Decidability for intersection of context free and regular languages

Question

I am wondering if the following are decidable or undecidable and why. L is a CFL and R is a regular language. How does the complement of the context-free language change the decidability of the intersection?

𝐿∩𝑅=∅ and $\bar{𝐿}$∩𝑅=∅

This is what I think but I am not sure so any help is appreciated.

For 𝐿, I think this is decidable because intersection of a CFL and a regular language is a CFL so you can build CFG for it. However, it can also be undecidable as we can check membership for all elements in Σ∗ which can take forever so we don't know if there is going to be an element that'll be in this intersection.

As for $\bar{𝐿}$, this one I don't really know at all.

Please excuse me if I made mistakes or if my explanation was unclear. I am new to the topic and trying to wrap my head around it and any help is appreciated.

Answer 1

The first one you have solved correctly. Emptiness of context-free languages is decidable.

For the second, we can rewrite $\bar L \cap R = \varnothing$ into $R\subseteq L$. This is undecidable (even for $R=\Sigma^*$).