# complexity theory – Is there a non-deterministic polynomial by time Turing machine such that: \$L(M)in NPC\$ and \$L(overline{M})in P\$

When $$overline{M}$$ is a non-deterministic polynomial by time Turing machine that final states switched: accept to reject and vice versa.
I’m thinking that this equal to $$P=NP$$, but I saw a solution (an example) that I disagree with:
$$M$$ is a non-deterministic polynomial by time Turing machine that decide $$SAT$$, if all that paths are rejected then $$L(overline{M})=Sigma^*in P$$

Is it a valid solution, or as I’m thinking $$L(M)in NPC$$ and $$L(overline{M})in P Leftrightarrow P=NP$$