The Connection between Concepts in Formal Concept Analysis and Concepts in a Dual Description Logic

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Formal concept analysis (FCA) and description logics (DLs) are two major formalisms around concepts. However, concepts in FCA and concepts in DLs are of different nature. The former are tuples/properties pairs in a given relation, the later are just subsets of domains in terminological interpretations. To represent the logical difference of tuples and attribute values in relations, a dual description logic (DDL) is introduced, in which constants are classified into two classes: tuple constants and value constants, and they are interpreted into two disjoint parts in domain, respectively. Given a model M for DDL, we can obtain a normal relation RM. This paper demonstrates how to connect FCA-concepts in RM and concepts in DDL. To do so, we introduce logical operations negation, disjunction and conjunction on values and on attribute-value pairs, and then build Φ-statements from attribute-value pairs inductively with these operations, and further define a kind of extended FCA-concepts, called Γ-concepts. Each Γ-concept consists of two parts: the extent (tuples the concept covers) and the intent (Φ-statements describing the concept), and there are the following results: the union of any number of extents is always an extent; the complement of an extent is always an extent. Consequently, we obtain two isomorphic lattices: one is LM=(Θ, ∪,∩,–), which is the lattice of subsets of Θ, and the other is concept lattice L(↑RM),  where Θ consists of the interpretations of DDL-concepts, and ↑RM is an extend relation from the normal relation RM.
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Formal Concept Analysis; Γ-Concepts; Dual Description Logics; Isomorphic Lattices

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