> The way I like to think about NP is: are there certain types of problems which we, as smart as we might be, cannot easily solve, not because we aren't smart enough but because of the nature of the problem itself?
That's not entirely accurate. There are many problems that are untractable (i.e. can't solve them because of the nature of the problem) but have nothing to do with P=NP. Consider the halting problem, for instance.
There are many problems that are untractable (i.e. can't solve them because of the nature of the problem) but have nothing to do with P=NP
Can you elaborate some more on this? I'd like to learn more about this area. Are you referring to problems for which no polynomial time solution exists, and there is also no polynomial time method for verifying a solution?
> The way I like to think about NP is: are there certain types of problems which we, as smart as we might be, cannot easily solve, not because we aren't smart enough but because of the nature of the problem itself?
If you read it one way, it sounds as if P=NP would allow us to solve every formal problem. It doesn't. One very interesting example:
The concept of Kolmogorov Complexity. I'm not a CS theorist, so I can only give my layman interpretation, but if I understand it well it goes like this: given a particular string of bits, the Kolmogorov Complexity relates to the shortest possible program that can describe this string. In effect, that means: the ultimate compression. Well, the whole theory of KC says that you can't find this shortest possible program for every string. Not because it takes too much time to check very possible combination (P=?=NP) but because of the nature of the problem.
That's not entirely accurate. There are many problems that are untractable (i.e. can't solve them because of the nature of the problem) but have nothing to do with P=NP. Consider the halting problem, for instance.