Solving Zadeh's Magnus challenge problem on linguistic probabilities via Linguistic Weighted Averages

Rajati, Mohammad Reza; Mendel, Jerry M.; Wu, Dongrui

doi:10.1109/fuzzy.2011.6007614

Cited by 14 publications

(5 citation statements)

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“…Note that F represents the set of all Type-1 Fuzzy Sets (T1 FSs) over the universe of discourse . The belief structure that represents the knowledge on = + , B 3 = B 1 ⊞ B 2 , 4 has focal elements { 1 , 2 , ⋅ ⋅ ⋅ , } and its probability mass assignments are determined by 3 : 4 Arithmetic operations other than summation on belief structures can also be envisioned, e.g. subtraction ⊟, multiplication ⊠, and inversion or division □ −1 .…”

Section: Our Solution To the Plpmentioning

confidence: 99%

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Advanced computing with words using syllogistic reasoning and arithmetic operations on linguistic belief structures

Rajati

Mendel

2013

2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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In this paper, we present solutions to an Advanced Computing with Words problem that is equivalent to one of Zadeh's challenge problems on linguistic probabilities. We use a syllogism based on the entailment principle to interpret the problem so that it yields two linguistic belief structures. Then we perform an addition of those linguistic belief structures to obtain a belief structure on the variable about which linguistic probabilities have to be inferred. We show that pessimistic (lower) and optimistic (upper) probabilities can be inferred from such a belief structure using Linguistic Weighted Averages and pessimistic and optimistic compatibility measures. Then, we choose vocabularies for linguistic attributes (lifetimes of products) and linguistic probabilities that are involved in the problem statement. The vocabularies are modeled using interval type-2 fuzzy sets. We calculate optimistic (upper) and pessimistic (lower) probabilities, and map them into words present in the vocabulary of linguistic probabilities, so that the results can be comprehended by a human.Index Terms-advanced computing with words, belief structures, compatibility measures, interval type-2 fuzzy sets, linguistic weighted average, lower and upper probabilities, operations on belief structures

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Section: Our Solution To the Plpmentioning

confidence: 99%

“…We have already proposed methodologies for solving some of those problems employing Linguistic Weighted Averages (LWAs) and IT2 FSs [4]- [7]. In this paper, we focus on an engineering problem that is similar to one of Zadeh's challenge problems (the Robert's Problem (RP)), which he has formulated as:…”

Section: Introductionmentioning

confidence: 99%

Advanced computing with words using syllogistic reasoning and arithmetic operations on linguistic belief structures

Rajati

Mendel

2013

2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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“…T1 FS theory was first introduced by Zadeh [66] in 1965 and has been successfully applied in many areas, including modeling and control [6], [47], [65], data mining [22], [40], [69], timeseries prediction [27], [29], [45], linguistic summarization [24], [54], [64], computing with words [41], [42], [68], etc.…”

Section: A Interval Type-2 Fuzzy Setsmentioning

confidence: 99%

On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers

2012

IEEE Trans. Fuzzy Syst.

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111

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Interval type-2 fuzzy logic controllers (IT2 FLCs) have recently been attracting a lot of research attention. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is the following: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and, hence, make better use of them. This paper explains two fundamental differences between IT2 and T1 FLCs: 1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and 2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase. We also present several methods to visualize and analyze the effects of these two fundamental differences, including the control surface, the P-map, the equivalent generalized T1 fuzzy sets, and the equivalent PI gains. Finally, we examine five alternative type reducers for IT2 FLCs and explain why they do not capture the fundamentals of IT2 FLCs.Index Terms-Continuity, equivalent generalized type-1 fuzzy sets (EGT1FSs), equivalent PI gains, interval type-2 fuzzy logic controller (IT2 FLC), P-map.

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“…Both were solved with the application of fuzzy arithmetic and mathematics. In the works of Rajati et al (2011) as well as Rajati and Mendel (2012), Zadeh's challenge problem of tall Swedes has been solved with two methods. These papers show how complicated the problems of autonomous thinking are.…”

Section: • Balls In a Boxmentioning

confidence: 99%

Computing with words with the use of inverse RDM models of membership functions

Piegat

Pluciński

2015

International Journal of Applied Mathematics and Computer Science

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Computing with words is a way to artificial, human-like thinking. The paper shows some new possibilities of solving difficult problems of computing with words which are offered by relative-distance-measure RDM models of fuzzy membership functions. Such models are based on RDM interval arithmetic. The way of calculation with words was shown using a specific problem of flight delay formulated by Lotfi Zadeh. The problem seems easy at first sight, but according to the authors' knowledge it has not been solved yet. Results produced with the achieved solution were tested. The investigations also showed that computing with words sometimes offers possibilities of achieving better problem solutions than with the human mind.

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Solving Zadeh's Magnus challenge problem on linguistic probabilities via Linguistic Weighted Averages

Cited by 14 publications

References 25 publications

Advanced computing with words using syllogistic reasoning and arithmetic operations on linguistic belief structures

Advanced computing with words using syllogistic reasoning and arithmetic operations on linguistic belief structures

On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers

Computing with words with the use of inverse RDM models of membership functions

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