Complexity analysis of propositional resolution with autarky pruning

Abstract

An algorithm called “Modoc”, which has been introduced elsewhere, enhances propositional model elimination with autarky pruning, and other features. The model elimination method is based on linear resolution, and is designed to produce refutations of formulas in conjunctive normal form (CNF). Informally, an autarky is a “self-sufficient” model for some clauses, but which does not affect the remaining clauses of the formula. Modoc finds a model if it fails to find a refutation, essentially by combining autarkies. Although the original motivation for autarky pruning was to extract a model when the refutation attempt failed, practical experience has shown that it also greatly increases the performance, by reducing the amount of search redundancy. This paper presents a worst-case analysis of Modoc as a function of the number of propositional variables in the formula. The analysis sheds light on why autarky pruning improves the performance, compared to “standard” model elimination. A worst-case analysis of the original algorithm of Davis, Putnam, Loveland and Logemann (DPLL) is also presented. The Modoc analysis yields a worst-case upper bound that is not as strong as the best known upper bound for model-searching satisfiability methods, on general propositional CNF. However, it is the first time a nontrivial upper bound on non-Horn formulas has been shown for any resolution-based refutation procedure.