CSCI 6610		  Automata and Formal Languages		     Spring 2004

	 Spring 2003 final exam, first 3 questions, for midterm review
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In general it is permissible to quote theorems and facts proved in class, in the
text through Chapter 7, or for written homework,  without additional 
justification.  When you do rely on such theorems or facts, state them before
going on to use them in your proof.	
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			     Notation Explanations
	(n)_2 denotes the string which is the binary representation of n.
	1^n denotes the string which consists of n copies of 1.
	N_0(w) denotes the number of 0's in the string w.
	N_1(w) denotes the number of 1's in the string w.
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1. (20 pts.)  Let C = {w in {0,1}* : N_1(w) > N_0(w)} and let 
E = {<M> : M is a T.M. with input alphabet {0,1} and L(M) intersect C = empty }.
	(a) Prove that A_{TM} is not mapping reducible to E. 
	(b) Prove that A_{TM} is Turing reducible to E. 
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2.  (20 pts.)  Assuming that P != NP, classify each of the following languages
    as being (A) in P, (B) NP-complete, (C) in NP but not known to be in P or to
    be NP-complete, or (D) not known to be in NP.  For each problem, mark (X)
    corresponding to its classification.  There is no penalty for guessing.

(a) {<G,k> : G is an undirected graph containing a k-clique}
    (A)( )  (B)( )  (C)( )  (D)( )

(b) {<G,k> : G is an undirected graph containing a k-clique but not a 
	     (k+1)-clique}
    (A)( )  (B)( )  (C)( )  (D)( )

(c) {(n)_2 : n is a prime number}
    (A)( )  (B)( )  (C)( )  (D)( )

(d) {1^n : n is a prime number}
    (A)( )  (B)( )  (C)( )  (D)( )

(e) {<D> : D is a directed graph containing a directed cycle}
    (A)( )  (B)( )  (C)( )  (D)( )

(f) {<D> : D is a directed graph containing a directed Hamilton cycle}
    (A)( )  (B)( )  (C)( )  (D)( )

(g) {<S, 1^t> : S is a sequence of positive integers (represented in binary) and 
		some subsequence of S adds up exactly to t}
    (A)( )  (B)( )  (C)( )  (D)( )

(h) {<S, (t)_2> : S is a sequence of positive integers (represented in binary) 
		  and some subsequence of S adds up exactly to t}
    (A)( )  (B)( )  (C)( )  (D)( )

(i) {<1^n, 1^m> : there is an exact divisor d of n such that 1 < d < m}
    (A)( )  (B)( )  (C)( )  (D)( )

(j) {<(n)_2, (m)_2> : there is an exact divisor d of n such that 1 < d < m}
    (A)( )  (B)( )  (C)( )  (D)( )
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3.  (10 pts.)  Prove that if 3SAT is in the class P, then P = NP.