Archivename: fuzzylogic/part1
Lastmodified: Fri Mar 14 11:38:33 1997 by Mark Kantrowitz Version: 1.27 Maintainer: Mark Kantrowitz et al <mkant+fuzzyfaq@cs.cmu.edu> URL: http://www.cs.cmu.edu/Web/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html Size: 85348 bytes, 2142 lines See reader questions & answers on this topic!  Help others by sharing your knowledge ;;; ***************************************************************** ;;; 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+fuzzyfaq@cs.cmu.edu. Note that the mkant+fuzzyfaq@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+fuzzyfaq@cs.cmu.edu. Thanks also go to Michael Arras <arras@forwiss.unierlangen.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.tumuenchen.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://ftp.cs.cmu.edu/user/ai/pubs/faqs/fuzzy/ [128.2.206.173] using username "anonymous" and password "name@host" (substitute your email address) or via AFS in the Andrew File System directory /afs/cs.cmu.edu/project/airepository/ai/pubs/faqs/fuzzy/ and get the file fuzzy.faq. You can also obtain a copy of the FAQ by sending a message to ai+query@cs.cmu.edu with Send Fuzzy FAQ in the message body. The FAQ postings are also archived in the periodic posting archive on rtfm.mit.edu:/pub/usenet/news.answers/fuzzylogic/ [18.181.0.24] If you do not have anonymous ftp access, you can access the archive by mail server as well. Send an Email message to mailserver@rtfm.mit.edu with "help" and "index" in the body on separate lines for more information. An automatically generated HTML version of the Fuzzy Logic FAQ is accessible by WWW as part of the AIrelated FAQs Mosaic page. The URL for this resource is http://www.cs.cmu.edu/Web/Groups/AI/html/faqs/top.html The direct URL for the Fuzzy FAQ is http://www.cs.cmu.edu/Web/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html If you need to cite the FAQ for some reason, use the following format: Mark Kantrowitz, Erik Horstkotte, and Cliff Joslyn, "Answers to Frequently Asked Questions about Fuzzy Logic and Fuzzy Expert Systems", comp.ai.fuzzy, <month>, <year>, ftp://ftp.cs.cmu.edu/user/ai/pubs/faqs/fuzzy/, mkant+fuzzyfaq@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, Mailservers 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: ;;; 26JUN95 mk Updated listing for American NeuraLogix; new name and ;;; contact information. ;;; ;;; 1.20: ;;; 13JUL95 mk Added email address to hyperlogic entry. ;;; ;;; 1.21: ;;; 31OCT95 mk Added URL to HyperLogic page. ;;; 15NOV95 mk Updated Technical Univ of Vienna Fuzzy mailing list entry. ;;; 20FEB96 mk Added entry on LPA's FLINT. ================================================================ Subject: [1] What is the purpose of this newsgroup? Date: 15APR93 The comp.ai.fuzzy newsgroup was created in January 1993, for the purpose of providing a forum for the discussion of fuzzy logic, fuzzy expert systems, and related topics. ================================================================ Subject: [2] What is fuzzy logic? Date: 15APR93 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 nonmembership, 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 nonmembership, 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 twodimensional 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. 16, 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 4:1330, 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 8:338353, 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 139. ================================================================ Subject: [3] Where is fuzzy logic used? Date: 15APR93 Fuzzy logic is used directly in very few applications. The Sony PalmTop apparently uses a fuzzy logic decision tree algorithm to perform handwritten (well, computer lightpen) Kanji character recognition. Most applications of fuzzy logic use it as the underlying logic system for fuzzy expert systems (see [4]). ================================================================ Subject: [4] What is a fuzzy expert system? Date: 21APR93 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 viceversa. 