Which of the following is/are undecidable?
Topic-wise GATE CS PYQs with verified steps
GATE CSE 2022 · Q46
Time
Select all that applyOne or more correct
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 UndecidabilityTo determine the undecidability of each option:
Option A: Deciding whether 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., , is also a non-trivial semantic property of RE languages, and by Rice's Theorem, it is undecidable.
Option D: In at most steps, a deterministic Turing machine can inspect at most the first characters of the input string. Therefore, its behavior within steps depends only on input prefixes of length . Since there are only finitely many such prefixes over a finite alphabet, we can simulate on all of them for steps. If takes more than steps on all these inputs, it will take more than steps on every string. Thus, this property is decidable.
Therefore, options A, B, and C are undecidable.
Try the question first, then check your work.
Examiner trap
Recall Rice's Theorem for semantic properties of Turing machine languages, and consider whether a property restricted to a bounded number of steps can be checked in finite time.
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.
Free diagnostic
Take the 20-minute diagnostic: the 3 topics that cost you the most marks, and a rank range. Or sit 5 related Turing Machines and Undecidability PYQs now with no account.
An official question by email at 7 AM IST, with a link to its verified solution. Unsubscribe in one click.
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.
How GATE tests Turing Machines and UndecidabilityIndependent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.