Attribute Generalized Metrics in Formal Contexts

  by   Martin Kovár, Marie Klimešová, Štěpán Křehlík

Abstract and usage:

Formal context is an structure introduced by B. Ganter and R. Wille at the end of the 70's as one of the key notions in Formal Concept Analysis. It is used in theoretical computer scince for various purposes, but especially in connection with data representation and organization. Framework is an algebraic structure whose purpose is capturing the topological or topology-like structure of a system from its externally observed properties. Formally, framework is a formal context whose incidence relation is the membership relation. These structures may be used, for example, for investigation of properties of various mathematical objects, including topological properties of causal structures, databases or information and dynamical systems. The generalized distance of two structures of the same type is an important characteristic of their mutual similarity.

Our experimental application Attribute Generalized Metrics checks if the formal context is correctly given by the input and then it calculates the tables of three modifications of the generalized distance between the attributes of the given formal context. The calcullation is based on the properties of the framework associated with the formal context on the set of its attributes. The notion of a generalized distance is inspired by the notion of partial metrics due to S. Matthews. A source of inspiration of this application is also the joint research of the first author with his former student, A. Chernikava during her doctoral study. The coauthors, M. Klimešová and Š. Křeklík are the current doctoral students of the Department of Mathematics.


The set of objects:

The set of attributes:

The incidence relation:


The generalized distances of attributes in the given formal context:

The attribute distance table of the first type:

The attribute distance table of the second type:

The attribute distance table of the third type:


The application Attribute Generalized Metrics is written in Java powered by Wolfram webMathematica 3.1. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. The results representing its theoretical background were presented by the authors on several scientific conferences as an integral part of their research. In case of interest in more detail, see [1] or contact the authors. For research and scientific activities the software is available free of charge. In all other cases, please contact RNDr. M. Novák, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, phone: +420 5 4114 3135. Acknowledgement: FEKT-S-14/2200 "Reprezentace řešení dynamických systémů, numerické algoritmy řešení"

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