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;;; *****************************************************************

;;; Answers to Questions about Fuzzy Logic and Fuzzy Expert Systems *

;;; *****************************************************************

;;; Written by Mark Kantrowitz, Erik Horstkotte, and Cliff Joslyn

;;; fuzzy.faq

Contributions and corrections should be sent to the mailing list

mkant+f...@cs.cmu.edu.

Note that the mkant+f...@cs.cmu.edu mailing list is for

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maintainers. It is not the place to ask questions about fuzzy logic

and fuzzy expert systems; use the newsgroup comp.ai.fuzzy for that. If

a question appears frequently in that forum, it will get added to the

FAQ list.

The original version of this FAQ posting was prepared by Erik

Horstkotte of SysSoft <er...@syssoft.com>, with significant

contributions by Cliff Joslyn <jos...@kong.gsfc.nasa.gov>. The FAQ is

maintained by Mark Kantrowitz <mk...@cs.cmu.edu> with advice from Erik

and Cliff. To reach us, send mail to mkant+f...@cs.cmu.edu.

Thanks also go to Michael Arras <ar...@forwiss.uni-erlangen.de> for

running the vote which resulted in the creation of comp.ai.fuzzy,

Yokichi Tanaka <tan...@til.com> for help in putting the FAQ together,

and Walter Hafner <haf...@informatik.tu-muenchen.de>, Satoru Isaka

<is...@oas.omron.com>, Henrik Legind Larsen <h...@ruc.dk>, Tom Parish

<tpa...@tpis.cactus.org>, Liliane Peters <pet...@borneo.gmd.de>, Naji

Rizk <m...@inco.com.lb>, Peter Stegmaier <pe...@ifr.ethz.ch>, Prof.

J.L. Verdegay <jver...@ugr.es>, and Dr. John Yen <y...@cs.tamu.edu> for

contributions to the initial contents of the FAQ.

This FAQ is posted once a month on the 13th of the month. In between

postings, the latest version of this FAQ is available by anonymous ftp

from CMU:

To obtain the files from CMU, connect by anonymous FTP to

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The direct URL for the Fuzzy FAQ is

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If you need to cite the FAQ for some reason, use the following format:

Mark Kantrowitz, Erik Horstkotte, and Cliff Joslyn, "Answers to

Frequently Asked Questions about Fuzzy Logic and Fuzzy Expert Systems",

comp.ai.fuzzy, <month>, <year>,

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mkant+f...@cs.cmu.edu.

*** Table of Contents:

[1] What is the purpose of this newsgroup?

[2] What is fuzzy logic?

[3] Where is fuzzy logic used?

[4] What is a fuzzy expert system?

[5] Where are fuzzy expert systems used?

[6] What is fuzzy control?

[7] What are fuzzy numbers and fuzzy arithmetic?

[8] Isn't "fuzzy logic" an inherent contradiction?

Why would anyone want to fuzzify logic?

[9] How are membership values determined?

[10] What is the relationship between fuzzy truth values and probabilities?

[11] Are there fuzzy state machines?

[12] What is possibility theory?

[13] How can I get a copy of the proceedings for <x>?

[14] Fuzzy BBS Systems, Mail-servers and FTP Repositories

[15] Mailing Lists

[16] Bibliography

[17] Journals and Technical Newsletters

[18] Professional Organizations

[19] Companies Supplying Fuzzy Tools

[20] Fuzzy Researchers

[21] Elkan's "The Paradoxical Success of Fuzzy Logic" paper

[22] Glossary

[24] Where to send calls for papers (cfp) and calls for participation

Search for [#] to get to topic number # quickly. In newsreaders which

support digests (such as rn), [CNTL]-G will page through the answers.

*** Recent changes:

;;; 1.19:

;;; 26-JUN-95 mk Updated listing for American NeuraLogix; new name and

;;; contact information.

;;;

;;; 1.20:

;;; 13-JUL-95 mk Added email address to hyperlogic entry.

;;;

;;; 1.21:

;;; 31-OCT-95 mk Added URL to HyperLogic page.

;;; 15-NOV-95 mk Updated Technical Univ of Vienna Fuzzy mailing list entry.

;;; 20-FEB-96 mk Added entry on LPA's FLINT.

================================================================

Subject: [1] What is the purpose of this newsgroup?

Date: 15-APR-93

The comp.ai.fuzzy newsgroup was created in January 1993, for the purpose

of providing a forum for the discussion of fuzzy logic, fuzzy expert

systems, and related topics.

================================================================

Subject: [2] What is fuzzy logic?

Date: 15-APR-93

Fuzzy logic is a superset of conventional (Boolean) logic that has been

extended to handle the concept of partial truth -- truth values between

"completely true" and "completely false". It was introduced by Dr. Lotfi

Zadeh of UC/Berkeley in the 1960's as a means to model the uncertainty

of natural language. (Note: Lotfi, not Lofti, is the correct spelling

of his name.)

Zadeh says that rather than regarding fuzzy theory as a single theory, we

should regard the process of ``fuzzification'' as a methodology to

generalize ANY specific theory from a crisp (discrete) to a continuous

(fuzzy) form (see "extension principle" in [2]). Thus recently researchers

have also introduced "fuzzy calculus", "fuzzy differential equations",

and so on (see [7]).

Fuzzy Subsets:

Just as there is a strong relationship between Boolean logic and the

concept of a subset, there is a similar strong relationship between fuzzy

logic and fuzzy subset theory.

In classical set theory, a subset U of a set S can be defined as a

mapping from the elements of S to the elements of the set {0, 1},

U: S --> {0, 1}

This mapping may be represented as a set of ordered pairs, with exactly

one ordered pair present for each element of S. The first element of the

ordered pair is an element of the set S, and the second element is an

element of the set {0, 1}. The value zero is used to represent

non-membership, and the value one is used to represent membership. The

truth or falsity of the statement

x is in U

is determined by finding the ordered pair whose first element is x. The

statement is true if the second element of the ordered pair is 1, and the

statement is false if it is 0.

Similarly, a fuzzy subset F of a set S can be defined as a set of ordered

pairs, each with the first element from S, and the second element from

the interval [0,1], with exactly one ordered pair present for each

element of S. This defines a mapping between elements of the set S and

values in the interval [0,1]. The value zero is used to represent

complete non-membership, the value one is used to represent complete

membership, and values in between are used to represent intermediate

DEGREES OF MEMBERSHIP. The set S is referred to as the UNIVERSE OF

DISCOURSE for the fuzzy subset F. Frequently, the mapping is described

as a function, the MEMBERSHIP FUNCTION of F. The degree to which the

statement

x is in F

is true is determined by finding the ordered pair whose first element is

x. The DEGREE OF TRUTH of the statement is the second element of the

ordered pair.

In practice, the terms "membership function" and fuzzy subset get used

interchangeably.

That's a lot of mathematical baggage, so here's an example. Let's

talk about people and "tallness". In this case the set S (the

universe of discourse) is the set of people. Let's define a fuzzy

subset TALL, which will answer the question "to what degree is person

x tall?" Zadeh describes TALL as a LINGUISTIC VARIABLE, which

represents our cognitive category of "tallness". To each person in the

universe of discourse, we have to assign a degree of membership in the

fuzzy subset TALL. The easiest way to do this is with a membership

function based on the person's height.

tall(x) = { 0, if height(x) < 5 ft.,

(height(x)-5ft.)/2ft., if 5 ft. <= height (x) <= 7 ft.,

1, if height(x) > 7 ft. }

A graph of this looks like:

1.0 + +-------------------

| /

| /

0.5 + /

| /

| /

0.0 +-------------+-----+-------------------

| |

5.0 7.0

height, ft. ->

Given this definition, here are some example values:

Person Height degree of tallness

--------------------------------------

Billy 3' 2" 0.00 [I think]

Yoke 5' 5" 0.21

Drew 5' 9" 0.38

Erik 5' 10" 0.42

Mark 6' 1" 0.54

Kareem 7' 2" 1.00 [depends on who you ask]

Expressions like "A is X" can be interpreted as degrees of truth,

e.g., "Drew is TALL" = 0.38.

Note: Membership functions used in most applications almost never have as

simple a shape as tall(x). At minimum, they tend to be triangles pointing

up, and they can be much more complex than that. Also, the discussion

characterizes membership functions as if they always are based on a

single criterion, but this isn't always the case, although it is quite

common. One could, for example, want to have the membership function for

TALL depend on both a person's height and their age (he's tall for his

age). This is perfectly legitimate, and occasionally used in practice.

It's referred to as a two-dimensional membership function, or a "fuzzy

relation". It's also possible to have even more criteria, or to have the

membership function depend on elements from two completely different

universes of discourse.

Logic Operations:

Now that we know what a statement like "X is LOW" means in fuzzy logic,

how do we interpret a statement like

X is LOW and Y is HIGH or (not Z is MEDIUM)

The standard definitions in fuzzy logic are:

truth (not x) = 1.0 - truth (x)

truth (x and y) = minimum (truth(x), truth(y))

truth (x or y) = maximum (truth(x), truth(y))

Some researchers in fuzzy logic have explored the use of other

interpretations of the AND and OR operations, but the definition for the

NOT operation seems to be safe.

Note that if you plug just the values zero and one into these

definitions, you get the same truth tables as you would expect from

conventional Boolean logic. This is known as the EXTENSION PRINCIPLE,

which states that the classical results of Boolean logic are recovered

from fuzzy logic operations when all fuzzy membership grades are

restricted to the traditional set {0, 1}. This effectively establishes

fuzzy subsets and logic as a true generalization of classical set theory

and logic. In fact, by this reasoning all crisp (traditional) subsets ARE

fuzzy subsets of this very special type; and there is no conflict between

fuzzy and crisp methods.

Some examples -- assume the same definition of TALL as above, and in addition,

assume that we have a fuzzy subset OLD defined by the membership function:

old (x) = { 0, if age(x) < 18 yr.

