✎ lfa:2022:lab08-the-pumping-lemma [books]

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====== 8. Proving languages are not regular ======

<note important>
** Pumping Lemma **

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$.

</note>

<note important>
** Complement of Pumping Lemma **

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.

</note>

===== 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.** $math[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.

/*
====== Homework ======

**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.


*/