Editing Cook–Levin theorem (section)

{{short description|Boolean satisfiability is NP-complete and therefore that NP-complete problems exist}}
In [[computational complexity theory]], the '''Cook–Levin theorem''', also known as '''Cook's theorem''', states that the [[Boolean satisfiability problem]] is [[NP-completeness|NP-complete]].  That is, it is in [[NP (complexity)|NP]], and any problem in NP can be [[reduction (complexity)|reduced]] in [[polynomial time]] by a [[deterministic Turing machine]] to the Boolean satisfiability problem.

The theorem is named after [[Stephen Cook]] and [[Leonid Levin]]. The proof is due to [[Richard Karp]], based on an earlier proof (using a different notion of reducibility) by Cook.<ref name="Karp"/>

An important consequence of this theorem is that if there exists a deterministic polynomial-time algorithm for solving Boolean satisfiability, then every [[NP (complexity)|NP]] problem can be solved by a deterministic polynomial-time algorithm.  The question of whether such an algorithm for Boolean satisfiability exists is thus equivalent to the [[P versus NP problem]], which is still widely considered the most important unsolved problem in [[theoretical computer science]].