For a Turing machine \(M, ⟨M⟩\) denotes an encoding of \(M\). Consider the…
2021
For a Turing machine \(M, ⟨M⟩\) denotes an encoding of \(M\). Consider the following two languages.
\(\begin{array}{ll} L_1 = \{ \langle M \rangle \mid M \text{ takes more than } 2021 \text{ steps on all inputs} \} \\ L_2 = \{ \langle M \rangle \mid M\text{ takes more than } 2021 \text{ steps on some input} \} \end{array}\)
Which one of the following options is correct?
- A.
Both
\(L_1\)and\(L_2\)are decidable. - B.
\(L_1\)is decidable and\(L_2\)is undecidable - C.
\(L_1\)is undecidable and\(L_2\)is decidable - D.
Both
\(L_1\)and\(L_2\)are undecidable
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Correct answer: A
Key observation: In 2021 steps a Turing machine can only read cells within distance 2021 from the start position, so its behavior during the first 2021 steps depends only on the input prefix of length at most 2021. Therefore, to decide any property about whether the machine halts within 2021 steps on inputs, it suffices to check a finite set of inputs (all inputs of length at most 2021).
Deciding the language of machines that take more than 2021 steps on all inputs:
Let Σ be the input alphabet. Enumerate every input string x in Σ of length at most 2021 (this is a finite set).
For each such x, simulate the given machine M on x for exactly 2021 steps.
If any simulation halts within those 2021 steps, then M does not take more than 2021 steps on all inputs, so reject.
If none of the finite simulations halts within 2021 steps, accept. This procedure always terminates, so the language is decidable.
Deciding the language of machines that take more than 2021 steps on some input:
Either observe this language is the complement of the previous property, and since the previous property is decidable its complement is decidable as well, or give a direct finite check:
Simulate M for 2021 steps on every input of length at most 2021. If any simulation is still running after 2021 steps, accept; otherwise reject. This also always terminates.
Conclusion: Both languages are decidable because the relevant behavior is determined by a finite set of inputs when the time bound is fixed at 2021 steps.
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