Michael Sipser Edition 3 Exercise 2 Question 48 (Page No. 159)

Question

Let $\Sigma = \{0,1\}$. Let $C_{1}$ be the language of all strings that contain a $1$ in their middle third. Let $C_{2}$ be the language of all strings that contain two $1s$ in their middle third.So $C_{1} = \{xyz \mid x,z\in \Sigma^{\ast}\: \text{and} \: y \in \Sigma^{\ast} 1 \Sigma^{\ast}, \text{where} \mid x \mid = \mid z \mid\: \geq\: \mid y \mid \}$ and $C_{2} = \{xyz \mid x,z\in \Sigma^{\ast}\:\text{and} \:y \in \Sigma^{\ast} 1 \Sigma^{\ast}1 \Sigma^{\ast}, \text{where} \mid x \mid = \mid z \mid\: \geq\: \mid y \mid  \}$.

1. Show that $C_{1}$ is a CFL.
2. Show that $C_{2}$ is not a CFL.