✎ lfa:lab07-the-pumping-lemma [books]

This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong.

====== The Pumping Lemma ======

**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} \: \} $

**2.5.** $  L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a power of two} \: \} $

**Exercise 3.** 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.


*/