# Acquiring Ontology Axioms through Mappings to Data Sources +
## 2 Higher-Order \(\textit{DL-Lite}_{\mathcal{R}}\)  
We start by recalling some notions on DLs in general and \(\textit{DL-Lite}_{\mathcal{R}}\) in particular. Then, we provide the syntax and the semantics of \(Hi(\textit{DL-Lite}_{\mathcal{R}})\), i.e., the higher-order version of \(\textit{DL-Lite}_{\mathcal{R}}\). DLs [3] are logics that represent the domain of interest in terms of _concepts_, denoting sets of objects, and _roles_, denoting binary relations between (instances of) concepts. Complex concept and role expressions are constructed starting from a set of atomic concepts and roles and by recursively applying suitable constructs. A DL knowledge base (KB) \(\mathcal{K}\) is constituted by two components, \(\mathcal{T}\) and \(\mathcal{A}\), i.e., \(\mathcal{K}=\mathcal{T}\cup\mathcal{A}\), where:* \(\mathcal{T}\), called TBox, is the _terminological component_ of \(\mathcal{K}\), which contains statements representing intensional knowledge, and
* \(\mathcal{A}\), called ABox, is the _assertional component_ of \(\mathcal{K}\), which contains assertions representing extensional knowledge. \(\textit{DL-Lite}_{\mathcal{R}}\) is a member of the _DL-Lite_ family of tractable DLs [22,23] and is the logical basis of OWL 2 QL, one of the profiles of OWL, the W3C standard for representing ontologies [24]. \(\textit{DL-Lite}_{\mathcal{R}}\) expressions are given by the following syntax:
* Concept expressions: \[ \begin{array}{ccccc}B&::=&A&|&\exists\,Q\\ C&::=&B&|&\
eg B\end{array} \]
* Role expressions: \[ \begin{array}{ccccc}Q&::=&P&|&P^{-}\\ R&::=&Q&|&\
eg Q\end{array} \]  
where \(A\) is an atomic concept (i.e., a unary predicate from the signature of the knowledge base), \(B\) is a basic concept (i.e, an atomic concept \(A\) or an existential restriction on a role \(\exists Q\), which denotes individuals occurring in the first component of the role \(Q\), which is called the _domain_ of \(Q\)), \(C\) is a general concept (i.e., a basic concept \(B\) or its negation \(\
eg B\)), \(P\) is an atomic role (i.e., a binary predicate from the signature of the knowledge base), \(Q\) is a basic role (i.e., an atomic role \(P\) or its inverse \(P^{-}\)), and \(R\) is a general role (i.e., a basic role \(Q\) or its negation \(\
eg Q\)). The domain of \(P^{-}\) (i.e., \(\exists P^{-}\)) is also called the _range_ of \(P\). A TBox in \(\textit{DL-Lite}_{\mathcal{R}}\) is a finite set of assertions in form:  
\[ \begin{array}{ccccc}B&\sqsubseteq&C&(\textit{concept inclusion assertion})\\ Q&\sqsubseteq&R&(\textit{role inclusion assertion})\end{array} \]  
When \(C\) and \(R\) above assume the form \(\
eg B^{\prime}\) and \(\
eg Q^{\prime}\), respectively, the above inclusions are called negative inclusions and allow us to specify _disjointness_ between concepts or between roles, respectively. Otherwise, the above inclusions are called positive inclusions and allow us to specify _is_a_ relations between concepts or roles, respectively. An ABox in \(\textit{DL-Lite}_{\mathcal{R}}\) is a finite set of membership assertions (i.e., facts) of the form:  
\[ \begin{array}{ll}A(a)&(\textit{concept membership assertion})\\ P(a,b)&(\textit{role membership assertion})\end{array} \]  
where \(a\) and \(b\) are constants, that is, names for individuals (i.e., predicates of arity 0 from the signature of the knowledge base). We are now ready to describe the higher-order DL \(\textit{Hi}(\textit{DL-Lite}_{\mathcal{R}})\). We start by observing that, as discussed in [18], every traditional DL \(\mathcal{L}\) can be characterized by a set \(\textit{OP}(\mathcal{L})\) of _operators_, used to form concept and role expressions and a set of \(\textit{MP}(\mathcal{L})\) of _meta-predicates_, used to form assertions. Each operator and each meta-predicate have an associated arity. Given a symbol \(T\), we write \(T/n\) to denote that \(T\) has arity \(n\).