The E-SHIQ Contextual Logic Framework

To deal with autonomous agents’ knowledge and subjective beliefs in the open, heterogeneous and inherently distributed settings concerned by Agreement Technologies, we need special formalisms that combine knowledge, taking also into account disagreements and heterogeneity from multiple interconnected contexts. For agents to reason jointly, they need to combine knowledge by means of correspondences and links between context elements. Each context contains a chunk of knowledge defining a logical theory, that we call ontology or ontology unit. While standard logics may be used, to deal with the issues of subjectiveness and heterogeneity in the semantic web, special knowledge representation formalisms based on web standards have been proposed, sometimes called contextual logics or modular ontology languages (e.g. [1] [2]). Among others, assumptions about the domains covered by the distinct ontology units affect the expressivity of the languages used for defining correspondences. Nevertheless, in distributed and open settings we may expect that different units should be combined in many different, subtle ways. Towards this goal, we have been motivated to propose the representation framework E HQ+ SHIQ (or simply E − SHIQ). The proposed framework alleviates the assumptions for overlapping or disjoint domains posed by DDL and E-connections, respectively. The E-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 (DL)[3]. SHIQ provides role transitivity and hierarchy, qualified cardinality restrictions and inverse roles. For each unit i ∈ I, let NCi , NRi and NOi be the sets of concept, role and individual names respectively. Each term C ∈ NCi is interpreted by an interpretation function Ii = 〈∆ Ii i , ·Ii〉 to a set of elements in the domain ∆Ii i modelled in the unit. Each term R ∈ NRi is interpreted as a relation of elements RIi ⊆ ∆Ii i ×∆ Ii i , and each x ∈ NOi as a single element xIi ∈ ∆Ii i . It holds that for each C v D ∈ Ti, CIi ⊆ DIi , where C,D are possibly complex concepts, and for each R v S ∈ Ri, RIi ⊆ SIi , where {R,S} ⊆ NRi . Each Ai contains concept and role assertions i.e. C(x), R(x, y), that are interpreted to membership xIi ∈ CIi and relation 〈xIi , yIi〉 ∈ RIi of elements in the domain. An element, axiom or assertion c in unit Mi is denoted by i : c. Towards combining knowledge of different units, the proposed framework allows: (a) concept-to-concept subjective correspondences [4], [1], specified by