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Is there an undecidable language A that is mapping reducible to its complement?

If it is possible, since A is an undecidable language, so A's complement must also be an undecidable language. But i don't know whether undecidable languages is closed under complement or not.

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Yes, it is possible. Enumerate the set of all possible Turing machines and let $H$ (resp. $\overline{H}$) be the set of indices of the Turing machines that halt (resp. do not halt) on empty input.

Let $L = \{ \langle T, 1 \rangle \, : T \in H \} \cup \{ \langle T, 0 \rangle \, : T \in \overline{H} \}$.

Clearly $L$ is not decidable but it is possible to reduce $L$ to $\overline{L}$ since $\langle T, r \rangle \in L \iff \langle T, 1-r \rangle \in \overline{L}$.

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