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