Prove that the class of CFG languages that are closed under reversal is undecidable
Let $G_1,G_2$ be two context-free grammars. We can construct a context-free grammar $G$ such that
$$
L(G) = \#L(G_1) \cup L(G_2)^r\#,
$$
where $\#$ is a new symbol. The language $L(G)$ is closed under reverse iff $L(G_1) = L(G_2)$.