Testing Satisfiability of CNF Formulas by Computing a Stable Set of Points

Abstract

We show that a conjunctive normal form (CNF) formula F is unsatisfiable if and only if there is a set of points of the Boolean space that is stable with respect to F. So testing the satisfiability of a CNF formula reduces to looking for a stable set of points (SSP). We give some properties of SSPs and describe a simple algorithm for constructing an SSP for a CNF formula. Building an SSP can be viewed as a "natural" way of search space traversal. This naturalness of search space examination allows one to make use of the regularity of CNF formulas to be checked for satisfiability. We illustrate this point by showing that if a CNF F formula is symmetric with respect to a group of permutations, it is very easy to make use of this symmetry when constructing an SSP. As an example, we show that the unsatisfiability of pigeon-hole CNF formulas can be proven by examining only a set of points whose size is quadratic in the number of holes. Finally, we introduce the notion of an SSP with excluded directions and sketch a procedure of satisfiability testing based on the construction of such SSPs.