A hierarchy of tractable satisfiability problems

Abstract

Propositional satisfiability (SAT) is a well-known NP-complete problem. We define a hierarchy Ω 0, Ω 1,… of classes of formulae such that for any class Ω k , SAT is solvable in O( n k + 1 ) time. The basic class Ω 0 contains all formulae in conjunctive normal form (CNF) where each conjunct is a Horn clause, and allformulae in CNF where each conjunct is a binary clause. This hierarchy improves upon the hierarchy defined by Gallo and Scutellá. We also present a sound and complete algorithm for solving SAT that takes time O( n k + 1 ) for any formula in the class Ω k .