Pushing the BoxEL Envelope
Amity Aharoni

Abstract:
This paper formalizes knowledge base embedding algorithms using categorical logic, focusing on box embeddings. We introduce a novel approach utilizing hyperdoctrines, a categorical construction, to analyze the relationship between a knowledge base and its embedding space. We provide a proof of the incom- pleteness of state-of-the-art box embedding approaches like BoxEL, and then use algebraic tools to extend box space embeddings to a novel MultiboxEL em- bedding approach. We establish that every EL++ knowledge base possesses a finite model and use that to show completeness of MultiboxEL with respect to EL++ . We further extend our embedding to ALC knowledge bases. Finally, we implement and compare our new embedding strategies against state-of-the- art box embedding models, providing empirical evidence for the usefulness and limitation of our approach. These contributions collectively offer a ro- bust, algebraic method for knowledge base embeddings, advancing the field and opening new avenues for the application of categorical logic in artificial intelligence and machine learning.