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The following exercise is taken from Chapter 17 of Languages and Machines by Thomas Sudkamp:

Let $Q$ be a language reducible to a language $L$ in polynomial time. Prove that $\overline{Q}$ is reducible to $\overline{L}$ in polynomial time.

I am not sure if my solution idea is correct. Anyone knows the correct solution of this exercise?

My idea is: I use the definition of NP-complete. Q is in NP class and L is NP-complete. The function f to reducible in polynomial time Q to L, is also the function f to reducible in polynomial time notQ to notL

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    $\begingroup$ Could you tell us your solution idea, and why you're not sure it's correct? $\endgroup$ – Yuval Filmus Nov 3 '19 at 16:44
  • $\begingroup$ I use the definition of NP-complete. Q is in NP class and L is NP-complete. The function f to reducible in polynomial time Q to L, is also the function f to reducible in polynomial time notQ to notL. $\endgroup$ – Gianni Spear Nov 3 '19 at 16:49
  • $\begingroup$ This seems completely fine. Perhaps you can add such an answer to your question. $\endgroup$ – Yuval Filmus Nov 3 '19 at 16:50
  • $\begingroup$ Thanks. updated my question $\endgroup$ – Gianni Spear Nov 3 '19 at 16:52
  • $\begingroup$ Wait.. In the question I see no assumption that $Q \in \textrm{NP}$ or that $L$ is $\textrm{NP}$-complete. In fact those assumptions are unnecessary. $\endgroup$ – Steven Nov 3 '19 at 17:20
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By hypothesis $Q \le_p L$, i.e., there exists a poly-time computable function $f(x)$ such that $x \in L \iff f(x) \in Q$. Then: $$ x \in \overline{L} \iff x \not\in L \iff f(x) \not\in Q \iff f(x) \in \overline{Q}. $$

Therefore $\overline{L} \le_p \overline{Q}$.

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