Mainstream research on Description Logics (DLs) usually treats DL knowledge bases as monolithic structures (cf. [1]). It has been pointed out, however, that in environments such as Semantic Web, distribution of ontological knowledge across various sources is expected and accepted [2]. Several use-cases can be provided for the envisioned distributed ontology environment: - Two distinct applications may use two different ontologies to refer to the same concept. Thanks to mapping between these ontologies, association of these concepts can be derived. - One particular application (e.g., a semantic annotation of a document) can use different ontologies to refer to two distinct concepts. These concepts, may be related by mapping, and so further semantic consequences can be derived. - Developers may choose to map from a foreign ontology instead of repeating a complex description of some concept. Reuse of concepts can be facilitated. As Kalfoglou et al. conclude in [3], “. . . ontology mapping nowadays faces some of the challenges we were facing ten years ago when the ontology field was at its infancy. We still do not understand completely the issues involved.” This suggests that accomplishing these scenarios is a long-term and incremental task. In our research, we focus on DLs, a formalism with precise, logical semantics, with encouraging computational properties and practical reasoner implementations [1]. Also, the most prominent ontology language suggested by W3C, OWL Web Ontology Language [4], is derived form DLs. From the DL point of view, the actual line of research suggests itself: 1. Describe useful syntactic constructs and intuitions behind them. 2. Provide formal model-theoretic semantics for these constructs, yielding a logic with reasoning tasks such as satisfiability of concepts and entailment of concept subsumption. 3. Provide reasoning algorithms for the decision tasks. 4. Develop and optimize implementations of reasoners. Moreover, each time different set of constructs is put together in Step 1, the following steps need to be repeated, in order to investigate properties and practical usability of thus constructed DL.
In Step 1, for a start, the existing research on ontology mapping (see [
There are several existing approaches. Distributed Description Logics (DDLs)
of Borgida, Serafini and Tamilin [
It is also noted in [
A DDL knowledge bases consist of a distributed TBox – a set of local TBoxes, each over its own DL language Li – and a set of bridge-rules B between these local TBoxes. Each of the local TBoxes Ti is a collection of GCIs of the form: i : C ⊑ D, where the prefix i identifies the TBox the GCI belongs to. Bridge-rules are of two forms, into bridge-rules and onto bridge-rules: i : A →⊑ j : G , i : B →⊒ j : H .
For precise formal semantics of DDLs, please refer to [
In [
And the distributed counterpart of O, divided into two ontologies A (on the left) and B (on the right):
While Penguin in O is not satisfiable, the corresponding PenguinB in B is. The
problem is that the DDL framework allows an interpretation to associate each
instance x of PenguinB with two distinct elements, say y1 and y2, one instance
of BirdA and the other one of NonFlyingA, even if BirdA and NonFlyingA are
disjoint. We agree with [
We address the problem outlined above by introducing so called conjunctive bridge-rules. We use the following syntax (into and onto form respectively): ⊑ i : C ։ j : G ,
⊒ i : D ։ j : H .
Recall from [
⊑ ⊑ 1. I |=d i : C ։ j : G if for each i : D ։ j : H, rij CIi ∩ DIi ⊆ GIj ∩ HIj , ⊒ ⊒ 2. I |=d i : C ։ j : G if for each i : D ։ j : H, rij CIi ∩ DIi ⊇ GIj ∩ HIj , where I |=d R is to be read I satisfies the bridge rule R.
Our choice of adding new kind of bridge-rules instead of replacing the old semantics by the new one is to underline that both kinds can co-exist and be used according to the intentions of the ontology editor. We continue with characterization of conjunctive bridge-rules. First, they are stronger than the original form in a sense: the semantic condition imposed by a bridge-rule of the original form is also imposed by the corresponding conjunctive form.
Theorem 1. Given a distributed TBox T with a set of bridge-rules B and some ⊑ ⊒ local TBoxes Ti and Tj such that i 6= j and i : C ։ j : G ∈ B (i : C ։ j : G ∈ B), for each distributed interpretation I such that I |=d T it holds that rij CIi ⊆ GIj (rij CIi ⊇ GIj ) respectively.
Next theorem shows that choosing conjunctive bridge-rules solves the problem outlined by the example above.
Theorem 2. Given a distributed TBox T with a set of bridge-rules B and some ⊑ local TBoxes Ti and Tj such that i 6= j, if for some n > 0 the bridge-rules i : C1 ։ ⊑ j : G1, . . . , i : Cn ։ j : Gn are all part of B then for every distributed interpretation I such that I |=d T it holds that rij (C1 ⊓ · · · ⊓ Cn)Ii ⊆ (G1 ⊓ · · · ⊓ Gn)Ij . ⊒ ⊒
Likewise, if for some n > 0 the bridge-rules i : C1 ։ j : G1, . . . , i : Cn ։ j : Gn are all part of B then for every distributed interpretation I such that I |=d T it holds that rij (C1 ⊓ · · · ⊓ Cn)Ii
⊇ (G1 ⊓ · · · ⊓ Gn)Ij .
And so, for concepts involved in conjunctive bridge-rules, the intersection is always “properly related” (subset/superset, w.r.t. the kind of the bridge-rules) to the image of the intersection of their mapped counterparts. In the example above, the concept PenguinB is mapped to both BirdA and NonFlyingA which are disjoint. If we would use conjunctive bridge rules in this example, we would have PenguinBIB ∩ PenguinBIB ⊆ rij BirdAIA ∩ NonFlyingAIA , which yields PenguinBIB ⊆ rij (∅) and so PenguinBIB ⊆ ∅. That is, in the distributed knowledge base, the concept PenguinB is unsatisfiable, according to our intuition.
In [
Simple subsumption propagation. Combination of into and onto bridgerules allows for propagation of subsumption across ontologies. Formally, if i : C →⊒ j : G ∈ B and i : D →⊑ j : H ∈ B then T |=d i : C ⊑ D =⇒ T |=d j : G ⊑ H.
Generalized subsumption propagation. If i : C →⊒ j : G ∈ B and i : Dk → ⊑ n j :nHk ∈ B, for 1 ≤ k ≤ n then T |=d i : C ⊑ Fk=1 Dk =⇒ T |=d j : G ⊑ Fk=1 Hk.
These desiderata are satisfied even in the presence of conjunctive bridgerules. Moreover, subsumption propagates over conjunctive bridge-rules even for intersection of concepts, as follows.
⊒ ⊑ Theorem 3. If i : C ։ j : G ∈ B and i : Dk ։ j : Hk ∈ B, 1 ≤ k ≤ n then n n T |=d i : C ⊑ dk=1 Dk =⇒ T |=d j : G ⊑ dk=1 Hk.
These results have not been published yet. We have submitted a technical
paper to DL-2007. Other interesting desiderata for DDLs published in [
We decided to pursue research for our Ph.D. dissertation in the area of
Distributed Ontologies based on Description Logics. Several approaches exist in
this field, most notably those of [
Acknowledgement. The author is a Ph.D. student in third year, however, he only switched to Distributed Ontologies a year ago. Support from Slovak Agency for Promotion of Research and Development projects, contract no. APVV-20P04805 and APVV-99-PO5005; and project VEGA of Slovak Ministry of Education and Slovak Academy of Sciences no. 1/0173/03 is acknowledged.