CSCI 6610		  Automata and Formal Languages		     Spring 2004

			       Homework set #1

		Due at the start of class on Thursday, January 22 
	     *** OR in 423 Boyd by 2:30 PM on Friday, January 23 ***

These exercises/problems are all from the text by Sipser except for the extra
credit problem at the end (decided on in class 01-15-04 after a question).

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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	4.10* (15 pts.)
	4.12* (15 pts.)
	4.13* (15 pts.)
	5.6   (10 pts.)
	5.12* (15 pts.)
	5.17* (15 pts.)
	5.19* (15 pts.)

    *For 4.10, let A = {<M>: M is a DFA WITH INPUT ALPHABET {0,1} which
     doesn't accept any string having an odd number of 1's}.
     HINT: Look at the proof of Thm. 1.12 to see how to construct
     algorithmically a DFA which recognizes the intersection of two regular 
     languages A and B, given as input individual DFAs for A and B.

    *For 4.12, HINT: If you are not familiar with the algorithm for finding the 
     generating symbols in a CFG then click on the "Two CFG Algorithms" link on
     the course web page and read the description.  You'll find this useful, but
     only after modifying the input CFG suitably.

    *For 4.13, HINT: Using thm. 2.19 AND its proof, you can algorithmically find
     a bound beta(G) such that 1* is contained in L(G) if and only if 1^k (the 
     string, of course) is in L(G) for all k in the range 0 <= k <= beta(G).  
     In fact, if p is a pumping length for L(G) then you should be able to prove 
     that beta(G) = p!+p will do. 

    *Modify 5.12 to show that S is not r.e. and also not co-r.e.; do this by 
     constructing direct reductions to S and to complement(S) from A_{TM}.

    *Hint for 5.17: Each domino in the set considered has top = 1^k and 
     bottom = 1^m for some non-negative integers k, m.  The key property for the
     PCP is the difference k-m.  What if this is 0 for some domino?  If no 
     domino has 0 difference, what if all the differences are > 0 ?  What if all 
     are < 0?  Finally, what if some are > 0 and some are < 0?

    *For 5.19 modify the construction suggested by the hint in the text so that
     you can give a direct reduction of MPCP (not PCP) to AMBIG_{CFG}.

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EXTRA CREDIT (25 pts.): prove that EQ_{TM} is mapping reducible to REG_{TM}.