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I need to prove that the following language is not recursively enumerable, while its complement is recursively enumerable: $L := \{w \in \{0,1\}^* |$ TM $M$ with $w = \langle $ M $\rangle$ does not accept any input $\}$.

Should I use a reduction from the halting problem in this case?

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You can directly use the extension of Rice's theorem to prove that $L\notin RE$

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