(age(x)-18 yr.)/42 yr., if 18 yr. <= age(x) <= 60 yr.

1, if age(x) > 60 yr. }

And for compactness, let

a = X is TALL and X is OLD

b = X is TALL or X is OLD

c = not (X is TALL)

Then we can compute the following values.

height age X is TALL X is OLD a b c

------------------------------------------------------------------------

3' 2" 65 0.00 1.00 0.00 1.00 1.00

5' 5" 30 0.21 0.29 0.21 0.29 0.79

5' 9" 27 0.38 0.21 0.21 0.38 0.62

5' 10" 32 0.42 0.33 0.33 0.42 0.58

6' 1" 31 0.54 0.31 0.31 0.54 0.46

7' 2" 45 1.00 0.64 0.64 1.00 0.00

3' 4" 4 0.00 0.00 0.00 0.00 1.00

For those of you who only grok the metric system, here's a dandy

little conversion table:

Feet+Inches = Meters

--------------------

3' 2" 0.9652

3' 4" 1.0160

5' 5" 1.6510

5' 9" 1.7526

5' 10" 1.7780

6' 1" 1.8542

7' 2" 2.1844

An excellent introductory article is:

Bezdek, James C, "Fuzzy Models --- What Are They, and Why?", IEEE

Transactions on Fuzzy Systems, 1:1, pp. 1-6, 1993.

For more information on fuzzy logic operators, see:

Bandler, W., and Kohout, L.J., "Fuzzy Power Sets and Fuzzy Implication

Operators", Fuzzy Sets and Systems 4:13-30, 1980.

Dubois, Didier, and Prade, H., "A Class of Fuzzy Measures Based on

Triangle Inequalities", Int. J. Gen. Sys. 8.

The original papers on fuzzy logic include:

Zadeh, Lotfi, "Fuzzy Sets," Information and Control 8:338-353, 1965.

Zadeh, Lotfi, "Outline of a New Approach to the Analysis of Complex

Systems", IEEE Trans. on Sys., Man and Cyb. 3, 1973.

Zadeh, Lotfi, "The Calculus of Fuzzy Restrictions", in Fuzzy Sets and

Applications to Cognitive and Decision Making Processes, edited

by L. A. Zadeh et. al., Academic Press, New York, 1975, pages 1-39.

================================================================

Subject: [3] Where is fuzzy logic used?

Date: 15-APR-93

Fuzzy logic is used directly in very few applications. The Sony PalmTop

apparently uses a fuzzy logic decision tree algorithm to perform

handwritten (well, computer lightpen) Kanji character recognition.

Most applications of fuzzy logic use it as the underlying logic system

for fuzzy expert systems (see [4]).

================================================================

Subject: [4] What is a fuzzy expert system?

Date: 21-APR-93

A fuzzy expert system is an expert system that uses a collection of

fuzzy membership functions and rules, instead of Boolean logic, to

reason about data. The rules in a fuzzy expert system are usually of a

form similar to the following:

if x is low and y is high then z = medium

where x and y are input variables (names for know data values), z is an

output variable (a name for a data value to be computed), low is a

membership function (fuzzy subset) defined on x, high is a membership

function defined on y, and medium is a membership function defined on z.

The antecedent (the rule's premise) describes to what degree the rule

applies, while the conclusion (the rule's consequent) assigns a

membership function to each of one or more output variables. Most tools

for working with fuzzy expert systems allow more than one conclusion per

rule. The set of rules in a fuzzy expert system is known as the rulebase

or knowledge base.

The general inference process proceeds in three (or four) steps.

1. Under FUZZIFICATION, the membership functions defined on the

input variables are applied to their actual values, to determine the

degree of truth for each rule premise.

2. Under INFERENCE, the truth value for the premise of each rule is

computed, and applied to the conclusion part of each rule. This results

in one fuzzy subset to be assigned to each output variable for each

rule. Usually only MIN or PRODUCT are used as inference rules. In MIN

inferencing, the output membership function is clipped off at a height

corresponding to the rule premise's computed degree of truth (fuzzy

logic AND). In PRODUCT inferencing, the output membership function is

scaled by the rule premise's computed degree of truth.

3. Under COMPOSITION, all of the fuzzy subsets assigned to each output

variable are combined together to form a single fuzzy subset

for each output variable. Again, usually MAX or SUM are used. In MAX

composition, the combined output fuzzy subset is constructed by taking

the pointwise maximum over all of the fuzzy subsets assigned tovariable

by the inference rule (fuzzy logic OR). In SUM composition, the

combined output fuzzy subset is constructed by taking the pointwise sum

over all of the fuzzy subsets assigned to the output variable by the

inference rule.

4. Finally is the (optional) DEFUZZIFICATION, which is used when it is

useful to convert the fuzzy output set to a crisp number. There are

more defuzzification methods than you can shake a stick at (at least

30). Two of the more common techniques are the CENTROID and MAXIMUM

methods. In the CENTROID method, the crisp value of the output variable

is computed by finding the variable value of the center of gravity of

the membership function for the fuzzy value. In the MAXIMUM method, one

of the variable values at which the fuzzy subset has its maximum truth

value is chosen as the crisp value for the output variable.

Extended Example:

Assume that the variables x, y, and z all take on values in the interval

[0,10], and that the following membership functions and rules are defined:

low(t) = 1 - ( t / 10 )

high(t) = t / 10

rule 1: if x is low and y is low then z is high

rule 2: if x is low and y is high then z is low

rule 3: if x is high and y is low then z is low

rule 4: if x is high and y is high then z is high

Notice that instead of assigning a single value to the output variable z, each

rule assigns an entire fuzzy subset (low or high).

Notes:

1. In this example, low(t)+high(t)=1.0 for all t. This is not required, but

it is fairly common.

2. The value of t at which low(t) is maximum is the same as the value of t at

which high(t) is minimum, and vice-versa. This is also not required, but

fairly common.

3. The same membership functions are used for all variables. This isn't

required, and is also *not* common.

In the fuzzification subprocess, the membership functions defined on the

input variables are applied to their actual values, to determine the

degree of truth for each rule premise. The degree of truth for a rule's

premise is sometimes referred to as its ALPHA. If a rule's premise has a

nonzero degree of truth (if the rule applies at all...) then the rule is

said to FIRE. For example,

x y low(x) high(x) low(y) high(y) alpha1 alpha2 alpha3 alpha4

------------------------------------------------------------------------------

0.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0

0.0 3.2 1.0 0.0 0.68 0.32 0.68 0.32 0.0 0.0

0.0 6.1 1.0 0.0 0.39 0.61 0.39 0.61 0.0 0.0

0.0 10.0 1.0 0.0 0.0 1.0 0.0 1.0 0.0 0.0

3.2 0.0 0.68 0.32 1.0 0.0 0.68 0.0 0.32 0.0

6.1 0.0 0.39 0.61 1.0 0.0 0.39 0.0 0.61 0.0

10.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 1.0 0.0

3.2 3.1 0.68 0.32 0.69 0.31 0.68 0.31 0.32 0.31

3.2 3.3 0.68 0.32 0.67 0.33 0.67 0.33 0.32 0.32

10.0 10.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 1.0

In the inference subprocess, the truth value for the premise of each rule is

computed, and applied to the conclusion part of each rule. This results in

one fuzzy subset to be assigned to each output variable for each rule.

MIN and PRODUCT are two INFERENCE METHODS or INFERENCE RULES. In MIN

inferencing, the output membership function is clipped off at a height

corresponding to the rule premise's computed degree of truth. This

corresponds to the traditional interpretation of the fuzzy logic AND

operation. In PRODUCT inferencing, the output membership function is

scaled by the rule premise's computed degree of truth.

For example, let's look at rule 1 for x = 0.0 and y = 3.2. As shown in the

table above, the premise degree of truth works out to 0.68. For this rule,

MIN inferencing will assign z the fuzzy subset defined by the membership

function:

rule1(z) = { z / 10, if z <= 6.8

0.68, if z >= 6.8 }

For the same conditions, PRODUCT inferencing will assign z the fuzzy subset

defined by the membership function:

rule1(z) = 0.68 * high(z)

= 0.068 * z

Note: The terminology used here is slightly nonstandard. In most texts,

the term "inference method" is used to mean the combination of the things

referred to separately here as "inference" and "composition." Thus

you'll see such terms as "MAX-MIN inference" and "SUM-PRODUCT inference"

in the literature. They are the combination of MAX composition and MIN

inference, or SUM composition and PRODUCT inference, respectively.

You'll also see the reverse terms "MIN-MAX" and "PRODUCT-SUM" -- these

mean the same things as the reverse order. It seems clearer to describe

the two processes separately.

In the composition subprocess, all of the fuzzy subsets assigned to each

output variable are combined together to form a single fuzzy subset for each

output variable.

MAX composition and SUM composition are two COMPOSITION RULES. In MAX

composition, the combined output fuzzy subset is constructed by taking

the pointwise maximum over all of the fuzzy subsets assigned to the

output variable by the inference rule. In SUM composition, the combined

output fuzzy subset is constructed by taking the pointwise sum over all

of the fuzzy subsets assigned to the output variable by the inference

rule. Note that this can result in truth values greater than one! For

this reason, SUM composition is only used when it will be followed by a

defuzzification method, such as the CENTROID method, that doesn't have a

problem with this odd case. Otherwise SUM composition can be combined

with normalization and is therefore a general purpose method again.

For example, assume x = 0.0 and y = 3.2. MIN inferencing would assign the

following four fuzzy subsets to z:

rule1(z) = { z / 10, if z <= 6.8

0.68, if z >= 6.8 }

rule2(z) = { 0.32, if z <= 6.8

1 - z / 10, if z >= 6.8 }

rule3(z) = 0.0

rule4(z) = 0.0

MAX composition would result in the fuzzy subset:

fuzzy(z) = { 0.32, if z <= 3.2

z / 10, if 3.2 <= z <= 6.8

0.68, if z >= 6.8 }

PRODUCT inferencing would assign the following four fuzzy subsets to z:

rule1(z) = 0.068 * z

rule2(z) = 0.32 - 0.032 * z

rule3(z) = 0.0

rule4(z) = 0.0

SUM composition would result in the fuzzy subset:

fuzzy(z) = 0.32 + 0.036 * z

Sometimes it is useful to just examine the fuzzy subsets that are the

result of the composition process, but more often, this FUZZY VALUE needs

to be converted to a single number -- a CRISP VALUE. This is what the

defuzzification subprocess does.

