GATE CS 2022 Set 1 — Question 46
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Theory of Computation → Turing Machines & Computability → Rice's Theorem
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Question
Which of the following is/are undecidable?
Correct 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.
Solution
Option (A) is the Equivalence Problem for Turing Machines, which is undecidable.
Option (B) asks if the language of a TM is regular. By Rice's Theorem, any non-trivial property of the RE languages is undecidable. Regularity is a non-trivial property, so this is undecidable.
Option (C) is the Totality Problem (), which is undecidable.
Option (D) is decidable. Since the machine runs for a fixed number of steps (1073), it can only scan a bounded prefix of the input (at most 1073 symbols). We only need to simulate on all strings of length up to 1073. If halts or loops within 1073 steps on any such string, we can determine the answer. Since the set of strings to check is finite and the simulation is bounded, this is decidable.
Option (B) asks if the language of a TM is regular. By Rice's Theorem, any non-trivial property of the RE languages is undecidable. Regularity is a non-trivial property, so this is undecidable.
Option (C) is the Totality Problem (), which is undecidable.
Option (D) is decidable. Since the machine runs for a fixed number of steps (1073), it can only scan a bounded prefix of the input (at most 1073 symbols). We only need to simulate on all strings of length up to 1073. If halts or loops within 1073 steps on any such string, we can determine the answer. Since the set of strings to check is finite and the simulation is bounded, this is decidable.
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