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Turing Machines and Undecidability

GATE CSE 2013 · Q41

MCQ2 marksEasy

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Which of the following is/are undecidable? 1. GG is a CFG. Is L(G)=ΦL(G) = \Phi? 2. GG is a CFG. Is L(G)=Σ∗L(G) = \Sigma^*? 3. MM is a Turing machine. Is L(M)L(M) regular? 4. AA is a DFA and NN is an NFA. Is L(A)=L(N)L(A) = L(N)?

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Official answer: D) 2 and 3 only

Turing Machines and Undecidability

Let us evaluate the decidability of each problem:

  1. Emptiness problem for CFGs (L(G)=∅L(G) = \emptyset): Decidable. We can test if any terminal string can be derived from the start symbol using the generating/reachable variables algorithm.

  2. **Universality problem for CFGs (L(G)=Σ∗L(G) = \Sigma^*): Undecidable**. Testing if a context-free grammar generates all strings over its alphabet is a well-known undecidable problem (can be reduced from the Post Correspondence Problem or TM halting problem).

  3. Regularity of a Turing Machine language (L(M)L(M) is regular): Undecidable. By Rice's Theorem, any non-trivial semantic property of the language accepted by a Turing machine is undecidable.

  4. Equivalence of a DFA and an NFA (L(A)=L(N)L(A) = L(N)): Decidable. We can convert the NFA NN to an equivalent DFA DND_N, construct the symmetric difference DFA Mdiff=(A∩DN‾)∪(A‾∩DN)M_{diff} = (A \cap \overline{D_N}) \cup (\overline{A} \cap D_N), and test if L(Mdiff)=∅L(M_{diff}) = \emptyset.

Therefore, only statements 2 and 3 are undecidable, which corresponds to option (D).

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This Turing Machines and Undecidability previous year question is Q41 of the GATE CSE 2013 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 2-mark MCQs like this one, and GATE CSE 2013 has 2.

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GateAI solution cardGATE CSE 2013 · Q41 · MCQ

Which of the following is/are undecidable? 1. G is a CFG. Is L(G) = ? 2. G is a CFG. Is L(G) = *? 3. M is a Turing machine. Is L(M) regular? 4. A is a DFA and N is an NFA. Is L(A) = L(N)?

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