There are more defuzzification methods than you can shake a stick at. A

couple of years ago, Mizumoto did a short paper that compared about ten

defuzzification methods. Two of the more common techniques are the

CENTROID and MAXIMUM methods. In the CENTROID method, the crisp value of

the output variable is computed by finding the variable value of the

center of gravity of the membership function for the fuzzy value. In the

MAXIMUM method, one of the variable values at which the fuzzy subset has

its maximum truth value is chosen as the crisp value for the output

variable. There are several variations of the MAXIMUM method that differ

only in what they do when there is more than one variable value at which

this maximum truth value occurs. One of these, the AVERAGE-OF-MAXIMA

method, returns the average of the variable values at which the maximum

truth value occurs.

For example, go back to our previous examples. Using MAX-MIN inferencing

and AVERAGE-OF-MAXIMA defuzzification results in a crisp value of 8.4 for

z. Using PRODUCT-SUM inferencing and CENTROID defuzzification results in

a crisp value of 5.6 for z, as follows.

Earlier on in the FAQ, we state that all variables (including z) take on

values in the range [0, 10]. To compute the centroid of the function f(x),

you divide the moment of the function by the area of the function. To compute

the moment of f(x), you compute the integral of x*f(x) dx, and to compute the

area of f(x), you compute the integral of f(x) dx. In this case, we would

compute the area as integral from 0 to 10 of (0.32+0.036*z) dz, which is

(0.32 * 10 + 0.018*100) =

(3.2 + 1.8) =

5.0

and the moment as the integral from 0 to 10 of (0.32*z+0.036*z*z) dz, which is

(0.16 * 10 * 10 + 0.012 * 10 * 10 * 10) =

(16 + 12) =

28

Finally, the centroid is 28/5 or 5.6.

Note: Sometimes the composition and defuzzification processes are

combined, taking advantage of mathematical relationships that simplify

the process of computing the final output variable values.

The Mizumoto reference is probably "Improvement Methods of Fuzzy

Controls", in Proceedings of the 3rd IFSA Congress, pages 60-62, 1989.

================================================================

Subject: [5] Where are fuzzy expert systems used?

Date: 15-APR-93

To date, fuzzy expert systems are the most common use of fuzzy logic. They

are used in several wide-ranging fields, including:

o Linear and Nonlinear Control

o Pattern Recognition

o Financial Systems

o Operation Research

o Data Analysis

================================================================

Subject: [6] What is fuzzy control?

Date: 17-MAR-95

The purpose of control is to influence the behavior of a system by

changing an input or inputs to that system according to a rule or

set of rules that model how the system operates. The system being

controlled may be mechanical, electrical, chemical or any combination

of these.

Classic control theory uses a mathematical model to define a relationship

that transforms the desired state (requested) and observed state (measured)

of the system into an input or inputs that will alter the future state of

that system.

reference----->0------->( SYSTEM ) -------+----------> output

^ |

| |

+--------( MODEL )<--------+feedback

The most common example of a control model is the PID (proportional-integral-

derivative) controller. This takes the output of the system and compares

it with the desired state of the system. It adjusts the input value based

on the difference between the two values according to the following

equation.

output = A.e + B.INT(e)dt + C.de/dt

Where, A, B and C are constants, e is the error term, INT(e)dt is the

integral of the error over time and de/dt is the change in the error term.

The major drawback of this system is that it usually assumes that the system

being modelled in linear or at least behaves in some fashion that is a

monotonic function. As the complexity of the system increases it becomes

more difficult to formulate that mathematical model.

Fuzzy control replaces, in the picture above, the role of the mathematical

model and replaces it with another that is build from a number of smaller

rules that in general only describe a small section of the whole system. The

process of inference binding them together to produce the desired outputs.

That is, a fuzzy model has replaced the mathematical one. The inputs and

outputs of the system have remained unchanged.

The Sendai subway is the prototypical example application of fuzzy control.

References:

Yager, R.R., and Zadeh, L. A., "An Introduction to Fuzzy Logic

Applications in Intelligent Systems", Kluwer Academic Publishers, 1991.

Dimiter Driankov, Hans Hellendoorn, and Michael Reinfrank,

"An Introduction to Fuzzy Control", Springer-Verlag, New York, 1993.

316 pages, ISBN 0-387-56362-8. [Discusses fuzzy control from a

theoretical point of view as a form of nonlinear control.]

C.J. Harris, C.G. Moore, M. Brown, "Intelligent Control, Aspects of

Fuzzy Logic and Neural Nets", World Scientific. ISBN 981-02-1042-6.

T. Terano, K. Asai, M. Sugeno, editors, "Applied Fuzzy Systems",

translated by C. Ascchmann, AP Professional. ISBN 0-12-685242-1.

================================================================

Subject: [7] What are fuzzy numbers and fuzzy arithmetic?

Date: 15-APR-93

Fuzzy numbers are fuzzy subsets of the real line. They have a peak or

plateau with membership grade 1, over which the members of the

universe are completely in the set. The membership function is

increasing towards the peak and decreasing away from it.

Fuzzy numbers are used very widely in fuzzy control applications. A typical

case is the triangular fuzzy number

1.0 + +

| / \

| / \

0.5 + / \

| / \

| / \

0.0 +-------------+-----+-----+--------------

| | |

5.0 7.0 9.0

which is one form of the fuzzy number 7. Slope and trapezoidal functions

are also used, as are exponential curves similar to Gaussian probability

densities.

For more information, see:

Dubois, Didier, and Prade, Henri, "Fuzzy Numbers: An Overview", in

Analysis of Fuzzy Information 1:3-39, CRC Press, Boca Raton, 1987.

Dubois, Didier, and Prade, Henri, "Mean Value of a Fuzzy Number",

Fuzzy Sets and Systems 24(3):279-300, 1987.

Kaufmann, A., and Gupta, M.M., "Introduction to Fuzzy Arithmetic",

Reinhold, New York, 1985.

================================================================

Subject: [8] Isn't "fuzzy logic" an inherent contradiction?

Why would anyone want to fuzzify logic?

Date: 15-APR-93

Fuzzy sets and logic must be viewed as a formal mathematical theory for

the representation of uncertainty. Uncertainty is crucial for the

management of real systems: if you had to park your car PRECISELY in one

place, it would not be possible. Instead, you work within, say, 10 cm

tolerances. The presence of uncertainty is the price you pay for handling

a complex system.

Nevertheless, fuzzy logic is a mathematical formalism, and a membership

grade is a precise number. What's crucial to realize is that fuzzy logic

is a logic OF fuzziness, not a logic which is ITSELF fuzzy. But that's

OK: just as the laws of probability are not random, so the laws of

fuzziness are not vague.

================================================================

Subject: [9] How are membership values determined?

Date: 15-APR-93

Determination methods break down broadly into the following categories:

1. Subjective evaluation and elicitation

As fuzzy sets are usually intended to model people's cognitive states,

they can be determined from either simple or sophisticated elicitation

procedures. At they very least, subjects simply draw or otherwise specify

different membership curves appropriate to a given problem. These

subjects are typcially experts in the problem area. Or they are given a

more constrained set of possible curves from which they choose. Under

more complex methods, users can be tested using psychological methods.

2. Ad-hoc forms

While there is a vast (hugely infinite) array of possible membership

function forms, most actual fuzzy control operations draw from a very

small set of different curves, for example simple forms of fuzzy numbers

(see [7]). This simplifies the problem, for example to choosing just the

central value and the slope on either side.

3. Converted frequencies or probabilities

Sometimes information taken in the form of frequency histograms or other

probability curves are used as the basis to construct a membership

function. There are a variety of possible conversion methods, each with

its own mathematical and methodological strengths and weaknesses.

However, it should always be remembered that membership functions are NOT

(necessarily) probabilities. See [10] for more information.

4. Physical measurement

Many applications of fuzzy logic use physical measurement, but almost

none measure the membership grade directly. Instead, a membership

function is provided by another method, and then the individual

membership grades of data are calculated from it (see FUZZIFICATION in [4]).

5. Learning and adaptation

For more information, see:

Roberts, D.W., "Analysis of Forest Succession with Fuzzy Graph Theory",

Ecological Modeling, 45:261-274, 1989.

Turksen, I.B., "Measurement of Fuzziness: Interpretiation of the Axioms

of Measure", in Proceeding of the Conference on Fuzzy Information and

Knowledge Representation for Decision Analysis, pages 97-102, IFAC,

Oxford, 1984.

================================================================

Subject: [10] What is the relationship between fuzzy truth values and

probabilities?

Date: 21-NOV-94

This question has to be answered in two ways: first, how does fuzzy

theory differ from probability theory mathematically, and second, how

does it differ in interpretation and application.

At the mathematical level, fuzzy values are commonly misunderstood to be

probabilities, or fuzzy logic is interpreted as some new way of handling

probabilities. But this is not the case. A minimum requirement of

probabilities is ADDITIVITY, that is that they must add together to one, or

the integral of their density curves must be one.

But this does not hold in general with membership grades. And while

membership grades can be determined with probability densities in mind (see

[11]), there are other methods as well which have nothing to do with

frequencies or probabilities.

Because of this, fuzzy researchers have gone to great pains to distance

themselves from probability. But in so doing, many of them have lost track

of another point, which is that the converse DOES hold: all probability

distributions are fuzzy sets! As fuzzy sets and logic generalize Boolean

sets and logic, they also generalize probability.

