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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