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From: [e-mail] (Mark Kantrowitz)
Subject: FAQ: Fuzzy Logic and Fuzzy Expert Systems 1/1 [Monthly posting]
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Archive-name: fuzzy-logic/part1
Last-modified: Mon Jun  7 16:11:26 1993 by Mark Kantrowitz
Version: 1.0

;;; *****************************************************************
;;; Answers to Questions about Fuzzy Logic and Fuzzy Expert Systems *
;;; *****************************************************************
;;; Written by Erik Horstkotte, Cliff Joslyn, and Mark Kantrowitz
;;; fuzzy-faq.text -- 59054 bytes

Contributions and corrections should be sent to the mailing list
[e-mail].

Note that the [e-mail] mailing list is for discussion
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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 Togai InfraLogic <[e-mail]>, with significant contributions by
Cliff Joslyn <[e-mail]>.  The FAQ is
maintained by Mark Kantrowitz <[e-mail]> with advice from Erik
and Cliff. To reach us, send mail to [e-mail].

Thanks also go to Michael Arras <[e-mail]> for
running the vote which resulted in the creation of comp.ai.fuzzy,
Yokichi Tanaka <[e-mail]> for help in putting the FAQ together,
and Walter Hafner <[e-mail]>, Satoru Isaka
<[e-mail]>, Henrik Legind Larsen <[e-mail]>, Tom Parish
<[e-mail]>, Liliane Peters <[e-mail]>, Naji
Rizk <[e-mail]>, Peter Stegmaier <[e-mail]>, Prof.
J.L. Verdegay <[e-mail]>, and Dr. John Yen <[e-mail]> for
contributions to the initial contents of the FAQ.

This FAQ is posted once a month on the 13th of the month. In between
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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
 [18] Professional Organizations
 [19] Companies Supplying Fuzzy Tools
 [20] Fuzzy Researchers

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:
;;; 21-APR-93 eh    Corrected crisp value of Centroid defuzzification and added
;;;                 example.
;;; 21-APR-93 mk    Two corrections from Dr. Aivars Celmins wrt IFSA entry.
;;; 20-MAY-93 mk    Minor changes to [7] and [10].
;;;  7-JUN-93 mk    Added pointer to Josef Benedikt's bibliography to 16.

================================================================
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 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    0.32
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.

================================================================
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: 15-APR-93

[Anybody want to write an answer?]

References:

    Yager, R.R., and Zadeh, L. A., "An Introduction to Fuzzy Logic
    Applications in Intelligent Systems", Kluwer Academic Publishers, 1991.

================================================================
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: 15-APR-93

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 is not the case 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 also
includes random sets, Demster-Shafer evidence theory, probability
intervals, possibility theory, fuzzy measures, and so on. 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]).

   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: [phone]
      Fax: [phone]
      Cable: SCINFO
      Telex: 84-5305

================================================================
Subject: [14] Fuzzy BBS Systems, Mail-servers and FTP Repositories
Date: 15-APR-93

Aptronix FuzzyNET BBS and Email Server:

   408-428-1883, 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 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 [e-mail]:

      begin
      help
      end

   If you don't receive a response within a day or two, or need help, contact
   Scott Irwin <[e-mail]> 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: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, and some demo programs.  Material to
    be included in the archive (e.g., papers and code) may be placed in the
    upload/ directory.  Send email to Asgeir Osterhus, <[e-mail]>.


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 [e-mail]. If you deposit anything in new/,
    please inform [e-mail]. The repository is maintained by
    Timothy Butler, [e-mail].

    The Fuzzy Logic Repository is also accessible through a mail server,
    [e-mail]. 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 [e-mail].
        help

    If you don't receive a response within a day or two, contact either
    [e-mail] or [e-mail] 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, <[e-mail]>.

================================================================
Subject: [15] Mailing Lists
Date: 15-APR-93


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

      [e-mail] -or- [e-mail]

    and the name of the mailing list itself is

      [e-mail] -or- [e-mail]

    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
    [e-mail]:
      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 [e-mail], 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 150 subscribers,
    located primarily in Europe, as one might expect.  In addition to the
    mailing list itself, the list server gives access to some files, including
    the "Who is Who in Fuzzy Logic" database that is currently under
    construction by Robert Fuller <[e-mail]>.

    Like many mailing lists, this one uses Anastasios Kotsikonas's LISTSERV
    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
      [e-mail]
    and the name of the mailing list is
      [e-mail]

    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
    [e-mail]:
      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 [e-mail], and any
    message that you send to that address will be sent to all of the other
    subscribers.

================================================================
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 in the directory
/afs/cs.cmu.edu/user/mkant/Public/AI/as the file fuzzy-bib.text.

Non-Mathematical Works:

   McNeill, Daniel, and Freiberger, Paul, "Fuzzy Logic", Simon and Schuster,
   1992.

   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.


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.

   Kosko, B., "Neural Networks and Fuzzy Systems", Prentice Hall, Englewood
   Cliffs, NJ, 1992.

   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.

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


Anthologies:

   Didier Dubois, Henri Prade, and Ronald R. Yager, editors, "Readings in
   Fuzzy Systems", Morgan Kaufmann Publishers, 1992.

   "A Quarter Century of Fuzzy Systems", Special Issue of the International
   Journal of General Systems, 17(2-3), June 1990.

