Fuzzy

logic was introduced in 1965 by Lofti A Zadeh in his paper “Fuzzy

Sets”. Zadeh and others continued to develop fuzzy logic at that time. The

idea of fuzzy sets and fuzzy logic were not accepted well within academic

circles because some of the underlying mathematics had not yet been explored. The

applications of fuzzy logic were slow to develop because of this, except in the

east. In Japan specifically fuzzy logic was fully accepted and implemented in

products simply because fuzzy logic worked, regardless of whether

mathematicians agreed or not. The success of many fuzzy logic based products in

Japan in the early 80s led to a revival in fuzzy logic in the US in the late

80s.

Some

of the objections that faced fuzzy logic in its early days are shown below:

“‘Fuzzification’

is a kind of scientific permissiveness. It tends to result in socially

appealing slogans unaccompanied by the discipline of hard scientific work and

patient observation.”

-Professor Rudolf Kalman UFlorida

“Fuzziness

is probability in disguise. I can design a controller with probability that

could do the same thing that you could do with fuzzy logic.” -Professor

Myron Tribus, on hearing of the fuzzy-logic control of the Sendai subway system

IEEE Institute, may 1988.

The theory of

fuzzy sets has advanced in a variety of ways and in many disciplines, since its

inception in 1965. Application of this theory can be found in artificial

intelligence, computer science, machine learning, robotics, pattern recognition

etc. Having capability to create algorithm which often can mimic the human

cognitive abilities like learning, reasoning, decision making and tolerance to

uncertainty due to vagueness, ambiguity, imprecision have paved the way of the

fuzzy system to be better suitable for the situation in which uncertainty

occurring in the system that cannot be accommodated for the rigorous

computation for modeling.

Let’s look at

the kind of problems in the history of mankind which paved the way for

development of fuzzy theory. The problems of uncertainty, imprecision and

vagueness have been discussed for many years. These problems have been major

topics in philosophical circles with much debate, in particular, about the

nature of vagueness and the ability of traditional Boolean logic to cope with

concepts and perceptions that are imprecise or vague. The Fuzzy Logic can be

considered a Multi-valued Logic, MVL. It is founded on, and is closely related

to-Fuzzy Sets Theory, and successfully applied on Fuzzy Systems. It may be

thought that fuzzy logic is quite recent and what has worked for a short time,

but its origins date back at least to the Greek philosophers and especially

Plato (428-347 B.C.). It even seems plausible to trace their origins in China

and India. It is because it seems that they were the first to consider that all

things need not be of a certain type or quit, but there are a stopover between.

That is, be the pioneers in considering that there may be varying degrees of

truth and falsehood. In case of colors, for example, between white and black

there is a whole infinite scale: the shades of gray. Some recent theorems show

that in principle fuzzy logic can be used to model any continuous system, be it

based in artificial intelligence or physics, or biology, or economics, etc.

Mathematical

development has advanced to a very high standard. The basic idea underlying all

these approaches is that of an intrinsic dichotomy between true and false. This

opposition implies the validity of two fundamental laws of classical logic: –

Principle of excluded middle: Every proposition is true or false, and there is

another possibility. – Principle of non-contradiction: No statement is true and

false simultaneously. This basic idea generates a series of paradoxes and

dissatisfaction that is based on the need to overcome this strict

truth-bivalence of classical logic. Accept that a proposition about a future

event is true or false becomes necessary or impossible, respectively, the event

expressed by the proposition. The solution proposed by Jan Lukasiewicz himself

in his classic 1920, is the acceptance of a logic with three truth values (or

three-valued), also called trivalent), which in addition to true and false,

accepts a value of indeterminate truth, which is ascribed a truth value or

grade of membership of 0.5. In the eighteenth century, David Hume (1711-1776)

and Immanuel Kant (1724-1804) were inquiring about such concepts. They

concluded that the reasoning is acquired through experiences throughout our

lives. Hume believed in the logic of common sense, and Kant thought that only

mathematicians could provide clear and precise definitions, both accepting that

there were conflicting principles that had no solution. In conclusion, both

were detecting conflicting principles within the so-called classical logic.

Then in the early twentieth century, the British philosopher and mathematician

Bertrand Russell reported the idea that classical logic inevitably leads to

contradictions. Also Charles Sanders Peirce (1839-1914) somewhat anticipated

this, but there are many who, like Bart Kosko, Bertrand Russell considered the

father of Fuzzy Logic. Because in the early twentieth century, the British

philosopher and mathematician Bertrand Russell (1872-1970) reported the idea

that classical logic inevitably leads to contradictions, making a study on the

vagaries of language, and concluding that the vagueness is precisely one

degree. According to this theory, we

have a transfer function derived from the characteristic function usually

called the “membership function”, which runs from the universe of discourse, U,

until the unit closed interval of reals, which is 0, 1. Not so in the sets

“classic” or “crisp sets”, where the range of the function is reduced to a set

consisting of only two elements, namely was the {0, 1}. Therefore, fuzzy set

theory is a generalization of classical set theory. The theory of “vague sets”

