Unraveling the P vs NP Problem in Computer Science

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SUMMARY

The P vs NP problem in computer science remains one of the most significant unsolved questions, with the consensus leaning towards P ≠ NP. P problems, such as matrix inversion, can be solved in polynomial time, while NP problems, like solving a minesweeper puzzle, do not have known polynomial-time solutions. The challenge lies in proving that no polynomial algorithm exists for NP problems, which is the crux of the Millennium Prize Problem, offering a reward of $1,000,000 for a valid proof. Discussions suggest that traditional mathematics may not suffice, and alternative approaches, such as boolean algebra and logic, could be necessary for breakthroughs.

PREREQUISITES
  • Understanding of polynomial time complexity
  • Familiarity with P and NP problem classifications
  • Basic knowledge of boolean algebra
  • Experience with algorithm design and analysis
NEXT STEPS
  • Research the implications of the P vs NP problem on algorithm design
  • Study boolean algebra and its applications in computational logic
  • Explore existing proofs and attempts regarding P vs NP
  • Investigate the significance of the Millennium Prize Problems in computer science
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Computer scientists, mathematicians, algorithm developers, and students interested in theoretical computer science and complexity theory will benefit from this discussion.

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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