In fact, from a mathematical perspective, fuzzy sets and probability exist

as parts of a greater Generalized Information Theory which includes many

formalisms for representing uncertainty (including random sets,

Demster-Shafer evidence theory, probability intervals, possibility theory,

general fuzzy measures, interval analysis, etc.). Furthermore, one can

also talk about random fuzzy events and fuzzy random events. This whole

issue is beyond the scope of this FAQ, so please refer to the following

articles, or the textbook by Klir and Folger (see [16]).

Semantically, the distinction between fuzzy logic and probability theory

has to do with the difference between the notions of probability and a

degree of membership. Probability statements are about the likelihoods of

outcomes: an event either occurs or does not, and you can bet on it. But

with fuzziness, one cannot say unequivocally whether an event occured or

not, and instead you are trying to model the EXTENT to which an event

occured. This issue is treated well in the swamp water example used by

James Bezdek of the University of West Florida (Bezdek, James C, "Fuzzy

Models --- What Are They, and Why?", IEEE Transactions on Fuzzy Systems,

1:1, pp. 1-6).

Delgado, M., and Moral, S., "On the Concept of Possibility-Probability

Consistency", Fuzzy Sets and Systems 21:311-318, 1987.

Dempster, A.P., "Upper and Lower Probabilities Induced by a Multivalued

Mapping", Annals of Math. Stat. 38:325-339, 1967.

Henkind, Steven J., and Harrison, Malcolm C., "Analysis of Four

Uncertainty Calculi", IEEE Trans. Man Sys. Cyb. 18(5)700-714, 1988.

Kamp`e de, F'eriet J., "Interpretation of Membership Functions of Fuzzy

Sets in Terms of Plausibility and Belief", in Fuzzy Information and

Decision Process, M.M. Gupta and E. Sanchez (editors), pages 93-98,

North-Holland, Amsterdam, 1982.

Klir, George, "Is There More to Uncertainty than Some Probability

Theorists Would Have Us Believe?", Int. J. Gen. Sys. 15(4):347-378, 1989.

Klir, George, "Generalized Information Theory", Fuzzy Sets and Systems

40:127-142, 1991.

Klir, George, "Probabilistic vs. Possibilistic Conceptualization of

Uncertainty", in Analysis and Management of Uncertainty, B.M. Ayyub et.

al. (editors), pages 13-25, Elsevier, 1992.

Klir, George, and Parviz, Behvad, "Probability-Possibility

Transformations: A Comparison", Int. J. Gen. Sys. 21(1):291-310, 1992.

Kosko, B., "Fuzziness vs. Probability", Int. J. Gen. Sys.

17(2-3):211-240, 1990.

Puri, M.L., and Ralescu, D.A., "Fuzzy Random Variables", J. Math.

Analysis and Applications, 114:409-422, 1986.

Shafer, Glen, "A Mathematical Theory of Evidence", Princeton University,

Princeton, 1976.

================================================================

Subject: [11] Are there fuzzy state machines?

Date: 15-APR-93

Yes. FSMs are obtained by assigning membership grades as weights to the

states of a machine, weights on transitions between states, and then a

composition rule such as MAX/MIN or PLUS/TIMES (see [4]) to calculate new

grades of future states. Refer to the following article, or to Section

III of the Dubois and Prade's 1980 textbook (see [16]).

Gaines, Brian R., and Kohout, Ladislav J., "Logic of Automata",

Int. J. Gen. Sys. 2(4):191-208, 1976.

================================================================

Subject: [12] What is possibility theory?

Date: 15-APR-93

Possibility theory is a new form of information theory which is related

to but independent of both fuzzy sets and probability theory.

Technically, a possibility distribution is a normal fuzzy set (at least

one membership grade equals 1). For example, all fuzzy numbers are

possibility distributions. However, possibility theory can also be

derived without reference to fuzzy sets.

The rules of possibility theory are similar to probability theory, but

use either MAX/MIN or MAX/TIMES calculus, rather than the PLUS/TIMES

calculus of probability theory. Also, possibilistic NONSPECIFICITY is

available as a measure of information similar to the stochastic

ENTROPY.

Possibility theory has a methodological advantage over probability theory

as a representation of nondeterminism in systems, because the PLUS/TIMES

calculus does not validly generalize nondeterministic processes, while

MAX/MIN and MAX/TIMES do.

For further information, see:

Dubois, Didier, and Prade, Henri, "Possibility Theory", Plenum Press,

New York, 1988.

Joslyn, Cliff, "Possibilistic Measurement and Set Statistics",

in Proceedings of the 1992 NAFIPS Conference 2:458-467, NASA, 1992.

Joslyn, Cliff, "Possibilistic Semantics and Measurement Methods in

Complex Systems", in Proceedings of the 2nd International Symposium on

Uncertainty Modeling and Analysis, Bilal Ayyub (editor), IEEE Computer

Society 1993.

Wang, Zhenyuan, and Klir, George J., "Fuzzy Measure Theory", Plenum

Press, New York, 1991.

Zadeh, Lotfi, "Fuzzy Sets as the Basis for a Theory of Possibility",

Fuzzy Sets and Systems 1:3-28, 1978.

================================================================

Subject: [13] How can I get a copy of the proceedings for <x>?

Date: 15-APR-93

This is rough sometimes. The first thing to do, of course, is to contact

the organization that ran the conference or workshop you are interested in.

If they can't help you, the best idea mentioned so far is to contact the

Institute for Scientific Information, Inc. (ISI), and check with their

Index to Scientific and Technical Proceedings (ISTP volumes).

Institute for Scientific Information, Inc.

3501 Market Street

Philadelphia, PA 19104, USA

Phone: +1.215.386.0100

Fax: +1.215.386.6362

Cable: SCINFO

Telex: 84-5305

================================================================

Subject: [14] Fuzzy BBS Systems, Mail-servers and FTP Repositories

Date: 24-AUG-93

Aptronix FuzzyNET BBS and Email Server:

408-261-1883, 1200-9600 N/8/1

This BBS contains a range of fuzzy-related material, including:

o Application notes.

o Product brochures.

o Technical information.

o Archived articles from the USENET newsgroup comp.ai.fuzzy.

o Text versions of "The Huntington Technical Brief" by Dr. Brubaker.

[The technical brief is no longer being updated, as Dr. Brubaker

now charges for subscriptions. See [17] for details.]

The Aptronix FuzzyNET Email Server allows anyone with access to Internet

email access to all of the files on the FuzzyNET BBS.

To receive instructions on how to access the server, send the following

message to fuzz...@aptronix.com:

begin

help

end

If you don't receive a response within a day or two, or need help, contact

Scott Irwin <ir...@aptronix.com> for assistance.

Electronic Design News (EDN) BBS:

617-558-4241, 1200-9600 N/8/1

Motorola FREEBBS:

512-891-3733, 1200-9600 E/7/1

Ostfold Regional College Fuzzy Logic Anonymous FTP Repository:

ftp.dhhalden.no:/pub/Fuzzy/ is a recently-started ftp site for

fuzzy-related material, operated by Ostfold Regional College in

Norway. Currently has files from the Togai InfraLogic Fuzzy Logic

Email Server, Tim Butler's Fuzzy Logic Anonymous FTP Repository, some

demo programs and source code, and lists of upcoming conferences,

articles, and literature about fuzzy logic. Material to be included

in the archive (e.g., papers and code) may be placed in the incoming/

directory. Send email to Randi Weberg <ran...@dhhalden.no>.

Tim Butler's Fuzzy Logic Anonymous FTP Repository & Email Server:

ntia.its.bldrdoc.gov:/pub/fuzzy contains information concerning fuzzy

logic, including bibliographies (bib/), product descriptions and demo

versions (com/), machine readable published papers (lit/), miscellaneous

information, documents and reports (txt/), and programs code and compilers

(prog/). You may download new items into the new/ subdirectory, or send

them by email to fu...@its.bldrdoc.gov. If you deposit anything in new/,

please inform fu...@its.bldrdoc.gov. The repository is maintained by

Timothy Butler, t...@its.bldrdoc.gov.

The Fuzzy Logic Repository is also accessible through a mail server,

rna...@its.bldrdoc.gov. For help on using the server, send mail to the

server with the following line in the body of the message:

@@ help

Togai InfraLogic Fuzzy Logic Email Server:

The Togai InfraLogic Fuzzy Logic Email Server allows anyone with access

to Internet email access to:

o PostScript copies of TIL's company newsletter, The Fuzzy Source.

o ASCII files for selected newsletter articles.

o Archived articles from the USENET newsgroup comp.ai.fuzzy.

o Fuzzy logic demonstration programs.

o Demonstration versions of TIL products.

o Conference announcements.

o User-contributed files.

To receive instructions on how to access the server, send the following

message, with no subject, to fuzzy-...@til.com.

help

If you don't receive a response within a day or two, contact either

er...@til.com or tan...@til.com for assistance.

Most of the contents of TIL's email server are mirrored by Tim Butler's

Fuzzy Logic Anonymous FTP Repository and the Ostfold Regional College

Fuzzy Logic Anonymous FTP Repository in Norway.

The Turning Point BBS:

512-219-7828/7848, DS/HST 1200-19,200 N/8/1

Fuzzy logic and neural network related files.

Miscellaneous Fuzzy Logic Files:

The "General Purpose Fuzzy Reasoning Library" is available by

anonymous FTP from utsun.s.u-tokyo.ac.jp:/fj/fj.sources/v25/2577.Z

[133.11.11.11]. This yields the "General-Purpose Fuzzy Inference

Library Ver. 3.0 (1/1)". The program is in C, with English comments,

but the documentation is in Japanese. Some English documentation has

been written by John Nagle, <na...@shasta.stanford.edu>.

CNCL is a C++ class library provides classes for simulation, fuzzy

logic, DEC's EZD, and UNIX system calls. It is available from

ftp.dfv.rwth-aachen.de:/pub/CNCL [137.226.4.111]. Contact Martin

Junius <m...@dfv.rwth-aachen.de> for more information.

