Let G1, G2 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?
Topic-wise GATE CS PYQs with verified steps
GATE CSE 2025 Set 2 · Q25
Time
Select one answerSingle correct
Solution
Official answer: D) Is L(G1) = ∅?
Turing Machines and UndecidabilityLet us check the decidability of the given decision problems for Context-Free Languages (CFLs):
Emptiness of a CFG (): This is decidable. It can be solved by determining whether any terminal string can be derived from the start variable (by iteratively marking variables that generate terminal strings).
Equivalence of two CFGs ($L(G_1) = L(G_2)$): This is undecidable.
Intersection-emptiness/Disjointness (): This is undecidable.
Equivalence of a CFG with a Regular Expression (): This is undecidable (since checking universality is a special case where , which is undecidable).
Therefore, only Is ? is decidable, which corresponds to option D.
Try the question first, then check your work.
Examiner trap
Recall the decidability table for CFLs: emptiness, finiteness, and membership are decidable; equivalence, intersection-emptiness, and universality are undecidable.
This Turing Machines and Undecidability previous year question is Q25 of the GATE CSE 2025 Set 2 paper. Find more GATE CSE PYQs on Turing Machines and Undecidability in Theory of Computation, or browse all GATE CSE PYQs.
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Turing Machines and Undecidability has 30 GATE CSE PYQs across 20 papers from 2012 to 2026. 13 of them are 1-mark MCQs like this one, and it is the only one in GATE CSE 2025 Set 2.
How GATE tests Turing Machines and UndecidabilityIndependent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.