A Fixpoint Characterization of Three-Valued Disjunctive Hybrid MKNF Knowledge Bases

TL;DR

This paper introduces a fixpoint method to characterize three-valued semantics of disjunctive hybrid MKNF knowledge bases, extending previous two-valued models and capturing partial stable models and partial information.

Contribution

It presents a novel fixpoint construction leveraging head-cuts to characterize three-valued models of disjunctive hybrid MKNF knowledge bases, extending prior two-valued semantics.

Findings

The fixpoint construction captures three-valued models with disjunctive rules.

It also characterizes partial stable models of disjunctive logic programs.

The approach relates to approximation fixpoint theory for hybrid MKNF knowledge bases.

Abstract

The logic of hybrid MKNF (minimal knowledge and negation as failure) is a powerful knowledge representation language that elegantly pairs ASP (answer set programming) with ontologies. Disjunctive rules are a desirable extension to normal rule-based reasoning and typically semantic frameworks designed for normal knowledge bases need substantial restructuring to support disjunctive rules. Alternatively, one may lift characterizations of normal rules to support disjunctive rules by inducing a collection of normal knowledge bases, each with the same body and a single atom in its head. In this work, we refer to a set of such normal knowledge bases as a head-cut of a disjunctive knowledge base. The question arises as to whether the semantics of disjunctive hybrid MKNF knowledge bases can be characterized using fixpoint constructions with head-cuts. Earlier, we have shown that head-cuts can be paired with fixpoint operators to capture the two-valued MKNF models of disjunctive hybrid MKNF knowledge bases. Three-valued semantics extends two-valued semantics with the ability to express partial information. In this work, we present a fixpoint construction that leverages head-cuts using an operator that iteratively captures three-valued models of hybrid MKNF knowledge bases with disjunctive rules. This characterization also captures partial stable models of disjunctive logic programs since a program can be expressed as a disjunctive hybrid MKNF knowledge base with an empty ontology. We elaborate on a relationship between this characterization and approximators in AFT (approximation fixpoint theory) for normal hybrid MKNF knowledge bases.