# Introduction

    Archive-name: fuzzy-logic/part1
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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
    discussion of the content of the FAQ posting only by the FAQ
    maintainers. It is not the place to ask questions about fuzzy logic
    and fuzzy expert systems; use the newsgroup comp.ai.fuzzy for that. If
    a question appears frequently in that forum, it will get added to the
    FAQ list.

    The original version of this FAQ posting was prepared by Erik
    Horstkotte of SysSoft <erik@syssoft.com>, with significant
    contributions by Cliff Joslyn <joslyn@kong.gsfc.nasa.gov>.  The FAQ is
    maintained by Mark Kantrowitz <mkant@cs.cmu.edu> with advice from Erik
    and Cliff. To reach us, send mail to mkant+fuzzy-faq@cs.cmu.edu.

    Thanks also go to Michael Arras <arras@forwiss.uni-erlangen.de> for
    running the vote which resulted in the creation of comp.ai.fuzzy,
    Yokichi Tanaka <tanaka@til.com> for help in putting the FAQ together,
    and Walter Hafner <hafner@informatik.tu-muenchen.de>, Satoru Isaka
    <isaka@oas.omron.com>, Henrik Legind Larsen <hll@ruc.dk>, Tom Parish
    <tparish@tpis.cactus.org>, Liliane Peters <peters@borneo.gmd.de>, Naji
    Rizk <mcs@inco.com.lb>, Peter Stegmaier <peter@ifr.ethz.ch>, Prof.
    J.L. Verdegay <jverdegay@ugr.es>, and Dr. John Yen <yen@cs.tamu.edu> for
    contributions to the initial contents of the FAQ.

    This FAQ is posted once a month on the 13th of the month. In between
    postings, the latest version of this FAQ is available by anonymous ftp
    from CMU:

       To obtain the files from CMU, connect by anonymous FTP to
          ftp.cs.cmu.edu:/user/ai/pubs/faqs/fuzzy/  [128.2.206.173]
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          /afs/cs.cmu.edu/project/ai-repository/ai/pubs/faqs/fuzzy/   and get the file fuzzy.faq.

    You can also obtain a copy of the FAQ by sending a message to
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    in the message body.

    The FAQ postings are also archived in the periodic posting archive on
       rtfm.mit.edu:/pub/usenet/news.answers/fuzzy-logic/ [18.181.0.24]
    If you do not have anonymous ftp access, you can access the archive by
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    An automatically generated HTML version of the Fuzzy Logic FAQ is
    accessible by WWW as part of the AI-related FAQs Mosaic page. The URL
    for this resource is
       http://www.cs.cmu.edu/Web/Groups/AI/html/faqs/top.htmlThe direct URL for the Fuzzy FAQ is
       http://www.cs.cmu.edu/Web/Groups/AI/html/faqs/ai/fuzzy/part1/faq.htmlIf you need to cite the FAQ for some reason, use the following format:
       Mark Kantrowitz, Erik Horstkotte, and Cliff Joslyn, "Answers to
       Frequently Asked Questions about Fuzzy Logic and Fuzzy Expert Systems",
       comp.ai.fuzzy, <month>, <year>,
       ftp.cs.cmu.edu:/user/ai/pubs/faqs/fuzzy/fuzzy.faq,
       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 <x>?
     [14] Fuzzy BBS Systems, Mail-servers and FTP Repositories
     [15] Mailing Lists
     [16] Bibliography
     [17] Journals and Technical Newsletters
     [18] Professional Organizations
     [19] Companies Supplying Fuzzy Tools
     [20] Fuzzy Researchers
     [21] Elkan's "The Paradoxical Success of Fuzzy Logic" paper
     [22] Glossary
     [24] Where to send calls for papers (cfp) and calls for participation

    Search for [#] to get to topic number # quickly. In newsreaders which
    support digests (such as rn), [CNTL]-G will page through the answers.

    *** Recent changes:

    ;;; 1.19:
    ;;; 26-JUN-95 mk    Updated listing for American NeuraLogix; new name and
    ;;;                 contact information.
    ;;;
    ;;; 1.20:
    ;;; 13-JUL-95 mk    Added email address to hyperlogic entry.
    ;;;
    ;;; 1.21:
    ;;; 31-OCT-95 mk    Added URL to HyperLogic page.
    ;;; 15-NOV-95 mk    Updated Technical Univ of Vienna Fuzzy mailing list entry.
    ;;; 20-FEB-96 mk    Added entry on LPA's FLINT.

