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lfa:2023:lab08 [2023/12/04 22:12] mihai.udubasa created |
lfa:2023:lab08 [2023/12/08 13:23] (current) alexandra.udrescu01 |
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| **8.1.1.** Show that the pumping lemma holds for finite languages. | **8.1.1.** Show that the pumping lemma holds for finite languages. | ||
| - | /* | + | |
| <hidden Solution> | <hidden Solution> | ||
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| </hidden> | </hidden> | ||
| - | */ | + | |
| **8.1.2.*** Find a language which is not regular for which the pumping lemma holds. | **8.1.2.*** Find a language which is not regular for which the pumping lemma holds. | ||
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| - | /* | + | |
| <hidden Solution> <note> | <hidden Solution> <note> | ||
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| </hidden> | </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) \: \} $ | ||
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| <hidden Solution> <note> | <hidden Solution> <note> | ||
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| </hidden> | </hidden> | ||
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| **8.2.3.** $math[L = \{(01)^n(10)^n \mid n > 0 \} ] | **8.2.3.** $math[L = \{(01)^n(10)^n \mid n > 0 \} ] | ||
| - | /* | + | |
| <hidden Solution> <note> | <hidden Solution> <note> | ||
| $ w_n = (01)^n(10)^n $ | $ w_n = (01)^n(10)^n $ | ||
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| </hidden> | </hidden> | ||
| - | */ | + | |
| **8.2.4.** $ L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $ | **8.2.4.** $ L = \{ \: w \in \{A,B\}^* \: | \: \text{w is a palindrome} \: \} $ | ||
| - | /* | ||
| <hidden Solution> <note> | <hidden Solution> <note> | ||
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| </hidden> | </hidden> | ||
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| **8.2.5.** $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $ | **8.2.5.** $ L = \{ \: w \in \{0\}^* \: | \: \text{the length of w is a prime number} \: \} $ | ||
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| - | /* | ||
| <hidden Solution> <note> | <hidden Solution> <note> | ||
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| </hidden> | </hidden> | ||
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| - | /* | ||
| <hidden Solution> <note> | <hidden Solution> <note> | ||
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| </hidden> | </hidden> | ||
| - | */ | + | |
| **8.2.7.** $ L = \{ \: ww^R \: | \: w\in \{0,1\}^* \} $ | **8.2.7.** $ L = \{ \: ww^R \: | \: w\in \{0,1\}^* \} $ | ||
| - | /* | ||
| <hidden Solution> <note> | <hidden Solution> <note> | ||
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| </hidden> | </hidden> | ||
| - | */ | + | |
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| **8.3.1.** Using the pumping lemma, prove that $ L = \{ \: A^nB^m \: | \: n \neq m \}$ is not a regular language. | **8.3.1.** Using the pumping lemma, prove that $ L = \{ \: A^nB^m \: | \: n \neq m \}$ is not a regular language. | ||
| - | /* | + | |
| <hidden Solution> | <hidden Solution> | ||
| <note> | <note> | ||
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| </hidden> | </hidden> | ||
| - | */ | ||
| - | /* | ||
| - | ====== 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. | + | ====== Nice to try at home ====== |
| + | |||
| + | **Exercise I** 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|} \: \}$ | $ \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. | + | **Exercise II** Prove that $ L = \{ \: A^nB^mC^{n-m} \: | \: n \geq m \geq 0 \: \}$ is not a regular language without using isomorphisms. |
| - | */ | ||