3.Show that there is a language A {0,1} with the following properties:

(a) For all x A, |x| 5.

(b) No DFA with fewer than 9 states recognizes A.

4. The length of a regex is the number of symbols in it (counting both special symbols such as and alphabet symbols such as 0; for instance, the length of (0|1)*0110 is 10. Show that there is a language A {0,1} with the following properties:

(a) For all x A, |x| 4.

(b) No regex with length smaller than 11 recognizes A. Assume that the only allowed symbols in a regex are ( ) * + | 0 1

Hint: You dont have to dene A explicitly; just show that it has to exist. Count the number of languages satisfying (4a) and the number of DFAs satisfying (4b), and then use the pigeonhole principle. To count the number of DFAs satisfying (4b), consider that a DFA behaves identically even if you rename all the states, so you can assume without loss of generality that any DFA with k states has the state set {q1,q2,...,qk}. Now think about how to count how many ways there are to choose the other parts of the DFA. To count the number of languages satisfying (4a), think about writing down all the strings of length at most 5, and then to dene such a language, you have to make a binary decision for each string about whether to include it in the language or not. How many ways are there to make these choices?