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

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# [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.

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

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# [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.

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

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# [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]).

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

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# [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.

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

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# [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

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

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# [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.

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

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# [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.

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

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# [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.

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

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# [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.

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

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# [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.

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

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# [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.

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

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# [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.

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

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# [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

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

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# [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 <irwin@aptronix.com> for assistance.

    Electronic Design News (EDN) BBS:

        617-558-4241, 1200-9600 N/8/1

    Motorola FREEBBS:

        512-891-3733, 1200-9600 E/7/1

    Ostfold Regional College Fuzzy Logic Anonymous FTP Repository:

        ftp.dhhalden.no:/pub/Fuzzy/ is a recently-started ftp site for
        fuzzy-related material, operated by Ostfold Regional College in
        Norway.  Currently has files from the Togai InfraLogic Fuzzy Logic
        Email Server, Tim Butler's Fuzzy Logic Anonymous FTP Repository, some
        demo programs and source code, and lists of upcoming conferences,
        articles, and literature about fuzzy logic.  Material to be included
        in the archive (e.g., papers and code) may be placed in the incoming/
        directory.  Send email to Randi Weberg <randiw@dhhalden.no>.

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

        ntia.its.bldrdoc.gov:/pub/fuzzy contains information concerning fuzzy
        logic, including bibliographies (bib/), product descriptions and demo
        versions (com/), machine readable published papers (lit/), miscellaneous
        information, documents and reports (txt/), and programs code and compilers
        (prog/). You may download new items into the new/ subdirectory, or send
        them by email to 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, <nagle@shasta.stanford.edu>.

       CNCL is a C++ class library provides classes for simulation, fuzzy
       logic, DEC's EZD, and UNIX system calls. It is available from
       ftp.dfv.rwth-aachen.de:/pub/CNCL [137.226.4.111]. Contact Martin
       Junius <mj@dfv.rwth-aachen.de> for more information.

       A demo version of Aptronix's FIDE 2.0 is available by anonymous ftp
       from ftp.cs.cmu.edu:/user/ai/areas/fuzzy/code/fide/. FIDE is a
       PC-based fuzzy logic design tool. It provides tools for the
       development, debugging, and simulation of fuzzy applications.
       For more information, contact 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 <nauck@ibr.cs.tu-bs.de>.

       The Fuzzy Arithmetic Library is a very simple C++ implementation of a
       fuzzy number representation using confidence intervals, together with
       the basic arithmetic operators and trigonometrical functions. It is
       available by anonymous FTP from
          mathct.dipmat.unict.it:fuzzy [151.97.252.1]
       [Note the system is a VAX running VMS.] For more information, write to
       Salvatore Deodato <deodato@dipmat.unict.it>.

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

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# [15] Mailing Lists

    Date: 15-APR-93

    The Fuzzy-Mail and NAFIPS-L mailing lists are now bidirectionally
    gatewayed to the comp.ai.fuzzy newsgroup.

    NAFIPS Fuzzy Logic Mailing List:

        This is a mailing list for the discussion of fuzzy logic, NAFIPS and
        related topics, located at the Georgia State University.  The last time
        that this FAQ was updated, there were about 225 subscribers, located
        primarily in North America, as one might expect.  Postings to the mailing
        list are automatically archived.

        The mailing list server itself is like most of those in use on the
        Internet.  If you're already familiar with Internet mailing lists, the
        only thing you'll need to know is that the name of the server is

           listproc@listproc.gsu.edu

        and the name of the mailing list itself is

           nafips-l@listproc.gsu.edu

        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@listproc.gsu.edu:
           help
        You will receive a brief set of instructions by email within
        a short time.

        Once you have subscribed, you will begin receiving a copy of each message
        that is sent by anyone to nafips-l@listproc.gsu.edu, and any message that
        you send to that address will be sent to all of the other subscribers.

