Knowledge Graph Embeddings with Ontologies: Reification for Representing Arbitrary Relations

Knowledge graph embeddings offer prospects to integrate machine learning and symbolic reasoning. Learning algorithms are designed that map constants, concepts, and relations to geometric entities in a real-valued domain R-n. By identifying logics that feature these geometric entities as their model, one is able to achieve a tight integration of logic reasoning with machine learning. However, interesting description logics are more expressive than current knowledge graph embeddings, as description logics allow concept definitions using arbitrary relations, such as non-functional relationships and partial ones. By contrast, geometric models of relations used so far in knowledge graph embeddings such as translations, rotations, or linear functions can only represent total functional relationships. In this paper we describe a new geometric model of the description logic ALC based on cones that exploits reification combined with linear functions to represent arbitrary relations. While this paper primarily describes reification in context of a particular model for ALC, the proposed reification technique is general and applicable with other ontology languages and knowledge graph embeddings.