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 "MAXMIN inference" and "SUMPRODUCT 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 "MINMAX" and "PRODUCTSUM"  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 AVERAGEOFMAXIMA method, returns the average of the variable values at which the maximum truth value occurs. For example, go back to our previous examples. Using MAXMIN inferencing and AVERAGEOFMAXIMA defuzzification results in a crisp value of 8.4 for z. Using PRODUCTSUM 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 6062, 1989. ================================================================ Subject: [5] Where are fuzzy expert systems used? Date: 15APR93 To date, fuzzy expert systems are the most common use of fuzzy logic. They are used in several wideranging fields, including: o Linear and Nonlinear Control o Pattern Recognition o Financial Systems o Operation Research o Data Analysis ================================================================ Subject: [6] What is fuzzy control? Date: 17MAR95 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 (proportionalintegral 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", SpringerVerlag, New York, 1993. 316 pages, ISBN 0387563628. [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 9810210426. T. Terano, K. Asai, M. Sugeno, editors, "Applied Fuzzy Systems", translated by C. Ascchmann, AP Professional. ISBN 0126852421. ================================================================ Subject: [7] What are fuzzy numbers and fuzzy arithmetic? Date: 15APR93 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 1:339, CRC Press, Boca Raton, 1987. Dubois, Didier, and Prade, Henri, "Mean Value of a Fuzzy Number", Fuzzy Sets and Systems 24(3):279300, 1987. Kaufmann, A., and Gupta, M.M., "Introduction to Fuzzy Arithmetic", Reinhold, New York, 1985. ================================================================ Subject: [8] Isn't "fuzzy logic" an inherent contradiction? Why would anyone want to fuzzify logic? Date: 15APR93 Fuzzy sets and logic must be viewed as a formal mathematical theory for the representation of uncertainty. Uncertainty is crucial for the management of real systems: if you had to park your car PRECISELY in one place, it would not be possible. Instead, you work within, say, 10 cm tolerances. The presence of uncertainty is the price you pay for handling a complex system. Nevertheless, fuzzy logic is a mathematical formalism, and a membership grade is a precise number. What's crucial to realize is that fuzzy logic is a logic OF fuzziness, not a logic which is ITSELF fuzzy. But that's OK: just as the laws of probability are not random, so the laws of fuzziness are not vague. ================================================================ Subject: [9] How are membership values determined? Date: 15APR93 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. Adhoc 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, 45:261274, 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 97102, IFAC, Oxford, 1984. ================================================================ Subject: [10] What is the relationship between fuzzy truth values and probabilities? Date: 21NOV94 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, DemsterShafer 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. 16). Delgado, M., and Moral, S., "On the Concept of PossibilityProbability Consistency", Fuzzy Sets and Systems 21:311318, 1987. Dempster, A.P., "Upper and Lower Probabilities Induced by a Multivalued Mapping", Annals of Math. Stat. 38:325339, 1967. Henkind, Steven J., and Harrison, Malcolm C., "Analysis of Four Uncertainty Calculi", IEEE Trans. Man Sys. Cyb. 18(5)700714, 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 9398, NorthHolland, Amsterdam, 1982. Klir, George, "Is There More to Uncertainty than Some Probability Theorists Would Have Us Believe?", Int. J. Gen. Sys. 15(4):347378, 1989. Klir, George, "Generalized Information Theory", Fuzzy Sets and Systems 40:127142, 1991. Klir, George, "Probabilistic vs. Possibilistic Conceptualization of Uncertainty", in Analysis and Management of Uncertainty, B.M. Ayyub et. al. (editors), pages 1325, Elsevier, 1992. Klir, George, and Parviz, Behvad, "ProbabilityPossibility Transformations: A Comparison", Int. J. Gen. Sys. 21(1):291310, 1992. Kosko, B., "Fuzziness vs. Probability", Int. J. Gen. Sys. 17(23):211240, 1990. Puri, M.L., and Ralescu, D.A., "Fuzzy Random Variables", J. Math. Analysis and Applications, 114:409422, 