Example for an undecidable language L such that L is reducible to its complement and vice versa

I am searching for an undecidable language $$L$$, such that $$L \leq \Sigma^* \setminus L$$ and $$\Sigma^* \setminus L \leq L$$, but I am not able to find a concrete language and reduction. Is there anything like this?

• Why don't you ask your teacher? Nov 24 '18 at 14:47

Let $$K$$ be some undecidable problem, and define $$L = \{0x : x \in K \} \cup \{1y : y \notin K \}.$$ (I'm assuming the alphabet is $$\{0,1\}$$.) Then $$\overline{L} = \{\epsilon\} \cup \{0x : x \notin K \} \cup \{1y : y \in K \}.$$ I'll let you take it from here.