P vs NP Was Solved Half A Century Ago

[odysseas]

Ladner made the assumption that P is not equal to NP and then went from there. This is not an accepted proof in mathematics (since you can't assume your answer before finding it yourself) but it did open the door for NP-intermediate problems which are in the complexity class NP but are neither in the class P nor NP-complete.

It's not that Ladner's problem was "unintersting" it is that it was "artificially" made by starting from the end and working his way backwords. By phrasing the problem assuming that P is not equal to NP he created a loophole that if his original statement is false ("if P=NP") then his whole construction is false. He basically built in shaky foundations.

There are non-artificial problems that are believed to be NP-intermediate. The key word is "believed" since nothing has been proven.

I think this is an issue somewhat similar to the Riemann Hypothesis: everyone believes it to be true, trillions of numbers have been checked BUT it has not been mathematically proven. I think if you ask any math professor if they had to guess they would say that P is not equal to NP. That is not a proof of course, it is just the academic consensus.

So, P-NP has definitely not been solved mathematically but there is good evidence to suggest that P is bot always equal to NP