Distributed Reasoning with

Abstract

To deal with autonomous agents' knowledge and subjective beliefs in open, heterogeneous 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 de ning 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 di erent ontology units should be combined in many di erent, subtle ways without making any assumption about the disjointness of the domains covered by di erent units. To address this issue we need to increase the expressivity of the language used for de ning correspondences.Towards this goal, we have been motivated to propose the representation framework EHDQD+L SHIQ (or simply E SHIQ).

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i : C !v j : G, where i 6= j 2 I. (b) Individual subjective correspondences i : ai 7 != j : bj , where ai 2 NOi and bj 2 NOj . The above mentioned subjective correspondences concern the point of view of Mj . (c) Link-properties [ 2 ](or ijproperties, i; j 2 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 2 I, are not necessarily pairwise disjoint, but disjoint with respect to NCi , and NOi . A set of ij -properties connecting concepts of Mi with concepts of Mj , is de ned as the set Eij = ij , i 6= j 2 I, and in case i = j, it is the set Eij = ij [ fInv(E)jE 2 jig, 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 2 Eij \ Eii, E is transitive in Mi and transitive ij-property. Transitivity axioms and the nite 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= fTigi2I . A distributed ABox A = fAigi2I , includes a collection of individual correspondences, and property assertions of the form (a Eij b), where Eij 2 Eij . A distributed knowledge base is composed as = hT; R; B; Ai, where B = fBij gi6=j2I is the collection of bridge rules between ontology units. Each Rij , is interpreted by a valuation function Iij that maps every ij -property to a subset of iIi jIj . Let Iij = h iIi ; jIj ; Iij i, i; j 2 I. It must be noted that, for a speci c i 2 I and a property E in the i-th unit, this property may be shared between di erent ij -property boxes (i.e. for di erent j's). In this case, the denotation of E is Sj2I EIij . A domain relation rij ; i 6= j from each d 2 iIi , rij (d) fd0jd0 2 rij (d2), then d1 = d2. For a subset D of iIi , rij (D) denotes [d2Drij (d). A domain relation represents only equalities, i.e. each d1 2 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 = hfIigi2I ; fIij gi;j2I ; frij gi6=j2I i.

We have speci ed 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 ].

iIi to jIj is a subset of iIi jIj , s.t. for

Ij j g, and in case d0 2 rij (d1) and d0 2 Acknowledgement: This research project is being supported by the project "IRAKLITOS 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. 3 The full paper describing the framework and the tableau algorithm can be found in http://ai-lab-webserver.aegean.gr/gsant/ESHIQ.Report

References

1. Borgida , A. , Sera ni , 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 epsilonconnections 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 )

4. Santipantakis , G. , Vouros , G.: Distributed Instance Retrieval in EHDQD+LSHIQ Representation Framework . In: Arti cial Intelligence: Theories and Applications . Volume 7297 of LNCS . Springer Berlin / Heidelberg ( 2012 ) 141 { 148