CS 3350 Automata, Computability, and Formal Languages
Fall 2015, Test 1, 12-1:20 pm version

Name: __________________________________________________________

1. Prove that the cubic root of 3 is irrational.

2-3. Use a general algorithm to design a non-deterministic finite automaton recognizing the language 1(0* U 1). After that, use the general algorithm to design a deterministic finite automaton recognizing this same language.

4. Use a general algorithm to transform the following finite automaton into the corresponding regular expression. This automaton has two states: a state q1 which is both a start state and a final state, and a state q2. In the state q1, 0 leads to q2, and 1 leads to q1. In the state q2, 0 leads to q2, and 1 leads to q1.

5. On the example of the automaton from Problem 4, explain, in detail, how the sequence 010101 will be presented as xyz according to the pumping lemma. For this sequence, check -- by tracing step-by-step -- that the sequence xyiz for i = 2 is indeed accepted by the automaton.

6. Use the Pumping Lemma to prove that the language L consisting of all repetitions ww, www, ..., is not regular.