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The Perceptual Computer (Per-C) is an application of Zadeh's theory of computing with words (CWW) in the field of assisting people to make subjective judgments. This paper describes an application of the Per-C to evaluate Learning Outcomes (LOs) in an outcome-based education (OBE) system. The evaluation problem in this paper is viewed as a hierarchical decision making problem. In the first level of...
In this paper, we present a solution to Zadeh's Swedes and Italians challenge problem which involves linguistic quantifiers and linguistic attributes. First, we argue that Zadeh's solution to this problem via the Generalized Extension Principle is very difficult to implement. Then, we use a syllogism based on the entailment principle to interpret the problem so that it can be solved via Linguistic...
Certain classes of fuzzy rule-based systems have been shown to be universal approximators, capable of approximating any continuous mapping on a compact subset of the domain. In previous cases, this property has been proven for fuzzy systems employing t-norms to determine the firing level of each rule. In this paper, we prove that use of the weighted power mean to determine rule firing levels also...
Karnik-Mendel (KM) algorithms used for type reduction (TR) followed by defuzzification are commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Up until now no closed-form solution has existed for KM+defuzzification. Our paper gives, for the first time, a Taylor-series approximation to KM+defuzzification. The first term in the approximation is the Nie-Tan (NT) formula, and the second-term...
Type-2 fuzzy logic systems that use general type-2 fuzzy sets (GT2 FSs) are now receiving renewed attention; hence, it is important to be able to communicate effectively about such fuzzy sets. While it is possible to generate the 3D membership functions (MFs) for these fuzzy sets using computer programs, this is not very useful for real-time on-the-fly discussions. Additionally, for most of us it...
In this paper, we present solutions to Zadeh's challenge problem on calculating linguistic probabilities. First, we argue that Zadeh's solution to this problem via the Generalized Extension Principle is very difficult to implement. Then, we use a syllogism based on the entailment principle to interpret the problem so that it can be solved by calculation of pessimistic (lower) and optimistic (upper)...
Removing Mixed Gaussian and Impulse Noise (MGIN) is considered to be very important in the domain of image restoration, but it is a somewhat more challenging topic than removing pure Gaussian or impulse noise. Therefore, relatively fewer works have been published in this area. This paper proposes a Non-Singleton Interval Type-2 (IT2) Fuzzy Logic System (FLS) for MGIN removal, explains how it can be...
In this paper, we present a solution to Zadeh's Magnus challenge problem on linguistic probabilities. First, we implement Zadeh's solution to this problem. Then, we use the intersection-product syllogism and a syllogism based on the entailment principle to interpret the problem so that it can be solved via Linguistic Weighted Averages. We show that the problem can be solved by calculation of pessimistic...
The union and intersection of general type-2 fuzzy sets (T2 FSs) are fundamental computations for such FSs. In the past, algorithms were developed for the union and intersection computations using vertical-slice and horizontal-slice representations of T2 FSs. The vertical-slice representation of a T2 FS traces its origins back to Zadeh [32] and requires computing the join or meet, whereas the horizontal-slice...
Computing the centroid of an interval type-2 fuzzy set is an important operation in a type-2 fuzzy logic system, and is usually implemented by Karnik-Mendel (KM) iterative algorithms. This paper proves that the centroid computation of an interval type-2 fuzzy set (IT2 FS) using KM algorithms is equivalent to the Newton-Raphson root-finding method in numerical analysis, and explains how continuous...
The Centroid Flow (CF) algorithm is a newly proposed approach for computing the centroid of a type-2 fuzzy set Ã, which normally can be obtained by taking the union of the centroids of all the α-planes of Ã. The CF algorithm utilizes the Karnik-Mendel (KM) or the Enhanced KM (EKM) algorithm only once at the α = 0 α-plane, and then lets its result “flows” stepwise to the α = 1 α-plane. The CF algorithm...
Summary form only given. Interval type-2 fuzzy logic systems (IT2 FLSs) fall into the category of "function approximation," and are widely used in fuzzy logic control, rule-based classification, signal processing, etc. On the other hand, a Perceptual Computer (Per-C) is an implementation of Zadeh's Computing With Words paradigm, and is used to aid people in making subjective judgments. This...
Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. It has been studied by many researchers; however, none of them has used it to generate IF-THEN rules, which can be added to a knowledge base for better understanding of the data, or be used in Perceptual Reasoning to infer the outputs for new scenarios. In this paper LS using IF-THEN...
The question “When will an IT2 FLS outperform a T1 FLS?” has been asked by many, and is acknowledged by many experts to be arguably the most important unanswered question in the T2 field. As a research problem, this question turns into: Establish when and by how much a type-2 fuzzy logic system (T2 FLS) will outperform a type-1 (T1) FLS. This paper provides first results on solving this problem. Its...
The international logistics centers choice problem is very important in International logistics. The location choice problem usually involves both numbers and words in which all of the criteria are weighted using words and the performance evaluations for all sub-criteria are either numbers or words. How to aggregate all of these data without losing information is a very daunting task using a type-1...
The centroid of a general type-2 fuzzy set (T2 FS) Ã can be obtained by taking the union of the centroids of all the α-planes (each raised to level α) of Ã. Karnik-Mendel (KM) or the Enhanced Karnik-Mendel (EKM) algorithms are used for computing the centroid of each α-plane. The iterative features in KM/EKM algorithms can be time-consuming, especially when the algorithms have to be repeated for...
Subsethood and similarity measures are important concepts in fuzzy set (FS) theory. There are many different definitions of them, for both type-1 (T1) FSs and interval type-2 (IT2) FSs. In this paper, Rickard et al.'s definition of IT2 FS subsethood measure, extended from Kosko's T1 FS subsethood measure using the Representation Theorem, and Nguyen and Kreinovich's IT2 FS similarity measure, extended...
The ordered weighted average (OWA) operator has been widely used in decision-making. In many situations, however, providing crisp numbers for either the sub-criteria or the weights is problematic (there could be uncertainties about them), and it is more meaningful to provide intervals, type-1 fuzzy sets (T1 FSs), interval type-2 fuzzy sets (IT2 FSs), or a mixture of all of these, for the sub-criteria...
This paper studies the continuity of the input-output mappings of fuzzy logic systems (FLSs), including both type-1 (T1) and interval type-2 (IT2) FLSs. We show that a T1 FLS being an universal approximator is equivalent to saying that a T1 FLS has a continuous input-output mapping. We also derive the condition under which a T1 FLS is discontinuous. For IT2 FLSs using Karnik-Mendel type-reduction...
Centroid, cardinality, fuzziness, variance and skewness are all important concepts for an interval type-2 fuzzy set (IT2 FS) because they are all measures of uncertainty, i.e. each of them is an interval, and the length of the interval is an indicator of the uncertainty. The centroid of an IT2 FS has been defined by Karnik and Mendel. In this paper, the other four concepts are defined. All definitions...
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