FUZZY CONCEPTUAL KNOWLEDGE PROCESSING
Christoph S. H e r r m a n n
F G In%ellektik, T H Darmstadt
Alexanderstr. 10, 64283 Darmstadt, G e r m a n y chri s~int ollektik, inf ormat ik. th- darmstadt, de

Abstract
W e introduce fuzziness to conceptual knowledge processing by using linguistic variables instead of a two-valued representation. The attribute/object table for conceptual lattices holds fuzzy membership values rather than
T R U E / F A L S E entries and can be mapped into a graph of dependencies. From this graph implications can be extracted together with the method to compute truth values for the inferibleconclusions. Hence, fuzzy conclusions can be drawn from interpreting the fuzzy concept values of the graph. W e demonstrate the feasibilityof our approach by computing patient data from a medical diagnosis example.

1

Introduction

The analysis of large data sets can be achieved in a way that is closer to human knowledge representation than simple lists of data by the technique of conceptual lattices [10]: Data given as relations between objects and attributes can be arranged to form concepts, which then can be organized in form of a lattice that makes the dependencies between the different concepts become obvious. Such lattices can not only be used to detect relations between concepts that were not known in advance, but they can also help to refine definitions or descriptions of concepts, if the lattice produced by the given definitions does not match the intended meaning.
Attributes of objects may be adjectives like tall, blond etc. Unfortunately, in most existing systems, these attributes must be assigned to the objects in a binary or at most three-valued manner [1]: For instance, a person may be described as tall or short or of unknown size. But it is not possible to express weak dependencies or vague information like a person being "rather tall". Fuzzy logic has been proposed to overcome this problem [12] and is well suited to express the membership of art object to such an attribute. "Permission to make digital/hard copy of all or part of this material without fee is granted provided that copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery, Inc.(ACM). To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee."
© 1996 ACM0-89791-820-7 96 0002 3.50

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Steffen H6Udobler, A n t j e S t r o h m a i e r
TU Dresden, FG Wissensverarbeitung
01062 D r e s d e n , G e r m a n y
{ s h , a s t } ~ i n f , t u - d r e s d e n , de

The attributes may be represented by linguistic variables and we have to take care that only consistent assignments of attributes to objects occur. We may then obtain a real-valued concept lattice rather than one with binary attributes. The generation of concepts from the resulting fuzzy lattice may be dynamically influenced by varying the criteria for the extraction of the concepts.
These criteria may also be adapted to the desired application. Also, we can draw conclusions from the resulting concepts In the following Section 2 we will shortly describe the idea and application of concept lattices and conceptual knowledge bases. In Section 3 we show how fuzziness can be introduced into this theory and point out possible appllcationsby an example. A comparison with the original conceptual lattices [101 and a different approach to concept fuzzilication [9] are pointed out in the last section where also possible extensions are outlined.

2

Conceptual Lattices

Conceptual lattices are built from a set of objects C9 which may be described with the help of the given attributes .4. The triple (0,.4,1) is called a (single. valued) contezt, where 27 is the incidence relation of attributes and objects. This relationmay be represented by
a