1986. Shafer, Glen, "A Mathematical Theory of Evidence", Princeton University, Princeton, 1976. ================================================================ Subject: [11] Are there fuzzy state machines? Date: 15APR93 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):191208, 1976. ================================================================ Subject: [12] What is possibility theory? Date: 15APR93 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 2:458467, 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 1:328, 1978. ================================================================ Subject: [13] How can I get a copy of the proceedings for <x>? Date: 15APR93 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: 845305 ================================================================ Subject: [14] Fuzzy BBS Systems, Mailservers and FTP Repositories Date: 24AUG93 Aptronix FuzzyNET BBS and Email Server: 4082611883, 12009600 N/8/1 This BBS contains a range of fuzzyrelated 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: 6175584241, 12009600 N/8/1 Motorola FREEBBS: 5128913733, 12009600 E/7/1 Ostfold Regional College Fuzzy Logic Anonymous FTP Repository: ftp.dhhalden.no:/pub/Fuzzy/ is a recentlystarted ftp site for fuzzyrelated 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: ftp://ntia.its.bldrdoc.gov/pub/ 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 Usercontributed files. To receive instructions on how to access the server, send the following message, with no subject, to fuzzyserver@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: 5122197828/7848, DS/HST 120019,200 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 ftp://utsun.s.utokyo.ac.jp/fj/fj.sources/v25/ [133.11.11.11]. This yields the "GeneralPurpose 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.rwthaachen.de:/pub/CNCL [137.226.4.111]. Contact Martin Junius <mj@dfv.rwthaachen.de> for more information. A demo version of Aptronix's FIDE 2.0 is available by anonymous ftp from ftp://ftp.cs.cmu.edu/user/ai/areas/fuzzy/code/fide/. FIDE is a PCbased 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 rulebased 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 Ccode) from sample data. FuNeGen 1.0 and the papers/reports describing the application and the theoretical background can be obtained by anonymous ftp from ftp://obelix.microelectronic.etechnik.thdarmstadt.de/pub/neurofuzzy/ NEFCONI (NEural Fuzzy CONtroller) is an X11 simulation environment based on Interviews designed to build and test neural fuzzy controllers. NEFCONI 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 NEFCONI communicates with another process, that implements a simulation of a dynamical process. NEFCONI 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. NEFCONI is available by anonymous ftp from ftp://ibr.cs.tubs.de/pub/local/nefcon/ [134.169.34.15] as the file nefcon_1.0.tar.gz. If you are using NEFCONI, please send an email message to the author, Detlef Nauck <nauck@ibr.cs.tubs.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>. ================================================================ Subject: [15] Mailing Lists Date: 15APR93 The FuzzyMail and NAFIPSL 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 nafipsl@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 nafipsl@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 twoway gatewayed to the aforementioned NAFIPSL 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 mailloop avoidance and duplicatemessage 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/fuzzymail/index.html as well as a ftp://mira.dbai.tuwien.ac.at/pub/ 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 fuzzymail@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 fuzzymail 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 fuzzymail@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.yamagatau.ac.jp To subscribe, send a message to aserver@ea5.yz.yamagatau.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, 4316 Jonan, Yonezawa 992 Japan, phone +81238225181, fax +81238242752, email nakatsu@ea5.yz.yamagatau.