91Ó°ÊÓ

Context-free grammar generates context-free grammar. The regular languages are the subset of the context-free languages that are an identical set of languages accepted by pushdown automata.

a.

Consider the given,$C_1=\left\{xyz|x,z∈∑*\text{andy}∈∑\text{*1}∑\text{*,where}\left|x\right|=\left|z\right|â‰\yen \left|y\right|\right\}$where $∑=\{0,1\}$.

Construct the pushdown automata p to determine the language $C_1$. The p can be defined as$(Q,∑,\mathrm{Γ},\mathrm{δ},q_0,F)$, where .$Q=\{q_0,q_1,q_2\},∑=\{0,1\},\mathrm{Γ}=\left\{x\right\},F=\left\{q_2\right\}$

The PDA is as follows,

Thus, the PDA accepts the given language $C_1$.

Therefore, $C_1$is a Context-free Language and it has been proved.

b.

Consider the given language$C_2$, $C_2=\left\{xyz|x,z∈∑*and\text{ }y∈∑*1∑*1∑*,\text{ â¶ÄŠ}where\text{ }\left|x\right|=\left|z\right|â‰\yen \left|y\right|\right\}$accepts all the string that has two 1’s in the middle of the string.

Consider the pumping lemma to show that $C_2$is not a context-free language. Assume that $C_2$is a CFL and prove its contradiction by the pumping lemma l.Consider the string ,S that is divided into pieces as$S=abcde$.

• For each $iâ‰\yen 0$,$a{b}^{i}c{d}^{i}e∈C_2$
• $\left|bd\right|>0$
  
• $\left|bcd\right|≤l$
  

Where, $a=0^{l+2},b=1,c=0^l,d=1,e=0^{l+2}$. Let $i=1$, after pumping $S=0^{l+2}{10}^2{10}^{l+2}$. The new string $\left|{10}^{l}1\right|≤l$fails, and thus it is not a CFL.

Therefore, the language$C_2$ is not a Context-free language.