c318f94f - CSc 318 Final Examination Monday 12 December...

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CSc 318 Final Examination Monday 12 December 1994 7 PM (Below ^ denotes a suprescript, e.g., 2^n denotes 2 raised to the nth power) 1. (10 pts) Find a CFG which is not left-recursive and which is equivalent to the grammar S--> Ax|By|c A-->Sy|By B-->a|b 2. (20 pts) Given R and S, binary relations on the set A, prove that if R is a subset of S, then R^* is a subset of S^*. 3. (10 pts) Find a lambda-free CFG, G', such that L(G')=L(G)-{lambda}, where G is the CFG given by S-->aBBaC|BC|bC B-->Ca|Cb|CC C-->abC|lambda 4. (10 pts) If A={a,b,c,d} and B={0,1,2, .... } prove that #B=#(A x B). 5. (20 pts) Prove that for every regular expression, E, there is a CFG, G, such that L(G)=L(E) but that there is a CFG, G, for which there is no regular expression, E, such that L(G)=L(E). 6. (10 pts) For the CFG below find an equivalent CFG which has all reachable and terminating symbols. S-->CA|CB|BC|Ad A-->AD|BD|aBD|DE B-->caS|b|c C-->Ab|Bc D-->aA|Ab|bD E-->a 7. (20 pts) Find a complete minimal DFA equivalent to the NFA given by M=({s,1,2,3,4}, {a,b,c}, {(s,a,1), (s,c,2), (1,a,1), (1,b,2), (1,b,3), (2,a,3), (2,b,2), (3,b,4), (3,c,2), (4,a,s)}, s, {3,4})
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