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: fuzzyjp@sys.es.osakau.ac.jp This is an unmoderated list, with mostly original contributions in Japanese (JIScode). To subscribe, send subscriptions to the listserv fuzzyjprequest@sys.es.osakau.ac.jp If you need to speak to a human being, send mail to the list owners, fuzzyadmin@tamlab.sys.es.osakau.ac.jp Itsuo Hatono and Motohide Umano of Osaka University. ================================================================ Subject: [16] Bibliography Date: 14AUG95 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://ftp.cs.cmu.edu/user/ai/pubs/bibs/ as the file fuzzybib.text. A short 1985 fuzzy systems tutorial by James Brule is available as http://life.anu.edu.au/complex_systems/fuzzy.html An ascii copy is also available in the gzipped tar file ftp://ftp.cs.cmu.edu/user/ai/areas/fuzzy/doc/intro/ Wolfgang Slany has compiled a BibTeX bibliography on fuzzy scheduling and related fuzzy techniques, including constraint satisfaction, linear programming, optimization, benchmarking, qualitative modeling, decision making, petrinets, production control, resource allocation, planning, design, and uncertainty management. It is available by anonymous ftp from ftp://mira.dbai.tuwien.ac.at/pub/slany/ as the file fuzzyscheduling.bib.Z (or .ps.Z), or by email from listproc@vexpert.dbai.tuwien.ac.at with GET LISTPROC fuzzyscheduling.bib in the message body. NonMathematical 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 ManMachine Studies 24:6575, 1986.] McNeill, Daniel, and Freiberger, Paul, "Fuzzy Logic: The Discovery of a Revolutionary Computer Technology", Simon and Schuster, 1992. ISBN 0671738437. [Mostly history, but many examples of applications.] Negoita, C.V., "Fuzzy Systems", Abacus Press, TunbridgeWells, 1981. Smithson, Michael, "Ignorance and Uncertainty: Emerging Paradigms", SpringerVerlag, New York, 1988. Brubaker, D.I., "Fuzzylogic 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 0121942708 ($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 0124859658 ($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 KnowledgeBased Systems", NorthHolland, Amsterdam, 1986. Kandel, Abraham, "Fuzzy Mathematical Techniques with Applications", AddisonWesley, 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 0133456382. Kosko, Bart, "Neural Networks and Fuzzy Systems", Prentice Hall, Englewood Cliffs, NJ, 1992. ISBN 0136114350. [Very good.] R. Kruse, J. Gebhardt, and F. Klawonn, "Foundations of Fuzzy Systems" John Wiley and Sons Ltd., Chichester, 1994. ISBN 047194243X ($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 0126852456. Translation of "Fajii shisutemu nyumon" (Japanese, 1987). Newly released as "Applied Fuzzy Systems", 1994, 320 pages, ISBN 0126852421 ($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 0442016212, $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 1558602577 paper ($49.95). "A Quarter Century of Fuzzy Systems", Special Issue of the International Journal of General Systems, 17(23), June 1990. R.J. Marks II, editor, "Fuzzy Logic Technology & Applications", IEEE, 1994. IEEE Order# 94CR01016PSP, $59.95 ($48.00 for IEEE members). Order from 1800678IEEE. [Selected papers from past IEEE conferences. Focus is on papers concerning applications of fuzzy systems. There are also some overview papers.] ================================================================ Subject: [17] Journals and Technical Newsletters Date: 24AUG93 INTERNATIONAL JOURNAL OF APPROXIMATE REASONING (IJAR) Official publication of the North American Fuzzy Information Processing Society (NAFIPS). Published 8 times annually. ISSN 0888613X. 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 2126333827, fax 2126333913, 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: 5183875155 Fax: 5183876845 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: 01650114 IEEE TRANSACTIONS ON FUZZY SYSTEMS ISSN 10636706 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 KNOWLEDGEBASED SYSTEMS (IJUFKS) Published 4 times annually. ISSN 02184885. 