1. [20 points] Do Exercise 2.11.
2. [20 points] Read the proof of Example 2.38 on page 127 of Sipser. Now completely rewrite the proof in such a way that every step is justified. You proof should be complete, i.e., no claim is made without justifation. You may draw pictures and/or use example strings to help clarify your ideas.
3. [20 points] First, read the solution to Problem 2.18(a) on page 133 of Sipser. Next, consider the PDA M1 in Figure 2.15 on page 113 and the DFA M2 in Figure 1.8 on page 37. However, change the names of the states of M2 from q1 to q'1, and from q2 to q'2. Now, use the algorithm described in the solution to Problem 2.18(a) to construct a PDA P that recognizes L(M1) ∩ L(M2). Your answer should be a state diagram of P. Do not simplify your machine, i.e., include in your diagram all edges obtainable by the algorithm.
Extra credit: Any of the starred problems on pages 130–132 of Sipser.