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Dual Box Embeddings for the Description Logic EL++

Mathias Jackermeier, Jiaoyan Chen, Ian Horrocks

TL;DR

Box$^\textsf{2}$EL introduces a principled geometric framework for embedding $\mathcal{EL^{++}}$ ontologies by representing concepts as axis-aligned boxes and roles as head/tail boxes connected via bump vectors, enabling faithful modeling of complex DL semantics including one-to-many relations and role inclusions. The method provides a semantics-preserving normalization and a suite of losses (NF1–NF7) plus role-inclusion and regularization terms, and proves a soundness theorem guaranteeing that zero-loss embeddings form a valid model of the ontology. Empirically, Box$^\textsf{2}$EL achieves state-of-the-art results on subsumption prediction, role assertion, and approximating deductive reasoning across multiple ontologies and a PPI dataset, outperforming prior box/translation-based approaches. The work demonstrates that combining box-based representation for concepts with a robust geometric treatment of roles yields both theoretical guarantees and practical improvements for ontology completion and inductive reasoning.

Abstract

OWL ontologies, whose formal semantics are rooted in Description Logic (DL), have been widely used for knowledge representation. Similar to Knowledge Graphs (KGs), ontologies are often incomplete, and maintaining and constructing them has proved challenging. While classical deductive reasoning algorithms use the precise formal semantics of an ontology to predict missing facts, recent years have witnessed growing interest in inductive reasoning techniques that can derive probable facts from an ontology. Similar to KGs, a promising approach is to learn ontology embeddings in a latent vector space, while additionally ensuring they adhere to the semantics of the underlying DL. While a variety of approaches have been proposed, current ontology embedding methods suffer from several shortcomings, especially that they all fail to faithfully model one-to-many, many-to-one, and many-to-many relations and role inclusion axioms. To address this problem and improve ontology completion performance, we propose a novel ontology embedding method named Box$^2$EL for the DL EL++, which represents both concepts and roles as boxes (i.e., axis-aligned hyperrectangles), and models inter-concept relationships using a bumping mechanism. We theoretically prove the soundness of Box$^2$EL and conduct an extensive experimental evaluation, achieving state-of-the-art results across a variety of datasets on the tasks of subsumption prediction, role assertion prediction, and approximating deductive reasoning.

Dual Box Embeddings for the Description Logic EL++

TL;DR

BoxEL introduces a principled geometric framework for embedding ontologies by representing concepts as axis-aligned boxes and roles as head/tail boxes connected via bump vectors, enabling faithful modeling of complex DL semantics including one-to-many relations and role inclusions. The method provides a semantics-preserving normalization and a suite of losses (NF1–NF7) plus role-inclusion and regularization terms, and proves a soundness theorem guaranteeing that zero-loss embeddings form a valid model of the ontology. Empirically, BoxEL achieves state-of-the-art results on subsumption prediction, role assertion, and approximating deductive reasoning across multiple ontologies and a PPI dataset, outperforming prior box/translation-based approaches. The work demonstrates that combining box-based representation for concepts with a robust geometric treatment of roles yields both theoretical guarantees and practical improvements for ontology completion and inductive reasoning.

Abstract

OWL ontologies, whose formal semantics are rooted in Description Logic (DL), have been widely used for knowledge representation. Similar to Knowledge Graphs (KGs), ontologies are often incomplete, and maintaining and constructing them has proved challenging. While classical deductive reasoning algorithms use the precise formal semantics of an ontology to predict missing facts, recent years have witnessed growing interest in inductive reasoning techniques that can derive probable facts from an ontology. Similar to KGs, a promising approach is to learn ontology embeddings in a latent vector space, while additionally ensuring they adhere to the semantics of the underlying DL. While a variety of approaches have been proposed, current ontology embedding methods suffer from several shortcomings, especially that they all fail to faithfully model one-to-many, many-to-one, and many-to-many relations and role inclusion axioms. To address this problem and improve ontology completion performance, we propose a novel ontology embedding method named BoxEL for the DL EL++, which represents both concepts and roles as boxes (i.e., axis-aligned hyperrectangles), and models inter-concept relationships using a bumping mechanism. We theoretically prove the soundness of BoxEL and conduct an extensive experimental evaluation, achieving state-of-the-art results across a variety of datasets on the tasks of subsumption prediction, role assertion prediction, and approximating deductive reasoning.
Paper Structure (24 sections, 1 theorem, 17 equations, 2 figures)

This paper contains 24 sections, 1 theorem, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Description Logic Ontologies
  4. Remark.
  5. Subsumption Inference
  6. Knowledge Graph Embeddings
  7. Semantic Ontology Embeddings
  8. Method
  9. Geometric Representation
  10. Concepts and individuals.
  11. Roles.
  12. Expressiveness.
  13. Model complexity.
  14. Training Procedure
  15. Generic inclusion loss.
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathcal{O} = (\mathcal{T}, \mathcal{A})$ be an $\mathcal{EL^{++}}$ ontology. If $\gamma\leq 0$ and there exist Box$^\textsf{2}$EL embeddings in $\mathbb R^n$ such that $\mathcal{L}(\mathcal{O}) = 0$, then these embeddings are a model of $\mathcal{O}$.

Figures (2)

  • Figure 1: An illustration of Box$^\textsf{2}$EL embeddings. Striped boxes represent concept embeddings, whereas role embeddings are shaded blue and labelled as $\mathsf{r}^{\mathsf{h}}$ or $\mathsf{r}^{\mathsf{t}}$ for the head or tail box of $\mathsf r$, respectively. Bump vectors are drawn as arrows and labelled with the corresponding concept. The illustrated embeddings form a logical model of the TBox in \ref{['ex:family']}.
  • Figure 2: Proof of concept ontology.

Theorems & Definitions (3)

  • Example 1
  • Example 2
  • Theorem 1: Soundness