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The X-mu representation of fuzzy sets

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Abstract

Fuzzy sets are a good model of the flexible definitions used in human language, but do not always give results in accordance with human reasoning. One reason is that operations on fuzzy sets do not obey all identities from crisp set algebra, such as the law of the excluded middle. Additionally, when used to represent fuzzy numbers they lead to a generalised interval arithmetic rather than a fuzzy version of standard arithmetic. In this paper, we outline the \(X\hbox {-}\mu \) approach, a new method of representing, visualizing and calculating functions of fuzzy values. This fuzzy representation retains standard Boolean operations and leads to standard arithmetic when applied to numerical quantities. It includes the idea of strict fuzzy values, which do not incorporate set or interval-based uncertainty. Using simple examples, we illustrate the \(X\hbox {-}\mu \) approach and outline the notion of a strict fuzzy value.

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Notes

  1. e.g. http://www.srh.noaa.gov/jetstream/global/preciptypes.htm.

  2. We use the notation X-mu instead of \(X\hbox {-}\mu \) where a pure ascii representation is required—for example, in the paper title, to facilitate automatic indexing.

  3. http://data.gov.uk/dataset/road-accidents-safety-data.

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Authors and Affiliations

  1. Machine Intelligence and Uncertainty Group, Intelligent Systems Lab, University of Bristol, Bristol , BS8 1TR, UK

    T. P. Martin

  2. Security Futures Practice, BT Research and Innovation, Adastral Park, Ipswich , IP5 3RE, UK

    T. P. Martin

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  1. T. P. Martin
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Correspondence to T. P. Martin.

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Communicated by Y. Jin.

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Martin, T.P. The X-mu representation of fuzzy sets. Soft Comput 19, 1497–1509 (2015). https://doi.org/10.1007/s00500-014-1302-0

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