# Acquiring Ontology Axioms through Mappings to Data Sources +
## 2 Higher-Order \(\textit{DL-Lite}_{\mathcal{R}}\)
: _If the names car, produced_in belong to the alphabet \(\mathcal{S}\), then the following are \(\mathit{Hi}(\mathit{DL\_Lite}_{R})\) terms: \(\mathit{Car}\), \(\mathit{Inv}(\mathit{produced\_in})\), \(\mathit{Exists}(\mathit{Inv}(\mathit{produced\_in}))\), which, intuitively, denote the concept representing the set of cars, the role produced_in, and the concept representing those individuals (e.g., countries) where something is produced._  
Ground terms, i.e., terms without variables, are called _expressions_, and the set of expressions is denoted by \(\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S})\). The terms in the example above are all expressions. A \(\mathit{Hi}(\mathit{DL\_Lite}_{R})\)_-atom_, or simply _atom_, over the alphabets \(\mathcal{S}\) and \(\mathcal{V}\) for \(\mathit{Hi}(\mathit{DL\_Lite}_{R})\) is a statement of the form \(p_{1}(e_{1},e_{2})\) or \(p_{2}(e_{1},e_{2},e_{3})\) where \(p_{1}/2\), \(p_{2}/3\) belong to \(\mathit{MP}(\mathit{DL\_Lite}_{R})\), and \(e_{1},e_{2},e_{3}\in\mathit{MP}(\mathit{DL\_Lite}_{R})\). \(\tau_{DL\text{-}Lite_{\mathcal{R}}}(\mathcal{S},\mathcal{V})\). Note that \(Hi(DL\text{-}Lite_{\mathcal{R}})\)-atoms (and even terms) are not required to respect the proviso we have introduced before to limit the usage of predicates from \(OP(DL\text{-}Lite_{\mathcal{R}})\) and \(MP(DL\text{-}Lite_{\mathcal{R}})\) so that through them it is possible to express (only) \(DL\text{-}Lite_{\mathcal{R}}\) knowledge bases. For instance, a \(Inst_{\mathcal{C}}(C,D)\) and \(Isa_{\mathcal{C}}(C,E)\) are both legal \(Hi(DL\text{-}Lite_{\mathcal{R}})\)-atoms, that is, differently from \(DL\text{-}Lite_{\mathcal{R}}\) we can instantiate a concept with another concept. If \(X\) is a subset of \(\mathcal{V}\), \(a\) is an atom, and all variables appearing in \(a\) belong to \(X\), then \(a\) is called an \(X\)-atom. Ground \(Hi(DL\text{-}Lite_{\mathcal{R}})\)-atoms, i.e., \(Hi(DL\text{-}Lite_{\mathcal{R}})\)-atoms without variables, are called \(Hi(DL\text{-}Lite_{\mathcal{R}})\)-_assertions_, or simply _assertions_. Thus, an assertion is simply an application of a meta-predicate to a set of expressions, which intuitively means that an assertion is an axiom that predicates over a set of individuals, concepts, or roles. A \(Hi(DL\text{-}Lite_{\mathcal{R}})\)\(KB\) over \(\mathcal{S}\) is a finite set of \(Hi(DL\text{-}Lite_{\mathcal{R}})\)-assertions over \(\mathcal{S}\). To agree with the usual terminology of DLs, we use the term TBox to denote a set of \(Isa_{\mathcal{C}}\), \(Isa_{R}\), \(Disj_{\mathcal{C}}\) and \(Disj_{R}\) assertions, also called intensional assertions, and the term ABox to denote a set of \(Inst_{\mathcal{C}}\) and \(Inst_{R}\) assertions, also called extensional assertions. Obviously, a \(DL\text{-}Lite_{\mathcal{R}}\) KB is also (a special case of) an \(Hi(DL\text{-}Lite_{\mathcal{R}})\) KB. **Example 3**. : _In Figure 4 we provide a graphical representation of a \(Hi(DL\text{-}Lite_{\mathcal{R}})\) KB modeling (a portion of) the domain of cars that we use throughout the paper in our examples. The ontology is drawn in Graphol, a diagrammatic language that allows OWL ontologies to be graphically specified [25; 26]. In Graphol, as in ER diagrams, concepts and roles are denoted by rectangles and diamonds, respectively, and solid directed arrows represent inclusions. The full hexagon denotes at the same time the union of the concepts connected to it through a dashed line and the disjunction of such concepts. Finally, each octagon represents an individual, and each arrow labeled instanceOf from and element \(A\) to an element \(B\) specifies that \(A\) is an instance of \(B\). In words, the ontology is saying that Coupe and Sedan are specific types of Cars (i.e., they are sub-concepts of CarType), and that they are disjoint.