A question about the Cook-Levin Theorem

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In summary, the Cook-Levin Theorem states that any NP-complete problem can be reduced to the Boolean Satisfiability Problem. It is also true that any NP-complete problem can be reduced to any other NP-complete problem. Additionally, a polynomial-time solution to one NP-complete problem can be converted to a polynomial-time solution to the Boolean Satisfiability Problem. This is due to a technique called "arithmetization", which was used to prove that IP = PSPACE and NEXP = MIP. However, there is a negative result that states arithmetization alone cannot be used to solve P vs. NP.
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David Carroll
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I know that the Cook-Levin Theorem states that any NP-complete problem can be reduced to the Boolean Satisfiability Problem. I haven't actually perused the Theorem and I know it probably contains language that's above my ken, but I was wondering if anyone knew whether it is also true that any NP-complete problem can be reduced to any other NP-complete problem. Also: if the NP-complete problems can be reduced to the Boolean Satisfiability Problem, does it also follow that any polynomial-time solution to one NP-complete problem is able to be converted to a polynomial-time solution to the Boolean Satisfiability Problem?
 
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Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
I don't seem to have any further information. A couple years ago, I tried to reformulate any statement in Boolean logic into terms of positive integers and then see if factoring any of those integers would solve the satisfiability of Boolean statements. The only problem was, at the time I had thought I came up with a polynomial time algorithm that would factorize any integer. Turns out I didn't, and the whole excursion was rendered moot:mad:
 
  • #4
The answers are "yes" to both of your questions. Any NP problem can be reduced to an instance of an NP-complete problem. A polynomial-time solution to any NP-complete problem means polynomial-time solutions to every NP problem.

What you tried to do in your second post is similar to something called "arithmetization", which was a technique used to prove that IP = PSPACE and NEXP = MIP. There is a negative result which states that arithmetization alone can't be used to solve P vs. NP.

More info on arithmetization can be found at http://www.scottaaronson.com/papers/alg.pdf
 
  • #5
:)Thank you very much, Dragonfall. That was very helpful and kind.
 

FAQ: A question about the Cook-Levin Theorem

What is the Cook-Levin Theorem?

The Cook-Levin Theorem is a fundamental result in theoretical computer science that states that any problem that can be solved by a computer can also be solved by a universal Turing machine.

Who developed the Cook-Levin Theorem?

The Cook-Levin Theorem was developed independently by Stephen Cook and Leonid Levin in the 1970s. Cook published his proof in 1971, while Levin published his in 1973.

What is the significance of the Cook-Levin Theorem?

The Cook-Levin Theorem is significant because it proves that there is a way to reduce any problem that can be solved by a computer to a single problem that can be solved by a Turing machine. This has major implications for the field of theoretical computer science and the study of computational complexity.

How does the Cook-Levin Theorem relate to the P vs. NP problem?

The P vs. NP problem is one of the most famous unsolved problems in computer science, and it asks whether there are problems that can be solved quickly (in polynomial time) by a computer but not verified quickly. The Cook-Levin Theorem suggests that if P equals NP, then the class of problems that can be solved quickly is the same as the class of problems that can be verified quickly.

What are the practical applications of the Cook-Levin Theorem?

The Cook-Levin Theorem has many practical applications in the field of computer science, including in the development of algorithms, complexity theory, and cryptography. It also has implications for the study of artificial intelligence and the limits of computation.

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