MHB Unraveling the P vs NP Problem in Computer Science

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The discussion centers on the P vs NP problem in computer science, a fundamental question regarding the relationship between problems that can be solved quickly (P) and those for which solutions can be verified quickly (NP). Participants express a belief that P does not equal NP (P ≠ NP) but acknowledge the difficulty in proving this assertion. One contributor suggests that traditional mathematics may not suffice to solve the problem, proposing that boolean algebra and logic could be key. The conversation highlights the complexity of the problem, noting that while some examples of P problems, like matrix inversion, exist, NP problems, such as solving a minesweeper puzzle, lack polynomial-time solutions. The challenge remains that no one has yet proven the absence of a polynomial algorithm for NP problems, which is the crux of the P vs NP millennium problem, offering a reward of $1 million for a solution.
shamieh
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Didn't know what forum to post this in..There are some brilliant people on this forum, just wanted to know your thoughts on the P vs NP problem in Computer Science? Do you think it will ever be solved? I think P != NP. That being said, I do believe it could be solved but not with normal mathematics. I believe that boolean algebra may be able to solve the problem through LOGIC. As far as statistical formulas creating a Computer AI, I just don't think it's possible. After all, we aren't machines.
 
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Of course $\text{P} \neq \text{NP}$ , but I believe this is more or less undecidable.
 
It seems like it is pretty obvious that P != NP, but how come no one can prove it? Can you give me a example? Like what are the problems people are running into? Or is it too advanced for me to even comprehend? I'm in Calculus II.
 
Most problems are more or less undecidable whether in P or NP, so I can't really give you some nontrivial ones. What background have you got in computer science?
 
shamieh said:
It seems like it is pretty obvious that P != NP, but how come no one can prove it? Can you give me a example? Like what are the problems people are running into? Or is it too advanced for me to even comprehend? I'm in Calculus II.

One example of a P problem is the inversion of a matrix.

One example of an NP problem is to solve a minesweeper puzzle.

P problems are those which have an algorithm to solve them which takes a number of steps that is a polynomial on the input. In the matrix case, the input are the numbers in the matrix.

NP problems are those which don't have an algorithm to solve them which takes a number of steps that is a polynomial on the input. But this special class of problems also has the property that if you could find a polynomial algorithm to solve one of them, you would solve all of them in polynomial time.

The thing is that you haven't still proved that there is no polynomial algorithm to solve an NP problem. That's the P vs NP millenium problem, which is worth $ 1 000 000.
 
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