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lfa:lab07-the-pumping-lemma [2020/11/15 15:39]
alexandru.poboranu created lab7 lfa 		lfa:lab07-the-pumping-lemma [2021/12/06 12:08] (current)
stefan.stancu [8.2. Languages which are not regular]
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-====== ​The Pumping Lemma ======+====== ​8. Proving languages are not regular ​======
  
-**Exercise ​1.** Consider the language $  L = L((A \cup BB^*)^*)$. ​+===== 8.1. The pumping lemma ===== 
  
-**1.1.** ​Suppose n = 4. Show that the pumping lemma holds for L.+**8.1.1.** Show that the pumping lemma holds for finite languages.
  
-**1.2.** ​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.
  
-**1.3.*** Find a language ​which is not regular ​for which the pumping lemma holds.+===== 8.2. Languages ​which are not regular ​=====
  
-**Exercise 2.**  Prove that L is not a regular ​language.+Show that each of the languages from the list below is not regular.
  
-**2.1.** $  L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $+**8.2.1.** $  L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $
  
-**2.2.** $  L = \{ \: w \in \{A,B\}^* \: | \: \#A(w) = \#B(w) \: \} $+**8.2.2.** $  L = \{ \: w \in \{A,B\}^* \: | \: \#A(w) = \#B(w) \: \} $
  
-**2.3.** $  L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $+**8.2.3.** $math[L = \{(01)^n(10)^n ​\mid n > 0 \} ]
  
-**2.4.** $  L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $+**8.2.4.** $  L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $
  
-**Exercise 3.** Using the pumping lemma indirectly, prove that $  L = \{ \: A^nB^m \: | \: n \neq m \}$ is not a regular language.+**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 ====== ====== Homework ======
  
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 **Exercise III** Prove that $  L = \{ \: A^nB^mC^{n-m} \: | \: n \geq m \geq 0 \: \}$ is not a regular language without using isomorphisms. **Exercise III** Prove that $  L = \{ \: A^nB^mC^{n-m} \: | \: n \geq m \geq 0 \: \}$ is not a regular language without using isomorphisms.
  
 +
 +*/