=Paper=
{{Paper
|id=None
|storemode=property
|title=Distributed Reasoning with EDDLHQ+ SHIQ
|pdfUrl=https://ceur-ws.org/Vol-875/short_paper_2.pdf
|volume=Vol-875
|dblpUrl=https://dblp.org/rec/conf/womo/VourosS12
}}
==Distributed Reasoning with EDDLHQ+ SHIQ==
https://ceur-ws.org/Vol-875/short_paper_2.pdf
DDL
Distributed Reasoning with EHQ + SHIQ.
George A. Vouros1 and George M.Santipantakis2
1
Digital Systems, University of Piraeus, Greece georgev@unipi.gr
2
ICSD, University of the Aegean, Greece gsant@aegean.gr
To deal with autonomous agents’ knowledge and subjective beliefs in open, het-
erogeneous and inherently distributed settings, we need special formalisms that
combine knowledge from multiple and potentially heterogeneous interconnected
contexts. Each context contains a chunk of knowledge defining a logical theory,
called ontology unit). While standard logics may be used, subjectiveness and
heterogeneity issues have been tackled by knowledge representation formalisms
called contextual logics or modular ontology languages (e.g. [1] [2]).Nevertheless,
in distributed and open settings we may expect that different ontology units
should be combined in many different, subtle ways without making any assump-
tion about the disjointness of the domains covered by different units. To address
this issue we need to increase the expressivity of the language used for defining
correspondences.Towards this goal, we have been motivated to propose the rep-
DDL
resentation framework EHQ + SHIQ (or simply E − SHIQ).
DDL
The EHQ + SHIQ framework. Given a finite index set of units’ identifiers
I, each unit Mi consists of a TBox Ti , RBox Ri , and ABox Ai in the SHIQ
fragment of Description Logics[3].
Given an i ∈ I, let NCi , NRi and NOi be the sets of concept, role and
individual names respectively. For some R ∈ NRi , Inv(R) denotes the inverse
role of R and (NR ∪ {Inv(R)|R ∈ NR }) is the set of SHIQ-roles. The set of
SHIQ-concepts is the smallest set constructed by the constructors in SHIQ.
Cardinality restrictions can be applied on R, given that R is a simple role. An
interpretation Ii = h∆Ii i , ·Ii i consists of a domain ∆Ii i 6= ∅ and the interpretation
function ·Ii which maps every C ∈ NCi to C Ii ⊆ ∆Ii i , every R ∈ NRi to
RIi ⊆ ∆Ii i × ∆Ii i and each a ∈ NOi to an element aIi ∈ ∆Ii i . Elements and
axioms in unit Mi are denoted by i : c. Each Tbox Ti contains generalized
concept inclusion axioms, RBox Ri contains role inclusion axioms, and ABox
Ai contains assertions for individuals and their relations [3].
Towards combining knowledge in different units, the proposed framework
allows the connection of units via: (a) concept-to-concept subjective correspon-
w
dences [1] specified by onto-bridge rules i : C → j : G, or into-bridge rules
v
i : C → j : G, where i 6= j ∈ I. (b) Individual subjective correspondences
=
i : ai 7→ j : bj , where ai ∈ NOi and bj ∈ NOj . The above mentioned subjective
correspondences concern the point of view of Mj . (c) Link-properties [2](or ij-
properties, i, j ∈ I ), which can be related via ij-property inclusion axioms, be
transitive and, if they are simple, be restricted by qualitative restrictions. The
sets of ij -properties’ names, i.e. the sets ij , i, j ∈ I, are not necessarily pair-
2
wise disjoint, but disjoint with respect to NCi , and NOi . A set of ij -properties
connecting concepts of Mi with concepts of Mj , is defined as the set Eij = ij ,
i 6= j ∈ I, and in case i = j, it is the set Eij = ij ∪ {Inv(E)|E ∈ ji }, where ij
is the set of (local to Mi ) role names. ij -properties are being used for specifying
concepts (so called i−concepts) in the Mi unit.
Transitive axioms are of the form T rans(E; (i, j)), where E ∈ Eij ∩ Eii , E
is transitive in Mi and transitive ij-property. Transitivity axioms and the finite
set of inclusion axioms for ij -properties form the ij -property box Rij (if i = j,
Rii = Ri ). The combined property box RBox R is a family of ij -property
boxes. A combined TBox is a family of TBoxes T= {Ti }i∈I . A distributed ABox
A = {Ai }i∈I , includes a collection of individual correspondences, and property
assertions of the form (a · Eij · b), where Eij ∈ Eij . A distributed knowledge base
Σ is composed as Σ = hT, R, B, Ai, where B = {Bij }i6=j∈I is the collection
of bridge rules between ontology units. Each Rij , is interpreted by a valuation
I
function ·Iij that maps every ij -property to a subset of ∆Ii i × ∆j j . Let Iij =
I
h∆Ii i , ∆j j , ·Iij i, i, j ∈ I. It must be noted that, for a specific i ∈ I and a property
E in the i-th unit, this property may be shared between differentS ij -property
boxes (i.e. for different j’s). In this case, the denotation of E is j∈I E Iij . A
I I
domain relation rij , i 6= j from ∆Ii i to ∆j j is a subset of ∆Ii i × ∆j j , s.t. for
I
each d ∈ ∆iIi , rij (d) ⊆ {d0 |d0 ∈ }, and in case d0 ∈ rij (d1 ) and d0 ∈
∆j j
rij (d2 ), then d1 = d2 . For a subset D of ∆Ii i , rij (D) denotes ∪d∈D rij (d). A
domain relation represents only equalities, i.e. each d1 ∈ rij (d) is equal to the
other individuals in rij (d). The distributed knowledge base is interpreted by a
Distributed Interpretation, I s.t. I = h{Ii }i∈I , {Iij }i,j∈I , {rij }i6=j∈I i.
We have specified a sound and complete distributed Tableau algorithm that
has been implemented by extending the Pellet reasoner 3 . The instance retrieval
algorithm for the framework has been presented in [4].
Acknowledgement: This research project is being supported by the project ”IRAK-
LITOS II” of the O.P.E.L.L. 2007 - 2013 of the NSRF (2007 - 2013), co-funded by the
European Union and National Resources of Greece.
References
1. Borgida, A., Serafini, L.: Distributed description logics: Assimilating information
from peer sources. Journal of Data Semantics 1 (2003) 153–184
2. Parsia, B., Cuenca Grau, B.: Generalized link properties for expressive epsilon-
connections of description logics. In: AAAI. (2005) 657–662
3. Baader, F.e.a., ed.: The Description Logic Handbook: Theory, Implementation, and
Applications, Cambridge University Press (2003)
DDL
4. Santipantakis, G., Vouros, G.: Distributed Instance Retrieval in EHQ + SHIQ Rep-
resentation Framework. In: Artificial Intelligence: Theories and Applications. Vol-
ume 7297 of LNCS. Springer Berlin / Heidelberg (2012) 141–148
3
The full paper describing the framework and the tableau algorithm can be found in
http://ai-lab-webserver.aegean.gr/gsant/ESHIQ.Report