    Technical University of Vienna Fuzzy Logic Mailing List:

        This is a mailing list for the discussion of fuzzy logic and related
        topics, located at the Technical University of Vienna in Austria.  The
        last time this FAQ was updated, there were about 980 subscribers.
        The list is slightly moderated (only irrelevant mails are rejected)
        and is two-way gatewayed to the aforementioned NAFIPS-L list and to
        the comp.ai.fuzzy internet newsgroup. Messages should therefore be
        sent only to one of the three media, although some mechanism for
        mail-loop avoidance and duplicate-message avoidance is activated.
        In addition to the mailing list itself, the list server gives
        access to some files, including archives and the "Who is Who in Fuzzy
        Logic" database that is currently under construction by Robert Fuller
        <rfuller@finabo.abo.fi>. There is also a WWW interface to the list
        at http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html as well
        as a ftp://mira.dbai.tuwien.ac.at/pub/mlowner site to access such
        files as the whoiswhoinfuzzy file mentioned above.

        Like many mailing lists, this one uses Anastasios Kotsikonas's LISTPROC
        system.  If you've used this kind of server before, the only thing you'll
        need to know is that the name of the server is
          listproc@dbai.tuwien.ac.at
        and the name of the mailing list is
          fuzzy-mail@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@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@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.

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

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# [16] Bibliography

    Date: 14-AUG-95

    A list of books compiled by Josef Benedikt for the FLAI '93 (Fuzzy
    Logic in Artificial Intelligence) conference's book exhibition is
    available by anonymous ftp from
       ftp.cs.cmu.edu:/user/ai/pubs/bibs/as the file fuzzy-bib.text.

    A short 1985 fuzzy systems tutorial by James Brule is available as
       http://life.anu.edu.au/complex_systems/fuzzy.htmlAn ascii copy is also available in the gzipped tar file
       ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/intro/tutorial.tgzWolfgang 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.]

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

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# [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, Farrer Road, PO Box 128,
       Singapore 9128, Rep. of Singapore.
       E-mail worldscp@singnet.com.sg, phone 65-382-5663, fax
       65-382-5919.

       Web pages for this journal:
       http://www.wspc.co.uk/wspc/Journals/ijufks/ijufks.html   Submissions: B Bouchon-Meunier, editor in chief, Laforia-IBP,
       Universite Paris VI, Boite 169, 4 Place Jussieu, 75252 Paris Cedex 05,
       FRANCE, phone 33-1-44-27-70-03, fax 33-1-44-27-70-00, e-mail
       bouchon@laforia.ibp.fr.

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

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# [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: <name>@fuzzy.or.jp

    NORTH AMERICAN FUZZY INFORMATION PROCESSING SOCIETY (NAFIPS)

        Holds a conference and a workshop in alternating years.

        President:
          Dr. Jim Keller
          President NAFIPS
          Electrical & Computer Engineering Dept
          University of Missouri-Col
          Columbia, MO 65211 USA
          Phone +1.314.882.7339
          Email: 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 <guptam@sask.usask.ca>
       Intelligent Systems Research Laboratory
       College of Engineering
       Sakatoon, Saskatchewan, S7N OWO
       Tel: 306-966-5451
       Fax: 306-966-8710

       Dr. Ralph O. Buchal <rbuchal@charon.engga.uwo.ca>
       Department of Mechanical Engineering
       Univ. of Western Ontario
       London, Ontario, N6A 5B9
       Tel: 519-679-2111, x8454
       Fax: 519-661-3375

       Dr. Martin Laplante
       RES Inc.
       Suite 501, 100 Sparks Street
       Ottawa, Ont. KIP-5B7
       Tel: 613-238-3690
       Fax: 613-235-5889

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

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# [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

    Adaptive Logic (formerly American NeuraLogix):

       Products:
          AL220       8 bit fuzzy microcontroller(18 pin DIP or 20 pin
                      SOIC) with A/D & D/A(4I/O).
          NLX221      4-8 bit digital I/O single chip fuzzy microcontroller
                      with EEPROM memory.
          NLX222      4-8 bit analog and digital I/O single chip fuzzy
                      microcontroller.
          NLX230      8 bit microcontroller utilizing fuzzy logic at 30 million
                      rules per second.
          NLX110      Fuzzy Pattern Comparator.
          NLX112/113  Fuzzy Data Correlators.
          INSiGHT IIe Real time emulator, programmer and development
                      software for AL220.
          INSiGHT     Development software for NLX22X family.
          INStANT     Programmer for NLX22X family.
          ADS230      Development System for NLX230.
          ADS110      Development System for NLX110

          Note: The AL220 was named Innovation Of the Year '94 by EDN
          Magazine in the microprocessor category.  Data sheets and
          application notes are available on the products plus local
          application assistance.