A demo version of Aptronix's FIDE 2.0 is available by anonymous ftp

from ftp.cs.cmu.edu:/user/ai/areas/fuzzy/code/fide/. FIDE is a

PC-based fuzzy logic design tool. It provides tools for the

development, debugging, and simulation of fuzzy applications.

For more information, contact in...@aptronix.com.

FuzzyCLIPS 6.02a is a version of the CLIPS rule-based expert system

shell with extensions for representing and manipulating fuzzy facts

and rules. In addition to the CLIPS functionality, FuzzyCLIPS can deal

with exact, fuzzy (or inexact), and combined reasoning, allowing fuzzy

and normal terms to be freely mixed in the rules and facts of an

expert system. The system uses two basic inexact concepts, fuzziness

and uncertainty. Versions are available for UNIX systems, Macintosh

systems and PC systems. There is no cost for the software, but please

read the terms for use in the FuzzyCLIPS documentation. FuzzyCLIPS is

available via WWW (World Wide Web). It can be accessed indirectly

through the Knowledge Systems Lab Server using the URL

http://ai.iit.nrc.ca/home_page.html

or more directly by using the URL

http://ai.iit.nrc.ca/fuzzy/fuzzy.html

or by anonymous ftp from

ai.iit.nrc.ca:/pub/fzclips/

For more information about FuzzyCLIPS send mail to fzc...@ai.iit.nrc.ca.

FuNeGen 1.0 is a fuzzy neural system capable of generating fuzzy

classification systems (as C-code) from sample data.

FuNeGen 1.0 and the papers/reports describing the application and the

theoretical background can be obtained by anonymous ftp from

obelix.microelectronic.e-technik.th-darmstadt.de:/pub/neurofuzzy/

NEFCON-I (NEural Fuzzy CONtroller) is an X11 simulation environment

based on Interviews designed to build and test neural fuzzy

controllers. NEFCON-I is able to learn fuzzy sets and fuzzy rules by

using a kind of reinforcement learning that is driven by a fuzzy error

measure. To do this NEFCON-I communicates with another process, that

implements a simulation of a dynamical process. NEFCON-I can optimize

the fuzzy sets of the antecedents and the conclusions of a given rule

base, and it can also create a rulebase from scratch. NEFCON-I is

available by anonymous ftp from

ibr.cs.tu-bs.de:/pub/local/nefcon/ [134.169.34.15]

as the file nefcon_1.0.tar.gz. If you are using NEFCON-I, please

send an email message to the author, Detlef Nauck <na...@ibr.cs.tu-bs.de>.

The Fuzzy Arithmetic Library is a very simple C++ implementation of a

fuzzy number representation using confidence intervals, together with

the basic arithmetic operators and trigonometrical functions. It is

available by anonymous FTP from

mathct.dipmat.unict.it:fuzzy [151.97.252.1]

[Note the system is a VAX running VMS.] For more information, write to

Salvatore Deodato <deo...@dipmat.unict.it>.

================================================================

Subject: [15] Mailing Lists

Date: 15-APR-93

The Fuzzy-Mail and NAFIPS-L mailing lists are now bidirectionally

gatewayed to the comp.ai.fuzzy newsgroup.

NAFIPS Fuzzy Logic Mailing List:

This is a mailing list for the discussion of fuzzy logic, NAFIPS and

related topics, located at the Georgia State University. The last time

that this FAQ was updated, there were about 225 subscribers, located

primarily in North America, as one might expect. Postings to the mailing

list are automatically archived.

The mailing list server itself is like most of those in use on the

Internet. If you're already familiar with Internet mailing lists, the

only thing you'll need to know is that the name of the server is

and the name of the mailing list itself is

If you're not familiar with this type of mailing list server, the

easiest way to get started is to send the following message to

list...@listproc.gsu.edu:

help

You will receive a brief set of instructions by email within

a short time.

Once you have subscribed, you will begin receiving a copy of each message

that is sent by anyone to nafi...@listproc.gsu.edu, and any message that

you send to that address will be sent to all of the other subscribers.

Technical University of Vienna Fuzzy Logic Mailing List:

This is a mailing list for the discussion of fuzzy logic and related

topics, located at the Technical University of Vienna in Austria. The

last time this FAQ was updated, there were about 980 subscribers.

The list is slightly moderated (only irrelevant mails are rejected)

and is two-way gatewayed to the aforementioned NAFIPS-L list and to

the comp.ai.fuzzy internet newsgroup. Messages should therefore be

sent only to one of the three media, although some mechanism for

mail-loop avoidance and duplicate-message avoidance is activated.

In addition to the mailing list itself, the list server gives

access to some files, including archives and the "Who is Who in Fuzzy

Logic" database that is currently under construction by Robert Fuller

<rfu...@finabo.abo.fi>. There is also a WWW interface to the list

at http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html as well

as a ftp://mira.dbai.tuwien.ac.at/pub/mlowner site to access such

files as the whoiswhoinfuzzy file mentioned above.

Like many mailing lists, this one uses Anastasios Kotsikonas's LISTPROC

system. If you've used this kind of server before, the only thing you'll

need to know is that the name of the server is

list...@dbai.tuwien.ac.at

and the name of the mailing list is

fuzzy...@dbai.tuwien.ac.at

If you're not familiar with this type of mailing list server, the easiest

way to get started is to send the following message to

list...@dbai.tuwien.ac.at:

get fuzzy-mail info

You will receive a brief set of instructions by email within a short time.

Once you have subscribed, you will begin receiving a copy of each message

that is sent by anyone to fuzzy...@dbai.tuwien.ac.at, and any

message that you send to that address will be sent to all of the other

subscribers.

Fuzzy Logic in Japan:

There are two mailing lists for fuzzy logic in Japan. Both forward

many articles from the international mailing lists, but the other

direction is not automatic.

Asian Fuzzy Mailing System (AFMS):

afu...@ea5.yz.yamagata-u.ac.jp

To subscribe, send a message to ase...@ea5.yz.yamagata-u.ac.jp

with your name and email address. Membership is restricted to

within Asia as a general rule.

The list is executed manually, and is maintained by Prof. Mikio

Nakatsuyama, Department of Electronic Engineering, Yamagata

University, 4-3-16 Jonan, Yonezawa 992 Japan, phone +81-238-22-5181,

fax +81-238-24-2752, email nak...@ea5.yz.yamagata-u.ac.jp.

All messages to the list have the Subject line replaced with "AFMS".

The language of the list is English.

Fuzzy Mailing List - Japan:

fuzz...@sys.es.osaka-u.ac.jp

This is an unmoderated list, with mostly original contributions

in Japanese (JIS-code).

To subscribe, send subscriptions to the listserv

fuzzy-jp...@sys.es.osaka-u.ac.jp

If you need to speak to a human being, send mail to the list

owners,

fuzzy...@tamlab.sys.es.osaka-u.ac.jp

Itsuo Hatono and Motohide Umano of Osaka University.

================================================================

Subject: [16] Bibliography

Date: 14-AUG-95

A list of books compiled by Josef Benedikt for the FLAI '93 (Fuzzy

Logic in Artificial Intelligence) conference's book exhibition is

available by anonymous ftp from

ftp.cs.cmu.edu:/user/ai/pubs/bibs/

as the file fuzzy-bib.text.

A short 1985 fuzzy systems tutorial by James Brule is available as

http://life.anu.edu.au/complex_systems/fuzzy.html

An ascii copy is also available in the gzipped tar file

ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/intro/tutorial.tgz

Wolfgang Slany has compiled a BibTeX bibliography on fuzzy

scheduling and related fuzzy techniques, including constraint satisfaction,

linear programming, optimization, benchmarking, qualitative

modeling, decision making, petri-nets, production control,

resource allocation, planning, design, and uncertainty management. It

is available by anonymous ftp from

mira.dbai.tuwien.ac.at:/pub/slany/

as the file fuzzy-scheduling.bib.Z (or .ps.Z), or by email from

list...@vexpert.dbai.tuwien.ac.at

with

GET LISTPROC fuzzy-scheduling.bib

in the message body.

Non-Mathematical Works:

Kosko, Bart, "Fuzzy Thinking: The New Science of Fuzzy Logic", Warner, 1993

[For technical details, see Kosko, Bart, "Fuzzy cognitive maps",

International Journal of Man-Machine Studies 24:65-75, 1986.]

McNeill, Daniel, and Freiberger, Paul, "Fuzzy Logic: The Discovery

of a Revolutionary Computer Technology", Simon and Schuster,

1992. ISBN 0-671-73843-7. [Mostly history, but many examples of

applications.]

Negoita, C.V., "Fuzzy Systems", Abacus Press, Tunbridge-Wells, 1981.

Smithson, Michael, "Ignorance and Uncertainty: Emerging Paradigms",

Springer-Verlag, New York, 1988.

Brubaker, D.I., "Fuzzy-logic Basics: Intuitive Rules Replace Complex Math,"

EDN, June 18, 1992.

Schwartz, D.G. and Klir, G.J., "Fuzzy Logic Flowers in Japan," IEEE

Spectrum, July 1992.

Earl Cox, "The Fuzzy Systems Handbook: A Practitioner's Guide to

Building, Using, and Maintaining Fuzzy Systems", Academic Press,

Boston, MA 1994. 615 pages, ISBN 0-12-194270-8 ($49.95). [Includes

disk with ANSI C++ source code. Very good.]

F. Martin McNeill and Ellen Thro, "Fuzzy Logic: A practical

approach", Academic Press, 1994. 350 pages, ISBN 0-12-485965-8 ($40).

[A good fuzzy logic primer.]

Textbooks:

Dubois, Didier, and Prade, H., "Fuzzy Sets and Systems: Theory and

Applications", Academic Press, New York, 1980.

Dubois, Didier, and Prade, Henri, "Possibility Theory", Plenum Press, New

York, 1988.

Goodman, I.R., and Nguyen, H.T., "Uncertainty Models for Knowledge-Based

Systems", North-Holland, Amsterdam, 1986.

