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Boolean satisfiability problem (SAT). [2] [3]: LO1 There are many variations that are also NP-complete. An important variant is where each clause has exactly three literals (3SAT), since it is used in the proof of many other NP-completeness results. [3]: p. 48 Circuit satisfiability problem; Conjunctive Boolean query [3]: SR31
There is often only a small difference between a problem in P and an NP-complete problem. For example, the 3-satisfiability problem, a restriction of the Boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted 2-satisfiability problem is in P (specifically, it is NL-complete), but the slightly more general max ...
The Boolean satisfiability problem is one of many NP-complete problems. If any NP-complete problem is in P, then it would follow that P = NP. However, many important problems are NP-complete, and no fast algorithm for any of them is known.
Karp showed each of his problems to be NP-complete by reducing another problem (already shown to be NP-complete) to that problem. For example, he showed the problem 3SAT (the Boolean satisfiability problem for expressions in conjunctive normal form (CNF) with exactly three variables or negations of variables per clause) to be NP-complete by ...
Tractability can be obtained by considering specific classes of constraint satisfaction problems. As an example, if the domain is binary and all constraints are binary, establishing satisfiability is a polynomial-time problem because this problem is equivalent to 2-SAT, which is a polynomial-time problem.
The problem of deciding the satisfiability of a given conjunction of Horn clauses is called Horn-satisfiability, or HORN-SAT. It can be solved in polynomial time by a single step of the unit propagation algorithm, which produces the single minimal model of the set of Horn clauses (w.r.t. the set of literals assigned to TRUE).
A problem related to satisfiability is that of finite satisfiability, which is the question of determining whether a formula admits a finite model that makes it true. For a logic that has the finite model property , the problems of satisfiability and finite satisfiability coincide, as a formula of that logic has a model if and only if it has a ...
Although originally conceived as a general method for solving the phase problem, the difference-map algorithm has been used for the boolean satisfiability problem, protein structure prediction, Ramsey numbers, diophantine equations, and Sudoku, [1] as well as sphere- and disk-packing problems. [2]