            Adaptive Logic Inc.
            800 Charcot Ave., Suite 112
            San Jose, CA 95131
            Phone: 408-383-7200
            Fax:       408-383-7201
            Email:   75471.2025@compuserve.com

            Europe:
            Applied Marketing & Technology Ltd.
            Saville Court, Saville Place, Clifton
            Bristol BS8 4EJ
            Phone: 117-9237594
            Fax:       117-9237598
            Email:    100435.1630@compuserve.com

            Japan:
            Nippon Precision Device
            Nichibei Time 24 Bldg.
            35 Tansu-cho
            Shinjuki-ku, Tokyo 162
            Phone: 332601411
            Fax:       332607100

            Adaptive Logic Inc.-R&D facility
            411 Central Park Drive
            Sanford, Fl 32771
            Phone: 407-322-5608
            Fax:   407-322-5609
            Email: 75471.2032@compuserve.com or info@adaptivelogic.com
            URL:   http://www.adaptivelogic.com/Aptronix:

       Products:
         Fide      A MS Windows-hosted graphical development environment for
                   fuzzy expert systems.  Code generators for Motorola's 6805,
                   68HC05, and 68HC11, and Omron's FP-3000 are available.  A
                   demonstration version of Fide is available.

       Aptronix, Inc.
       2150 North First Street, Suite 300
       San Jose, Ca. 95131 USA
       Phone: 408-261-1888
       Fax:   408-261-1897
       Fuzzy Net BBS: 408-261-1883, 8/n/1

    Aria Ltd.:

       Products:
         DB-fuzzy         A library of fuzzy information retrieval for CA-Clipper.
                          See ftp.cs.cmu.edu:/user/ai/areas/fuzzy/com/aria/ for
                          more information.

       Aria Ltd.
       Dubravska 3
       842 21 Bratislava
       SLOVAKIA
       Phone: (+42 7) 3709 286
       Fax:   (+42 7) 3709 232
       Email: 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.txtFujitsu:

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

    FuziWare:

       Products:
         FuziCalc  An MS-Windows-based fuzzy development system based on a
                   spreadsheet view of fuzzy systems.

       FuziWare, Inc.
       316 Nancy Kynn Lane, Suite 10
       Knoxville, Tn. 37919 USA
       Phone: 800-472-6183, 615-588-4144
       Fax:   615-588-9487

    FuzzySoft AG:

       Product:
          FuzzySoft     Fuzzy Logic Operating System runs under MS-Windows,
                        generates C-code, extended simulation capabalities.

       Selling office for Germany, Switzerland and Austria (all product
       inquiries should be directed here)

       GTS Trautzl GmbbH
       Gottlieb-Daimler-Str. 9
       W-2358 Kaltenkirchen/Hamburg
       Germany
       Phone: (49) 4191 8711
       Fax:   (49) 4191 88665

    Fuzzy Systems Engineering:

       Products:
         Manifold Editor           ?
         Manifold Graphics Editor  ?

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

       Fuzzy Systems Engineering
       P. O. Box 27390
       San Diego, CA 92198 USA
       Phone: 619-748-7384
       Fax:   619-748-7384 (?)

    HyperLogic Corporation:

       Products:
          CubiCalc            Windows-based Fuzzy Logic Shell. Includes
                              fuzzy and plant simulation, plots, file
                              I/O, DDE.

          CubiCalc RTC        Windows-based Fuzzy Logic Development
                              Environment. Superset of CubiCalc includes
                              run-time generator, code libraries, DLL for
                              Windows Applications (incl Visual Basic).

          CubiCard            Superset of CubiCalc RTC with data acquisition
                              capabilties via hardware interface board.

          CubiQuick           Inexpensive version of CubiCalc with limited
                              capabilties for classroom and small projects.
                              Academic discounts available.

          Rule Maker          Add-on to CubiCalc and higher products for
                              automatic rulebase generation. Provides four
                              different generation strategies.

       HyperLogic Corporation
       P.O. Box 300010
       Escondido, CA 92030-0010
       Tel: 619-746-2765
       Fax: 619-746-4089
       Email: prodinfo@hyperlogic.com

       The URL for their home page is http://www.hyperlogic.com/hl. It includes
       product descriptions, pricing information, their Tech Notes on various
       subjects, and several downloadable demonstration programs.

    Inform:

       Products:
         fuzzyTECH 3.0     A graphical fuzzy development environment.  Versions
                           are available that generate either C source code or
                           Intel MCS-96 assembly source code as output.  A
                           demonstration version is available. Runs under MS-DOS.

