Let \(𝐺_1, 𝐺_2 \) be Context Free Grammars (CFGs) and \(𝑅\) be a regular…

Answer: The only decidable question among the list is "Is L(G1) = ∅?"

Why this is decidable:

• There is a standard algorithm to decide emptiness of a context-free grammar.

• Algorithm (marking/generating-variable method):

  • Initialize a set M of nonterminals that have a production whose right-hand side is composed entirely of terminals.

  • Repeat: add any nonterminal A to M if there is a production A → α where every symbol of α is either a terminal or already in M.

  • When no new nonterminals can be added, check whether the start symbol is in M. If it is, the grammar generates at least one string (so L(G) ≠ ∅); otherwise L(G) = ∅.

Why the other questions are not decidable (sketches):

• Equivalence of two arbitrary context-free grammars is undecidable. There is no algorithm that decides for every pair of CFGs whether they generate the same language.

• Emptiness of the intersection of two context-free languages is undecidable. Deciding whether L(G1) ∩ L(G2) is empty is known to be an undecidable problem.

• Equality between an arbitrary context-free language and a given regular language is not generally decidable; there is no uniform algorithm that solves this equality test in all cases.

Therefore the decidable question among the provided choices is the emptiness test for a single CFG.