Alternating Fixpoint Operator for Hybrid MKNF Knowledge Bases as an Approximator of AFT

TL;DR

This paper demonstrates that the alternating fixpoint operator for hybrid MKNF knowledge bases is an approximation of approximation fixpoint theory (AFT), enabling richer semantics and improved computation of well-founded semantics.

Contribution

It reveals the operator as an AFT approximator, extends AFT to handle inconsistencies, and proposes an improved approximator for hybrid MKNF knowledge bases.

Findings

The operator characterizes multiple semantics for hybrid MKNF knowledge bases.

An improved approximator yields richer information than previous constructions.

Enhanced computation methods for well-founded semantics are proposed.

Abstract

Approximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices and has found its applications in characterizing semantics for various classes of logic programs and nonmonotonic languages. In this paper, we show one more application of this kind: the alternating fixpoint operator by Knorr et al. for the study of the well-founded semantics for hybrid MKNF knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the power of abstraction of AFT, characterizes not only the well-founded semantics but also two-valued as well as three-valued semantics for hybrid MKNF knowledge bases. Furthermore, we show an improved approximator for these knowledge bases, of which the least stable fixpoint is information richer than the one formulated from Knorr et al.'s construction. This leads to an improved computation for the well-founded semantics. This work is built on an extension of AFT that supports consistent as well as inconsistent pairs in the induced product bilattice, to deal with inconsistencies that arise in the context of hybrid MKNF knowledge bases. This part of the work can be considered generalizing the original AFT from symmetric approximators to arbitrary approximators.