       Inform Software Corp
       1840 Oak Street, Suite 324
       Evanston, Il. 60201 USA
       Phone:  708-866-1838

       INFORM GmbH
       Geschaeftsbereich Fuzzy--Technologien
       Pascalstraese 23
       W-5100 Aachen
       Tel: (02408) 6091
       Fax: (02408) 6090

    IIS:

       IIS specializes in offering short courses on soft computing.  They
       also perform research and development in fuzzy logic, fuzzy control,
       neural networks, adaptive fuzzy systems, and genetic algorithms.

       Intelligent Inference Systems Corp.
       P.O. Box 2908
       Sunnyvale, CA 94087
       Phone: (408) 730-8345
       Fax:   (408) 730-8550
       email: iiscorp@netcom.com

    LPA, Ltd.:

       FLINT, a Fuzzy Logic INferencing Toolkit, is a versatile fuzzy logic
       inferencing system that makes fuzzy logic technology and fuzzy rules
       available within a sophisticated programming environment. FLINT
       supports standard and user-defined membership functions, linear and
       curved membership lines, automatic propagation of fuzzy values, range
       of and/or/not combinators, configurable linguistic hedges, standard
       and user-defined defuzzification algorithms. FLINT is available as a
       versatile programming toolkit for LPA Prolog running Windows 95/3.1/NT
       or Macintosh and as an extension to LPA's popular expert system
       toolkit, Flex.

       For further information contact:
       Logic Programming Associates Ltd.,
       Studio 4, R.V.P.B., Trinity Road,
       London, SW18 3SX, UK.

       Web: http://www.lpa.co.uk
       US Toll Free: 1-800-949-7567
       Tel: +44 181 871 2016
       Fax: +44 181 874 0449
       Email: lpa@cix.compulink.co.uk

    Metus Systems Group:

       Products:
         Metus Fuzzy Library       A library of fuzzy processing routines for
                                   C or C++.  Source code is available.

       The Metus Systems Group
       1 Griggs Lane
       Chappaqua, Ny. 10514 USA
       Phone: 914-238-0647

    Modico:

       Products:
         Fuzzle 1.8        A fuzzy development shell that generates either ANSI
                           FORTRAN or C source code.

       Modico, Inc.
       P. O. Box 8485
       Knoxville, Tn. 37996 USA
       Phone: 615-531-7008

    National Semiconductor, Santa Clara CA, USA
    http://www.commerce.net/directories/participants/ns/home.html  NeuFuz is aimed at low end controls applications in automotive,
      industrial, and appliance areas.  NeuFuz is a neural-fuzzy technology
      which uses backpropagation techniques to initially select fuzzy rules
      and membership functions.  Initial stages of design using NeuFuz
      technology are performed using training data and backpropagation.
      The result is a fuzzy associative memory (FAM) which implements an
      approximation of the training data.  By implementing a FAM, rather
      than a multi-layer perceptron, the designer has a solution which can
      be understood and tuned to a particular application using Fuzzy Logic
      design techniques.

           NeuFuz4 Learning Kit, Product ordering code (NSID): NF2-C8A-KIT
           - NeuFuz2 Neural Network Learning Software
           - Up to 2 inputs, 1 output
           - 50 training patterns
           - Up to 3 membership functions
           - COP8 Code Generator (COP8 is National's family of 8-bit
                microcontrollers)

          NeuFuz4 Software Package, Product ordering code (NSID): NF4-C8A
           - NeuFuz4 Software
           - Neural Network Learning Software - Up to 4 inputs, 1 output and
              1200 training patterns
           - Up to 7 membership functions
           - COP8 Code Generator

         The NeuFuz4 Development System, Product ordering code: (NSID):
         NF4-C8A-SYS.
           - Neural Network Learning Software - Up to 4 inputs, 1 output and
             1200 training patterns
           - Up to 7 membership functions
           - COP8 Code Generator
           - COP8 In-Circuit Emulator "Debug Module"
             - Real-Time Emulation Microcontroller EPROM Programming
             - Real-Time Trace
             - Complete Source/Symbolic Debug
          - One-Day Training on Customer Request
          - Access to Factory Expert via Telephone (Maximum 16 hrs.)

