Reference paper name: Fuzzy sets and their applications to cognitive and decision processes
Edited by: Lotfi A. Zadeh, King-Sun Fu, Kokichi Tanaka, Masamichi Shimura
Published by: Academic Press, 1974
Background: Proceedings of the US-Japan Seminar on Fuzzy sets and their applications, held at the University of California, Berkeley, Ca
July 1-4,1974
ISBN: 0-12-775260-9
1.History of fuzzy logic
Fuzzy systems were first proposed by Lukasiewicz in 1920s. His research led to a formal inexact reasoning technique aptly named possibility theory.
In 1965,Zadeh, professor of UC, Berkeley, author of this paper, extended the work on the possibility theory into a formal system of mathematical logic. This new logic tool for representing and manipulating fuzzy terms was called Fuzzy Logic.
2.Topic area: Calculus of fuzzy restrictions
A fuzzy restriction may be visualized as elastic constraint on the values that may be assigned to a variable. In terms of such restrictions, the meaning of a proposition of the form "x is P," where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x))=P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R.
For example, "Tosi is young." can be expressed as an assignment equation R(Age(Tosi))=young, in which the fuzzy set young is assigned to the restriction on the variable Age(Tosi). Age(Tosi) denote a numerically valued variable which ranged over the interval [0-100]. With this interval regarded as our universe of discourse U, young may be interpreted as the label of a fuzzy subset of U which is characterized by membership function, myoung whose value is between 0 and 1.
.Concentration (Very)
Further reduce the membership values of those elements that have smaller membership values.
mCon(A)(x)=( mA(x))2
.Dilation (More or Less)
Dilates the fuzzy elements by increasing the membership values of those
elements with small membership values more than those elements with
high membership values.
mDil(A)(x)=(mA(x))0.5
.Intersection L is used to represent the "min" operation
mALB(x)=min(mA(x), mB(x))
.Union V is used to represent the "max" operation
mAVB(x)=max(mA(x), mB(x))
.Complement (Not)
m~A(x)=1-mA(x)
We can derive a variety of other fuzzy sets from the above ones like not
very tall, not very tall and not very short.
Approximate reasoning is an important application of the calculus of fuzzy restrictions. That is, a type of reasoning which is neither very exact nor very inexact. Such reasoning plays a basic role in human decision-making because it provides a way of dealing with problems which are too complex for precise solution. However, approximate reasoning is more than a method of last recourse for coping with insurmountable complexities. It is, also, a way of simplifying the performance of tasks in which a high degree of precision is neither needed nor required. Such tasks pervade much of what we do on both conscious and subconscious levels.
The calculus of fuzzy restrictions provides a basis for a systematic approach to approximate reasoning by interpreting such reasoning as the process of approximate solution of a system of relational assignment equations.
The two most popular fuzzy inference techniques used in practice are max-min inference and max-product inference.
3.Paper Value
Calculus of fuzzy restrictions and its application to approximate reasoning
are the basic for the applications in human cognition, communication,
decision-making, and engineering systems analysis.
Approximate reasoning will become an important area of study and
research in artificial intelligence, psychology and related fields.
4.Strength and Weakness
Fuzzy logic provides the means to both represent and reason with common
sense knowledge in a computer. This ability is extremely valuable to the
knowledge engineer responsible for building an expert system who is
confronted with an expert that explains the problem-solving tasks in
common sense terms. Vague terms or rules can be represented and
manipulated numerically to provide results that are consistent with the
expert.
Fuzzy logic has particular value in those control applications where it is
difficult or impossible to develop a traditional control system.
To date, Japan has been the leaders in developing fuzzy logic control
systems for such diverse applications as washing machines, video
camcorders, and railway systems.
The theory of fuzzy sets has developed in a variety of directions, finding
applications in such diverse fields as taxonomy, topology, linguistics,
automata theory, logic, control theory, game theory, information theory,
psychology, pattern recognition, medicine, law, decision analysis, system
theory and information retrieval.
It is a new technology, still under-development.