Combining Two Formalism for Reasoning about Concepts

There are two major formalisms that are developed around concepts. The first one is Formal Concept Analysis (FCA) by R. Wille and B. Ganter. Roughly speaking, FCA is an extension of algebraic Lattice Theory for knowledge representation. The second formalism, Description Logic (DL), goes back to the universal terminological logic by P.F. Patel-Schneider. It is closely related to modal and program logics. DL is widely used for ontology research, design, and implementation. Since both formalism use concepts and are used for closely related purposes, it is very natural to compare and combine them. In this paper we introduce and study variants of DL extended by three constructs motivated by FCA. Intentional semantics of two of the new constructs are new modalities that correspond to ‘intent’ and ‘extent’ (two major algebraic constructions of FCA). The third new construct is a connective that is designed to express the ‘formal concept’ property. If L is a variant of DL then we call L extended by these new constructs by L for FCA and denote this logic by L/FCA. We compare expressive powers of L/FCA and L(¬,−) – another variant of L extended by role complement ¬ and inverse − simultaneously. We demonstrate that L/FCA can be expressed in L(¬,−). It implies that for the basic description logic ALC, ALC/FCA is decidable. 1 Basic Description Logics Description logics [2] has originated from the universal terminological logic [9]. There exists many variants of description logics, but we will define only some of them in this section. Definition 1. Syntax of every description logic is constructed from disjoint alphabets of concept, role, and object symbols CS, RS, and OS, respectively. The ? This research is supported in parts by joint grant RFBR 05-01-04003-a DFG project COMO, GZ: 436 RUS 113/829/0-1, by grant RFBR 06-01-00464-a, and Integration Grant n.14 Siberia Branch, Russian Academy of Science. 1 We give detailed definition of description logics for avoiding ambiguities, since there exists some difference in syntax notation between different research groups. sets of concept terms (or concepts) CT and role terms (or roles) RT are defined by induction. The usual definition admits the following clauses. – (Concept terms) • the top concept > and the bottom concept ⊥ are concept terms; • any concept symbol is a concept term; • for any concepts X and Y their union (X tY ) and intersection (X uY ) are concept terms; • for any concept X its complement (¬X) is a concept term; • for any role R and any concept X the universal (∀R. X) and existential (∃R. X) restrictions are concept terms; – (Role terms) • the top role ∇ and the bottom role 4 are role terms; • any role symbol is a role term; • for any roles R and S their union (R t S), intersection (R u S), and composition (R ◦ S) are terms; • for any role R its complement (¬R), inverse (R), and transitive closure (R) are role terms. Concept and role terms altogether form the set of terminological expressions.