# Acquiring Ontology Axioms through Mappings to Data Sources +
## 2 Higher-Order \(\textit{DL-Lite}_{\mathcal{R}}\)
For \(\textit{DL-Lite}_{\mathcal{R}}\), we simply have:  
* \(\textit{OP}(\textit{DL-Lite}_{\mathcal{R}})=\{\textit{Inv}/1,\textit{ Exists}/1\}\),
* \(\textit{MP}(\textit{DL-Lite}_{\mathcal{R}})=\{\textit{Inst}_{C}/2,\textit{Inst}_{ R}/3,\textit{Isa}_{C}/2,\textit{Isa}_{R}/2,\textit{Disj}_{C}/2,\textit{Disj}_{R}/2\}\),  
provided that _Inv_ and _Exists_ take only a role as argument, \(\textit{Inst}_{C}\) takes an individual as first argument and a concept as second argument, \(\textit{Inst}_{R}\) takes an individual as first and second argument and a role as third argument, \(\mathit{Isa}_{C}\) and \(\mathit{Disj}_{C}\) take two concepts as arguments, and \(\mathit{Isa}_{R}\) and \(\mathit{Disj}_{R}\) take two roles as arguments. **Example 1**. : _Consider the following DL-Lite\({}_{R}\) knowledge base:_  
\[ \begin{array}{lcl}\mathit{Ford}&\sqsubseteq&\mathit{Car}\\ \exists\mathit{produced\_in}&\sqsubseteq&\mathit{Car}\\ \exists\mathit{produced\_in}^{-}&\sqsubseteq&\mathit{ProdCountry}\\ \mathit{Ford}(\textsc{interceptor})&\\ \mathit{produced\_in}(\textsc{interceptor},\textsc{australia}\ )\end{array} \]  
_In words, the knowledge base says that \(\mathit{Ford}\) is a concept that specializes the concept \(\mathit{Car}\), produced \(\_in\) is a role typed on \(\mathit{Car}\) and \(\mathit{ProdCountry}\), which is a concept denoting production countries. More precisely, the domain of produced_in can be instantiated only with instances of \(\mathit{Car}\), as stated by the second inclusion, whereas the range of produced_in can be instantiated only with instances of \(\mathit{ProdCountry}\), as stated by the third inclusion. The above knowledge base can be reformulated in the high-order style syntax for \(\mathit{DL\_Lite}_{R}\) as follows:_  
\[ \begin{array}{l}\mathit{Isa}_{C}(\mathit{Ford},\mathit{Car})\\ \mathit{Isa}_{C}(\mathit{Exists}(\mathit{produced\_in}),\mathit{Car})\\ \mathit{Isa}_{C}(\mathit{Exists}(\mathit{Inv}(\mathit{produced\_in})),\mathit{ ProdCountry})\\ \mathit{Inst}_{C}(\textsc{interceptor},\mathit{Ford})\\ \mathit{Inst}_{R}(\textsc{interceptor},\textsc{australia},\mathit{produced\_in}) \end{array} \]  
_We finally note that in both syntaxes given above, the first three assertions constitute the TBox of the KB, whereas the last two assertions constitute the ABox of the KB._  
Let us turn our attention to the definition of the syntax of \(\mathit{Hi}(\mathit{DL\_Lite}_{R})\). Hereinafter we assume the existence of two disjoint, countably infinite alphabets: \(\mathcal{S}\), the set of predicates, also called _names_ and \(\mathcal{V}\), the set of _variables_. Intuitively, the names in \(\mathcal{S}\) are the symbols denoting the atomic elements of a \(\mathit{Hi}(\mathit{DL\_Lite}_{R})\) knowledge base. The building blocks of such a knowledge base are _assertions_, which in turn are based on _terms_ and _atoms_. We inductively define the set of _terms_, denoted by \(\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S},\mathcal{V})\), over the alphabets \(\mathcal{S}\) and \(\mathcal{V}\) for \(\mathit{Hi}(\mathit{DL\_Lite}_{R})\) as follows:  
* if \(e\in\mathcal{S}\cup\mathcal{V}\) then \(e\in\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S},\mathcal{V})\);
* if \(e\in\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S},\mathcal{V})\), and \(e\) is not of the form \(\mathit{Inv}(e^{\prime})\) (where \(e^{\prime}\) is any term), then \(\mathit{Inv}(e)\in\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S},\mathcal{V})\) (Differently from [18], we avoid the construction of terms of the form \(\mathit{Inv}(\mathit{Inv}(e))\) which, as roles, are equivalent to \(e\). Under this assumption, we do not have safe-range issues when dealing with queries, thus, differently form [18], we consider here non-Boolean queries. );
* if \(e\in\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S},\mathcal{V})\) then \(\mathit{Exists}(e)\in\tau_{\mathit{DL\_Lite}_{R}}(\mathcal{S},\mathcal{V})\). Intuitively, a term denotes either an atomic element, the inverse of an atomic element, or the projection of an atomic element on either the first or the second component. **Example 2**.