(today, so-called Fuzzy Sets) proceeds from the quantum physicist and German

philosopher Max Black (1937) analyzed the problem of modeling “vagueness”. He

differs from Russell in that he proposes that traditional logic can be used by

representing vagueness at an appropriate level of detail and suggests that

Russell’s definition of vagueness confuses vagueness with generality. He

discusses vagueness of terms or symbols by using borderline cases where it is

unclear whether the term can be used to describe the case. When discussing

scientific measurement he points out “the indeterminacy which is characteristic

in vagueness is present also in all scientific measurement”. To the fuzzy logic

researcher of today these curves bear a strong resemblance to the membership

functions of (type-1)-fuzzy sets. At the beginning of its brainstorm, the

papers published by Lotfi A. Zadeh was not well received in the West, even in

many cases were bitterly dismissed by the more conservative elements of the

scientific community, as mentioned before. However, over time began to gain

enough supporters, which led to these theories were being extended again and

again, settling firmly among the most innovative scientists, and especially

among the best professionals, more than anywhere else, initially in Japan and

then South Korea, China and India. Europe and the States have been incorporated

into this new math, but more slowly. As a matter picturesque, if you will, but

true, we can tell that the now recognized by many as “the father of Fuzzy

Logic”, Lotfi A. Zadeh, in his time met with executives from IBM, which told

him that his “discovery” had no interest or no utility. Of course, it will be

considered a very clear model of intelligence and vision.

Some Important Land

marks in the Advancement of Fuzzy theory

Year

Event

1971

Zadeh introduces the idea of Type-n fuzzy sets and, therefore,

Type-2 fuzzy sets. 1975. Zadeh presents the definition of Type n Fuzzy Sets.

1976

Grattan-Guinness presents the notion of Set Values Fuzzy Sets

as well as some operations based on previous developments for many-valued algebras.

1983

Atanassov presents the definition of Atanassov Intuitionistic

Fuzzy Sets

1990

Dubois and Prade introduce the definition of Fuzzy Rough Sets

1996

Zhang presents the definition of Bipolar Valued Fuzzy Sets of

Zhang.

2000

Liang and Mendel introduce the idea of Interval Type-2 Fuzzy

Sets.

2000

Lee introduces a new concept with the name of bipolar-valued

Fuzzy Sets.

2002

Smaradache introduces the concept of Nutrosophic sets.

2013

Yager gives the idea of Pythagorean Fuzzy Sets.

2014

Mesiarova-Zemankova et

al. present the concept of m-Polar Valued Fuzzy Sets.

Applications:

The

analysis of system reliability often requires the use of subjective judgements,

uncertain data and approximate system models. By allowing imprecision and

approximate analysis fuzzy logic provides effective tools for characterizing

system reliability. Indeed, the applications of fuzzy logic,

once thought to be an obscure mathematical curiosity, can be found in many

engineering and scientific works.

Fuzzy logic has been used in numerous

applications such as facial pattern recognition, air conditioners, washing

machines, vacuum cleaners, antiskid braking systems, transmission systems,

control of subway systems and unmanned helicopters, knowledge-based systems for

multi-objective optimization of power systems, weather forecasting systems,

models for new product pricing or project risk assessment, medical diagnosis

and treatment plans, and stock trading. Fuzzy logic has been successfully used

in numerous fields such as control systems engineering, image processing, power

engineering, industrial automation, robotics, consumer electronics, and

optimization. This branch of mathematics has instilled new life into scientific

fields that have been dormant for a long time. One of the most famous applications of fuzzy logic is that of the

Sendai Subway system in Sendai, Japan. This control of the Nanboku line,

developed by Hitachi, used a fuzzy controller to run the train all day long.

This made the line one of the smoothest running subway systems in the world and

increased efficiency as well as stopping time. This is also an example of the

earlier acceptance of fuzzy logic in the east since the subway went into

operation in 1988. In “Detection and

elimination of a potential fire in engine and battery compartments of hybrid

electric vehicles” by M. S. Dattathreya et al, the authors present a novel

fuzzy deterministic non-controller type (FDNCT) system and an FDNCT inference

algorithm (FIA). The FDNCT is used in an intelligent system for detecting and

eliminating potential fires in the engine and battery compartments of a hybrid

electric vehicle. They also present the simulation results of the comparison

between the FIA and singleton inference algorithms for detecting potential

fires and determining the actions for eliminating them. In “Comparison of

detection and classification algorithms using boolean and fuzzy techniques” by

R. Dixit and H. Singh, the authors compare various logic analysis methods and

present results for a hypothetical target classification scenario. They show

how preprocessing can reasonably preserve result confidence and compare the

results between Boolean, multi-quantization Boolean, and fuzzy techniques.

In conclusion, a word about the methodology of computing

with words (CWW) which is rooted in the concept of a linguistic variable. CWW

opens the door to construction of mathematical solutions of computational

problems which are stated in natural language. In coming years, CWW is likely

to play an increasingly important role in origination and development of

real-life applications of fuzzy logic.