CS 390 Final Exam

August 2006

1. For the folloswing NFA-$\Lambda$ answer the questions given below: [16]









(a) Find



(b) Find



(c) Find



2. Convert the following NFA to DFA that accepts the same language:









3. Using Myhill-Nerode theorem, prove that the following languages are non-regular:

(a) {0ⁿ1²ⁿ: n is a natural number.}















(b) {ww^rw : w is a string.}, where w^r denotes the reversal of string w.

















4. For each of the following grammars answer the questions given below:

(1) S -> aS | bS | a
(2) S -> SaS | b

(a) What kind of grammar is it ? Regular, context-free, context-sensitive or phrase structure ? [4]

(1)


(2)


(b) Describe in simple English the strings generated by the grammar. [7]

(1)





(2)





5. For each of the following statements answer whether or not it is true:

(a) If the language of yes instances of a decision problem is accepted by a Turing machine, then the decision problem is solvable.


(b) If a language is decided by a Turing machine, then the Turing machine stops on every string.


(c) If the language of yes instances of a decision problem is not accepted by any Turing machine, then the decision problem is unsolvable.


(d) If a decision problem is undecidable, then a computer can solve it.


(e) If a language is accepted by a Turing machine, then the Turing machine stops on every string.


(f) If a decision problem is "reducible" to another decision problem, then their corresponding instances have the same answer.


(g) If a language is accepted by a Turing machine, then its complement is also accepted by a Turing machine.


(h) Every regular language is decidable.


(i) The intersection of regular languages is decidable.


(j) The complement of a context-free language is regular.



6. Design a Turing machine that accepts the language aⁿb²ⁿ, where n is a positive integer. You may use basic Turing machines such as T_a, T_R, T_Ra etc.