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--- lfa:2022:lab08-the-pumping-lemma [2022/11/27 00:32]
alexandra.udrescu01
+++ lfa:2022:lab08-the-pumping-lemma [2022/12/09 08:55] (current)
pdmatei
@@ Line 18: / Line 18: @@
 **8.1.1.** Show that the pumping lemma holds for finite languages.
 <hidden Solution>
@@ Line 28: / Line 29: @@
 </hidden>
 **8.1.2.*** Find a language which is not regular for which the pumping lemma holds.
@@ Line 36: / Line 38: @@
 **8.2.1.** $  L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $
 <hidden Solution> <note>
@@ Line 60: / Line 64: @@
 </hidden>
-**8.2.2.** $  L = \{ \: w \in \{A,B\}^* \: | \: \#A(w) = \#B(w) \: \} $
+**8.2.2.** $  L = \{ \: w \in \{A,B\}^* \: | \: \#_A(w) = \#_B(w) \: \} $
 <hidden Solution> <note>
@@ Line 84: / Line 90: @@
 </hidden>
 **8.2.3.** $math[L = \{(01)^n(10)^n \mid n > 0 \} ]
 <hidden Solution> <note>
@@ Line 142: / Line 151: @@
 </hidden>
 **8.2.4.** $  L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $
 <hidden Solution> <note>
@@ Line 168: / Line 180: @@
 **8.2.5.** $  L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $
@@ Line 192: / Line 206: @@
 </hidden>
 **8.2.6.** $  L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a power of two} \: \} $
@@ Line 222: / Line 238: @@
 </hidden>
-**8.2.7.** $  L = \{ \: ww^R  \: | \: w\in \{0,1\}^* \}
-$
+**8.2.7.** $  L = \{ \: ww^R  \: | \: w\in \{0,1\}^* \} $
+<hidden Solution> <note>
+For a fixed n, pick a word $ w_n \in L $ and $ w_n = xyz $:
+$ w_n = 0^n110^n $
+$ x = 0^a $
+$ y = 0^b, b \ge 1 $
+$ z = 0^{n-a-b}10^n $
+Find k such that $ xy^kz \notin L $:
+$ w_k = 0^{n+(k-1)b}110^n $
+Pick $ k = 2 $ => $ w_2 = 0^{n+b}110^n \notin L $ because $ b \ge 1 $
+=> Complement of Pumping Lemma holds => L is not a regular language
+</note>
+</hidden>