Bernhard Ganter

Formal Concept Analysis

Formal concept analysis (FCA) is a mathematical theory about concepts and concept hierarchies. Based on lattice theory, it allows to derive concept hierarchies from datasets. In this survey, we recall the basic notions of FCA, including its relationship to folksonomies. The survey is concluded by a list of FCA based knowledge engineering solutions.

Pattern Structures and Their Projections

Pattern structures consist of objects with descriptions (called patterns) that allow a semilattice operation on them. Pattern structures arise naturally from ordered data, e.g., from labeled graphs ordered by graph morphisms. It is shown that pattern structures can be reduced to formal contexts, however sometimes processing the former is often more efficient and obvious than processing the latter. Concepts, implications, plausible hypotheses, and classifications are defined for data given by pattern structures. Since computation in pattern structures may be intractable, approximations of patterns by means of projections are introduced. It is shown how concepts, implications, hypotheses, and classifications in projected pattern structures are related to those in original ones.

Two basic algorithms in concept analysis

We describe two algorithms for closure systems. The purpose of the first is to produce all closed sets of a given closure operator. The second constructs a minimal family of implications for the ”logic” of a closure system. These algorithms then are applied to problems in concept analysis: Determining all concepts of a given context and describing the dependencies between attributes. The problem of finding all concepts is equivalent, e.g., to finding all maximal complete bipartite subgraphs of a bipartite graph.

Conceptual Structures: Logical, Linguistic, and Computational Issues

We generally think of conceptual structures (and their mathematical representations: conceptual graphs) as a technical framework providing a sound basis for research and design work in knowledge representation and related areas of computer science. In this article, I suggest that conceptual structure predates computer science by some three-million years. In particular, I argue that the human brain (or rather, the brain of our hominid ancestors) acquired conceptual structure long before it acquired language, and that the acquisition of conceptual structure was the key cognitive development that led to the emergence of contemporary humans. Moreover, human language was the result of the addition of grammatical structure to an already developed conceptual structure.

Discovering shared conceptualizations in folksonomies

Social bookmarking tools are rapidly emerging on the Web. In such systems users are setting up lightweight conceptual structures called folksonomies. Unlike ontologies, shared conceptualizations are not formalized, but rather implicit. We present a new data mining task, the mining of all frequent tri-concepts, together with an efficient algorithm, for discovering these implicit shared conceptualizations. Our approach extends the data mining task of discovering all closed itemsets to three-dimensional data structures to allow for mining folksonomies. We provide a formal definition of the problem, and present an efficient algorithm for its solution. Finally, we show the applicability of our approach on three large real-world examples.

Conceptual Structures for Knowledge Creation and Communication

In Contextual Judgment Logic, Sowa’s conceptual graphs (understood as graphically structured judgments) are made mathematically explicit as concept graphs which represent information formally based on a power context family and rhetorically structured by relational graphs. The conceptual content of a concept graph is viewed as the information directly represented by the graph together with the information deducible from the direct information by object and concept implications coded in the power context family. The main result of this paper is that the conceptual contents can be derived as extents of the so-called conceptual information context of the corresponding power context family. In short, the conceptual contents of judgments are formally derivable as concept extents. 1 Information in Contextual Judgment Logic In this paper, information in the scope of Conceptual Knowledge Processing shall be understood in the same way as in Devlin’s book “Infosense Turning Information into Knowledge” [De99]. Devlin briefly summarizes his understanding of information and knowledge by the formulas: Information = Data + Meaning Knowledge = Internalized information + Ability to utilize the information Through Devlin’s understanding it becomes clear that Formal Concept Analysis [GW99] enables to support the representation and processing of information and knowledge as outlined in [Wi02a]. Since Contextual Logic [Wi00] with its semantics is based on Formal Concept Analysis, it is desirable to make also explicit why and how Contextual Logic may support the representation and processing of information and knowledge. In this paper, we concentrate on approaching this aim for Contextual Judgment Logic [DK03]. In Contextual Judgment Logic, judgments understood as asserting propositions are formally mathematized by so-called concept graphs which have been semantically introduced in [Wi97] as mathematizations of Sowa’s conceptual graphs [So84]. Since judgments are philosophically conceived as assertional combinations of concepts, their mathematizatzion is based on formal concepts of formal contexts (introduced in [Wi82]): The semantical base is given by a power A. de Moor, W. Lex, and B. Ganter (Eds.): ICCS 2003, LNAI 2746, pp. 1–15, 2003. c

TRIAS--An Algorithm for Mining Iceberg Tri-Lattices

In this paper, we present the foundations for mining frequent tri-concepts, which extend the notion of closed item-sets to three-dimensional data to allow for mining folk-sonomies. We provide a formal definition of the problem, and present an efficient algorithm for its solution as well as experimental results on a large real-world example.

Attribute exploration with background knowledge

We give the mathematical theory of a simple knowledge acquisition procedure based on implications and counter examples, and show examples of application.

Finding all closed sets: A general approach

We present a unifying theoretical and algorithmic approach to the problems to determine all closed sets of a closure operator, to do this up to isomorphism, and to determine the elements of certain ideals of a power set. This will be done by generalizing the concept of closure operators using the interplay of several orders of a power set.

Contextual Attribute Logic

Contextual Attribute Logic is part of Contextual Concept Logic. It may be considered as a contextual version of the Boolean Logic of Signs and Classes. In this paper we survey basic notions and results of a Contextual Attribute Logic. Main themes are the clause logic and the implication logic of formal contexts. For algorithmically computing bases of those logics, a common theory of cumulated clauses is presented.