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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Last Updated :
11 Oct, 2021
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