On the tractability of minimal model computation for some CNF theories

Designing algorithms capable of efficiently constructing minimal models of Conjunctive Normal Form theories (CNFs) is an important task in AI. This paper provides new results along this research line and presents new algorithms for performing minimal model finding and checking over positive propositional CNFs and model minimization over propositional CNFs. A CNF is positive if each of its clauses has at least a positive literal. An algorithmic schema, called the Generalized Elimination Algorithm (GEA) is presented, that computes a minimal model of any positive CNF. The schema generalizes the Elimination Algorithm (EA) [5], which computes a minimal model of positive head-cycle-free (HCF) CNF theories. While the EA always runs in polynomial time in the size of the input HCF CNF, the complexity of the GEA depends on the complexity of the specific eliminating operator invoked therein, which may in general turn out to be exponential. Therefore, a specific eliminating operator is defined by which the GEA computes, in polynomial time, a minimal model for a class of CNF that strictly includes head-elementary-set-free (HEF) CNF theories [14], which form, in their turn, a strict superset of HCF theories. Furthermore, in order to deal with the high complexity associated with recognizing HEF theories, an “incomplete” variant of the GEA (called IGEA) is proposed: the resulting schema, once instantiated with an appropriate elimination operator, always constructs a model of the input CNF, which is guaranteed to be minimal if the input theory is HEF. In the light of the above results, the main contribution of this work is the enlargement of the tractability frontier for the minimal model finding and checking and the model minimization problems.