Object Generalized Metrics in Formal Contexts

  by   Martin Kovár, Marie Klimešová, Štěpán Křehlík

Abstract and usage:

Formal context is used in theoretical computer scince for various purposes, but especially in connection with data representation and organization. It was introduced by B. Ganter and R. Wille at the end of the 70's as one of the key notions in Formal Concept Analysis. Framework is an algebraic, formal context structure whose purpose is capturing the topological or topology-like structure of a system from its externally observed properties. The objects of a framework are called places and the attributes are collections of places connected by a possible presence of some physical object, for example, a particle. The set of attributes is called a framology. The generalized distance may be used for study of mutual similarities between two objects of the same type, which, however may vary from various topological and algebraic structures, structures carrying infomation, like databases, to, for instance, dynamical systems.

Our experimental application Object 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 objects 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 objects. 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 objects in the given formal context:

The object distance table of the first type:

The object distance table of the second type:

The object distance table of the third type:


The application Object 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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