Kandel, Abraham, "Fuzzy Mathematical Techniques with Applications",

Addison-Wesley, 1986.

Kandel, Abraham, and Lee, A., "Fuzzy Switching and Automata", Crane

Russak, New York, 1979.

Klir, George, and Folger, Tina, "Fuzzy Sets, Uncertainty, and

Information", Prentice Hall, Englewood Cliffs, NJ, 1987. ISBN 0-13-345638-2.

Kosko, Bart, "Neural Networks and Fuzzy Systems", Prentice Hall, Englewood

Cliffs, NJ, 1992. ISBN 0-13-611435-0. [Very good.]

R. Kruse, J. Gebhardt, and F. Klawonn, "Foundations of Fuzzy Systems"

John Wiley and Sons Ltd., Chichester, 1994. ISBN 0471-94243-X ($47.95).

[Theory of fuzzy sets.]

Toshiro Terano, Kiyoji Asai, and Michio Sugeno, "Fuzzy Systems Theory

and its Applications", Academic Press, 1992, 268 pages.

ISBN 0-12-685245-6. Translation of "Fajii shisutemu nyumon"

(Japanese, 1987). Newly released as "Applied Fuzzy Systems", 1994,

320 pages, ISBN 0-12-685242-1 ($40).

Wang, Paul P., "Theory of Fuzzy Sets and Their Applications", Shanghai

Science and Technology, Shanghai, 1982.

Wang, Zhenyuan, and Klir, George J., "Fuzzy Measure Theory", Plenum

Press, New York, 1991.

Yager, R.R., (editor), "Fuzzy Sets and Applications", John Wiley

and Sons, New York, 1987.

Yager, Ronald R., and Zadeh, Lofti, "Fuzzy Sets, Neural Networks,

and Soft Computing", Van Nostrand Reinhold, 1994.

ISBN 0-442-01621-2, $64.95.

Zimmerman, Hans J., "Fuzzy Set Theory", Kluwer, Boston, 2nd edition, 1991.

Anthologies:

Didier Dubois, Henri Prade, and Ronald R. Yager, editors,

"Readings in Fuzzy Sets for Intelligent Systems", Morgan Kaufmann

Publishers, 1993. 916 pages, ISBN 1-55860-257-7 paper ($49.95).

"A Quarter Century of Fuzzy Systems", Special Issue of the International

Journal of General Systems, 17(2-3), June 1990.

R.J. Marks II, editor, "Fuzzy Logic Technology & Applications", IEEE,

1994. IEEE Order# 94CR0101-6-PSP, $59.95 ($48.00 for IEEE members).

Order from 1-800-678-IEEE. [Selected papers from past IEEE

conferences. Focus is on papers concerning applications of fuzzy

systems. There are also some overview papers.]

================================================================

Subject: [17] Journals and Technical Newsletters

Date: 24-AUG-93

INTERNATIONAL JOURNAL OF APPROXIMATE REASONING (IJAR)

Official publication of the North American Fuzzy Information Processing

Society (NAFIPS).

Published 8 times annually. ISSN 0888-613X.

Subscriptions: Institutions $282, NAFIPS members $72 (plus $5 NAFIPS dues)

$36 mailing surcharge if outside North America.

For subscription information, write to David Reis, Elsevier Science

Publishing Company, Inc., 655 Avenue of the Americas, New York, New York

10010, call 212-633-3827, fax 212-633-3913, or send email to

74740...@compuserve.com.

Editor:

Piero Bonissone

Editor, Int'l J of Approx Reasoning (IJAR)

GE Corp R&D

Bldg K1 Rm 5C32A

PO Box 8

Schenectady, NY 12301 USA

Email: boni...@crd.ge.com

Voice: 518-387-5155

Fax: 518-387-6845

Email: Boni...@crd.ge.com

INTERNATIONAL JOURNAL OF FUZZY SETS AND SYSTEMS (IJFSS)

The official publication of the International Fuzzy Systems Association.

Subscriptions: Subscription is free to members of IFSA.

ISSN: 0165-0114

IEEE TRANSACTIONS ON FUZZY SYSTEMS

ISSN 1063-6706

Editor in Chief: James Bezdek

THE HUNTINGTON TECHNICAL BRIEF

Technical newsletter about fuzzy logic edited by Dr. Brubaker. It is

mailed monthly, is a single sheet, front and back, and rotates among

tutorials, descriptions of actual fuzzy applications, and discussions

(reviews, sort of) of existing fuzzy tools and products.

[The Huntington Technical Brief was discontinued in December 1994.]

INTERNATIONAL JOURNAL OF

UNCERTAINTY, FUZZINESS AND KNOWLEDGE-BASED SYSTEMS (IJUFKS)

Published 4 times annually. ISSN 0218-4885.

Intended as a forum for research on methods for managing imprecise,

vague, uncertain and incomplete knowledge.

Subscriptions: Individuals $90, Institutions $180. (add $25 for airmail)

World Scientific Publishing Co Pte Ltd, Farrer Road, PO Box 128,

Singapore 9128, Rep. of Singapore.

E-mail worl...@singnet.com.sg, phone 65-382-5663, fax

65-382-5919.

Web pages for this journal:

http://www.wspc.co.uk/wspc/Journals/ijufks/ijufks.html

Submissions: B Bouchon-Meunier, editor in chief, Laforia-IBP,

Universite Paris VI, Boite 169, 4 Place Jussieu, 75252 Paris Cedex 05,

FRANCE, phone 33-1-44-27-70-03, fax 33-1-44-27-70-00, e-mail

bou...@laforia.ibp.fr.

================================================================

Subject: [18] Professional Organizations

Date: 15-APR-93

INSTITUTION FOR FUZZY SYSTEMS AND INTELLIGENT CONTROL, INC.

Sponsors, organizes, and publishes the proceedings of the International

Fuzzy Systems and Intelligent Control Conference. The conference is

devoted primarily to computer based feedback control systems that rely on

rule bases, machine learning, and other artificial intelligence and soft

computing techniques. The theme of the 1993 conference was "Fuzzy Logic,

Neural Networks, and Soft Computing."

Thomas L. Ward

Institution for Fuzzy Systems and Intelligent Control, Inc.

P. O. Box 1297

Louisville KY 40201-1297 USA

Phone: +1.502.588.6342

Fax: +1.502.588.5633

Email: TLWa...@ulkyvm.louisville.edu, TLWa...@ulkyvm.bitnet

INTERNATIONAL FUZZY SYSTEMS ASSOCIATION (IFSA)

Holds biannual conferences that rotate between Asia, North America,

and Europe. Membership is $232, which includes a subscription to the

International Journal of Fuzzy Sets and Systems.

Prof. Philippe Smets

University of Brussels, IRIDIA

50 av. F. Roosevelt

CP 194/6

1050 Brussels, Belgium

LABORATORY FOR INTERNATIONAL FUZZY ENGINEERING (LIFE)

Laboratory for International Fuzzy Engineering Research

Siber Hegner Building 3FL

89-1 Yamashita-cho, Naka-ku

Yokohama-shi 231 Japan

Email: <name>@fuzzy.or.jp

NORTH AMERICAN FUZZY INFORMATION PROCESSING SOCIETY (NAFIPS)

Holds a conference and a workshop in alternating years.

President:

Dr. Jim Keller

President NAFIPS

Electrical & Computer Engineering Dept

University of Missouri-Col

Columbia, MO 65211 USA

Phone +1.314.882.7339

Email: ec...@mizzou1.missouri.edu, ec...@mizzou1.bitnet

Secretary/Treasurer:

Thomas H. Whalen

Sec'y/Treasurer NAFIPS

Decision Sciences Dept

Georgia State University

Atlanta, GA 30303 USA

Phone: +1.404.651.4080

Email: qmd...@gsuvm1.gsu.edu, qmd...@gsuvm1.bitnet

SPANISH ASSOCIATION FOR FUZZY LOGIC AND TECHNOLOGY

Prof. J. L. Verdegay

Dept. of Computer Science and A.I.

Faculty of Sciences

University of Granada

18071 Granada (Spain)

Phone: +34.58.244019

Tele-fax: +34.58.243317, +34.58.274258

Email: jver...@ugr.es

CANADIAN SOCIETY FOR FUZZY INFORMATION AND NEURAL SYSTEMS (CANS-FINS)

Dr. Madan M. Gupta, Director <gup...@sask.usask.ca>

Intelligent Systems Research Laboratory

College of Engineering

Sakatoon, Saskatchewan, S7N OWO

Tel: 306-966-5451

Fax: 306-966-8710

Dr. Ralph O. Buchal <rbu...@charon.engga.uwo.ca>

Department of Mechanical Engineering

Univ. of Western Ontario

London, Ontario, N6A 5B9

Tel: 519-679-2111, x8454

Fax: 519-661-3375

Dr. Martin Laplante

RES Inc.

Suite 501, 100 Sparks Street

Ottawa, Ont. KIP-5B7

Tel: 613-238-3690

Fax: 613-235-5889

================================================================

Subject: [19] Companies Supplying Fuzzy Tools

Date: 15-APR-93

*** Note: Inclusion in this list is not an endorsement for the product. ***

Accel Infotech Spore Pte Ltd:

Accel Infotech is a distributor for FUZZ-C from Byte Craft.

FUZZ-C generates C code that may be cross-compiled to the 6805, Z8C

and COP8C microprocessors using separate compilers.

FUZZ-C was reviewed in the March 1993 issue of AI Expert.

For more information, send email to ac...@solomon.technet.sg, call

+65-7446863 (Richard) or fax +65-7492467.

Adaptive Informations Systems:

This is a new company that specializes in fuzzy information systems.

