# Cosine-Based Embedding for Completing Lightweight Schematic Knowledge in DL-Lite\({}_{core}\)+
###### Abstract
Recently, several works have tried to encode the properties of relations for completing schematic knowledge, such as EmbedS [20], TransC [21], HAKE [22], RotatE [23], EL Embeddings [24], and OWL2 Vec* [25]. Most of them have attempted to encode the concepts and instances into a spherical semantic space so that the transitivity and other logical properties of relations could be preserved. Benefiting from this idea, more potential axioms and triples could be predicted. Although the methods of KG embedding have achieved great success in KG completion, most of them suffer from two limitations. On the one hand, the mainstream models of KG embedding mainly consider the triples derived from factual knowledge, but few of them pay attention to the modeling of the logical properties of relations. Hence, these methods can hardly be applied to the related tasks of schematic knowledge (e.g., completion, reasoning, and repairing). For example, given one axiom \((C_{i},r,C_{j})\) with two concepts \(C_{i}\), \(C_{j}\), and a symmetry relation \(r\), for the translation-based KG embedding method TransE [13], if two concepts and a relation are projected into its defined semantic space, the axiom's score of \(||\mathbf{C_{i}+r-C_{j}}||_{2}\) is not equal to \(||\mathbf{C_{j}+r-C_{i}}||_{2}\). Therefore, the symmetry of relation \(r\) is lost in the defined semantic space of TransE. On the other hand, existing embedding models for schematic knowledge mainly focus on modeling the properties of relations such as transitivity, (anti)-symmetry, inversion, and composition. It remains a challenge for them to simultaneously preserve the transitivity (e.g., \(subclassOf\)) and symmetry (e.g., \(disjointWith\)) of axioms well. As the schematic knowledge of KG usually has its logical foundations based  
Figure 1: An example of the missing one \(subclassOf\) axiom for schematic knowledge completion. on ontology languages, such as the Resource Description Framework (Schema) (RDF(S)) ([[https://www.w3.org/TR/rdf-schema/](https://www.w3.org/TR/rdf-schema/)]([https://www.w3.org/TR/rdf-schema/](https://www.w3.org/TR/rdf-schema/)) accessed on 5 September 2022) and the Ontology Web Language (OWL) ([[https://www.w3.org/OWL/](https://www.w3.org/OWL/)]([https://www.w3.org/OWL/](https://www.w3.org/OWL/)) accessed on 5 September 2022), so it is important to improve the completion of \(subclassOf\) axioms and \(disjointWith\) ones, because both of them are basic axioms asserted in schematic knowledge or ontologies, which can ensure the quality of KGs and infer more implicit knowledge. To solve the problems, we attempt to propose a cosine-based embedding method, namely CosE (**C**osine-based **E**mbedding), for learning the vector representations of concepts in lightweight schematic knowledge that corresponds to the ontology expressed in DL-Lite\({}_{core}\). DL-Lite\({}_{core}\) is a lightweight language of Description Logics (DL), which can capture basic ontology languages and maintain a low complexity of reasoning [26]. In the previous study [27], the authors demonstrated that all the axioms asserted in DL-Lite\({}_{core}\) could be reduced to the ones with \(subclassOf\) and \(disjointWith\) relations. Therefore, our proposed model is mainly designed to learn the representations of axioms that are defined by these two relations. In order to better preserve the properties of relations and to measure the confidence of axioms in schematic knowledge, we implement CosE by projecting concepts into the angle-based semantic space and translation-based semantic space according to the type of relations. In the angle-based semantic space, each concept is encoded with one vector and a valid length, which are utilized to preserve the properties of these two kinds of relations. In the translation-based semantic space, the vector representations of concepts are employed to measure the confidence of related axioms.