- #1
Jimmy84
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I was reading about the P = NP problem, and it seems very intersting because it has deep implications for math and for all of science if proven true.
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http://en.wikipedia.org/wiki/P_=_NP_problem#Consequences_of_proof"
One of the reasons the problem attracts so much attention is the consequences of the answer.
A proof of P = NP could have stunning practical consequences, if the proof leads to efficient methods for solving some of the important problems in NP. Various NP-complete problems are fundamental in many fields. There are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP-complete, such as some types of integer programming, and the traveling salesman problem, to name two of the most famous examples. Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in Protein structure prediction are also NP-complete;[11] if these problems were efficiently solvable it could spur considerable advances in biology.
But such changes may pale in significance compared to the revolution an efficient method for solving NP-complete problems would cause in mathematics itself. According to Stephen Cook,[12]
...it would transform mathematics by allowing a computer to find a formal proof of any theorem which has a proof of a reasonable length, since formal proofs can easily be recognized in polynomial time. Example problems may well include all of the CMI prize problems.
Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated – for instance, Fermat's Last Theorem took over three centuries to prove. A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, would essentially end this struggle.
A proof that showed that P ≠ NP, while lacking the practical computational benefits of a proof that P = NP, would also represent a massive advance in computational complexity theory and provide guidance for future research. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place.
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I don't know much about computer science, I was wondering which fields of computer science and mathematics do I have to learn in order to get aquainted with the problem? I by no means would attempt to solve it but I am very curious about it because of its vast implications.
(Quote)
http://en.wikipedia.org/wiki/P_=_NP_problem#Consequences_of_proof"
One of the reasons the problem attracts so much attention is the consequences of the answer.
A proof of P = NP could have stunning practical consequences, if the proof leads to efficient methods for solving some of the important problems in NP. Various NP-complete problems are fundamental in many fields. There are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP-complete, such as some types of integer programming, and the traveling salesman problem, to name two of the most famous examples. Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in Protein structure prediction are also NP-complete;[11] if these problems were efficiently solvable it could spur considerable advances in biology.
But such changes may pale in significance compared to the revolution an efficient method for solving NP-complete problems would cause in mathematics itself. According to Stephen Cook,[12]
...it would transform mathematics by allowing a computer to find a formal proof of any theorem which has a proof of a reasonable length, since formal proofs can easily be recognized in polynomial time. Example problems may well include all of the CMI prize problems.
Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated – for instance, Fermat's Last Theorem took over three centuries to prove. A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, would essentially end this struggle.
A proof that showed that P ≠ NP, while lacking the practical computational benefits of a proof that P = NP, would also represent a massive advance in computational complexity theory and provide guidance for future research. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place.
(EndQuote)
I don't know much about computer science, I was wondering which fields of computer science and mathematics do I have to learn in order to get aquainted with the problem? I by no means would attempt to solve it but I am very curious about it because of its vast implications.
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