Exercise 1. Consider the language $ L = L((A \cup BB^*)^*)$.

1.1. Suppose n = 4. Show that the pumping lemma holds for L.

1.2. Show that the pumping lemma holds for finite languages.

1.3.* Find a language which is not regular for which the pumping lemma holds.

Exercise 2. Prove that L is not a regular language.

2.1. $ L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $

2.2. $ L = \{ \: w \in \{A,B\}^* \: | \: \#A(w) = \#B(w) \: \} $

2.3. $ L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $

2.4. $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $

Exercise 3. Using the pumping lemma indirectly, prove that $ L = \{ \: A^nB^m \: | \: n \neq m \}$ is not a regular language.

Exercise I $ L = \{ \: w \in \{0\}^* \: | \: \text{|w| is a power of 2} \: \} $

Exercise II Show that $ \text{snd(L)}$ is a closure property for regular languages.

$ \text{snd(L)} = \{ \: w \: | \: xw \in L \: \text{, for some x such that |x| = |w|} \: \}$

Exercise III Prove that $ L = \{ \: A^nB^mC^{n-m} \: | \: n \geq m \geq 0 \: \}$ is not a regular language without using isomorphisms.