Counting Models using Extension Rules

In recent years we have viewed tremendous improvements in the field of Propositional satisfiability (SAT). Many NPcomplete problems from a variety of domains, such as classic planning problems (Kautz 1999), have been compiled into SAT instances, and solved effectively by SAT solvers (Zhang et al 2001). On the other hand, the problem of counting the number of models of a prepositional formula (#SAT) is an important extension of satisfiability testing (Bacchus et al 2003). Recent research has also shown that model counting corresponds to numerous #P-complete problems such as performing inference in Bayesian networks (Sang et al 2005) and conformant probabilistic planning (Domshlak and Hoffman. 2006). Resolution principle is the rule of inference at the basis of most procedures for both SAT and #SAT, though a number of techniques, such as clause learning, variable selection, can be integrated to improve performance tremendously. The aim of this paper is to challenge the traditional idea by using the inverse rule of resolution, which we called extension rule (Lin et al 2003). Specifically, the basic idea is to deduce the set of all the maximum terms for counting models and to use the inclusion-exclusion principle to circumvent the problem of space complexity. Our contributions are: (1) We use the inverse of resolution together with the inclusion-exclusion principle for counting models. This may be considered a novel framework for model counting. (2) Our method leads to a new target logical language, which permits model counting to be solved in linear time. (3) We propose a revised method to lift our method from model counting to weighted model counting (WMC).