Let G1 ,G2 be Context Free Grammars (CFGs) and RR be a regular expression. For a grammar G, let L(G) denote the language generated by G. Which ONE among the following questions is decidable?

The correct option is: Is $L\left(G_1\right)=\varnothing$?

To understand why this question is decidable, let us analyze the properties of Context-Free Grammars (CFGs) and the languages they generate, and examine the decidability of each of the options provided.

1. Decidability of Emptiness for CFGs (Is $L\left(G_1\right)=\varnothing$?)
The emptiness problem for Context-Free Grammars asks whether a given CFG $G_1$ generates any string at all, i.e., whether $L\left(G_1\right)=\varnothing$.
This problem is decidable. We can determine if the language is empty by using a simple algorithm that finds all generating variables in the grammar:
- A variable $A$ is generating if it can derive a terminal string (i.e., $A\Rightarrow *w$ where $w\in {\Sigma }^{*}$).
- We start by marking all terminal symbols as generating.
- We then iteratively mark a variable $A$ as generating if there is a production rule $A\to X$ where all symbols in $X$ are already marked as generating.
- This process continues until no new variables can be marked. If the start symbol of the grammar is marked as generating, then $L\left(G_1\right)\ne \varnothing$; otherwise, $L\left(G_1\right)=\varnothing$.
Since this algorithm is guaranteed to terminate in finite steps, the emptiness problem for CFGs is decidable.

2. Why the other options are undecidable:
- Is $L\left(G_1\right)=L\left(G_2\right)$?: The equivalence problem for two CFGs is a classic undecidable problem. There is no algorithm that can determine if two arbitrary CFGs generate the same language.
- Is $L\left(G_1\right)\cap L\left(G_2\right)=\varnothing$?: The intersection-emptiness problem for two CFGs is also undecidable. This can be proven by reduction from the Post Correspondence Problem (PCP).
- Is $L\left(G_1\right)=L\left(R\right)$?: Testing whether a context-free language is equal to a regular language is undecidable. Specifically, testing whether $L\left(G_1\right)={\Sigma }^{*}$ (the universality problem for CFGs, where $L\left(R\right)={\Sigma }^{*}$ is a regular language) is undecidable.

Therefore, the only question among the choices that can be decided by an algorithm in all cases is "Is $L\left(G_1\right)=\varnothing$?".