Non—Monotonic Reasoning Based on Minimal Models and its Efficient Implementation

Abstract

A common approach to non-monotonic reasoningis to choose a set S of models (e.g. minimal, perfect or stable models) for a logic program P. Then, the meaning of P is defined as the set of all formulas which are implied logically by all models in S.

For a disjunctive logic program P,we consider the set S = MMp of minimal models and get the non-montonic consequences NM p of ground disjunctions over positive or negative literals. The set NMp consists of three subsets: The minimal model state MSp (ground disjunctions over positive literals), the negated extended generalized closed world assumption ¬ εgCWAp (ground disjunctions over negative literals), and the rest (mixed disjunctions over positive and negative literals). We show that the mixed disjunctions are implied logically by MSp U ¬ εgCWAp. Thus the interesting sets are the logical consequences MSp and the non-monotonic consequences ¬εgCWAp. As described in [16], MSp consists of the positive ground disjunctions, which are true in all minimal models, and εgCWAp consists of the positive ground conjunctions, which are false in all minimal models.

We will present a compact data structure, called clause tree, for disjunctive Herbrand states (ground disjunctions over positive literals). It is used for deriving MSp as the least fixpoint of the disjunctive consequence operator T ^S_P on disjunctive Herbrand states, and, on the other hand, for deriving εgCWAp based on the support-for-negation sets, which are certain disjunctive Herbrand states.