Context Free Languages

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 \ldots \alpha_n $?

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\}^*\ |\ \#_{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.

3. Show that the following are closed under CF languages:

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.