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# GATE | GATE CS 2021 | Set 1 | Question 49

• Last Updated : 24 May, 2021

For a Turing machine M, ⟨M⟩ denotes an encoding of M. Consider the following two languages.

• L1 = { ⟨M⟩ ∣ M takes more than 2021 steps on all inputs }
• L2 = { ⟨M⟩ ∣ M takes more than 2021 steps on some input }
• Which one of the following options is correct?
(A) Both L1 and L2 are decidable
(B) L1 is decidable and L2 is undecidable
(C) L1 is undecidable and L2 is decidable
(D) Both L1 and L2 are undecidable

Answer: (A)

Explanation:

• For L1: check if Turing machine M, halts within 2021 steps, for all inputs of length less than or equal to 2021, if its not halting within 2021 steps, definitely it is taking more than 2021 steps, so it is decidable.
• For L2: similarly we can check if Turing machine M, halts within 2021 steps, for some inputs of length less than or equal to 2021. If its not halting within 2021 steps for some inputs, it is taking more than 2021 steps, so it is also decidable.
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