

			        Homework set #3

		       Due by 2:30 PM on Friday, February 20  

            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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	7.2*     ( 6 pts.)
	7.5*     ( 9 pts.)
	7.6      (20 pts.)
	7.8      (15 pts.)
	7.11*    (10 pts.)
	7.13*    (20 pts.)
	7.20*    (20 pts.)


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

For 7.2:  NO EXPLANATION is needed!

For 7.5 and the rest:  DO EXPLAIN your answers.

For 7.11: the assumption is that the vertices of a graph with n nodes are the 
	  set {1,2,...,n}, so a "reodering" of the vertices is a permutation on
	  {1,2,...,n}.

For 7.13: Sipser's hint is intended to suggest dynamic programming, which is
	  necessary here. 

For 7.20: Sipser's hint is intended to lead you to repeated squaring in case t
	  is a power of 2, which can then be extended to an efficient method of
	  exponentiation to the power t for any natural number t.  Powers of a 
	  permutation follow all of the laws you are familiar with for powers of
	  a real number (except that the 0 power is the identity permutation
	  instead of the number 1).

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Extra Credit I (20 pts., but no easy partial credit; if you have an idea, please
discuss it with the instructor before carefully writing it up to turn in):
	prove that EQ_{TM} is mapping equivalent to INFINITE_{TM}.
(Here INFINITE_{TM} = {<M> | M is a TM with L(M) infinite}.  Mapping equivalent
means that each is mapping reducible to the other.) 

Extra Credit II (30 pts., but do EC I first and again please discuss with the
instructor before carefully writing it up to turn in):
prove that EQ_{TM} is mapping reducible to REG_{TM} and to complement(REG_{TM}).