Find an equivalent NFA for the following regular expression in the alphabet S = {0, 1}: 0S+1(0 ? 10) ? 0(1 ? e)S* 0 * 2. (10 points) Convert the DFA below into a regular expression that describes exactly the same language. Eliminate states in the following order: q3, q1, q2. Show your work. start q0 q1 q2 q3 0 1 1 0,1 0 1 3. (10 points) Convert the NFA below to its equivalent DFA: start q0 q1 q2 0 e 0 0,1 4. (25 points) Find a regular expression which recognizes each of the following languages (in parts (b) and (c), assume that S is an arbitrary non-empty finite set of symbols and n is an arbitrary positive integer): (a) w ? {a, b, c}|w contains at most two a’s and at most two b’s (b) w ? S| the length of w is at most n (c) w ? S| the length of w is not equal to n (d) {w ? {a, b, c}| (every odd position of w is b or c) ? (every even position of w is a)} (e) {w ? 0, 1|w 6= 0100 ? w 6= 101} 5. (5 points) Formally prove that every finite language is regular (Hint: Use proof by induction on the cardinality of an arbitrary finite language, and the theorem that say every regular language can be described by a regular expression). 6. (5 points) Use pumping lemma to show that {0 n1 m|0 = n = m} is not a regular language. 7. (25 points) Are the following languages regular? (Prove your answers) (a) n a ( n 2) ? {a} * n ? Z=2 o (b) n a p b qa b vpqc ? {a, b} * p, q ? Z + o (c) n a p b 2+qa 2 ? {a, b} * p, q ? Z + o 8. (10 points) Let f : R + 7? R + be a differentiable function and ?x ? R +, f0 (x) = dxe. Prove that {1 df(n)e |n ? Z +} is not regular (Hint: use pumping lemma)