Approximate nature of traditional fuzzy methodology naturally leads to complex-valued fuzzy degrees

Explainable AI and Other Applications of Fuzzy Techniques

2021

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We show that a natural formalization of the process of changing one's mind leads to such seemingly non-intuitive ideas as square root of "not" and complex-valued fuzzy degrees. Formulation of the ProblemEverything is a Matter of Degree -At Least Our Opinions. It is rare that we immediately become convinced in the truth of some statement. Ok, this happens in mathematics: once we hear (or read) and understand the proof, we are 100% convinced. However, outside mathematical proofs, our degree of belief in a statement changes gradually. We may eventually reach the stage when we are absolutely convinced that a given statement is true (or when we are absolutely convinced that a given statement is false), but before we reach this stage, we go through intermediate stages, in which we only have a partial degree of belief in the given statement.This idea of intermediate degrees of belief is one of the main ideas behind Zadeh's fuzzy logic (see, e.g., [2,6,10,12,14,18]) -that everything is a matter of degree.

Section: Example Of Changing One's Mindmentioning

confidence: 97%

A Natural Formalization of Changing-One’s-Mind Leads to Square Root of “Not” and to Complex-Valued Fuzzy Logic

Explainable AI and Other Applications of Fuzzy Techniques

2021

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“…Another useful 2-D distributive extension of the usual fuzzy logic is the complex-valued fuzzy logic, in which the degrees of belief can take any complex values a + b · i, with i def = √ −1; see, e.g., [1], [2], [3], [8], [9], [15]. While it is empirically successful, it is not as widely used an interval-valued fuzzy logic, since it lacks a clear justification and clear interpretation.…”

Section: Introductionmentioning

confidence: 99%

From 1-D to 2-D fuzzy: A proof that interval-valued and complex-valued are the only distributive options

Servin

2015 Annual Conference of the North American Fuzzy Information Processing Society (NAFIPS) Held Jointly With 2015 5th World Con

2015

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Abstract-While the usual 1-D fuzzy logic has many successful applications, in some practical cases, it is desirable to come up with a more subtle way of representing expert uncertainty. A natural idea is to add additional information, i.e., to go from 1-D to 2-D (and multi-D) fuzzy logic. At present, there are two main approaches to 2-D fuzzy logic: interval-valued and complexvalued. At first glance, it may seem that many other options are potentially possible. We show, however, that, under certain reasonable conditions, interval-valued and complex-valued are the only two possible options.

“…Section 6 (expanding [15]) explains how the hierarchical character of discrete data leads to computing with words. Section 7 (expanding [14]) shows that an approximate character of fuzzy values leads to complexvalued fuzzy sets. Finally, Section 8 (expanding [17]) speculates on how fuzzy techniques can properly take into account the dynamic character of objects and processes.…”

Section: Fuzzy Techniques As An Easier-to-compute Continuous Approximmentioning

confidence: 99%

50 Years of fuzzy: from discrete to continuous to – Where?

Nguyen

et al. 2015

IFS

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Abstract. While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes. We show that the need for such an approximation explains many features of fuzzy techniques, and we speculate on to which promising future directions of fuzzy research this need can lead us.Keywords: fuzzy techniques, discrete, continuous, interval-valued fuzzy, complex-valued fuzzy, computing with words, dynamical fuzzy logic, chemical kinetics, non-additive measures, symmetry Fuzzy Techniques as an Easier-to-Compute Continuous Approximation for Difficult-to-Compute Discrete Objects and ProcessesDiscrete objects and processes are ubiquitous. Many real-life objects are processes are discrete. On the macro level, there is an abrupt transition in space between physical bodies, there is an abrupt transition in time when, e.g., a glass breaks or a person changes his/her opinion. On the micro level, matter consists of discrete atoms and molecules, with abrupt transitions between different states of an atom.Continuous problems are easier to compute. While discrete objects and processes are ubiquitous in nature, from the computational viewpoint, it is often much easier to handle continuous problems. This may sound * Corresponding author. E-mail: vladik@utep.edu.counter-intuitive, since intuitively, if we restrict our search or optimization to only integer values, the problem would become easier -but it is not. For example, in the continuous case, it is relatively easy to find a solution x 1 , . . . , x n to a system of linearmany known feasible algorithms for that), the problem becomes NP-hard (computationally intractable) if we only allow discrete values of x i ; see, e.g., [9,28].Similarly, in the continuous case, it is relatively easy to find the values x 1 , . . . , x n that minimize a given quadratic function f (x 1 , . . . , x n ): it is sufficient to solve the corresponding system of linear equations ∂f ∂x i = 0. However, optimization of quadratic functions for discrete inputs, e.g., for x i ∈ {0, 1}, is NPhard [9,28]. Continuous approximations of discrete objects and processes are ubiquitous in physics. Because dealing with discrete objects and processes is often computationally complicated, physicists often approximate discrete objects with continuous ones. For example, it is not feasible to describe the changes in atmosphere by tracing all 10 23 molecules, but approximate equations that describe the atmosphere as a continuous field leads to many useful weather predictions. Similar, a solid body -in effect, a collection of atoms -is well described by a continuous density field, and an atomic nucleus -a collection of protons and neutrons -is well described by a continuous (liquid) model; see, e.g., [8].Such approximations are also useful in analyzing social phenomena. For example, in analyzing how epidemics spread, ...