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Kolmogorov’s Theorem and Its Impact on Soft Computing

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The Ordered Weighted Averaging Operators

Abstract

In this chapter, we describe various applications of the Kolmogorov’s theorem on representing continuous functions of several variables (as superpositions of functions of one and two variables) to soft computing. Kolmogorov’s theorem leads to a theoretical justification, as well as to design methodologies, for neural networks. In the design of intelligent systems, Kolmogorov’s theorem is used to show that general logical operators can be expressed in terms of basic fuzzy logic operations.

In the area of reliable computing (i.e., computing that takes into consideration the accuracy of the input data), an extended version of Kolmogorov’s theorem justifies the need to use operations with three or more operands in soft computing. Such operations have already been actively used in soft computing; the simplest (and, so far, most used) of such operations are ordered weighted averaging (OWA) operators proposed by R. R. Yager.

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Nguyen, H.T., Kreinovich, V. (1997). Kolmogorov’s Theorem and Its Impact on Soft Computing. In: Yager, R.R., Kacprzyk, J. (eds) The Ordered Weighted Averaging Operators. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6123-1_1

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  • DOI: https://doi.org/10.1007/978-1-4615-6123-1_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7806-8

  • Online ISBN: 978-1-4615-6123-1

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