Is fuzzy logic passé?

Lotfi A. Zadeh
to BISC-Group

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Dear members of the BISC Group:

    For your information, following is an updated version of a brief report on the impact of fuzzy logic. Comments are welcome.

     Regards,

     Lotfi
Factual Information about the Impact of Fuzzy Logic
L.A. ZADEH

COUNT of PUBLICATIONS

•    Total number of papers with “fuzzy” in title (Google Scholar): 360,000

•    Count of publications containing the word “fuzzy” in title, as cited in INSPEC and MATH.SCI.NET databases.
Compiled on September 11, 2014.

INSPEC Database

1970-1979:   569
1980-1989:   2,375
1990-1999:   21,572
2000-2009: 44,695
2010-present: 31,826
Total:   101,037

MathSciNet Database

1970-1979:   446
1980-1989:   2,474
1990-1999:   5,526
2000-2009: 10,295
2010-present: 7,131
Total:   25,872


Count of citations
•    Number of citations/results of papers by L. Zadeh (Google Scholar): 138,270
•    Number of citations of L. Zadeh’s paper “Fuzzy sets,” Information and Control, 1965 (Google Scholar): 52,484
•    Number of members of the BISC Group (subscribers on BISC mailing list)
worldwide: 990

PATENTS

•    Number of fuzzy-logic-related patents issued: 560,000


JOURNALS

Fuzzy in title
1.    Fuzzy Sets and Systems
2.    IEEE Transactions on Fuzzy Systems
3.    International Journal of Fuzzy Logic and Intelligent Systems
4.    Fuzzy Optimization and Decision Making
5.    Journal of Intelligent & Fuzzy Systems
6.    Fuzzy Economic Review
7.    International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
8.    Journal of Japan Society for Fuzzy Theory and Systems
9.    International Journal of Fuzzy Systems
10.    International Review of Fuzzy Mathematics
11.    Fuzzy Systems and Soft Computing
12.    Turkish Journal of Fuzzy Systems
13.    Annals of Fuzzy Sets, Fuzzy Logic and Fuzzy Systems
14.    Iranian Journal of Fuzzy Systems
15.    Fuzzy Information and Engineering
16.    Advances in Fuzzy Systems
17.    International Journal of Fuzzy System Applications
18.    Advances in Fuzzy Sets and Systems
19.    International Journal of Fuzzy Systems and Rough Systems
20.    International Journal of Fuzzy Logic Systems
21.    Journal of Biomedical Fuzzy Systems Association
22.    Advances in Fuzzy Mathematics
23.    Journal of Fuzzy Mathematics
24.    Journal of Advanced Research in Fuzzy and Uncertain
25.    Fuzzy Systems & AI—Reports & Letters
26.    Neural and Fuzzy Modeling Technology in Economics
27.    International Journal of Fuzzy Systems and Advanced Applications
28.    International Journal of Fuzzy Computation and Modelling
29.    International Journal of Fuzzy Information and Engineering

Soft Computing in title (soft computing=fuzzy logic, neurocomputing and evolutionary computing)
1.    Soft Computing
2.    Applied Soft Computing
3.    Mathware & Soft Computing
4.    Journal of Multiple-Valued Logic and Soft Computing
5.    Applied Computational Intelligence and Soft Computing
6.    Autosoft Journal. Intelligent Automation & Soft Computing
7.    International Journal of Advances in Soft Computing and Its Applications
8.    International Journal of Artificial Intelligence and Soft Computing
9.    International Journal of Soft Computing Applications
10.    International Journal on Soft Computing
11.    International Journal of Soft Computing
12.    International Journal of Mathematics and Soft Computing
13.    International Journal of Soft Computing Simulation and Software Engineering
14.    International Journal of Soft Computing and Bioinformatics
15.    Journal of Artificial Intelligence and Soft Computing Research
16.    International Journal of Soft Computing and Engineering
17.    Fuzzy Systems and Soft Computing
18.    International Journal of Research and Reviews in Soft and Intelligent Computing
19.    International Journal of Factory Automation, Robotics and Soft Computing
20.    International Journal of Biomedical Soft Computing and Human Sciences
21.    Archives for the Philosophy and History of Soft Computing

Impact on pure mathematics
•    Total number of papers with “fuzzy topology” or “fuzzy topological spaces” in title (Google Scholar): 1417
•    Total number of books with "fuzzy topology" or "fuzzy topological spaces" in title (Google Books): 148

--
Lotfi A. Zadeh
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC)
Address:
729 Soda Hall #1776
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720-1776
zadeh@eecs.berkeley.edu
Tel.(home): (510) 526-2569
URL: http://www.cs.berkeley.edu/~zadeh/

Precise reasoning about reasoning with fuzzy words

Comment by Tillers: legal scholars might avoid uttering a great deal of nonsense about imprecise legal concepts if they took the trouble to study fuzzy logic.

