Schaefer's dichotomy theorem

In computational complexity theory, a branch of computer science, Schaefer's dichotomy theorem states necessary and sufficient conditions under which a finite set S of relations over the Boolean domain yields polynomial-time or NP-complete problems when the relations of S are used to constrain some of the propositional variables. It is called a dichotomy theorem because the complexity of the problem defined by S is either in P or NP-complete as opposed to one of the classes of intermediate complexity that is known to exist (assuming P ≠ NP) by Ladner's theorem.

Special cases of Schaefer's dichotomy theorem include the NP-completeness of SAT (the Boolean satisfiability problem) and its two popular variants 1-in-3 SAT and Not-All-Equal 3SAT (often denoted by NAE-3SAT). In fact, for these two variants of SAT, Schaefer's dichotomy theorem shows that their monotone versions (where negations of variables are not allowed) are also NP-complete.

Schaefer defines a decision problem that he calls the Generalized Satisfiability problem for S (denoted by SAT(S)), where ${\displaystyle S=\{R_{1},\ldots ,R_{m}\}}$ is a finite set of relations over propositional variables. An instance of the problem is an S-formula, i.e. a conjunction of constraints of the form ${\displaystyle R_{j}(x_{i_{1}},\ldots ,x_{i_{n}})}$ where ${\displaystyle R_{j}\in S}$ and the ${\displaystyle x_{i_{j}}}$ are propositional variables. The problem is to determine whether the given formula is satisfiable, in other words if the variables can be assigned values such that they satisfy all the constraints as given by the relations from S.

...
Wikipedia