1. In Chapter 1, we worked on a procedure to convert a regular expression to a DFA. This time, we would like to convert a regular expression to a CFG. Sketch a procedure to construct a CFG that generates the same language as a regular expression R. Make sure that you include all three base cases, and the cases for union, concatenation, and star as in Definition 1.52 of regular expressions.
2. (Problem 2.14) Convert the following CFG into an equivalent CFG in Chomsky normal form, using the procedure given in Theorem 2.9.
3. Construct a PDA that recognizes the languages below. Be sure to give a brief, English description of how your PDA works. In all parts the alphabet Σ is {0,1}.
a. { w | w contains at least three 1s}.
b. { w | w contains more 1s than 0s}.
4. (Exercise 2.12) Convert the CFG G4 given in
Exercise 2.3 to
an equivalent PDA using the procedure given in Theorem 2.20.