Main products of AIS:

- Consultancy and application development in fuzzy information retrieval

and flexible querying systems

- Development of a fuzzy querying application for value added network

services

- A fuzzy solution for utilization of a large (lexicon based)

terminological knowledge base for NL query evaluation

Adaptive Informations Systems

Hoestvej 8 B

DK-2800 Lyngby

Denmark

Phone: 45-4587-3217

Email: h...@dat.ruc.dk

Adaptive Logic (formerly American NeuraLogix):

Products:

AL220 8 bit fuzzy microcontroller(18 pin DIP or 20 pin

SOIC) with A/D & D/A(4I/O).

NLX221 4-8 bit digital I/O single chip fuzzy microcontroller

with EEPROM memory.

NLX222 4-8 bit analog and digital I/O single chip fuzzy

microcontroller.

NLX230 8 bit microcontroller utilizing fuzzy logic at 30 million

rules per second.

NLX110 Fuzzy Pattern Comparator.

NLX112/113 Fuzzy Data Correlators.

INSiGHT IIe Real time emulator, programmer and development

software for AL220.

INSiGHT Development software for NLX22X family.

INStANT Programmer for NLX22X family.

ADS230 Development System for NLX230.

ADS110 Development System for NLX110

Note: The AL220 was named Innovation Of the Year '94 by EDN

Magazine in the microprocessor category. Data sheets and

application notes are available on the products plus local

application assistance.

Adaptive Logic Inc.

800 Charcot Ave., Suite 112

San Jose, CA 95131

Phone: 408-383-7200

Fax: 408-383-7201

Email: 75471...@compuserve.com

Europe:

Applied Marketing & Technology Ltd.

Saville Court, Saville Place, Clifton

Bristol BS8 4EJ

Phone: 117-9237594

Fax: 117-9237598

Email: 10043...@compuserve.com

Japan:

Nippon Precision Device

Nichibei Time 24 Bldg.

35 Tansu-cho

Shinjuki-ku, Tokyo 162

Phone: 332601411

Fax: 332607100

Adaptive Logic Inc.-R&D facility

411 Central Park Drive

Sanford, Fl 32771

Phone: 407-322-5608

Fax: 407-322-5609

Email: 75471...@compuserve.com or in...@adaptivelogic.com

URL: http://www.adaptivelogic.com/

Aptronix:

Products:

Fide A MS Windows-hosted graphical development environment for

fuzzy expert systems. Code generators for Motorola's 6805,

68HC05, and 68HC11, and Omron's FP-3000 are available. A

demonstration version of Fide is available.

Aptronix, Inc.

2150 North First Street, Suite 300

San Jose, Ca. 95131 USA

Phone: 408-261-1888

Fax: 408-261-1897

Fuzzy Net BBS: 408-261-1883, 8/n/1

Aria Ltd.:

Products:

DB-fuzzy A library of fuzzy information retrieval for CA-Clipper.

See ftp.cs.cmu.edu:/user/ai/areas/fuzzy/com/aria/ for

more information.

Aria Ltd.

Dubravska 3

842 21 Bratislava

SLOVAKIA

Phone: (+42 7) 3709 286

Fax: (+42 7) 3709 232

Email: ar...@softec.sk

ClippArt Ltd. is the exclusive distributor of DB-fuzzy.

Any additional information about DB-fuzzy you can obtain

from this company.

ClippArt Ltd. Polianky 15 Tel. (+42 7) 786 160

841 02 Bratislava Fax (+42 7) 786 160

Slovakia

ByteCraft, Ltd.:

Products:

Fuzz-C "A C preprocessor for fuzzy logic" according to the cover of

its manual. Translates an extended C language to C source

code.

Byte Craft Limited

421 King Street North

Waterloo, Ontario

Canada N2J 4E4

Phone: 519-888-6911

Fax: 519-746-6751

Support BBS: 519-888-7626

Fril Systems Ltd:

FRIL (Fuzzy Relational Inference Language) is a logic-programming

language that incorporates a consistent method for handling

uncertainty, based on Baldwin's theories of support logic, mass

assignments, and evidential reasoning. Mass assignments give a

consistent way of manipulating fuzzy and probabilistic uncertainties,

enabling different forms of uncertainty to be integrated within a

single framework. Fril has a list-based syntax, similar to the early

micro-Prolog from LPA. Prolog is a special case of Fril, in which

programs involve no uncertainty. Fril runs on Unix, Macintosh,

MS-DOS, and Windows 3.1 platforms.

For further information, write to

Dr B.W. Pilsworth

Fril Systems Ltd

Bristol Business Centre,

Maggs House,

78 Queens Rd,

Bristol BS8 1QX, UK.

A longer description is available as

ftp.cs.cmu.edu:/user/ai/areas/fuzzy/com/fril/fril.txt

Fujitsu:

Products:

MB94100 Single-chip 4-bit (?) fuzzy controller.

FuziWare:

Products:

FuziCalc An MS-Windows-based fuzzy development system based on a

spreadsheet view of fuzzy systems.

FuziWare, Inc.

316 Nancy Kynn Lane, Suite 10

Knoxville, Tn. 37919 USA

Phone: 800-472-6183, 615-588-4144

Fax: 615-588-9487

FuzzySoft AG:

Product:

FuzzySoft Fuzzy Logic Operating System runs under MS-Windows,

generates C-code, extended simulation capabalities.

Selling office for Germany, Switzerland and Austria (all product

inquiries should be directed here)

GTS Trautzl GmbbH

Gottlieb-Daimler-Str. 9

W-2358 Kaltenkirchen/Hamburg

Germany

Phone: (49) 4191 8711

Fax: (49) 4191 88665

Fuzzy Systems Engineering:

Products:

Manifold Editor ?

Manifold Graphics Editor ?

[These seem to be membership function & rulebase editors.]

Fuzzy Systems Engineering

P. O. Box 27390

San Diego, CA 92198 USA

Phone: 619-748-7384

Fax: 619-748-7384 (?)

HyperLogic Corporation:

Products:

CubiCalc Windows-based Fuzzy Logic Shell. Includes

fuzzy and plant simulation, plots, file

I/O, DDE.

CubiCalc RTC Windows-based Fuzzy Logic Development

Environment. Superset of CubiCalc includes

run-time generator, code libraries, DLL for

Windows Applications (incl Visual Basic).

CubiCard Superset of CubiCalc RTC with data acquisition

capabilties via hardware interface board.

CubiQuick Inexpensive version of CubiCalc with limited

capabilties for classroom and small projects.

Academic discounts available.

Rule Maker Add-on to CubiCalc and higher products for

automatic rulebase generation. Provides four

different generation strategies.

HyperLogic Corporation

P.O. Box 300010

Escondido, CA 92030-0010

Tel: 619-746-2765

Fax: 619-746-4089

Email: prod...@hyperlogic.com

The URL for their home page is http://www.hyperlogic.com/hl. It includes

product descriptions, pricing information, their Tech Notes on various

subjects, and several downloadable demonstration programs.

Inform:

Products:

fuzzyTECH 3.0 A graphical fuzzy development environment. Versions

are available that generate either C source code or

Intel MCS-96 assembly source code as output. A

demonstration version is available. Runs under MS-DOS.

Inform Software Corp

1840 Oak Street, Suite 324

Evanston, Il. 60201 USA

Phone: 708-866-1838

INFORM GmbH

Geschaeftsbereich Fuzzy--Technologien

Pascalstraese 23

W-5100 Aachen

Tel: (02408) 6091

Fax: (02408) 6090

IIS:

IIS specializes in offering short courses on soft computing. They

also perform research and development in fuzzy logic, fuzzy control,

neural networks, adaptive fuzzy systems, and genetic algorithms.

Intelligent Inference Systems Corp.

P.O. Box 2908

Sunnyvale, CA 94087

Phone: (408) 730-8345

Fax: (408) 730-8550

email: iis...@netcom.com

LPA, Ltd.:

FLINT, a Fuzzy Logic INferencing Toolkit, is a versatile fuzzy logic

inferencing system that makes fuzzy logic technology and fuzzy rules

available within a sophisticated programming environment. FLINT

supports standard and user-defined membership functions, linear and

curved membership lines, automatic propagation of fuzzy values, range

of and/or/not combinators, configurable linguistic hedges, standard

and user-defined defuzzification algorithms. FLINT is available as a

versatile programming toolkit for LPA Prolog running Windows 95/3.1/NT

or Macintosh and as an extension to LPA's popular expert system

toolkit, Flex.

For further information contact:

Logic Programming Associates Ltd.,

Studio 4, R.V.P.B., Trinity Road,

London, SW18 3SX, UK.

Web: http://www.lpa.co.uk

US Toll Free: 1-800-949-7567

Tel: +44 181 871 2016

Fax: +44 181 874 0449

Email: l...@cix.compulink.co.uk

Metus Systems Group:

Products:

Metus Fuzzy Library A library of fuzzy processing routines for

C or C++. Source code is available.

The Metus Systems Group

1 Griggs Lane

Chappaqua, Ny. 10514 USA

Phone: 914-238-0647

Modico:

Products:

Fuzzle 1.8 A fuzzy development shell that generates either ANSI

FORTRAN or C source code.

Modico, Inc.

P. O. Box 8485

Knoxville, Tn. 37996 USA

Phone: 615-531-7008

National Semiconductor, Santa Clara CA, USA

http://www.commerce.net/directories/participants/ns/home.html

NeuFuz is aimed at low end controls applications in automotive,

industrial, and appliance areas. NeuFuz is a neural-fuzzy technology

which uses backpropagation techniques to initially select fuzzy rules

and membership functions. Initial stages of design using NeuFuz

technology are performed using training data and backpropagation.

The result is a fuzzy associative memory (FAM) which implements an

approximation of the training data. By implementing a FAM, rather

than a multi-layer perceptron, the designer has a solution which can

be understood and tuned to a particular application using Fuzzy Logic

design techniques.