================================================================
Subject: [17] Journals
Date: 15-APR-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 [phone], or send email to
   [e-mail].

    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: [e-mail]
        Voice: 518-387-5155
        Fax:   [phone]
        Email: [e-mail]


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

================================================================
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: [phone]
    Fax: [phone]
    Email: [e-mail], [e-mail]


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 [phone]
      Email: [e-mail], [e-mail]

    Secretary/Treasurer:
      Thomas H. Whalen
      Sec'y/Treasurer NAFIPS
      Decision Sciences Dept
      Georgia State University
      Atlanta, GA 30303 USA
      Phone: [phone]
      Email: [e-mail], [e-mail]


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: [phone]
    Tele-fax: [phone]
    Email: [e-mail]

================================================================
Subject: [19] Companies Supplying Fuzzy Tools
Date: 15-APR-93

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: [phone]
    Email: [e-mail]


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: [phone]
   Fax:   [phone]


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: [phone]
   Fax:   [phone]
   Fuzzy Net BBS: 408-428-1883, 8/n/1


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: [phone]
   Fax:   [phone]
   Support BBS: 519-888-7626


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: [phone]
   Fax:   [phone]

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: [phone]
   Fax:   [phone]


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: [phone]
   Fax:   [phone] (?)


HyperLogic, Inc.:

   Products:
     CubiCalc          An MS-Windows-based fuzzy development environment.
     CubiCalc RTC      C source-code generator addon for CubiCalc.

   HyperLogic Corp
   1855 East Valley Parkway, Suite 210
   P. O. Box 3751
   Escondido, Ca. 92027 USA
   Phone: [phone]
   Fax:   [phone]


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:  [phone]

   INFORM GmbH
   Geschaeftsbereich Fuzzy--Technologien
   Pascalstraese 23
   W-5100 Aachen
   Tel: [phone]
   Fax: [phone]


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: [phone]


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: [phone]


Oki Electric:

   Products:
     MSM91U111         A single-chip 8-bit fuzzy controller.

   Europe:

     Oki Electric Europe GmbH.
     Hellersbergstrasse 2
     D-4040 Neuss, Germany
     Phone: [phone]
     Fax:   [phone]

   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: [phone]
     Fax:   [phone]

   Japan:

     Oki Electric Industry Co., Ltd.
     Head Office Annex
     7-5-25 Nishishinjuku
     Shinjuku-ku Tokyo 160 JAPAN
     Phone: [phone]
     Fax:   [phone]

   USA:

     Oki Semiconductor
     785 North Mary Avenue
     Sunnyvale, Ca. 94086 USA
     Phone: [phone]
     Fax:   [phone]


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: [phone]
     Fax:   [phone]

   USA Sales (all product inquiries should be directed here)

     Pat Murphy
     OMRON Electronics, Inc.
     One East Commerce Drive
     Schaumburg, IL 60173 USA
     Phone: [phone]
     Fax:   [phone]

   USA Research

     Satoru Isaka
     OMRON Advanced Systems, Inc.
     3945 Freedom Circle, Suite 410
     Santa Clara, CA 95054
     Phone: [phone]
     Fax: [phone]
     Email: [e-mail]


Togai InfraLogic, Inc.:

   Togai InfraLogic (TIL for short) supplies software development tools,
   board-, chip- and core-level fuzzy hardware, and engineering services.
   Contact [e-mail] 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: [phone]
     Fax: [phone]
     Email: [e-mail]


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: [phone]
   Fax:   [phone]

================================================================
Subject: [20] Fuzzy Researchers
Date: 15-APR-93

This is a *partial* list of some of the researchers and research
organizations in the field of fuzzy logic and fuzzy expert systems.

Center for Fuzzy Logic and Intelligent Systems Research (Texas A&M):

   This group publishes a Technical Report Series, in addition to the
   proceedings and video tapes of the first and second International Workshop
   on Industrial Fuzzy Control and Intelligent Systems (IFIS 91/92).

   Dr. John Yen
   Center for Fuzzy Logic and Intelligent Systems Research
   Texas A&M University
   MS 3112
   Harvey R. Bright Building
   Texas A&M University
   College Station, TX 77843 USA
   Phone: [phone]
   Fax:   [phone]
   Email: [e-mail]


German National Research Center for Computer Science (GMD):

   The GMD supports a fuzzy logic group which does research in

   - adaptive control
   - VLSI design
   - image processing

   Liliane E. Peters
   GMD-SET
   P. O. Box 1316
   D-5205 St. Augustin 1, Germany
   Phone: [phone]
   Fax:   [phone]
   Email: [e-mail]


Swiss Federal Institute of Technology (SFIT):

  Email: [e-mail], [e-mail]

Tokyo Institute of Technology (TITech):

  TITech's Department of Systems Science support Dr. Michio Sugeno's
  laboratory, which does research in practical applications of fuzzy
  logic and fuzzy expert systems.

  Tokyo Institute of Technology
  Department of Systems Science
  4259 Nagatsuta, Midori-ku
  Yokohama 227 Japan
  Phone: [phone] x2641
  Fax:   [phone]
  Email: <name>@sys.titech.ac.jp

  [According to Dr. Michael Griffin ([e-mail]),
  "Don't bother sending e-mail to Professor M. Sugeno, he doesn't use it."]

================================================================
;;; *EOF*