Which of the following languages are undecidable? Note that ⟨M⟩ indicates encoding of the Turing machine M.

- L
_{1}= { ⟨M⟩ ∣ L(M)=∅ } - L
_{2}= { ⟨M,w,q⟩ ∣ M on input w reaches state q in exactly 100 steps } - L
_{3}= { ⟨M⟩ ∣ L(M) is not recursive } - L
_{4}= { ⟨M⟩ ∣ L(M) contains at least 21 members }

**(A)** L_{1}, L_{3}, and L_{4} only

**(B)** L_{1} and L_{3} only

**(C)** L_{2} and L_{3} only

**(D)** L_{2}, L_{3}, and L_{4} only

**Answer:** **(A)** **Explanation:**

L_{1} = { ⟨M⟩ ∣ L(M)=∅ } is emptiness problem of TM, which is undecidable, by Rice’s theorem since it is a non-trivial problem.

L_{2} = { ⟨M,w,q⟩ ∣ M on input w reaches state q in exactly 100 steps } is decidable as we can run the TM for 100 steps and see if it reaches state q.

L_{3} = { ⟨M⟩ ∣ L(M) is not recursive } is undecidable according to Rice theorem.

L_{4} = { ⟨M⟩ ∣ L(M) contains at least 21 members } is undecidable. It may or may not halt.

Only L_{2} is decidable.

Option (A) is correct.

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