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From: mkant+@cs.cmu.edu (Mark Kantrowitz)
Newsgroups: comp.ai.fuzzy,comp.answers,news.answers
Subject: FAQ: Fuzzy Logic and Fuzzy Expert Systems 1/1 [Monthly posting]
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Summary: Answers to Frequently Asked Fuzzy Questions. Read before posting.
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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+fuzzy-faq@cs.cmu.edu.
Note that the mkant+fuzzy-faq@cs.cmu.edu mailing list is for
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The original version of this FAQ posting was prepared by Erik
Horstkotte of SysSoft , with significant
contributions by Cliff Joslyn . The FAQ is
maintained by Mark Kantrowitz with advice from Erik
and Cliff. To reach us, send mail to mkant+fuzzy-faq@cs.cmu.edu.
Thanks also go to Michael Arras for
running the vote which resulted in the creation of comp.ai.fuzzy,
Yokichi Tanaka for help in putting the FAQ together,
and Walter Hafner , Satoru Isaka
, Henrik Legind Larsen , Tom Parish
, Liliane Peters , Naji
Rizk , Peter Stegmaier , Prof.
J.L. Verdegay , and Dr. John Yen 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
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Mark Kantrowitz, Erik Horstkotte, and Cliff Joslyn, "Answers to
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mkant+fuzzy-faq@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 ?
[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:
;;; 1.14:
;;; 16-SEP-94 mk Updated [20] -- listserver is now a listproc.
;;;
;;; 1.15:
;;; 13-OCT-94 mk Added entry on Slany's bibliography to [16].
;;; 13-OCT-94 mk Added Fuzzy Arithmetic Library to [14].
;;; 3-NOV-94 mk Updated Hyperlogic entry.
;;; 1-DEC-94 mk Added [24] Where to send calls for papers (cfp) and calls
;;; for participation, thanks to Wolfgang Slany.
;;; 1-DEC-94 mk Updated text of [10] and deleted [23], thanks to Cliff.
;;; 1-DEC-94 mk Aptronix telephone exchange changed from 428 to 261.
;;;
;;; 1.16:
;;; 14-DEC-94 mk Huntington Technical Brief discontinued December 1994.
;;; 18-JAN-95 mk Added national semiconductor to [19].
;;;
;;; 1.17:
;;; 17-MAR-95 mk Added answer to [6] Fuzzy Control, supplied by Mike Ellims
;;; .
================================================================
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 ?
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 fuzzynet@aptronix.com:
begin
help
end
If you don't receive a response within a day or two, or need help, contact
Scott Irwin 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 .
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 fuzzy@its.bldrdoc.gov. If you deposit anything in new/,
please inform fuzzy@its.bldrdoc.gov. The repository is maintained by
Timothy Butler, tim@its.bldrdoc.gov.
The Fuzzy Logic Repository is also accessible through a mail server,
rnalib@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-server@til.com.
help
If you don't receive a response within a day or two, contact either
erik@til.com or tanaka@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, .
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 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 info@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 fzclips@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 .
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 .
================================================================
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 150 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
listserv@gsuvm1.gsu.edu -or- listserv@gsuvm1.bitnet
and the name of the mailing list itself is
nafips-l@gsuvm1.gsu.edu -or- nafips-l@gsuvm1.bitnet
Use the "gsuvm1.gsu.edu" addresses if you're on the Internet, and the
"gsuvm1.bitnet" addresses if you're on BITNET. If you're on some other
network, try to figure out which is "closer" to you, and use that one. 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
listserv@gsuvm1.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 nafips-l@gsuvm1.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 720 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
.
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
listproc@vexpert.dbai.tuwien.ac.at
and the name of the mailing list is
fuzzy-mail@vexpert.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
listproc@vexpert.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-mail@vexpert.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):
afuzzy@ea5.yz.yamagata-u.ac.jp
To subscribe, send a message to aserver@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 nakatsu@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:
fuzzy-jp@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-request@sys.es.osaka-u.ac.jp
If you need to speak to a human being, send mail to the list
owners,
fuzzy-admin@tamlab.sys.es.osaka-u.ac.jp
Itsuo Hatono and Motohide Umano of Osaka University.
================================================================
Subject: [16] Bibliography
Date: 7-JUN-93
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 as
ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/fuzzytut.txt
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
listproc@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
74740.2600@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: bonissone@crd.ge.com
Voice: 518-387-5155
Fax: 518-387-6845
Email: Bonissone@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, Farer Road, PO Box 128,
SINGAPORE 9128, e-mail phua@ictp.trieste.it, phone 65-382-5663, fax
65-382-5919.
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
bouchon@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: TLWard01@ulkyvm.louisville.edu, TLWard01@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: @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: ecejk@mizzou1.missouri.edu, ecejk@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: qmdthw@gsuvm1.gsu.edu, qmdthw@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: jverdegay@ugr.es
CANADIAN SOCIETY FOR FUZZY INFORMATION AND NEURAL SYSTEMS (CANS-FINS)
Dr. Madan M. Gupta, Director
Intelligent Systems Research Laboratory
College of Engineering
Sakatoon, Saskatchewan, S7N OWO
Tel: 306-966-5451
Fax: 306-966-8710
Dr. Ralph O. Buchal
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 accel@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: hll@dat.ruc.dk
American NeuraLogix:
Products:
NLX110 Fuzzy Pattern Comparator.
NLX230 8-bit single-chip fuzzy microcontroller.
NLX20xC 8- and 16-bit VLSI Core elements for fuzzy processing.
Others Other nonfuzzy and quasi-fuzzy devices.
[American NeuraLogix describes these chips and cores as "fuzzy"
processing devices, but as far as I can tell, they're not really
fuzzy. The NLX110 is a Hamming-distance calculator, and the
NLX230 and NLX20xC are based on a winner-take-all inference
strategy that discards most of the advantages of fuzzy expert
systems. Read the data sheets carefully before deciding.]
American NeuraLogix, Inc.
411 Central Park Drive
Sanford, FL 32771 USA
Phone: 407-322-5608
Fax: 407-322-5609
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: aria@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
1855 East Valley Parkway, Suite 210
P.O. Box 300010
Escondido, CA 92030-0010
Tel: 619-746-2765
Fax: 619-746-4089
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: iiscorp@netcom.com
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: isaka@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 info@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: info@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.
Rebenring 33
W-3300 Braunschweig, Germany
Phone: 49-531-3801139
Fax: 49-531-3801152
================================================================
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
listproc@vexpert.dbai.tuwien.ac.at
with
GET LISTPROC WHOISWHOINFUZZY
in the message body. New entries and corrections should be sent to
Robert Fuller .
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: 18-NOV-94
Fuzzy related calls for papers and calls for participation should be
sent to nac@sparky.sterling.com. Please keep Subject lines informative;
if space permits, mention the topic and location there, and avoid
acronyms unless very widely known. The message will then be distributed
to the internet news-group news.announce.conferences. 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 nac@sparky.sterling.com is normally enough to
distribute the message efficiently to all the other media.
================================================================
;;; *EOF*