Rudolf Wille

RESTRUCTURING LATTICE THEORY: AN APPROACH BASED ON HIERARCHIES OF CONCEPTS

Lattice theory today reflects the general Status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

Concept lattices and conceptual knowledge systems

Abstract “Concept Lattice” is the central notion of “Formal Concept Analysis”, a new area of research which is based on a set-theoretical model for concepts and conceptual hierarchies. This model yields not only a new approach to data analysis but also methods for formal representation of conceptual knowledge. These methods are outlined on three levels. First, basics on concept lattices are explained starting from simple data contexts which consist of a binary relation between objects and attributes indicating which object has which attribute. On the second level, conceptual relationships are discussed for data matrices which assign attribute values to each of the given objects. Finally, a mathematical model for conceptual knowledge systems is described. This model allows us to study mathematically the representation, inference, acquisition, and communication of conceptual knowledge.

Formal concept analysis as mathematical theory of concepts and concept hierarchies

Formal Concept Analysis has been originally developed as a subfield of Applied Mathematics based on the mathematization of concept and concept hierarchy. Only after more than a decade of development, the connections to the philosophical logic of human thought became clearer and even later the connections to Piagets cognitive structuralism which Thomas Bernhard Seiler convincingly elaborated to a comprehensive theory of concepts in his recent book [Se01]. It is the main concern of this paper to show the surprisingly rich correspondences between Seilers multifarious aspects of concepts in the human mind and the structural properties and relationships of formal concepts in Formal Concept Analysis. These correspondences make understandable, what has been experienced in a great multitude of applications, that Formal Concept Analysis may function in the sense of transdisciplinary mathematics, i.e., it allows mathematical thought to aggregate with other ways of thinking and thereby to support human thought and action.

A Triadic Approach to Formal Concept Analysis

Formal Concept Analysis, developed during the last fifteen years, has been based on the dyadic understanding of a concept constituted by its extension and its intension. The pragmatic philosophy of Charles S. Peirce with his three universal categories, and experiences in data analysis, have suggested a triadic approach to Formal Concept Analysis. This approach starts with the primitive notion of a triadic context defined as a quadruple (G, M, B, Y) where G, M, and B are sets and Y is a ternary relation between G, M, and B, i.e. Y ⊑ G×M×B; the elements of G, M, and B are called objects, attributes, and conditions, respectively, and (g, m,b) e Y is read: the object g has the attribute m under (or according to) the condition b. A triadic concept of a triadic context (G, M, B, Y) is defined as a triple (A1, A2, A3) with A1× A2sx A3 ⊑ Y which is maximal with respect to component-wise inclusion. The triadic concepts are structured by three quasiorders given by the inclusion order within each of the three components. In analogy to the dyadic case, we discuss how the ordinal structure of the triadic concepts of a triadic context can be analysed and graphically represented. A basic result is that those structures can be understood order-theoretically as “complete trilattices” up to isomorphism.

Conceptual Graphs and Formal Concept Analysis

It is shown how Conceptual Graphs and Formal Concept Analysis may be combined to obtain a formalization of Elementary Logic which is useful for knowledge representation and processing. For this, a translation of conceptual graphs to formal contexts and concept lattices is described through an example. Using a suitable mathematization of conceptual graphs, basics of a unified mathematical theory for Elementary Logic are proposed.

TOSCANA - a Graphical Tool for Analyzing and Exploring Data

TOSCANA is a computer program which allows an online interaction with larger data bases to analyse and explore data conceptually. It uses labelled line diagrams of concept lattices to communicate knowledge coded in given data. The basic problem to create online presentations of concept lattices is solved by composing prepared diagrams to nested line diagrams. A larger number of applications in different areas have already shown that TOSCANA is a useful tool for many purposes.

Conceptual Knowledge Discovery in Databases Using Formal Concept Analysis Methods

In this paper we discuss Conceptual Knowledge Discovery in Databases (CKDD) as it is developing in the field of Conceptual Knowledge Processing (cf. [29],[30]). Conceptual Knowledge Processing is based on the mathematical theory of Formal Concept Analysis which has become a successful theory for data analysis during the last 18 years. This approach relies on the pragmatic philosophy of Ch.S. Peirce [15] who claims that we can only analyze and argue within restricted contexts where we always rely on pre-knowledge and common sense. The development of Formal Concept Analysis led to the software system TOSCANA, which is presented as a CKDD tool in this paper. TOSCANA is a flexible navigation tool that allows dynamic browsing through and zooming into the data. It supports the exploration of large databases by visualizing conceptual aspects inherent to the data. We want to clarify that CKDD can be understood as a human-centered approach of Knowledge Discovery in Databases. The actual discussion about human-centered Knowledge Discovery is therefore briefly summarized in Section 1.

The Basic Theorem of triadic concept analysis

Experiences with applications of concept lattices and the pragmatic philosophy founded by Ch. S. Peirce have suggested a triadic approach to formal concept analysis. It starts with the notion of atriadic context combiningobjects, attributes, andconditions under which objects may have certain attributes.The Basic Theorem of triadic concept analysis clarifies the class of structures which are formed by thetriadic concepts of triadic contexts: These structures are exactly thecomplete trilattices up to isomorphism.

Lattices in Data Analysis: How to Draw Them with a Computer

There is a growing demand for lattice diagrams in concept analysis and for this computer programs are needed. The diagrams should not only reflect the structure of the concept lattices but also unfold views for interpreting the data. Therefore different methods of decomposing and representing concept lattices are discussed to clarify how computer programs can be designed for drawing such lattices. Examples are given too.

Why can concept lattices support knowledge discovery in databases

Knowledge discovery should be understood as information discovery combined with knowledge creation. The creation of knowledge from information can be promoted by proper representations of information which make the inherent logical structure of the information transparent. Since concepts are the basic units of human thought and hence the basic structures of logic, the logical structure of information is based on concepts and concept systems. Therefore, concept lattices as mathematical abstraction of concept systems can support humans to discover information and then to create knowledge. The TOSCANA software even allows the navigation through a network of concept lattices and thereby information discovery in databases which may further lead to knowledge creation.