I am trying to solve this problem: Given context-free languages A, B, and C find if the language $ (A\cap B)\cup (B\cap C)$ is empty. Is this problem decidable.
I know the CFG is not closed under intersection so we do not know that $ A\cap B$ is also CFG. If it was CFG, I know how to prove decidability. Since we can’t determine about $ A\cap B$ , is there a way to prove whether or not this is decidable?