Proving decidability and undecidablity of CFL DCFL problems [closed]


I am trying to understand how can I prove various problems of CFLs and DCFLs are undecidable or undecidable.

For context free grammars $ G, G_1, G_2$ , how can I prove following problems are undecidable:

  1. Whether $ L(G)$ is a regular language?
  2. Whether $ L(G)$ is a DCFL?
  3. Whether $ L(G)^c$ is CFL?
  4. Whether $ L(G_1)\cap L(G_2)$ is CFL?

For deterministic context free grammar $ D$ and regular grammar $ R$ , how can I prove following problems are decidable:

  1. Whether $ L(D)\subseteq L(R)$
  2. Whether $ L(D)=L(R)$
  3. Whether $ L(R)\subseteq L(D)$

I gave following attempts:

  1. I know whether $ L(G)=\Sigma^*$ is undecidable. $ \Sigma^*$ is regular language. So, problem 1 is undecidable.
  2. Undecidability of this problem follows from undecidability of 1st problem, since set of regular languages is proper subset of set of DCFLs.
  3. I am unable to come up with any logic for this.
  4. Given that I know whether $ L(G_1)\cap L(G_2)=\emptyset$ is undecidable. $ \emptyset$ is CFL. So, problem 4 is undecidable.
  5. Given that I know, $ L(G)\subset L(R)$ is decidable, $ L(D)\subset L(R)$ is also decidable as set of DCFLs are proper subset of set of CFLs.
  6. Decidability of this problem follows from 5th problem.
  7. I am unable to come up with any logic for this.

Was I correct with my attempt? Also can someone help me out for problem 3 and 7?