GATE | Sudo GATE 2020 Mock III (24 January 2019) | Question 50

Which of the following language is Recursive Enumerable but not recursive ?
(A) {<M> | M is a TM and there exists an input on which M halts in less than |<M>| steps}
(B) {<M>| M is a TM and |L(M)| ≤ 3}
(C) {<M>| M is a TM and |L(M)| ≥ 3}
(D) {<M>| M is a TM that accepts all even numbers}

Answer: (C)

Explanation: I. Recursive. The number of possible inputs is finite, and the number of steps M runs on each input is finite, therefore M is guaranteed to halt and decide the language.
II. Not Recursive Enumerable.
III. Recursive Enumerable but not recursive.
IV. Not Recursive Enumerable.

Option (C) is correct.

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