Schaefer's Dichotomy Theorem - Original Presentation

Original Presentation

Schaefer defines a decision problem that he calls the Generalized Satisfiability problem for S (denoted by SAT(S)). An instance of the problem is an S-formula, i.e. a conjunction of constraints of the form where R is a relation in S and the 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.

Schaefer identifies six classes of sets of Boolean relations for which SAT(S) is in P and proves that all other sets of relations generate an NP-complete problem. A finite set of relations S over the Boolean domain defines a polynomial time computable satisfiability problem if any one of the following conditions holds:

1. all relations which are not constantly false are true when all its arguments are true;
2. all relations which are not constantly false are true when all its arguments are false;
3. all relations are equivalent to a conjunction of binary clauses;
4. all relations are equivalent to a conjunction of Horn clauses;
5. all relations are equivalent to a conjunction of dual-Horn clauses;
6. all relations are equivalent to a conjunction of affine formulae (e.g., ).

Otherwise, the problem SAT(S) is NP-complete.