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. Email worldscp@singnet.com.sg, phone 653825663, fax 653825919. Web pages for this journal: http://www.wspc.co.uk/wspc/Journals/ijufks/ijufks.html Submissions: B BouchonMeunier, editor in chief, LaforiaIBP, Universite Paris VI, Boite 169, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE, phone 33144277003, fax 33144277000, email bouchon@laforia.ibp.fr. ================================================================ Subject: [18] Professional Organizations Date: 15APR93 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 402011297 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 891 Yamashitacho, Nakaku Yokohamashi 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 MissouriCol 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 Telefax: +34.58.243317, +34.58.274258 Email: jverdegay@ugr.es CANADIAN SOCIETY FOR FUZZY INFORMATION AND NEURAL SYSTEMS (CANSFINS) Dr. Madan M. Gupta, Director <guptam@sask.usask.ca> Intelligent Systems Research Laboratory College of Engineering Sakatoon, Saskatchewan, S7N OWO Tel: 3069665451 Fax: 3069668710 Dr. Ralph O. Buchal <rbuchal@charon.engga.uwo.ca> Department of Mechanical Engineering Univ. of Western Ontario London, Ontario, N6A 5B9 Tel: 5196792111, x8454 Fax: 5196613375 Dr. Martin Laplante RES Inc. Suite 501, 100 Sparks Street Ottawa, Ont. KIP5B7 Tel: 6132383690 Fax: 6132355889 ================================================================ Subject: [19] Companies Supplying Fuzzy Tools Date: 15APR93 *** Note: Inclusion in this list is not an endorsement for the product. *** Accel Infotech Spore Pte Ltd: Accel Infotech is a distributor for FUZZC from Byte Craft. FUZZC generates C code that may be crosscompiled to the 6805, Z8C and COP8C microprocessors using separate compilers. FUZZC was reviewed in the March 1993 issue of AI Expert. For more information, send email to accel@solomon.technet.sg, call +657446863 (Richard) or fax +657492467. 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 DK2800 Lyngby Denmark Phone: 4545873217 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 48 bit digital I/O single chip fuzzy microcontroller with EEPROM memory. NLX222 48 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: 4083837200 Fax: 4083837201 Email: 75471.2025@compuserve.com Europe: Applied Marketing & Technology Ltd. Saville Court, Saville Place, Clifton Bristol BS8 4EJ Phone: 1179237594 Fax: 1179237598 Email: 100435.1630@compuserve.com Japan: Nippon Precision Device Nichibei Time 24 Bldg. 35 Tansucho Shinjukiku, Tokyo 162 Phone: 332601411 Fax: 332607100 Adaptive Logic Inc.R&D facility 411 Central Park Drive Sanford, Fl 32771 Phone: 4073225608 Fax: 4073225609 Email: 75471.2032@compuserve.com or info@adaptivelogic.com URL: http://www.adaptivelogic.com/ Aptronix: Products: Fide A MS Windowshosted graphical development environment for fuzzy expert systems. Code generators for Motorola's 6805, 68HC05, and 68HC11, and Omron's FP3000 are available. A demonstration version of Fide is available. Aptronix, Inc. 2150 North First Street, Suite 300 San Jose, Ca. 95131 USA Phone: 4082611888 Fax: 4082611897 Fuzzy Net BBS: 4082611883, 8/n/1 Aria Ltd.: Products: DBfuzzy A library of fuzzy information retrieval for CAClipper. 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 DBfuzzy. Any additional information about DBfuzzy 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: FuzzC "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: 5198886911 Fax: 5197466751 Support BBS: 5198887626 Fril Systems Ltd: FRIL (Fuzzy Relational Inference Language) is a logicprogramming 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 listbased syntax, similar to the early microProlog from LPA. Prolog is a special case of Fril, in which programs involve no uncertainty. Fril runs on Unix, Macintosh, MSDOS, 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://ftp.cs.cmu.edu/user/ai/areas/fuzzy/com/fril/ Fujitsu: Products: MB94100 Singlechip 4bit (?) fuzzy controller. FuziWare: Products: FuziCalc An MSWindowsbased fuzzy development system based on a spreadsheet view of fuzzy systems. FuziWare, Inc. 316 Nancy Kynn Lane, Suite 10 Knoxville, Tn. 37919 USA Phone: 8004726183, 6155884144 Fax: 6155889487 FuzzySoft AG: Product: FuzzySoft Fuzzy Logic Operating System runs under MSWindows, generates Ccode, extended simulation capabalities. Selling office for Germany, Switzerland and Austria (all product inquiries should be directed here) GTS Trautzl GmbbH GottliebDaimlerStr. 