Lotfi A. Zadeh:

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Dear members of the BISC Group: 

    The concept of FL-generalization was introduced in my 2008 paper "Is there a need for fuzzy logic?" Information Sciences. The basic importance of FL-generalization has not been fully recognized as yet. For those who are not familiar with FL-generalization, a brief explanation is provided in the following.  

    In large measure, science -- including mathematics -- is based on the classical, Aristotelian, bivalent logic. Bivalent-logic-based science has achieved brilliant successes. But what is striking is that bivalent-logic-based science ignores a basic reality. In human cognition, almost all classes have unsharp (fuzzy) boundaries. Bivalent logic is not the right logic for dealing with such classes, nor is bivalent-logic-based probability theory. What is needed for this purpose is fuzzy set theory and, more broadly, fuzzy logic, FL. Informally, fuzzy logic is a system of reasoning and computation in which the objects of reasoning and computation are classes with unsharp (fuzzy) boundaries.  

    The point of departure in fuzzy set theory is a generalization of the concept of a set to the concept of a fuzzy set. A fuzzy set, A, in a space, U, is a graduated class of elements of U. Graduation involves association of each element, u, of U with its grade of membership in A. This very simple generalization has wide-ranging ramifications. 

    Let T be a bivalent-logic-based theory, formalism, algorithm, concept, etc. T is FL-generalized by adding to T the concept of a fuzzy set along with associated concepts and techniques. The result of FL-generalization is fuzzy T.Examples. Fuzzy arithmetic, fuzzy linear programming, fuzzy control, fuzzy stability, fuzzy support vector machine, fuzzy group theory, fuzzy topology, fuzzy convex set, fuzzy back-propagation algorithm, fuzzy probability, etc. T may be viewed as a special case of fuzzy T. FL-generalization is a matter of degree, reflecting the extent to which sets in T are replaced by fuzzy sets. In the limit, FL-generalization of T involves a shift in the foundations of T from bivalent logic to fuzzy logic. 

    What is gained by FL-generalization? There are two principal rationales. First, FL-generalization opens the door to construction of better models of reality. There is a fundamental conflict between two realities. In the world of human cognition, almost all concepts are classes with unsharp (fuzzy) boundaries, that is, are a matter of degree. In the world of science, almost all definitions are bivalent, with no degrees allowed. Here are a few examples. In economics, the official definition of recession is bivalent. Specifically, economy is in a state of recession if the GDP declined in two successive quarters. Realistically, recession is not a bivalent concept -- it is a matter of degree. In probability theory, stationarity is defined as a bivalent concept. Realistically, stationarity is a matter of degree. In stability theory, stability is defined as a bivalent concept. Realistically, stability is a matter of degree, and so on, and on and on. FL-generalization of definitions, serves an important purpose--replacement of bivalent definitions with fuzzy-logic-based definitions which are better models of reality. 

    The second rationale has a position of centrality in applications of fuzzy logic. Specifically, the second rationale involves an exploitation of tolerance for imprecision through replacement of numbers with precisiated words. A word is precisiated by representing it as a label of a fuzzy set which has a specified membership function. A striking example of exploitation of a tolerance for imprecision is the problem of stabilization of an inverted pendulum. The traditional approach starts with formulation of differential equations of motion, followed by application of stability theory. In the fuzzy-logic-based approach, a small number of very simple fuzzy if-then rules, with precisiated words in the antecedents and consequents, are employed to describe the dynamics of the inverted pendulum. This is the essence of what is called the Fuzzy Logic Gambit. Fuzzy Logic Gambit is an essential ingredient of the formalism of Computing with Words (CWW). Basically, CWW may be viewed as a progression from the use of numbers to the use of precisiated words. 

