A Survey on how Description Logic Ontologies Benefit from Formal Concept Analysis <s> 3 <s> Designing ontologies and specifying axioms of the described domains is an expensive and error-prone task. Thus, we propose a method originating from Formal Concept Analysis which uses empirical data to systematically generate hypothetical axioms about the domain, which are represented to an ontology engineer for decision. In this paper, we focus on axioms that can be expressed as entailment statements in the description logic FLE . The proposed technique is an incremental one, therefore, in every new step we have to reuse the ax- iomatic information acquired so far. We present a sound and complete deduction calculus for FLE entailment statements. We give a detailed description of this multistep algorithm including a technique called empirical attribute reduction and demonstrate the pro- posed technique using an example from mathematics. We give a completeness result on the explored information and address the question of algorithm termination. Finally, we discuss possible appli- cations of our method. <s> BIB001 </s> A Survey on how Description Logic Ontologies Benefit from Formal Concept Analysis <s> 3 <s> Formal Concept Analysis (FCA) can be used to analyze data given in the form of a formal context. In particular, FCA provides efficient algorithms for computing a minimal basis of the implications holding in the context. In this paper, we extend classical FCA by considering data that are represented by relational structures rather than formal contexts, and by replacing atomic attributes by complex formulae defined in some logic. After generalizing some of the FCA theory to this more general form of contexts, we instantiate the general framework with attributes defined in the Description Logic (DL) EL, and with relational structures over a signature of unary and binary predicates, i.e., models for EL. In this setting, an implication corresponds to a so-called general concept inclusion axiom (GCI) in EL. The main technical result of this paper is that, in EL, for any finite model there is a finite set of implications (GCIs) holding in this model from which all implications (GCIs) holding in the model follow. <s> BIB002 </s> A Survey on how Description Logic Ontologies Benefit from Formal Concept Analysis <s> 3 <s> In a previous ICFCA paper we have shown that, in the De- scription Logics EL and ELgfp, the set of general concept inclusions hold- ing in a finite model always has a finite basis. In this paper, we address the problem of how to compute this basis efficiently, by adapting methods from formal concept analysis. <s> BIB003 </s> A Survey on how Description Logic Ontologies Benefit from Formal Concept Analysis <s> 3 <s> This paper is the successor to two previous papers published at the ICFCA conference. In the first paper we have shown that in the Description Logics $\mathcal{EL}$ and $\mathcal{EL}_{\rm gfp}$, the set of general concept inclusions holding in a finite model always has a finite basis. An exploration formalism that can be used to obtain this basis was presented in the second paper. In this paper we show how this formalism can be modified such that counterexamples to GCIs can be provided in the form of ABox-individuals. In a second part of the paper we examine which description logics can be used for this ABox. <s> BIB004
Exploring Finite Models in the DL EL gf p In BIB002 Baader and Distel have extended classical FCA in order to provide support for analyzing relational structures by using efficient FCA algorithms. In this approach the atomic attributes are replaced by complex formulae in some logical language, and data is represented using relational structures rather than just formal contexts. This extension is later instantiated with atrributes defined in the DL EL, and with relational structures defined over a signature of unary and binary predicates, i.e., models for EL. In this setting an implication corresponds to a GCI in EL. This approach at the first sight seems to be very close to the approach introduced in BIB001  . One of the main differences between these approaches is that in BIB002 the authors use one context with infinitely many complex attributes, whereas in  Rudolph uses an infinite family of contexts, each having finitely many attributes that are obtained by restricting the role depth of concepts. In BIB002 the authors additionally show that for the DLs EL and EL gf p , which extends EL with cyclic concept definitions interpreted with greatest fixpoint semantics, the set of GCIs holding in a finite model always has a finite basis. That is, there is always a finite subset of the infinitely many GCIs from which the rest follows. Later in BIB003 the authors have shown how to compute this basis efficiently by using methods from FCA. In a follow-up paper BIB004 , Distel has described how this method can be modified to allow ABox individuals as counterexamples to GCIs.