GATE CS 2020 Set 1 — Question 36
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Theory of Computation → Turing Machines & Computability → Rice's Theorem
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Question
Which of the following languages are undecidable? Note that indicates encoding of the Turing machine M.
Correct answer
(A) L₁, L₃, and L₄ only
Solution
We analyze each language:
1.: This is the Emptiness problem for Turing Machines, which is a well-known undecidable problem (Rice's Theorem applies as emptiness is a non-trivial property of RE languages).
2.: We can simulate the Turing machine on input for exactly 100 steps. Since the number of steps is fixed and finite, the simulation will halt and give a yes/no answer. Thus, this is decidable.
3.: This asks whether the language accepted by is not recursive (i.e., it is Recursively Enumerable but not Recursive). This is a non-trivial property of the language accepted by the TM. By Rice's Theorem, any non-trivial property of RE languages is undecidable.
4.: This asks about the cardinality of the language accepted by . Specifically, . This is a non-trivial property of RE languages (some have members, some have ). By Rice's Theorem, this is undecidable.
Therefore, and are undecidable.Continue learning with Success Tracker
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