GATE CS 2025 Set 2 Q.25: Decidability, Reducibility and Undecidability

Let $G_1, G_2$ be Context Free Grammars (CFGs) and $R$ 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?

Is $L(G_1) = L(G_2)$ ?

Is $L(G_1) \cap L(G_2) = \emptyset$ ?

Is $L(G_1) = L(R)$ ?

Is $L(G_1) = \emptyset$ ?

Official answer and solution

Official answer: D
From: Official GATE 2025 answer key

Solution

Recall that emptiness, finiteness, and membership are decidable for CFGs, whereas equivalence, disjointness, and universality ( $L(G_1) = L(R)$ ) are undecidable.

Emptiness of a CFG ( $L(G_1) = \emptyset$ ?) is decidable: We can determine in polynomial time whether the start symbol of $G_1$ can derive any terminal string by iteratively marking all generating non-terminals.
Equivalence ( $L(G_1) = L(G_2)$ ), intersection emptiness ( $L(G_1) \cap L(G_2) = \emptyset$ ), and equality to a regular language ( $L(G_1) = L(R)$ , which includes universality $L(G_1) = \Sigma^*$ ) are all undecidable for CFGs.

Final answer

Option (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 25 of 65
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source