# An Embedding-Based Approach to Repairing OWL Ontologies
## 2 Background Knowledge
### Description Logics  
A DL-based ontology could describe three kinds of entities: concepts, roles and individuals in a domain. An individual describes an instance in a specific domain, and an individual name means a single individual. For example, [PERSON] is an individual name describing a person, and Nanjing is an individual name describing a place. A concept indicates a set of individuals, and a role represents a binary relation between the individuals and individual names. The roles can be further divided into abstract roles and datatype roles. The domain and range of an abstract role are concepts. If the range of a role is a datatype literal such as string or integer, this role is called a datatype role [28]. The concepts and roles can be atomic or complex. A complex one is constructed based on the atomic ones by using various kinds of constructors such as existential restriction (\(\exists\)) and universal restriction (\(\forall\)). With different constructors, many DL languages can be formed, such as \(\mathcal{ALC}\) allowing transitive roles, role hierarchy and inverse operation, and \(\mathcal{SHL}\) further allowing role hierarchy and inverse operation. A DL-based ontology could also describe the characteristics of some properties and the relationships among the entities by various axioms. Such axioms can be divided into TBox (terminology axioms) and ABox (assertion axioms). A TBox mainly describes those relationships among concepts and properties such as concept inclusion axioms in the form of \(C\sqsubseteq D\) and disjoint role axioms in the form of \(C\sqsubseteq\
eg D\), where \(C\) and \(D\) are concepts. An ABox describes those relationships that are relevant to individuals, such as concept assertions in the form of \(C(a)\) and equality assertions in the form of \(a=b\), where \(a\) and \(b\) are individuals. It is noted that the concepts, abstract roles and datatype roles in DL-based ontologies correspond to the classes, object properties and data properties in OWL, respectively. In an ontology, if there exist unsatisfiable concepts, minimal unsatisfiability-preserving sub-TBoxes (abbreviated as MUPS) are often computed to explain the unsatisfiability of such a concept. **Definition 1** ((Mups) [16]). : _For an unsatisfiable concept name \(C\) in a TBox \(\mathcal{T}\), a sub-TBox \(\mathcal{T}^{\prime}\subseteq\mathcal{T}\) is a MUPS of \(\mathcal{T}\) with respect to \(C\) if \(C\) is unsatisfiable in \(\mathcal{T}^{\prime}\) and there is no sub-TBox \(\mathcal{T}^{\prime\prime}\subset\mathcal{T}^{\prime}\) such that \(C\) is unsatisfiable in \(\mathcal{T}^{\prime\prime}\)._  
To explain the incoherence of an ontology, a set of minimal incoherence-preserving sub-TBoxes (abbreviated as MIPS) can be calculated. **Definition 2** ((Mips) [16]). : _For an incoherent TBox \(\mathcal{T}\), its sub-TBox \(\mathcal{T}^{\prime}\subseteq\mathcal{T}\) is a MIPS of \(\mathcal{T}\) if \(\mathcal{T}^{\prime}\) is incoherent and every sub-TBox \(\mathcal{T}^{\prime\prime}\subset\mathcal{T}^{\prime}\) is coherent._  
From Definitions 1 and 2 we can observe that a MIPS is a MUPS, but not vice versa. When repairing an incoherent ontology, a diagnosis is often computed. That is, removing or modifying the axioms in the diagnosis could regain coherence. The formal definition of a diagnosis is described below. **Definition 3** (Diagnosis). : _For an incoherent ontology, \(O\) and a sub-ontology \(O^{\prime}\subseteq O\), \(O^{\prime}\) is a diagnosis of \(O\) if \(O\setminus O^{\prime}\) is coherent. \(O\setminus O^{\prime}\) indicates removing all axioms in \(O^{\prime}\) from \(O\)._  
To repair an ontology automatically, it is often desired to achieve some kind of minimal change [22] and compute a minimal diagnosis by keeping information as much as possible. **Definition 4** ((**Minimal Diagnosis**) (**[**29**]****)).