1. [20 points] Do Exercise 1.16 part a only.
2. [20 points] Give regular expressions generating the languages of Exercise 1.6 parts a, b, d, e, f, g, i, j, l, n only.
3. [20 points] Do Exercise 1.19 part b only. Your answer should be the NFA that results from following the algorithm exactly. You should not change the algorithm or simplify your machine.
4. [20 points] Do Exercise 1.21 part b only by following the algorithm exactly. Clearly describe each step. Do not skip any step and do not simplify the intermediate or the final regular expressions.
5. [20 points] Prove that the language { ak : k is a perfect square } is not regular by “pumping down.” You may model your proof after this proof by “pumping up.”
Extra credit: For any string w,
define even(w) to be the string that
results from deleting all the letters that occur
in odd positions of w. For example,
even(a)=ε,
even(ab)=b,
even(acb) = c,
and
even(acbd) = cd.
Now extend the definition of even
to languages as follows.
For any language L, define even(L)
to be the language
{ even(w) : w ∈ L }.
Prove that if L is regular then even(L) is regular.