Does the proof of the widely believed result P $\neq$ NP depend on the proof of NP $\neq$ Co-NP ?

  • $\begingroup$ Which proofs are you referring to? What do you mean by depending? $\endgroup$
    – Kaveh
    Jan 10, 2013 at 2:44

1 Answer 1


Only in one direction. As $\mathsf{P}=\text{co-}\mathsf{P}$, if $\mathsf{NP}\neq\text{co-}\mathsf{NP}$ then we would know that $\mathsf{P}\neq\mathsf{NP}$. However the reverse implication doesn't hold. If $\mathsf{P}\neq\mathsf{NP}$ then it's possible that either $\mathsf{NP}\neq\text{co-}\mathsf{NP}$ or $\mathsf{NP}=\text{co-}\mathsf{NP}$.


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