GATE | GATE CS 2020 | Question 36

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

• L₁ = { ⟨M⟩ ∣ L(M)=∅ }
• L₂ = { ⟨M,w,q⟩ ∣ M on input w reaches state q in exactly 100 steps }
• L₃ = { ⟨M⟩ ∣ L(M) is not recursive }
• L₄ = { ⟨M⟩ ∣ L(M) contains at least 21 members }

(A) L₁, L₃, and L₄ only
(B) L₁ and L₃ only
(C) L₂ and L₃ only
(D) L₂, L₃, and L₄ only

Answer: (A)
Explanation: L₁ = { ⟨M⟩ ∣ L(M)=∅ } is emptiness problem of TM, which is undecidable, by Rice’s theorem since it is a non-trivial problem.

L₂ = { ⟨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.

L₃ = { ⟨M⟩ ∣ L(M) is not recursive } is undecidable according to Rice theorem.

L₄ = { ⟨M⟩ ∣ L(M) contains at least 21 members } is undecidable. It may or may not halt.

Only L₂ is decidable.

Option (A) is correct.

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