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 **of** → **and** 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 I**of** 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 k**and** N k I k. An ij-property axiom is**of** the form P 1 ⊑ P 2 , where P 1 , P 2 ∈ ɛ ij . An ij-property box R ij is a finite set**of** ij-property axioms. The combined property box R contains all the propertyboxes for each i, j ∈ I.