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--- lfa:lab07-the-pumping-lemma [2020/11/19 16:49]
pdmatei
+++ lfa:lab07-the-pumping-lemma [2021/12/06 12:08] (current)
stefan.stancu [8.2. Languages which are not regular]
@@ Line 1: / Line 1: @@
-====== 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} \: \} $
-**2.5.** $  L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a power of two} \: \} $
+**8.2.5.** $  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.
+**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.
 /*