system
Given the following post, respond as if you're commenting on Reddit:
human
Can someone PLEASE explain the P = NP logic problem.

I have come across this a number of times but don't fully understand what is being asked. Can someone help out?
gpt
I wouldn't call it a "logic problem", it's an unsolved problem in Computer Science, with relevance to such things as cryptography.

Different problems are categorised into "Classes", based on the minimum complexity (ie rate of growth of the time taken to solve as a function of the size of the input).

For example, P is the class of problems that can be solved in polynomial time (ie for a problem size N, as N grows, the problem will get closer to the shape of N^x where x is a constant).

NP is the class of problems that can be solved in polynomial time (like P) **if** you use a hypothetical "Non-deterministic Turing Machine". Practically, that means you can test multiple (or all) possible solutions for each step at once.

NP problems can be verified (you can check whether it is correct) in polynomial time, but cannot be conclusively solved in polynomial time (and are in fact out of range of what is considered reasonable for computation time).

NP problems tend to be of the kind that are easier to work out by hand (by trial and error) than to automate.

The P = NP problem asks whether we can prove whether or not all NP problems can somehow be reduced to P problems, which would obviously make them easier to solve, revolutionising CS. The consensus is the P =/= NP, but a proof either that it is or that it isn't has not yet been found.

The $1000000 Millenium Prize would be awarded to anyone who could provide such proof (one way or the other).