Proof Idea: How to prove the intersection of regular language and CFL is a CFL?

Question

I know that the intersection of a regular language and a context free language is a context free language. I've seen this fact proven on here and other websites. However, after spending hours reading through proofs, I don't really understand the general idea of the proof itself. I'm not looking for a formal proof, but I am more concerned with understanding the high level idea. This is my basic idea for a proof, but there are some issues:

Given a regular language $R$ and a context free language $L$, the intersection of $R$ and $L$ is a context free language $L_1$. We know that there exists a DFA $D$ that recognizes $R$, as $R$ is a regular language. We also know that there exists a PDA $P$ that recognizes $L$, as $L$ is a context free language. Since it is a fact that $D$ can be converted into a PDA $P_1$, we know that $R$ is also a context free language. Next, we create a PDA with states as pairs of states from $P$ and $P_1$.

This is where I am getting stuck. How are the transitions modified to simulate both PDAs? And what happens to the stack?

Answer 1

1. Look at the product construction for DFAs, it will give you a hint of what the state space of a PDA accepting the intersection will look like.

2. You can convert $D$ to a PDA, but you have more information than that. It doesn't need to touch the stack at all, since it can be accepted by a DFA.

These should get you close to the answer.

Answer 2

Don't convert $D$ to a PDA.

Instead, you can construct a PDA that accepts $R \cap L$ by using the product construction on $D$ and $P$ -- check your favorite textbook on automata theory; it should be covered there. Basically, you take $P$, and then modify its control state using the product construction. See Intersection of context free with regular languages for more details.

---

You don't want to convert $D$ to a PDA, because that would require you to take a product construction of two PDAs, and there's no good way to do that (the intersection of two context-free languages is not context-free in general).