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Turing Machines and Undecidability

GATE CSE 2022 · Q46

MSQ2 marksMedium

Time

Which of the following is/are undecidable?

Select all that applyOne or more correct

Pick every correct option

Solution

Step-by-step verified solution

Official answer: A) Given two Turing machines M1 and M2, decide if L(M1) = L(M2).; B) Given a Turing machine M, decide if L(M) is regular.; C) Given a Turing machine M, decide if M accepts all strings.

Turing Machines and Undecidability

To determine the undecidability of each option:

  • Option A: Deciding whether L(M1)=L(M2)L(M_1) = L(M_2) for two Turing machines is the equivalence problem for TMs, which is undecidable (in fact, not Turing-recognizable / not semi-decidable).

  • Option B: Testing if the language of a TM, $L(M)$, is regular is a non-trivial property of Turing-recognizable languages. ByRice's Theorem, this property is undecidable.

  • Option C: Deciding if a TM accepts all strings, i.e., L(M)=Σ∗L(M) = \Sigma^*, is also a non-trivial semantic property of RE languages, and by Rice's Theorem, it is undecidable.

  • Option D: In at most 10731073 steps, a deterministic Turing machine MM can inspect at most the first 10731073 characters of the input string. Therefore, its behavior within 10731073 steps depends only on input prefixes of length ≤1073\le 1073. Since there are only finitely many such prefixes over a finite alphabet, we can simulate MM on all of them for 10741074 steps. If MM takes more than 10731073 steps on all these inputs, it will take more than 10731073 steps on every string. Thus, this property is decidable.

Therefore, options A, B, and C are undecidable.

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This Turing Machines and Undecidability previous year question is Q46 of the GATE CSE 2022 paper. Find more GATE CSE PYQs on Turing Machines and Undecidability in Theory of Computation, or browse all GATE CSE PYQs.

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Turing Machines and Undecidability in GATE CSE

Turing Machines and Undecidability has 30 GATE CSE PYQs across 20 papers from 2012 to 2026. This is the only 2-mark MSQ among them, and GATE CSE 2022 has 2.

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Which of the following is/are undecidable?

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