         NeuFuz4-C Learning Kit, Product ordering code (NSID): NF2-C-KIT
          - Up to 2 inputs, 1 output 50 training patterns
          - Up to 3 membership functions
          - ANSI Standard C Language Code Generator
          - Tutorial Examples for Neural Network Learning and Fuzzy Rule
            Generation

        NeuFuz4-C Software Package, Product ordering code (NSID): NF4-C
          - Up to 4 inputs, 1 output and 1200 training patterns
          - Up to 7 membership functions
          - ANSI Standard C Language Code Generator
          - One-Day Training on Customer Request
          - Access to Factory Expert via Telephone (Maximum 16 hrs.)

    Oki Electric:

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

       Europe:

         Oki Electric Europe GmbH.
         Hellersbergstrasse 2
         D-4040 Neuss, Germany
         Phone: 49-2131-15960
         Fax:   49-2131-103539

       Hong Kong:

         Oki Electronics (Hong Kong) Ltd.
         Suite 1810-4, Tower 1
         China Hong Kong City
         33 Canton Road, Tsim Sha Tsui
         Kowloon, Hong Kong
         Phone: 3-7362336
         Fax:   3-7362395

       Japan:

         Oki Electric Industry Co., Ltd.
         Head Office Annex
         7-5-25 Nishishinjuku
         Shinjuku-ku Tokyo 160 JAPAN
         Phone: 81-3-5386-8100
         Fax:   81-3-5386-8110

       USA:

         Oki Semiconductor
         785 North Mary Avenue
         Sunnyvale, Ca. 94086 USA
         Phone: 408-720-1900
         Fax:   408-720-1918

    OMRON Corporation:

       Products:
         C500-FZ001        Fuzzy logic processor module for Omron C-series PLCs.
         E5AF              Fuzzy process temperature controller.
         FB-30AT           FP-3000 based PC AT fuzzy inference board.
         FP-1000           Digital fuzzy controller.
         FP-3000           Single-chip 12-bit digital fuzzy controller.
         FP-5000           Analog fuzzy controller.
         FS-10AT           PC-based software development environment for the
                           FP-3000.

       Japan

         Kazuaki Urasaki
         Fuzzy Technology Business Promotion Center
         OMRON Corporation
         20 Igadera, Shimokaiinji
         Nagaokakyo Shi, Kyoto 617  Japan
         Phone: 81-075-951-5117
         Fax:   81-075-952-0411

       USA Sales (all product inquiries should be directed here)

         Pat Murphy
         OMRON Electronics, Inc.
         One East Commerce Drive
         Schaumburg, IL 60173 USA
         Phone: 708-843-7900
         Fax:   708-843-7787/8568

       USA Research

         Satoru Isaka
         OMRON Advanced Systems, Inc.
         3945 Freedom Circle, Suite 410
         Santa Clara, CA 95054
         Phone: 408-727-6644
         Fax: 408-727-5540
         Email: 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
       Cyriaksring 9A
       D-38118 Braunschweig
       Germany
       Tel: +49 531 890255
       Fax: +49 531 890355
       Email: info@transfertech.de
       URL: http://www.transfertech.de

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

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# [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 <rfuller@finabo.abo.fi>.

    A copy of this list is also available by anonymous ftp from
       mira.dbai.tuwien.ac.at:/pub/mlowner/whoiswhoinfuzzyor
       ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/whos_who/whos_who.txt================================================================

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# [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.txtA 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.

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

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# [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.

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

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

    Date: 15-MAY-95

    Fuzzy related calls for papers and calls for participation should be
    sent by email to conferences@iao.fhg.de, or posted to the moderated
    newsgroup news.announce.conferences. Both actions will have the same
    effect. Please keep Subject lines informative; if space permits,
    mention the topic and location there, and avoid acronyms unless very
    widely known. Submissions will simultaneously appear in the newsgroup
    news.announce.conferences and on the WorldWideWeb server of
    Fraunhofer-IAO at <URL:http://www.iao.fhg.de/Library/conferences> as
    soon as they have been processed.  The fuzzy-mail mailing list (see
    [15]) scans this news-group for items related to fuzzy and uncertainty.
    Matching messages will be moderated like any other message sent to the
    mailing list, and if selected, will be forwarded to the Asian
    fuzzy-mailing list (see [15]), NAFIPS-L (see [15]), as well as the
    internet news-group comp.ai.fuzzy (see [1]).  Sending it only to
    conferences@iao.fhg.de is normally enough to distribute the message
    efficiently to all the other media.

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

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