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Let $A$ be a problem that I want to show it is coNP-complete. I know I could just show its complement $\bar{A}$ is NP-complete or that $\bar{A}$ is in NP and for some coNP-complete problem $Q$, show that $Q$ can be Karp-reduced to $A$.

But I wonder if the following steps are sufficient to show that A is coNP-complete?

  1. Show that $A$ is in $coNP$ by showing that its complement is in NP
  2. Choose one coNP-complete problem $Q$ and Cook-reduce it to $A$.

Does the coNP-complete class still be distinguishable (under the assumption $P\neq NP$) from the NP-complete class? since it seems that any coNP-complete problem is Turing-polynomially equivalent to any other NP-Complete problem.

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  • $\begingroup$ Please make "2" more precise. That was rather loose there. Which type of reduction are you talking about? Karp reduction, or Cook reduction? $\endgroup$ – D.W. Sep 24 '18 at 0:02
  • $\begingroup$ 2 is a Cook reduction. $\endgroup$ – RTK Sep 24 '18 at 0:05
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    $\begingroup$ Note that at the very least you have to assume $P\not = NP$ because otherwise taking $A$ to be the empty language would falsify your statement. $\endgroup$ – Tom van der Zanden Sep 24 '18 at 7:58
  • $\begingroup$ I think if $P=NP$ and $A$ is the empty language then (2) will still hold since the chosen coNP-complete problem $P$ will be still solvable in poly-time; but with no call of the algorithm solving $A$. Am I missing something? $\endgroup$ – RTK Sep 24 '18 at 8:19
  • $\begingroup$ Your logic would show $A$ to be $coNP$-complete even though it isn't. $\endgroup$ – Tom van der Zanden Sep 24 '18 at 10:11
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No, the steps are insufficient. You need to use a Karp reduction, since that's the type of reduction used in the definition of coNP-completeness.

SAT is NP-complete, but probably not coNP-complete (unless NP=coNP); and coSAT is coNP-complete, but probably not NP-complete. This distinction would be lost if you used Cook reductions.

Some questions on this site which are relevant are this one and that one, as well as a few others.

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  • $\begingroup$ Thanks! Can I with (1) and (2) conclude that $A$ is Cook-coNP-complete? considering the Cook-coNP completeness defined similarly as the coNP completeness but with the Cook reduction. $\endgroup$ – RTK Sep 25 '18 at 16:50
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    $\begingroup$ If you define a Cook-coNP-complete problem to be one that satisfies (1) and (2), then any problem that satisfies (1) and (2) is Cook-coNP-complete, by definition. $\endgroup$ – Yuval Filmus Sep 25 '18 at 17:13
  • $\begingroup$ Please according to my second preoccupation when using only CooK reduction does it matter if the chosen problem to be reduced is p-complete or coNP complete? In other words when showing that a problem $A$ is Cook NP-hard* could I Cook reduce a coNP-complete (such as TAUT, UNSAT) problem to $A$ instead of an NP-complete (SAT) to A? $\endgroup$ – RTK Sep 25 '18 at 18:00
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    $\begingroup$ A problem is Cook-NP-hard iff it is Cook-coNP-hard. $\endgroup$ – Yuval Filmus Sep 25 '18 at 18:00

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