# Cosine-Based Embedding for Completing Lightweight Schematic Knowledge in DL-Lite\({}_{core}\)+
## 2 Related Work
### Schematic Knowledge Embedding  
The studies of schematic knowledge embedding are primarily composed of logical rules embedding, logical properties embedding, and ontology embedding [35].  
As the RDF(S) and schematic knowledge can be transformed into logical rules, several studies have tried to encode them into embedding models and enhance the performances for knowledge completion [36]. [PERSON] et al. [37] designed a joint model named KALE, which could simultaneously encode the triples of factual knowledge and their related logical rules. Furthermore, the authors proposed an improved model called RUGE [38] that could integrate the labeled triples, soft rules, and unlabeled triples into an iterative framework for learning their vector representations in the semantic space. Similarly, [PERSON] et al. [39] proposed a model named IterE. Different from above methods that mainly proposed to learn rules, the authors devoted to learning the embeddings of entities and rules at the same time, making full of their advantages complementing each other during the processing of model learning.  
To further maintain the logical properties of relations, some works have been proposed for schematic knowledge and lightweight ontologies called RDF Schema. On2 Vec [40] was a translation-based method for embedding the population of ontologies, in which the matrices were introduced to encode the transitivity of several relations. In order to encode concepts and instances into the same semantic space, EmbedS [20] and TransC [21] tried to encode concepts as spheres and instances as vectors so that the transitivity of the is-A relations could be preserved. To model the semantic hierarchies of KGs, [PERSON] et al. [22] proposed a method called HAKE for modeling semantic hierarchies. It is inspired by the fact that concentric circles can naturally reflect the hierarchy in the polar coordinate system. To further model composition among relations, RotatE [23] encoded entities and relations into a complex vector space, in which each relation was treated as a rotation from its related head entity to the tail entity. Furthermore, it could persevere the (anti)-symmetry and inversion of relations at the same time.  
Recently, embedding models for ontologies have received attention. EL Embedding [24] and Quantum Embedding [41] were two representative algorithms based on the end-toend paradigm, in which loss functions and score functions were designed tailored for logical axioms expressed by \(\mathcal{EL^{++}}\) and \(\mathcal{ALC}\), respectively. These two embedding models encoded the semantics of the logical constructors by transforming the relations into geometric relations so that they could complete some kinds of axioms in ontologies very well. [PERSON] et al. [25] proposed an ontology embedding model combined with word embedding and random walk algorithm, called OWL2 Vec\({}^{\star}\), which took in the lexical information, logical constructors, and graph structures of ontologies so that it could preserve the semantics of ontologies well.  
Although the above models are enabled to encode the logical properties of relations in their designed semantic spaces, it is still a challenge for them to preserve the transitivity (e.g., \(subclassOf\)) and symmetry (e.g., \(disjointWith\)) of axioms at the same time. To the best of our knowledge, our model is the first work for completing lightweight schematic knowledge expressed in DL-Lite\({}_{core}\), by which the transitivity and symmetry of relations in axioms can be simultaneously preserved well.