An Iterative Fixpoint Semantics for MKNF Hybrid Knowledge Bases with Function Symbols

TL;DR

This paper introduces an iterative fixpoint semantics for hybrid knowledge bases with function symbols, extending previous semantics to handle more expressive logic programs and enabling future probabilistic extensions.

Contribution

It proposes a new iterated fixpoint semantics for MKNF hybrid knowledge bases with function symbols, extending existing semantics and maintaining consistency with prior work in the function-free case.

Findings

Semantics extends existing fixpoint semantics to function-symbols.

Ensures consistency with prior semantics for function-free cases.

Lays groundwork for probabilistic extensions of hybrid knowledge bases.

Abstract

Hybrid Knowledge Bases based on Lifschitz's logic of Minimal Knowledge with Negation as Failure are a successful approach to combine the expressivity of Description Logics and Logic Programming in a single language. Their syntax, defined by Motik and Rosati, disallows function symbols. In order to define a well-founded semantics for MKNF HKBs, Knorr et al. define a partition of the modal atoms occurring in it, called the alternating fixpoint partition. In this paper, we propose an iterated fixpoint semantics for HKBs with function symbols. We prove that our semantics extends Knorr et al.'s, in that, for a function-free HKBs, it coincides with its alternating fixpoint partition. The proposed semantics lends itself well to a probabilistic extension with a distribution semantic approach, which is the subject of future work.