# Towards Formal Comparison of Ontology Linking, Mapping and ...

Towards Formal Comparison of Ontology Linking, Mapping and ...

otherwise it is a foreign term in P i . If t is a foreign term in P j , and home (t) = i,ttthen we write P i → P j . If P i → P j for any term t, then also the package P jimports the package P i (denoted P i → P j ). By → ∗ we denote the transitiveclosure ofand by Pj∗ the set P j ∪ {P i |i → ∗ j}.When P i → P j , the symbol ⊤ i occurring within P j represents the importeddomain of P i . Also in this case a novel contextualized negation constructor ¬ i isapplicable within P j ¬ i (it is however a mere syntactic sugar and can be ruledout as we always have ¬ i C ≡ ⊤ i ⊓ ¬ j C).〉A distributed interpretation of P is a pair I =〈{I i } i∈I , {r ij } ∗ i→j , such thateach I i = 〈 〉∆ Ii , ·Ii is an interpretation of the local package Pi and each r ij ⊆∆ Ii ×∆ Ij is a domain relation between ∆ Ii and ∆ Ij . A distributed interpretationI is a model of {P i } i∈I , if the following conditions hold:1. there is at least one i ∈ I such that ∆ Ii ≠ ∅;2. I i |= P i ;3. r ij is an injective partial function, and r ii is the identity function;4. if i ∗ → j and j ∗ → k, then r ik = r ij ◦ r jk (compositional consistency);5. if i t → j, then r ij (t Ii ) = t Ij ;6. if i R → j and (x, y) ∈ R Ii than r ij (x) ≠ ∅ =⇒ r ij (y) ≠ ∅ (role preserving).The three main reasoning tasks for P-DL are consistency of KB, conceptsatisfiability and concept subsumption entailment with respect to a KB. Theseare always defined with respect to a so called witness package P w ∈ P. A packagebasedontology P is consistent as witnessed by a package P w of P, if there exists amodel I of Pw ∗ such that ∆ Iw ≠ ∅. In this case we also say that P is w-consistent.A concept C is satisfiable as witnessed by a package P w of P, if there exists amodel I of Pw ∗ such that C Iw ≠ ∅. A subsumption formula C ⊑ D is valid aswitnessed by a package P w of P (denoted P |= C ⊑ w D), if for every model Iof Pw ∗ we have C Iw ⊆ D Iw .4 E-connectionsThe E-connections framework represents the ontology linking paradigm. Althoughmany flavours of E-connections are known, for sake of simplicity weintroduce the language C E (SHIQ) [6,21,22]. This language allows to connectmultiple ontologies expressed in SHIQ [16] with links.Assume a finite index set I. For i ∈ I let N Ci and N Ii be pairwise disjointsets of concepts names and individual names respectively. For i, j ∈ I, i and jnot necessarily distinct, let ɛ ij be sets of properties, not necessarily mutuallymdisjoint, but disjoint with respect to N kmC kand N k I k. An ij-property axiom isof the form P 1 ⊑ P 2 , where P 1 , P 2 ∈ ɛ ij . An ij-property box R ij is a finite setof ij-property axioms. The combined property box R contains all the propertyboxes for each i, j ∈ I.

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