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Turing Machines and Undecidability

GATE CSE 2025 Set 2 · Q25

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Let G1,G2G_1, G_2 be Context Free Grammars (CFGs) and RR be a regular expression. For a grammar GG, let L(G)L(G) denote the language generated by GG. Which ONE among the following questions is decidable?

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Solution

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Official answer: D) Is L(G1) = ∅?

Turing Machines and Undecidability

Let us check the decidability of the given decision problems for Context-Free Languages (CFLs):

  1. Emptiness of a CFG (L(G1)=∅L(G_1) = \emptyset): 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).

  2. Equivalence of two CFGs ($L(G_1) = L(G_2)$): This is undecidable.

  3. Intersection-emptiness/Disjointness (L(G1)∩L(G2)=∅L(G_1) \cap L(G_2) = \emptyset): This is undecidable.

  4. Equivalence of a CFG with a Regular Expression (L(G1)=L(R)L(G_1) = L(R)): This is undecidable (since checking universality L(G1)=Σ∗L(G_1) = \Sigma^* is a special case where R=Σ∗R = \Sigma^*, which is undecidable).

Therefore, only Is L(G1)=∅L(G_1) = \emptyset? is decidable, which corresponds to option D.

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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 in GATE CSE

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.

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GateAI solution cardGATE CSE 2025 Set 2 · Q25 · MCQ

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?

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