Would an exponential lower bound on an NP-complete language prove P does not equal NP?

Question

If someone were able to prove an exponential lower bound for a NP-complete problem, would that prove that P ≠ NP?

Answer 1

Yes, that would prove that P is not equal to NP. All polynomials are bounded from above by any exponential function, so an exponential lower bound on any NP problem would prove that the problem is not in P, and thus would prove that P cannot equal NP.

Hope this helps!

Answer 2

You are absolutely right. If you prove an exponential lower bound for A, you have shown that A cannot lie in P. If A did lie in P, it would be decidable in polynomial time which is asymptotically faster than your just proven lower bound - we have a contradiction!

However, you don't have to pick an NP-complete problem. You can pick any language A in NP. By proving that A does not lie in P, you have also proven that P does not equal NP. Why? Because if P did equal NP, A would also lie in P, since we just picked A from NP.