GATE CS 2021 Set 1 — Question 49
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Theory of Computation → Turing Machines & Computability → Recursive & RE Languages
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Question
For a Turing machine , denotes an encoding of . Consider the following two languages.Which one of the following options is correct?
Correct answer
(A) Both L₁ and L₂ are decidable.
Solution
To determine if a Turing machine takes more than 2021 steps on an input , we only need to simulate on for at most 2022 steps. Crucially, if takes more than 2021 steps, it can only read at most the first 2021 symbols of the input tape. Therefore, the behavior of for the first 2021 steps depends only on the first 2021 symbols of the input. Since the input alphabet is finite, there are only a finite number of input prefixes of length 2021 (specifically, ).
- For : We can check all inputs of length up to 2021. If takes steps on all of them, it will take steps on any input (as any longer input shares a prefix that was already checked). This is a finite check, so is decidable.
- For : Similarly, we can check all inputs of length up to 2021. If takes steps on at least one of them, then is satisfied. This is also a finite check, so is decidable.
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