Joachim Hereth Correia

The toscanaj suite for implementing conceptual information systems

For over a decade, work on Formal Concept Analysis has been accompanied by the development of the Toscana software. Toscana was implemented to realize the idea of Conceptual Information Systems which allow the analysis of data using concept-oriented methods. Over the years, many ideas from Formal Concept Analysis have been tested in Toscana systems while the real-world problems encountered led to new theoretical research. After ten years of development, the ToscanaJ project was initiated to solve some outstanding problems of the older Toscana versions. The ToscanaJ suite provides programs for creating and using Conceptual Information Systems. The experience with older Toscana implementations has been applied to the design of the programs. A workflow that developed through many Toscana projects has now been integrated into the tools to make them easier to use. Implemented as an Open-Source project and embedded into the larger Tockit project, ToscanaJ is also a starting point for creating a common base for software development for Formal Concept Analysis. In this paper, we present the features of the ToscanaJ suite and how they can be used to implement Conceptual Information Systems.

Conceptual knowledge discovery--a human-centered approach

In this paper, we discuss Conceptual Knowledge Discovery in Databases (CKDD) as it is developing in the field of Conceptual Knowledge Processing. 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 two decades. CKDD aims to support a human-centered process of discovering knowledge from data by visualizing and analyzing the conceptual structure of the data. Basic to the philosophy of CKDD is the idea that only the human analyst is able to make meaningful decisions, and that the analysis tools should not oversimplify the data, but help the analyst to deal with the complex situation. We discuss how the management system TOSCANA for conceptual information systems is supporting those goals of CKDD, and illustrate it by two applications in database marketing and flight movement analysis. Finally, we present a new tool for conceptual deviation discovery, C HIANTI .

The Power of Peircean Algebraic Logic (PAL)

The existential graphs devised by Charles S. Peirce can be understood as an approach to represent and to work with relational structures long before the manifestation of relational algebras as known today in modern mathematics. Robert Burch proposed in [Bur91] an algebraization of the existential graphs and called it the Peircean Algebraic Logic (PAL). In this paper, we show that the expressive power of PAL is equivalent to the expressive power of Krasner-algebras (which extend relational algebras). Therefore, from the mathematical point of view these graphs can be considered as a two-dimensional representation language for first-order formulas. Furthermore, we investigate the special properties of the teridentity in this framework and Peirce’s thesis, that to build all relations out of smaller ones we need at least a relation of arity three (for instance, the teridentity itself).

A Mathematical Model for TOSCANA-Systems: Conceptual Data Systems

The development of the theory of Formal Concept Analysis has been accompanied from its beginning by applications of the theory to real-world problems. Those applications gave rise to the implementation of the software Toscana and the creation of Toscana-systems. In this paper, we provide a mathematical model for these systems. This model – called Conceptual Data System – enables us to describe Toscana-systems and to discuss possible extensions in mathematical terminology.

Protoconcept Graphs: The Lattice of Conceptual Contents

Protoconcept graphs are part of Contextual Judgment Logic. Generalizing the well-developed theory of concept graphs, they express judgments with a negation on the level of concepts and relations by representing information given in a power context family in a rhetorically structured way. The conceptual content of a protoconcept graph is understood as the information which is represented in the graph directly, enlarged by the information deducible from it by protoconcept implications of the power context family. The main result of this paper is that conceptual contents of protoconcept graphs of a given power context family can be derived as extents of the so-called conceptual information context of the power context family, thus a generalization of the Basic Theorem on \(\overrightarrow{{\mathbb K}}-\)Conceptual Contents in [Wi03].

The teridentity and peircean algebraic logic

A main source of inspiration for the work on Conceptual Graphs by John Sowa and on Contextual Logic by Rudolf Wille has been the Philosophy of Charles S. Peirce and his logic system of Existential Graphs invented at the end of the 19th century. Although Peirce has described the system in much detail, there is no formal definition which suits the requirements of contemporary mathematics. In his book A Peircean Reduction Thesis: The Foundations of topological Logic, Robert Burch has presented the Peircean Algebraic Logic (PAL) which aims to reconstruct in an algebraic precise manner Peirces logic system. Using a restriction on the allowed constructions, he is able to prove the Peircean Reduction Thesis, that in PAL all relations can be constructed from ternary relations, but not from unary and binary relations alone. This is a mathematical version of Peirces central claim that the category of thirdness cannot be decomposed into the categories of firstness and secondness. Removing Burchs restriction from PAL makes the system very similar to the system of Existential Graphs, but the proof of the Reduction Thesis becomes extremely complicated. In this paper, we prove that the teridentity relation is – as also elaborated by Burch – irreducible, but we prove this without the additional restriction on PAL. This leads to a proof of the Peircean Reduction Thesis.

Protoconceptual contents and implications

The development of a mathematical model for judgments understood as compositions of concepts and relations has been an important branch of research in recent years. It led to the definitions of concept and protoconcept graphs which are based on information contained in a power context family, where incidence relations between objects (or tuples of objects) and attributes are stored. A theory of the information those graphs represent (called conceptual content) has been developed for concept graphs in [PW99] and [Wi03]. In [HK04], an extension of this theory to protoconcept graphs not considering object implications (as it is done for concept graphs) has been established. The first part of this paper concentrates on the investigation of the protoconceptual content of protoconcept graphs respecting both protoconceptual and object implications. The second part compares the different structures of conceptual and protoconceptual contents of a given power context family, showing how more background information (using object implications and concepts instead of protoconcepts) reduces the number of possible contents. The third and final part analyzes how the different approaches can be generalized. Here we will concentrate on the (generalized) conceptual content of a formal context. In each part an information context will be defined, which provides an accessible representation of the lattice of (proto-)conceptual closures.