Edit this page Backlinks This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== Context Free Languages ====== === Context-Free Grammar to Pushdown Automata === ** 1.** Consider the following CFG: $ S \leftarrow X\ |\ Y $ \\ $ X \leftarrow YXY\ |\ 0X\ |\ 0 $ \\ $ Y \leftarrow YY\ |\ 1\ |\ \epsilon $ //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? **1.3** In our PDA, $ (p, 111100, XZ0) \mapsto^* (p, e, Z0) $. Write-down the steps. How is $ \alpha $ split into $ \alpha_1 ... \alpha_n $? === Closure Properties of Context-Free Languages === == **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 \} $ \\ **2.2.** $ 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 \} $ \\ **2.4.** $ L = \{w \in \{a, b\}^*\ |\ a\ and\ b\ can\ be\ matched\ in\ sequences,\ in\ any\ order \} $. Example : $ aabbbbaaaabb \in L $ \\ **2.5.** $ L = \{w \in \{a, b\}^*\ |\ w=a^nb^n \ and\ |w|\ \%\ 3\ =\ 0 \} $ \\ **2.6.** $ L = \{w \in \{a, b\}^*\ |\ w=a^nb^n \ and\ |w|\ \%\ 3\ =\ 0 \} $ \\ **2.7.** Give an example of two context-free languages whose intersection is context-free. == **3.** Show that the following are closed under CF languages: = **2.1.** $ 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?