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lfa:lab10-cfl-closures [2020/12/07 09:14] lfa |
lfa:lab10-cfl-closures [2020/12/08 09:12] (current) lfa |
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| === Context-Free Grammar to Pushdown Automata === | === Context-Free Grammar to Pushdown Automata === | ||
| - | **1.** Consider the following CFG: | + | ** 1.** Consider the following CFG: |
| $ S \leftarrow X\ |\ Y $ \\ | $ S \leftarrow X\ |\ Y $ \\ | ||
| Line 9: | Line 9: | ||
| $ Y \leftarrow YY\ |\ 1\ |\ \epsilon $ | $ Y \leftarrow YY\ |\ 1\ |\ \epsilon $ | ||
| - | **1.1** Write a PDA which accepts L(G) | + | //1.1// Write a PDA which accepts L(G) |
| - | **1.2** Write a sequence of derivations which yeilds $ S\ \Rightarrow\ 110X1Y $ . What is it's corresponding sequence of transitions in the PDA? | + | //1.2// Write a sequence of derivations which yeilds $ S\ \Rightarrow\ 110X1Y $ . What is it's corresponding sequence of transitions in the PDA? |
| - | **1.3** In our PDA, $ (p, 111100, XZ0) \mapsto^* (p, e, Z0) $. Write-down the steps. How is $ \alpha $ split into $ \alpha_1 ... \alpha_n $? | + | //1.3// In our PDA, $ (p, 111100, XZ0) \mapsto^* (p, e, Z0) $. Write-down the steps. How is $ \alpha $ split into $ \alpha_1 ... \alpha_n $? \\ \\ |
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| **2.** Which of the following languages are Context-Free? Argue for your answer. | **2.** Which of the following languages are Context-Free? Argue for your answer. | ||
| + | |||
| + | //2.1.// $ L = \{a^{n}b^{2n}c^{2m}d^{m}\ |\ n, m \geq 0 \} $ \\ | ||
| - | **2.1.** $ L = \{a^{n}b^{2n}c^{2m}d^{m}\ |\ n, m \geq 0 \} $ \\ | + | //2.2.// $ L = \{w \in \{a, b\}^*\ |\ each\ sequence\ of\ consecutive\ As\ is\ followed\ by\ the\ same\ number\ of\ Bs\} $ \\ |
| - | **2.2.** $ L = \{w \in \{a, b\}^*\ |\ each\ sequence\ of\ consecutive\ As\ is\ followed\ by\ the\ same\ number\ of\ Bs\} $ \\ | + | //2.3.// $ L = \{a^{n}b^{2n}c^{m}\ |\ n, m \geq 0 \} \cap \{a^{n}b^{m}c^{2m}\ |\ n, m \geq 0 \} $ \\ |
| - | **2.3.** $ L = \{a^{n}b^{2n}c^{m}\ |\ n, m \geq 0 \} \cap \{a^{n}b^{m}c^{2m}\ |\ n, m \geq 0 \} $ \\ | + | //2.4.// $ L = \{w \in \{a, b\}^*\ |\ a\ and\ b\ can\ be\ matched\ in\ sequences,\ in\ any\ order \} $. Example : $ aabbbbaaaabb \in L $ \\ |
| - | **2.4.** $ L = \{w \in \{a, b\}^*\ |\ \#_a(w) \eq \#_b(w) \} $ \\ | + | //2.5.// $ L = \{w \in \{a, b\}^*\ |\ w=a^nb^n \ and\ |w|\ \%\ 3\ =\ 0 \} $ \\ |
| - | **2.5.** $ L = \{w \in \{a, b\}^*\ |\ w=a^nb^n \|w\|\%3\leq0 $ \\ | + | //2.6.// $ L = \{w \in \{a, b\}^*\ |\ \#_{a}(w)=\#_{b}(w) \ and\ no\ b\ should\ be\ followed\ by\ two\ a \} $ \\ |
| + | //2.7.// Give an example of two context-free languages whose intersection is context-free. | ||
| - | **2.1.** | + | **3.** Show that the following are closed under CF languages: |
| - | + | ||
| - | $ S \leftarrow aA | A $ \\ | + | |
| - | $ A \leftarrow aA | B $ \\ | + | |
| - | $ B \leftarrow bB | \epsilon $ | + | |
| - | + | ||
| - | **2.2.** | + | |
| - | + | ||
| - | $ S \leftarrow AS | \epsilon $ \\ | + | |
| - | $ A \leftarrow 0A1 | 01 | B $\\ | + | |
| - | $ B \leftarrow B1 | \epsilon $ | + | |
| - | + | ||
| - | **2.3.** | + | |
| - | + | ||
| - | $ S \leftarrow ASB | BSA | \epsilon $\\ | + | |
| - | $ A \leftarrow aA | \epsilon $\\ | + | |
| - | $ B \leftarrow bB | \epsilon $ | + | |
| - | + | ||
| - | **3.** Write an ambiguous grammar for $ L(a^*) $. | + | |
| - | + | ||
| - | === Regular Grammars === | + | |
| - | + | ||
| - | **4.** Is the language described by the following grammar regular? If so, write a regular expression for it. | + | |
| - | + | ||
| - | $ S \leftarrow aA $\\ | + | |
| - | $ A \leftarrow aA | B $\\ | + | |
| - | $ B \leftarrow Bb | \epsilon $ | + | |
| - | + | ||
| - | **5.** Write a regular expression for the language described by: | + | |
| - | + | ||
| - | $ S \leftarrow aX $\\ | + | |
| - | $ X \leftarrow bY | S $\\ | + | |
| - | $ Y \leftarrow aX | bS | \epsilon $ | + | |
| - | + | ||
| - | **6.** Write a regular grammar for $ L((0 \cup 1^*)^*01^*) $. | + | |
| - | + | ||
| - | === Chomsky Normal Form === | + | |
| - | + | ||
| - | **7.** Remove "$ \epsilon \text{-rules} $" from the following grammar: | + | |
| - | + | ||
| - | $ A \leftarrow \epsilon | B $\\ | + | |
| - | $ B \leftarrow b $\\ | + | |
| - | $ B \leftarrow ABC | BAC $\\ | + | |
| - | $ C \leftarrow AC | c$ | + | |
| - | + | ||
| - | **8.** Remove the "unit rules" from the previous grammar, after "$ \epsilon \text{-rules} $" have been removed. | + | |
| - | + | ||
| - | **9.** Apply the CNF conversion rules to the solution for **1.1.** Does the accepted language stay the same? | + | |
| + | //3.1// Reversal \\ | ||
| + | //3.2// $ init(L) = \{w\ \in\ \Sigma^*\ |\ \exists x\ such\ that\ wx\ \in\ L\}. $ //Hint//: Write a CNF grammar for init L, starting from a grammar for L. | ||