9 W2358 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: 6197487384 Fax: 6197487384 (?) HyperLogic Corporation: Products: CubiCalc Windowsbased Fuzzy Logic Shell. Includes fuzzy and plant simulation, plots, file I/O, DDE. CubiCalc RTC Windowsbased Fuzzy Logic Development Environment. Superset of CubiCalc includes runtime 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 Addon to CubiCalc and higher products for automatic rulebase generation. Provides four different generation strategies. HyperLogic Corporation P.O. Box 300010 Escondido, CA 920300010 Tel: 6197462765 Fax: 6197464089 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 MCS96 assembly source code as output. A demonstration version is available. Runs under MSDOS. Inform Software Corp 1840 Oak Street, Suite 324 Evanston, Il. 60201 USA Phone: 7088661838 INFORM GmbH Geschaeftsbereich FuzzyTechnologien Pascalstraese 23 W5100 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) 7308345 Fax: (408) 7308550 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 userdefined membership functions, linear and curved membership lines, automatic propagation of fuzzy values, range of and/or/not combinators, configurable linguistic hedges, standard and userdefined 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: 18009497567 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: 9142380647 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: 6155317008 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 neuralfuzzy 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 multilayer 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): NF2C8AKIT  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 8bit microcontrollers) NeuFuz4 Software Package, Product ordering code (NSID): NF4C8A  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): NF4C8ASYS.  Neural Network Learning Software  Up to 4 inputs, 1 output and 1200 training patterns  Up to 7 membership functions  COP8 Code Generator  COP8 InCircuit Emulator "Debug Module"  RealTime Emulation Microcontroller EPROM Programming  RealTime Trace  Complete Source/Symbolic Debug  OneDay Training on Customer Request  Access to Factory Expert via Telephone (Maximum 16 hrs.) NeuFuz4C Learning Kit, Product ordering code (NSID): NF2CKIT  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 NeuFuz4C Software Package, Product ordering code (NSID): NF4C  Up to 4 inputs, 1 output and 1200 training patterns  Up to 7 membership functions  ANSI Standard C Language Code Generator  OneDay Training on Customer Request  Access to Factory Expert via Telephone (Maximum 16 hrs.) Oki Electric: Products: MSM91U111 A singlechip 8bit fuzzy controller. Europe: Oki Electric Europe GmbH. Hellersbergstrasse 2 D4040 Neuss, Germany Phone: 49213115960 Fax: 492131103539 Hong Kong: Oki Electronics (Hong Kong) Ltd. Suite 18104, Tower 1 China Hong Kong City 33 Canton Road, Tsim Sha Tsui Kowloon, Hong Kong Phone: 37362336 Fax: 37362395 Japan: Oki Electric Industry Co., Ltd. Head Office Annex 7525 Nishishinjuku Shinjukuku Tokyo 160 JAPAN Phone: 81353868100 Fax: 81353868110 USA: Oki Semiconductor 785 North Mary Avenue Sunnyvale, Ca. 94086 USA Phone: 4087201900 Fax: 4087201918 OMRON Corporation: Products: C500FZ001 Fuzzy logic processor module for Omron Cseries PLCs. E5AF Fuzzy process temperature controller. FB30AT FP3000 based PC AT fuzzy inference board. FP1000 Digital fuzzy controller. FP3000 Singlechip 12bit digital fuzzy controller. FP5000 Analog fuzzy controller. FS10AT PCbased software development environment for the FP3000. Japan Kazuaki Urasaki Fuzzy Technology Business Promotion Center OMRON Corporation 20 Igadera, Shimokaiinji Nagaokakyo Shi, Kyoto 617 Japan Phone: 810759515117 Fax: 810759520411 USA Sales (all product inquiries should be directed here) Pat Murphy OMRON Electronics, Inc. One East Commerce Drive Schaumburg, IL 60173 USA Phone: 7088437900 Fax: 7088437787/8568 USA Research Satoru Isaka OMRON Advanced Systems, Inc. 3945 Freedom Circle, Suite 410 Santa Clara, CA 95054 Phone: 4087276644 Fax: 4087275540 Email: isaka@oas.omron.com Togai InfraLogic, Inc.: Togai InfraLogic (TIL for short) supplies software development tools, board, chip and corelevel fuzzy hardware, and engineering services. Contact info@til.com for more detailed information. Products: FC110 (the FC110(tm) Digital Fuzzy Processor (DFPtm)). An 8bit 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 (FPLtm) compiler. FCA VLSI Cores based on Fuzzy Computational Acceleration (FCAtm). FCA10AT FC110based fuzzy accelerator board for PC/ATcompatibles. FCA10VME