On the KLM properties of a fuzzy DL with Typicality

TL;DR

This paper explores the properties of a fuzzy description logic with a typicality operator, extending fuzzy ALC, and analyzes its preferential consequence relations and model properties.

Contribution

It introduces a monotonic extension of fuzzy ALC with typicality and studies its logical properties and model semantics.

Findings

Most KLM properties are satisfied under certain conditions.

A new notion of faithful model for weighted knowledge bases is proposed.

The approach generalizes previous models of conditional knowledge bases.

Abstract

The paper investigates the properties of a fuzzy logic of typicality. The extension of fuzzy logic with a typicality operator was proposed in recent work to define a fuzzy multipreference semantics for Multilayer Perceptrons, by regarding the deep neural network as a conditional knowledge base. In this paper, we study its properties. First, a monotonic extension of a fuzzy ALC with typicality is considered (called ALC^FT) and a reformulation the KLM properties of a preferential consequence relation for this logic is devised. Most of the properties are satisfied, depending on the reformulation and on the fuzzy combination functions considered. We then strengthen ALC^FT with a closure construction by introducing a notion of faithful model of a weighted knowledge base, which generalizes the notion of coherent model of a conditional knowledge base previously introduced, and we study its properties.