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GATE CSE 2021 Set 1 Q49: Decidability of TMs taking more than 2021 steps on all/some inputs

Question

For a Turing machine MM, ⟨M⟩\langle M \rangle denotes an encoding of MM. 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}\begin{aligned} L_1 &= \{\langle M \rangle \mid M \text{ takes more than 2021 steps on all inputs}\} \\ L_2 &= \{\langle M \rangle \mid M \text{ takes more than 2021 steps on some input}\} \end{aligned}

Which one of the following options is correct?

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Answer: A (official GATE 2021 final answer key)

Concept used: In 2021 steps a TM reads at most the first 2021 input symbols, so only finitely many input prefixes matter.

First step: Simulate MM for 2022 steps on every string of length at most 2021; this finite check decides both properties.

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key

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