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An Algebraic Foundation for Linguistic Reasoning

Huynh. V.N., Murai T., Nakamori Y.

Fundamenta Informaticae

2007

Vol. 78, nr 2

271-294

It is well known that algebraization has been successfully applied to classical and non-classical logics (Rasiowa and Sikorski, 1968). Following this direction, an ordered-based approach to the problem of finding out a tool to describe algebraic semantics of Zadeh's fuzzy logic has been introduced and developed by Nguyen Cat-Ho and colleagues during the last decades. In this line of research, RH_algebra has been introduced in [20] as a unified algebraic approach to the natural structure of linguistic domains of linguistic variables. It was shown that every RH_algebra of a linguistic variable with a chain of the primary terms is a distributive lattice. In this paper we will examine algebraic structures of RH_algebras corresponding to linguistic domains having exactly two distinct primary terms, one being an antonym of the other, called symmetrical RH_algebras. Computational results for the relatively pseudo-complement operation in these algebras will be given.

An alternative extension of the k-means algorithm for clustering categorical data

San O. M., Huynh V. N., Nakamori Y.

International Journal of Applied Mathematics and Computer Science

2004

Vol. 14, no 2

241-247

Most of the earlier work on clustering has mainly been focused on numerical data whose inherent geometric properties can be exploited to naturally define distance functions between data points. Recently, the problem of clustering categorical data has started drawing interest. However, the computational cost makes most of the previous algorithms unacceptable for clustering very large databases. The k-means algorithm is well known for its efficiency in this respect. At the same time, working only on numerical data prohibits them from being used for clustering categorical data. The main contribution of this paper is to show how to apply the notion of "cluster centers'' on a dataset of categorical objects and how to use this notion for formulating the clustering problem of categorical objects as a partitioning problem. Finally, a k-means-like algorithm for clustering categorical data is introduced. The clustering performance of the algorithm is demonstrated with two well-known data sets, namely, soybean disease and nursery databases.