     In summary, FL-generalization may be viewed as an important instrument of generalization in which the point of departure is replacement of the concept of a set with the concept of a fuzzy set. In large measure, scientific progress is driven by a quest for better models of reality. What I see in my crystal ball is the following. In coming years, more and more theories, formalisms, algorithms and concepts will be FL-generalized. This is likely to be the case even in mathematics--a discipline in which the word "fuzzy" strikes a dissonant note. What should be recognized is that shifting foundations of a theory from bivalent logic to fuzzy logic need not involve a loss of rigor and precision. Example. Fuzzy topology is every bit as rigorous and precise as classical topology. Comments are welcome.

                 Regards,

                 Lotfi

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Lotfi A. Zadeh
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC)

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How is fuzzy logic doing? Is it passé?

Professor Lotfi Zadeh sent the following message today to his discussion list (BISC):

How is fuzzy logic doing? A significant measure is the number of publications with "fuzzy" in title (annually). My administrative assistant, Ixel Chavez, has compiled the information which follows. Comments are welcome.

    Regards,

    Lotfi

 Annual number of publications with "fuzzy" in title (Google Scholar)1993: 5,030
1994: 5,700
1995: 6,340
1996: 6,620
1997: 6,810
1998: 7,130
1999: 7,650
2000: 7,620
2001: 8,260
2002: 8,650
2003: 9,240
2004: 10,900
2005: 12,300
2006: 13,900
2007: 14,800
2008: 16,000
2009: 17,900
2010: 18,700
2011: 18,900
2012: 18,700
2013: 17,000

Total: 238,150 (20 year total)

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An Important Message about Fuzzy Logic

The recent award of a major scientific prize to Lotfi Zadeh provoked many admirers from around the world to send along their congratulations on a discussion list devoted to fuzzy logic and soft computing. In so doing, these well-wishers also made a variety of comments about fuzzy logic. Professor Zadeh eventually responded with a substantive comment. (See below.) His comment is important and enlightening in several different ways. Although I will let his comment speak for itself, I do wish to emphasize the astonishing reach of fuzzy logic that Zadeh highlights. It is also worth mentioning that the use of fuzzy logic is apparently accelerating rather than plateauing. There is thus reason to think and to hope that an increasing number of legal scholars (in addition to luminaries such as Kevin Clermont and Lothar Phillips) will decide to use fuzzy logic to explore reasoning about and in law. It is high time that they do so!   

The message from Professor Zadeh: 

Dear members of the BISC Group:

    The BBVA Award has rekindled discussions and debates regarding what fuzzy logic is and what it has to offer. The discussions and debates brought to the surface many misconceptions and misunderstandings. A major source of misunderstanding is rooted in the fact that fuzzy logic has two different meanings -- fuzzy logic in a narrow sense, and fuzzy logic in a wide sense. Informally, narrow-sense fuzzy logic is a logical system which is a generalization of multivalued logic. An important example of narrow-sense fuzzy logic is fuzzy modal logic. In multivalued logic, truth is a matter of degree. A very important distinguishing feature of fuzzy logic is that in fuzzy logic everything is, or is allowed to be, a matter of degree. Furthermore, the degrees are allowed to be fuzzy. Wide-sense fuzzy logic, call it FL, is much more than a logical system. Informally, FL is a precise system of reasoning and computation in which the objects of reasoning and computation are classes with unsharp (fuzzy) boundaries. The centerpiece of fuzzy logic is the concept of a fuzzy set. More generally, FL may be a system of such systems. Today, the term fuzzy logic, FL, is used preponderantly in its wide sense. This is the sense in which the term fuzzy logic is used in the sequel. It is important to note that when we talk about the impact of fuzzy logic, we are talking about the impact of FL. Intellectually, narrow-sense fuzzy logic is an important part of FL, but volume-wise it is a very small part. In fact, most applications of fuzzy logic involve no logic in its traditional sense. 

    What is not widely recognized within the scientific community and the general public, is that fuzzy logic has become a vast enterprise.There are over 280,000 papers in the literature with fuzzy in title. There are 25 journals with fuzzy in title. There are close to 25,000 fuzzy-logic-related patents issued or applied for in the United States and Japan. There is a long list of applications ranging from digital cameras to fraud detection systems. Particularly worthy of note, on one end, is the fuzzy logic subway system in Sendai, a city of over 1 million in Japan. On the other end, numerically, is Omron's 120 million fuzzy logic blood pressure meters.

