Bernhard Ganter

Bernhard Ganter, Tom Hanika, Johannes Hirth

Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic definitions are in flux. Our contribution was triggered by the study of concept lattices for tree classifiers and the scaling methods used there. We extend some basic notions, give precise mathematical definitions for them and introduce the concept of scaling dimension. In addition to a detailed discussion of its properties, including an example, we show theoretical bounds related to the order dimension of concept lattices. We also study special subclasses, such as the ordinal and the interordinal scaling dimensions, and show for them first results and examples.

Marcel Nöhre, Dominik Dürrschnabel, Bernhard Ganter et al.

The visualization of concept lattices is a central problem in the field of Formal Concept Analysis. Force-directed algorithms, as popular in graph drawing, are a promising approach, treating lattice diagrams as physical models, optimizing node positions based on forces derived from the lattice structure. We build on the work of Zschalig, who, however, limited himself to attribute-additive diagrams. We use a more general additivity, in which both the attributes and the objects contribute to the positions of the concept nodes. We replace the planarity enhancer used by Zschalig to obtain a starting diagram for force-directed optimization with the DimDraw algorithm, which generates structured order diagrams on its own. The combination results in DimFlux, an algorithm that leverages the advantages of DimDraw but generates additive diagrams in which readability is increased by maximizing the conflict distance between nodes and non-incident edges.

Daniel Borchmann, Bernhard Ganter

Attribute exploration has been investigated in several studies, with particular emphasis on the algorithmic aspects of this knowledge acquisition method. In its basic version the method itself is rather simple and transparent. But when background knowledge and partially described counter-examples are admitted, it gets more difficult. Here we discuss this case in an abstract, somewhat "axiomatic" setting, providing a terminology that clarifies the abstract strategy of the method rather than its algorithmic implementation.