Strong Faithfulness for ELH Ontology Embeddings

TL;DR

This paper proves that normalized ELH ontologies can be embedded into convex geometric models while preserving all entailments, ensuring accurate reasoning within a continuous vector space.

Contribution

It introduces a formal proof of strong faithfulness for normalized ELH ontologies and presents a region-based geometric embedding model that captures all axioms.

Findings

Normalized ELH has the strong faithfulness property.

A convex geometric model can embed ELH ontologies exactly.

The construction leverages the finite canonical model of ELH.

Abstract

Ontology embedding methods are powerful approaches to represent and reason over structured knowledge in various domains. One advantage of ontology embeddings over knowledge graph embeddings is their ability to capture and impose an underlying schema to which the model must conform. Despite advances, most current approaches do not guarantee that the resulting embedding respects the axioms the ontology entails. In this work, we formally prove that normalized ${\cal ELH}$ has the strong faithfulness property on convex geometric models, which means that there is an embedding that precisely captures the original ontology. We present a region-based geometric model for embedding normalized ${\cal ELH}$ ontologies into a continuous vector space. To prove strong faithfulness, our construction takes advantage of the fact that normalized ${\cal ELH}$ has a finite canonical model. We first prove the statement assuming (possibly) non-convex regions, allowing us to keep the required dimensions low. Then, we impose convexity on the regions and show the property still holds. Finally, we consider reasoning tasks on geometric models and analyze the complexity in the class of convex geometric models used for proving strong faithfulness.