An Almost Quadratic Class of Satisfiability Problems

Many reasoning problems in Knowledge Representation can be reduced to the satisfiability problem of propositional logic. Since Cook proved that satisfiability is NP-Complete, several classes of propositional formulas have been identified for which satisfiability is polynomially solvable. These polynomial classes form the basis of several tractable knowledge representation systems, for example, those based on Horn clauses. In this paper, we present a new class Quad of formulas for which satisfiability is solvable in O(n2k) time, where n is the size of the formula and k is the size of the longest clause in the formula. Formulas in class Quad can be also recognized in O(n2k) time. Thus, for any restriction on satisfiability that forces a fixed upper bound on clause length, for example, 3-SAT, the restricted class Quad is recognizable and solvable in quadratic time. The class Quad strictly contains all Horn and binary formulas.