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From computing with numbers to computing with words - From manipulation of measurements to manipulation of perceptions

Lotfi Zadeh (2002)

International Journal of Applied Mathematics and Computer Science

Computing, in its usual sense, is centered on manipulation of numbers and symbols. In contrast, computing with words, or CW for short, is a methodology in which the objects of computation are words and propositions drawn from a natural language, e.g., small, large, far, heavy, not very likely, the price of gas is low and declining, Berkeley is near San Francisco, it is very unlikely that there will be a significant increase in the price of oil in the near future, etc. Computing with words is inspired...

Functions Equivalent to Borel Measurable Ones

Andrzej Komisarski, Henryk Michalewski, Paweł Milewski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β ( f ) ≤ ω ξ ₁ · ω ξ ₂ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions...

Fundamentals of a mathematical theory of fuzzy sets

Jindřich Spal (1982)

Aplikace matematiky

Fuzzy sets establish a mapping from the interval of values of a criterial function onto a system of subsets of a basic set. In the paper, a system of definitions and theorems is introduced, which is aimed at an adequate expression of this point of view. The criterial function, with an arbitrary interval of values, serves for expressing the really existing objective property, forming the basis for defining a fuzzy set.

Fuzzy clustering of fuzzy data considering the shape of the membership functions using a novel representation learning technique

Alireza Khastan, Elham Eskandari (2025)

Kybernetika

Most existing distance measures for fuzzy data do not capture differences in the shapes of the left and right tails of membership functions. As a result, they may calculate a distance of zero between fuzzy data even when these differences exist. Additionally, some distance measures cannot compute distances between fuzzy data when their membership functions differ in type. In this paper, inspired by human visual perception, we propose a fuzzy clustering method for fuzzy data using a novel representation...

Fuzzy data in statistics

Milan Mareš (2007)

Kybernetika

The development of effective methods of data processing belongs to important challenges of modern applied mathematics and theoretical information science. If the natural uncertainty of the data means their vagueness, then the theory of fuzzy quantities offers relatively strong tools for their treatment. These tools differ from the statistical methods and this difference is not only justifiable but also admissible. This relatively brief paper aims to summarize the main fuzzy approaches to vague data...

Fuzzy distances

Josef Bednář (2005)

Kybernetika

In the paper, three different ways of constructing distances between vaguely described objects are shown: a generalization of the classic distance between subsets of a metric space, distance between membership functions of fuzzy sets and a fuzzy metric introduced by generalizing a metric space to fuzzy-metric one. Fuzzy metric spaces defined by Zadeh’s extension principle, particularly to ℝ n are dealt with in detail.

Fuzzy equality and convergences for F -observables in F -quantum spaces

Ferdinand Chovanec, František Kôpka (1991)

Applications of Mathematics

We introduce a fuzzy equality for F -observables on an F -quantum space which enables us to characterize different kinds of convergences, and to represent them by pointwise functions on an appropriate measurable space.

Fuzzy Markov chains: uncertain probabilities.

James J. Buckley, Esfandiar Eslami (2002)

Mathware and Soft Computing

We consider finite Markov chains where there are uncertainties in some of the transition probabilities. These uncertainties are modeled by fuzzy numbers. Using a restricted fuzzy matrix multiplication we investigate the properties of regular, and absorbing, fuzzy Markov chains and show that the basic properties of these classical Markov chains generalize to fuzzy Markov chains.

Fuzzy numbers, definitions and properties.

Miguel Delgado, José Luis Verdegay, M. Amparo Vila (1994)

Mathware and Soft Computing

Two different definitions of a Fuzzy number may be found in the literature. Both fulfill Goguen's Fuzzification Principle but are different in nature because of their different starting points.The first one was introduced by Zadeh and has well suited arithmetic and algebraic properties. The second one, introduced by Gantner, Steinlage and Warren, is a good and formal representation of the concept from a topological point of view.The objective of this paper is to analyze these definitions and discuss...

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