Let
$$
P=\left\{\frac{u_i}{c_i} \mid i=1,2, \ldots, k\right\}
$$
be a set of dominoes on the alphabet $A$. Let $B=\left\{c_1, \ldots, c_k\right\}$ be an alphabet such that $A \cap B=\varnothing$. Let $c \notin A \cup B$. Let
$$
\begin{aligned}
R & =\left\{y c y^R \mid y \in A^* B^*\right\}, \\
L_1 & =\left\{u_{i_1} u_{i_2} \cdots u_{i_n} c_{i_n} c_{i_{n-1}} \cdots c_{i_2} c_{i_1}\right\}, \\
L_2 & =\left\{v_{i_1} v_{i_2} \cdots v_{i_n} c_{i_n} c_{i_{n-1}} \cdots c_{i_2} c_{i_1}\right\}, \\
S_p & =\left\{y c z^R \mid y \in L_1, z \in L_2\right\} .
\end{aligned}
$$
Recall that by Theorem 6.5 , the Post correspondence problem $P$ has a solution if and only if $L_1 \cap L_2 \neq \varnothing$.
(a) Show that the Post correspondence problem $P$ has no solution if and only if $R \cap S_p=\varnothing$.
(b) Show that $(A \cup B \cup\{c\})^*-R$ and $(A \cup B \cup\{c\})^*-S_P$ are both context-free. (Hint: Construct push-down automata.)
(c) From (a) and (b) show how to conclude that there is no algorithm that can determine for a given context-free grammar $\Gamma$ with terminals $T$ whether $L(\Gamma) \cup\{0\}=T^*$.
(d) Now show that there is no algorithm that can determine for a given context-free grammar $\Gamma_1$ and regular grammar $\Gamma_2$ whether
(i) $L\left(\Gamma_1\right)=L\left(\Gamma_2\right)$,
(ii) $L\left(\Gamma_1\right) \supseteq L\left(\Gamma_2\right)$.