Final answer:
L is undecidable and unrecognizable. We can prove this using Rice's Theorem and the Halting Problem.
Step-by-step explanation:
To prove that L is undecidable and unrecognizable, we can use Rice's Theorem and the Halting Problem.
A) Rice's Theorem:
L is undecidable because it satisfies the requirements of Rice's Theorem, which states that any non-trivial property of a language defined by a Turing machine is undecidable. In this case, the property of L(m) = empty set is non-trivial, as it cannot be determined whether a given Turing machine M will ever halt on an input.
C) Halting Problem:
L is unrecognizable because it is equivalent to the Halting Problem, which is proven to be undecidable. The Halting Problem asks whether a given Turing machine will halt on a specific input. Since L(m) = empty set means that a Turing machine does not halt on any input, determining emptiness of L(m) is equivalent to the Halting Problem, making it unrecognizable.