Computing Minimal Models, Stable Models, and Answer Sets

Abstract

We propose and study algorithms for computing minimal models, stable models and answer sets of 2- and 3-CNF theories, and normal and disjunctive 2- and 3-programs. We are especially interested in algorithms with non-trivial worst-case performance bounds. We show that one can find all minimal models of 2-CNF theories and all answer sets of disjunctive 2-programs in time O(m 1.4422..ⁿ) (n is the number of atoms in an input theory or program and m is its size). Our main results concern computing stable models of normal 3-programs, minimal models of 3-CNF theories and answer sets of disjunctive 3-programs. We design algorithms that run in time O(m1.6701..ⁿ), in the case of the first problem, and in time O(mn ²2.2720..ⁿ), in the case of the latter two. All these bounds improve by exponential factors the best algorithms known previously. We also obtain closely related upper bounds on the number of minimal models, stable models and answer sets a 2- or 3-theory or program may have.