# Acquiring Ontology Axioms through Mappings to Data Sources +
## 1 Introduction  
Ontology-based data access (OBDA) [1; 2] is a paradigm for accessing data using a conceptual representation of the domain of interest expressed as an ontology. An OBDA system relies on a three-level architecture, consisting of the data layer, the ontology, and the mapping between the two. More in detail,  
* the data layer is constituted by the existing data sources that are relevant for the organization,
* the ontology is a declarative and explicit representation of the domain of interest for the organization, formulated in a description logic (DL) [3] so as to take advantage of various reasoning capabilities in accessing data,
* the mapping is a set of declarative assertions specifying how the sources in the data layer relate to the ontology. Several OBDA projects have been carried out in recent years [4; 5; 6; 7], and various OBDA management systems have been designed to support OBDA applications, e.g., [8; 9; 10]. In current OBDA systems the ontology is expressed as a DL TBox, i.e., a set of assertions on the relevant concepts and roles (i.e., binary relationships between concepts) of the domain of interest, constituting the intensional level of the representation, and the mapping assertions are used to specify how the data atthe sources correspond to the instances of the concepts and the relations, which form the extensional level of the representation. Thus, the mapping assertions, together with the source data, determine the so-called virtual ABox, in the sense that the instance assertions are not explicitly given but are specified through the relationships between the data and the elements of the TBox. From the above observations it should be clear that current works on OBDA share the idea, originally stemmed in data integration and in data exchange [11; 12; 13; 14], that mappings are used to (virtually) retrieve extensional information of the ontology from the sources, as shown in Figure 1, while the intensional level information, represented by the TBox, remains fixed a priori, once and for all. In this paper, we challenge this preconception and propose to adopt a virtual approach to the specification of the TBox, thus virtualizing both the extensional and the intensional level of the ontology. In other words, we propose a setting where not only the ABox but also the TBox is specified through mappings linking the data to the ontology, as illustrated in Figure 2. Our approach is based on addressing two issues. Figure 1: Ontology-Based Data Access (OBDA): virtualizing extensional information. The first issue is related to looking at the content of the data sources to identify concepts and roles that are relevant to the domain of interest but that have not been modeled in the TBox, because they were not known at design time. As an example, the database \(\mathcal{D}\) in Figure 3 stores data about different models and different types of cars manufactured by motor companies (table T-CarTypes), as well as various cars of such types (table T-Cars). If we look carefully at the semantics of the data in \(\mathcal{D}\), we realize that such database not only stores information about the instances of the concepts in the domain of interest (e.g., the first row of table T-Cars collects data about an instance of the concept Car) but contains also pieces of data denoting new concepts of the domain. In particular, table T-CarTypes contains data denoting concepts such as 1973 FALCONKBGT, 1967 MUSTANG SHELBY, 1973 MUSTANGACH1, and so on. Considering the context where these concepts appear, it is not difficult to conclude that they are all mutually disjoint subconcepts of Car. Table T-Cars, on the other hand, provides information about the instances of the various concepts, as well as other properties about them (i.e., Color, ProdCountry). We observe that, in order to acquire the knowledge about the concepts mentioned in the data sources, we need a flexible mechanism that is able to map data at the sources to concepts in the ontology.