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8. Proving languages are not regular
Let L be an infinite regular language. Then, for $\forall w \in L$, $\exists n \in \mathbf{N}$, $ |w| \ge n $, $ w = xyz $, $ |xy| \le n $ and $ y \neq \varepsilon $, such that $ \forall k \ge 0, w_{k} = xy^{k}z \in L$.
Let L be an infinite language. If $\forall n \in \mathbf{N}$, $\exists w_{n} \in L $ with $ |w| \ge n $ such that regardless of how $ w_{n} $ is split into $ w_{n} = xyz $ with $ |xy| \le n $ and $ y \neq \varepsilon $, $\exists k \ge 0 $ such that $ w_{n} = xy^{k}z \notin L $, then L in not a regular language.
8.1. The pumping lemma
8.1.1. Show that the pumping lemma holds for finite languages.
8.1.2.* Find a language which is not regular for which the pumping lemma holds.
8.2. Languages which are not regular
Show that each of the languages from the list below is not regular.
8.2.1. $ L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $
8.2.2. $ L = \{ \: w \in \{A,B\}^* \: | \: \#_A(w) = \#_B(w) \: \} $
8.2.3. $ L = \{(01)^n(10)^n \mid n > 0 \} $
8.2.4. $ L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $
8.2.5. $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $
8.2.6. $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a power of two} \: \} $
8.2.7. $ L = \{ \: ww^R \: | \: w\in \{0,1\}^* \} $
8.3. Combining the pumping lemma with closure properties
8.3.1. Using the pumping lemma, prove that $ L = \{ \: A^nB^m \: | \: n \neq m \}$ is not a regular language.