FC110based fourprocessor VME fuzzy accelerator. FCD10SA FC110based fuzzy processing module. FCD10SBFC FC110based single board fuzzy controller module. FCD10SBus FC110based twoprocessor SBus fuzzy accelerator. FCDS (the FuzzyC 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: 7149758522 Fax: 7149758524 Email: info@til.com Toshiba: Products: T/FC150 10bit fuzzy inference processor. LFZY1 FC150based NEC PC fuzzy logic board. T/FT Fuzzy system development tool. TransferTech GmbH: Products: Fuzzy Control Manager (FMC) Fuzzy shell, runs under MSWindows TransferTech GmbH Cyriaksring 9A D38118 Braunschweig Germany Tel: +49 531 890255 Fax: +49 531 890355 Email: info@transfertech.de URL: http://www.transfertech.de ================================================================ Subject: [20] Fuzzy Researchers Date: 23AUG94 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 ftp://mira.dbai.tuwien.ac.at/pub/mlowner/ or ftp://ftp.cs.cmu.edu/user/ai/areas/fuzzy/doc/whos_who/ ================================================================ Subject: [21] Elkan's "The Paradoxical Success of Fuzzy Logic" paper The presentation of Elkan's AAAI93 paper Charles Elkan, "The Paradoxical Success of Fuzzy Logic", in Proceedings of the Eleventh National Conference on Artificial Intelligence, 698703, 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):68, Spring 1994.) Elkan proves that for a particular set of axiomatizations of fuzzy logic, fuzzy logic collapses to twovalued logic. The proof is correct in the sense that the conclusion follows from the premises. The disagreement concerns the relevance of the premises to fuzzy logic. At issue are the logical equivalence axioms. Elkan has shown that if you include any of several plausible equivalences, such as not(A and not B) == (not A and not B) or B with the min, max, and 1 axioms of fuzzy logic, then fuzzy logic reduces to binary logic. The fuzzy logic community states that these logical equivalence axioms are not required in fuzzy logic, and that Elkan's proof requires the excluded middle law, a law that is commonly rejected in fuzzy logic. Fuzzy logic researchers must simply take care to avoid using any of these equivalences in their work. It is difficult to do justice to the issues in so short a summary. Readers of this FAQ should not assume that this summary is the last word on this topic, but should read Elkan's paper and some of the other correspondence on this topic (some of which has appeared in the comp.ai.fuzzy newsgroup). Two responses to Elkan's paper, one by Enrique Ruspini and the other by Didier Dubois and Henri Prade, may be found as ftp.cs.cmu.edu:/user/ai/areas/fuzzy/doc/elkan/response.txt A final version of Elkan's paper, together with responses from members of the fuzzy logic community, will appear in an issue of IEEE Expert sometime in 1994. A paper by Dubois and Prade will be presented at AAAI94. ================================================================ Subject: [22] Glossary Hedge A hedge is a oneinput truth value manipulation operation. It modifies the shape of the truth function, in a manner analogous to the function of adjectives and adverbs in English. Some examples that are commonly seen in the literature are intensifiers like "very", detensifiers like "somewhat", and complementizers like "not". One might define "very x" as the square of the truth value of x, and define "somewhat x" as the square root of the truth value of x. Then you can make fuzzy logic statements like: y is very low which would evaluate to (y is low) * (y is low). One can think of "not x" as being a hedge in the same sense, defining "not x" as one minus the truth value of x. ================================================================ Subject: [24] Where to send calls for papers (cfp) and calls for participation Date: 15MAY95 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 FraunhoferIAO at <URL:http://www.iao.fhg.de/Library/conferences> as soon as they have been processed. The fuzzymail mailing list (see [15]) scans this newsgroup 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 fuzzymailing list (see [15]), NAFIPSL (see [15]), as well as the internet newsgroup 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* User Contributions:
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