Consider a language $L_1$ that is recursively enumerate, $L_2$ that is regular, and $L_3$ that is context-free.
Are the following problems algorithmically decidable?
- Is $L_1 \cap L_2 = L_3$?
- Is $L_1 \cap L_3 = L_2$?
I think problem 1 is undecidable since $L_1 \cap L_2$ is a recursively enumerable language it can be expressed as a Turing machine and since $L_3$ is context free it can be expressed as a Grammar or Turing machine. Since we cannot determine the equivalence of two Turing machines, the problem is undecidable.
I think problem 2 is also undecidable for similar reasons, only the right hand side ($L_2$) is regular instead of context-free.
Am I correct?