# Acquiring Ontology Axioms through Mappings to Data Sources +
## 3 Mapping-Based Knowledge Bases
Definition 2: An interpretation \(\mathcal{I}\) satisfies a mapping assertion \(\Phi(\vec{x})\leadsto\psi\) with respect to a database \(DB\), if for every tuple of values \(\vec{v}\in\mathit{ans}(\Phi,DB)\), the ground atom \(\psi[\vec{x}/\vec{v}]\) is satisfied by \(\mathcal{I}\). The interpretation \(\mathcal{I}\) is called a _model_ of \(\mathcal{K}=\langle DB,\mathcal{M}\rangle\) if it satisfies every assertion in \(\mathcal{M}\) with respect to \(DB\). We now present an example of MKB, with the goal of illustrating how such a notion can capture real world situations by introducing a notable flexibility in modeling the domain of interest. Example 4: Consider the \(\mathit{Hi}(\mathit{DL}\text{-}\mathit{Lite}_{\mathcal{R}})\) MKB \(\mathcal{K}=\langle\mathcal{D},\mathcal{M}\rangle\), where \(\mathcal{D}\) is the database shown in the introduction (cf. Figure 3) and \(\mathcal{M}\) is the following set of mapping assertions:  
* M1: \(\{m,b\mid\textsf{T-CarTypes}(a,m,b,c)\}\leadsto\mathit{Isa}_{C}(m,b)\)
* M2: \(\{b\mid\textsf{T-CarTypes}(a,m,b,c)\}\leadsto\mathit{Isa}_{C}(b,\text{Car})\)
* M3: \(\{t\mid\textsf{T-CarTypes}(a,m,b,t)\}\leadsto\mathit{[PERSON]}_{C}(t,\text{CarType})\)
* M4: \(\{t1,t2\mid\textsf{T-CarTypes}(a,b,c,t1)\land\textsf{T-CarTypes}(d,e,f,t2) \land t1\
eq t2\}\leadsto\mathit{Disj}_{C}(t1,t2)\)
* M5: \(\{m1,m2\mid\textsf{T-CarTypes}(c1,m1,a,b)\land\textsf{T-CarTypes}(c2,m2,d,e) \land m1\
eq m2\}\leadsto\mathit{Disj}_{C}(m1,m2)\)
* M6: \(\mathit{true}\leadsto\mathit{Isa}_{C}(\text{Car},\mathit{Exists}(\text{ produced\_in}))\)
* M7: \(\mathit{true}\leadsto\mathit{Isa}_{C}(\text{ProdCountry},\mathit{Exists}( \text{Inv}(\text{produced\_in})))\)
* M8: \(\mathit{true}\leadsto\mathit{Isa}_{C}(\mathit{Exists}(\text{produced\_in} ),\text{Car})\)
* M9: \(\mathit{true}\leadsto\mathit{Isa}_{C}(\mathit{Exists}(\text{Inv}(\text{ produced\_in})),\text{ProdCountry})\)
* M10: \(\{x,p\mid\textsf{T-Carys}(x,a,b,p)\}\leadsto\mathit{Inst}_{R}(x,p,\text{produced\_in})\)
* M11: \(\{p\mid\textsf{T-Carys}(a,b,c,p)\}\leadsto\mathit{Inst}_{C}(p,\text{ProdCountry})\)
* M12: \(\{m,t\mid\textsf{T-CarTypes}(a,m,b,t)\}\leadsto\mathit{Inst}_{C}(m,t)\)
* M13: \(\{x,y\mid\textsf{T-Carys}(x,c1,a,b)\land\textsf{T-CarTypes}(c1,y,d,e)\} \leadsto\mathit{Inst}_{C}(x,y)\)  
M1 asserts that for every tuple \(t\) of the T-CarTypes table, the value appearing in the second column of \(t\) denotes a subconcept of the concept denoted by the value in the third column of the same tuple. Thus, for example, considering the fifth tuple \(t_{5}\) of T-CarTypes, M1 states that 1973 MUSTANG MACH 1 is a subconcept of the concept Ford. M2 asserts that every value appearing in the third column of T-CarTypes is a subconcept of Car. For example, by referring to \(t_{5}\) again, such tuple states that Ford is a subconcept of Car. Analogously, M3 asserts that every value appearing in the fourth column of T-CarTypes is a subconcept of CarType. For example, \(t_{5}\) states that Coupe is a subconcept of CarType. M4 (resp., M5) asserts that the values appearing in the fifth column (resp., second column) of T-CarTypes denote concepts that are pairwise disjoint. M6-M9 assert properties the relation _produced_in_, between the concepts Car and _ProdCountry_. M10 populates the relation _produced_in_, and M11 does the same for the concept _ProdCountry_. Mapping M12 exploits the meta-modeling capabilities of \(\mathit{Hi}(\mathit{DL}\text{-}\mathit{Lite}_{\mathcal{R}})\) and relates the different car models to their specific type. For example, looking at tuple \(t_{5}\) again, we can infer by M12 that 1973 FALCON XB GT COUIPE is an instance of _the concept Coupe. Note that M1 asserted that 1973 FALCON XB GT COUIPE is a concept, and therefore, we are taking advantage of the possibility provided by Hi(DL-Lite\({}_{\mathcal{R}}\)) of defining a concept to be an instance of another concept (a metaconcept). Finally, M13 allows us to correctly assign the instances stored in the T-Cars table to the concepts corresponding to the different car models.