NeuFuz4 Learning Kit, Product ordering code (NSID): NF2-C8A-KIT

- NeuFuz2 Neural Network Learning Software

- Up to 2 inputs, 1 output

- 50 training patterns

- Up to 3 membership functions

- COP8 Code Generator (COP8 is National's family of 8-bit

microcontrollers)

NeuFuz4 Software Package, Product ordering code (NSID): NF4-C8A

- NeuFuz4 Software

- Neural Network Learning Software - Up to 4 inputs, 1 output and

1200 training patterns

- Up to 7 membership functions

- COP8 Code Generator

The NeuFuz4 Development System, Product ordering code: (NSID):

NF4-C8A-SYS.

- Neural Network Learning Software - Up to 4 inputs, 1 output and

1200 training patterns

- Up to 7 membership functions

- COP8 Code Generator

- COP8 In-Circuit Emulator "Debug Module"

- Real-Time Emulation Microcontroller EPROM Programming

- Real-Time Trace

- Complete Source/Symbolic Debug

- One-Day Training on Customer Request

- Access to Factory Expert via Telephone (Maximum 16 hrs.)

NeuFuz4-C Learning Kit, Product ordering code (NSID): NF2-C-KIT

- Up to 2 inputs, 1 output 50 training patterns

- Up to 3 membership functions

- ANSI Standard C Language Code Generator

- Tutorial Examples for Neural Network Learning and Fuzzy Rule

Generation

NeuFuz4-C Software Package, Product ordering code (NSID): NF4-C

- Up to 4 inputs, 1 output and 1200 training patterns

- Up to 7 membership functions

- ANSI Standard C Language Code Generator

- One-Day Training on Customer Request

- Access to Factory Expert via Telephone (Maximum 16 hrs.)

Oki Electric:

Products:

MSM91U111 A single-chip 8-bit fuzzy controller.

Europe:

Oki Electric Europe GmbH.

Hellersbergstrasse 2

D-4040 Neuss, Germany

Phone: 49-2131-15960

Fax: 49-2131-103539

Hong Kong:

Oki Electronics (Hong Kong) Ltd.

Suite 1810-4, Tower 1

China Hong Kong City

33 Canton Road, Tsim Sha Tsui

Kowloon, Hong Kong

Phone: 3-7362336

Fax: 3-7362395

Japan:

Oki Electric Industry Co., Ltd.

Head Office Annex

7-5-25 Nishishinjuku

Shinjuku-ku Tokyo 160 JAPAN

Phone: 81-3-5386-8100

Fax: 81-3-5386-8110

USA:

Oki Semiconductor

785 North Mary Avenue

Sunnyvale, Ca. 94086 USA

Phone: 408-720-1900

Fax: 408-720-1918

OMRON Corporation:

Products:

C500-FZ001 Fuzzy logic processor module for Omron C-series PLCs.

E5AF Fuzzy process temperature controller.

FB-30AT FP-3000 based PC AT fuzzy inference board.

FP-1000 Digital fuzzy controller.

FP-3000 Single-chip 12-bit digital fuzzy controller.

FP-5000 Analog fuzzy controller.

FS-10AT PC-based software development environment for the

FP-3000.

Japan

Kazuaki Urasaki

Fuzzy Technology Business Promotion Center

OMRON Corporation

20 Igadera, Shimokaiinji

Nagaokakyo Shi, Kyoto 617 Japan

Phone: 81-075-951-5117

Fax: 81-075-952-0411

USA Sales (all product inquiries should be directed here)

Pat Murphy

OMRON Electronics, Inc.

One East Commerce Drive

Schaumburg, IL 60173 USA

Phone: 708-843-7900

Fax: 708-843-7787/8568

USA Research

Satoru Isaka

OMRON Advanced Systems, Inc.

3945 Freedom Circle, Suite 410

Santa Clara, CA 95054

Phone: 408-727-6644

Fax: 408-727-5540

Email: is...@oas.omron.com

Togai InfraLogic, Inc.:

Togai InfraLogic (TIL for short) supplies software development tools,

board-, chip- and core-level fuzzy hardware, and engineering services.

Contact in...@til.com for more detailed information.

Products:

FC110 (the FC110(tm) Digital Fuzzy Processor (DFP-tm)). An

8-bit microprocessor/coprocessor with fuzzy acceleration.

FC110DS (the FC110 Development System) A software development package

for the FC110 DFP, including an assembler, linker and Fuzzy

Programming Language (FPL-tm) compiler.

FCA VLSI Cores based on Fuzzy Computational Acceleration (FCA-tm).

FCA10AT FC110-based fuzzy accelerator board for PC/AT-compatibles.

FCA10VME FC110-based four-processor VME fuzzy accelerator.

FCD10SA FC110-based fuzzy processing module.

FCD10SBFC FC110-based single board fuzzy controller module.

FCD10SBus FC110-based two-processor SBus fuzzy accelerator.

FCDS (the Fuzzy-C Development System) An FPL compiler that emits

K&R or ANSI C source to implement the specified fuzzy system.

MicroFPL An FPL compiler and runtime module that support using fuzzy

techniques on small microcontrollers by several companies.

TILGen A tool for automatically constructing fuzzy expert systems from

sampled data.

TILShell+ A graphical development and simulation environment for fuzzy

systems.

USA

Togai InfraLogic, Inc.

5 Vanderbilt

Irvine, CA 92718 USA

Phone: 714-975-8522

Fax: 714-975-8524

Email: in...@til.com

Toshiba:

Products:

T/FC150 10-bit fuzzy inference processor.

LFZY1 FC150-based NEC PC fuzzy logic board.

T/FT Fuzzy system development tool.

TransferTech GmbH:

Products:

Fuzzy Control Manager (FMC) Fuzzy shell, runs under MS-Windows

TransferTech GmbH

Cyriaksring 9A

D-38118 Braunschweig

Germany

Tel: +49 531 890255

Fax: +49 531 890355

Email: in...@transfertech.de

URL: http://www.transfertech.de

================================================================

Subject: [20] Fuzzy Researchers

Date: 23-AUG-94

A list of "Who's Who in Fuzzy Logic" (researchers and research

organizations in the field of fuzzy logic and fuzzy expert systems)

may be obtained by sending a message to

list...@vexpert.dbai.tuwien.ac.at

with

GET LISTPROC WHOISWHOINFUZZY

in the message body. New entries and corrections should be sent to

Robert Fuller <rfu...@finabo.abo.fi>.

A copy of this list is also available by anonymous ftp from

mira.dbai.tuwien.ac.at:/pub/mlowner/whoiswhoinfuzzy

or

ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/whos_who/whos_who.txt

================================================================

Subject: [21] Elkan's "The Paradoxical Success of Fuzzy Logic" paper

The presentation of Elkan's AAAI-93 paper

Charles Elkan, "The Paradoxical Success of Fuzzy Logic", in

Proceedings of the Eleventh National Conference on Artificial

Intelligence, 698-703, 1993.

has generated much controversy. The fuzzy logic community claims that

the paper is based on some common misunderstandings about fuzzy logic, but

Elkan still maintains the correctness of his proof. (See, for

instance, AI Magazine 15(1):6-8, Spring 1994.)

Elkan proves that for a particular set of axiomatizations of fuzzy

logic, fuzzy logic collapses to two-valued logic. The proof is correct

in the sense that the conclusion follows from the premises. The

disagreement concerns the relevance of the premises to fuzzy logic.

At issue are the logical equivalence axioms. Elkan has shown that if

you include any of several plausible equivalences, such as

not(A and not B) == (not A and not B) or B

with the min, max, and 1- axioms of fuzzy logic, then fuzzy logic

reduces to binary logic. The fuzzy logic community states that these

logical equivalence axioms are not required in fuzzy logic, and that

Elkan's proof requires the excluded middle law, a law that is commonly

rejected in fuzzy logic. Fuzzy logic researchers must simply take care

to avoid using any of these equivalences in their work.

It is difficult to do justice to the issues in so short a summary.

Readers of this FAQ should not assume that this summary is the last

word on this topic, but should read Elkan's paper and some of the

other correspondence on this topic (some of which has appeared in the

comp.ai.fuzzy newsgroup).

Two responses to Elkan's paper, one by Enrique Ruspini and the other

by Didier Dubois and Henri Prade, may be found as

ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/elkan/response.txt

A final version of Elkan's paper, together with responses from members

of the fuzzy logic community, will appear in an issue of IEEE Expert

sometime in 1994. A paper by Dubois and Prade will be presented at AAAI-94.

================================================================

Subject: [22] Glossary

Hedge

A hedge is a one-input truth value manipulation operation. It modifies

the shape of the truth function, in a manner analogous to the function

of adjectives and adverbs in English. Some examples that are commonly seen

in the literature are intensifiers like "very", detensifiers like

"somewhat", and complementizers like "not". One might define "very x"

as the square of the truth value of x, and define "somewhat x" as the

square root of the truth value of x. Then you can make fuzzy logic

statements like:

y is very low

which would evaluate to (y is low) * (y is low). One can think of

"not x" as being a hedge in the same sense, defining "not x" as one

minus the truth value of x.

================================================================

Subject: [24] Where to send calls for papers (cfp) and calls for participation

Date: 15-MAY-95

Fuzzy related calls for papers and calls for participation should be

sent by email to confe...@iao.fhg.de, or posted to the moderated

newsgroup news.announce.conferences. Both actions will have the same

effect. Please keep Subject lines informative; if space permits,

mention the topic and location there, and avoid acronyms unless very

widely known. Submissions will simultaneously appear in the newsgroup

news.announce.conferences and on the WorldWideWeb server of

Fraunhofer-IAO at <URL:http://www.iao.fhg.de/Library/conferences> as

soon as they have been processed. The fuzzy-mail mailing list (see

[15]) scans this news-group for items related to fuzzy and uncertainty.

Matching messages will be moderated like any other message sent to the

mailing list, and if selected, will be forwarded to the Asian

fuzzy-mailing list (see [15]), NAFIPS-L (see [15]), as well as the

internet news-group comp.ai.fuzzy (see [1]). Sending it only to

confe...@iao.fhg.de is normally enough to distribute the message

efficiently to all the other media.

================================================================

;;; *EOF*

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