Computing Minimal Models by Positively Minimal Disjuncts

Abstract

In this paper, we consider a method of computing minimal models in propositional logic. We firstly show that positively minimal disjuncts in DNF (Disjunctive Normal Form) of the original axiom corresponds with minimal models. A disjunct D is positively minimal if there is no disjunct which contains less positive literal than D. We show that using superset query and membership query which were used in some learning algorithms in computational learning theory, we can compute all the minimal models.

We then give a restriction and an extension of the method. The restriction is to consider a class of positive (sometimes called monotone) formula where minimization corresponds with diagnosis and other important problems in computer science. Then, we can replace superset query with sampling to give an approximation method. The algorithm itself has been already proposed by [Valiant84], but we show that the algorithm can be used to approximate a set of minimal models as well.

On the other hand, the extension is to consider circumscription with varied propositions. We show that we can compute equivalent formula of circumscription using a similar technique to the above.