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In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP , and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.
As noted above, this is the Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete ...
The concept of NP-completeness was introduced in 1971 (see Cook–Levin theorem), though the term NP-complete was introduced later. At the 1971 STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine.
For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", [1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete [2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction ...
Conway circle theorem (Euclidean plane geometry) Cook's theorem (computational complexity theory) Corners theorem (arithmetic combinatorics) Corona theorem (complex analysis) Courcelle's theorem (graph theory) Cousin's lemma (real analysis) Cox's theorem (probability) Craig's theorem (mathematical logic) Craig's interpolation theorem ...
Satisfiability, in turn, was proved NP-complete in the Cook–Levin theorem. From a given CNF formula, Karp forms a graph that has a vertex for every pair (v,c), where v is a variable or its negation and c is a clause in the formula that contains v. Two of these vertices are connected by an edge if they represent compatible variable assignments ...
The circuit on the left is satisfiable but the circuit on the right is not. In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. [1]