    Most, but not all of the constituents of fuzzy logic are what are called FL-generalizations of traditional, bivalent-logic-based systems of reasoning and computation. Examples. Fuzzy arithmetic, fuzzy cluster analysis, fuzzy differential equations, fuzzy control, fuzzy linear programming, etc. FL-generalization of a theory or a formalism, T, involves introduction into T of the concept of a fuzzy set, followed by addition of related concepts and techniques. FL-generalization may be applied to any field, any theory, any system, any formalism and any algorithm. The  fundamental importance of FL-generalization derives from the fact that in the real world almost all classes have unsharp (fuzzy) boundaries. As a consequence, FL-generalization opens the door to construction of better models of reality.

    It is of interest to observe that the impact of FL-generalization is growing in visibility and importance in mathematics -- a field in which precision plays a quintessential role. We see a growing number of papers with fuzzy in title in many branches of mathematics, among them topology, algebra, differential equations, group theory, set theory, and functional analysis. What may come as a surprise to many is that Math.Sci.Net database lists over 22,383 papers with fuzzy in title. I did not anticipate that this will happen when I wrote my first paper on fuzzy sets. My expectation was that the concept of a fuzzy set would find its main applications in the realm of soft, human-centered sciences. 

    When it comes to practical application of fuzzy logic, there is a major source of misunderstanding. Fundamentally, fuzzy logic is aimed at precisiation of what is imprecise. But in many of its applications fuzzy logic is used, paradoxically to imprecisiate what is precise. In such applications, there is a tolerance for imprecision, which is exploited through the use of fuzzy logic. Precisiation carries a cost. Imprecisiation reduces cost and enhances tractability. This is what I call the Fuzzy Logic Gambit. What is important to note is  that precision has two different meanings: precision in value and precision in meaning. In the Fuzzy Logic Gambit what is sacrificed is precision in value, but not precision in meaning. More concretely, in the Fuzzy Logic Gambit imprecisiation in value is followed by precisiation in meaning. An example is Yamakawa's inverted pendulum. In this case, differential equations are replaced by fuzzy if-then rules in which words are used in place of numbers. What is precisiated is the meaning of words.

        Some critics have been saying that fuzzy logic is a passing fad. This assessment of fuzzy logic fails to recognize that the world we live in is, in large measure, a world of fuzzy classes, and that science has much to gain from shifting its foundation from classicalAristotelian logic to fuzzy logic. Comments are welcome.
                         Regards to all,

                         Lotfi

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Fuzzy Logic and Fuzzy Thinking in Law

I have been thinking about Lotfi Zadeh and his fuzzy logic. Indeed, I have been thinking, off and on, about Zadeh's fuzzy logic for decades. The law is full of fuzzy thinking. If there is a precise way to think about fuzzy ideas, legal scholars, judges, etc., should use it. On reflection and re-reflection and re-re-reflection etc., I do believe that Zadeh has given us important tools for radically better ways of thinking about fuzzy thinking.

It has often been said that the standard probability calculus can do all that fuzzy logic does. But such claims sometimes have the air of gratuitous dicta: one wants to see the proof in such probabilistic pudding. The fuzziness and vagueness of legal concepts (and many other types of concepts) usually can only awkwardly be characterized in terms of uncertainty. Legal principles, concepts, etc., often employ vague notions. There is a difference between a vague concept and an uncertain proposition. Probability theory sometimes deals well with uncertain propositions - but not so well with vague and fuzzy concepts.   In many of its iterations, Zadeh's fuzzy logic does not seem to reach beneath the surface of our fuzzy thinking (e.g., our conventional legal thinking), but seems just to accept the fuzziness of ordinary thinking. It might be said that fuzzy logic, in some its iterations, just describes the natural behavior of fuzzy terms, operators ("and" "or" etc.), and the like, and does not attempt to identify the causes, bases, or foundations of the fuzzy ideas we have and use.  This creates a puzzle: How can a logic that does not purport to identify the foundations, sources, or causes of fuzzy thinking give us useful new thoughts? (This question assumes that fuzzy thinking is a sort of disease.)  

The answer, I think, lies in the notion of tacit knowledge. There is genuine knowledge buried in some or much of our "ordinary" fuzzy thinking. (If that were not the case, few of us would survive even for one day.) Fuzzy logic's proven successes suggest that fuzzy logic may offer a way to uncover, or display, much "innate," or tacit, human knowledge.

These are admittedly deep and possibly murky waters, and I confess I do not have the ability to swim through them easily. I console myself with the thought that the acquisition of knowledge is a collective human enterprise and that there will be others who may be able to build on some of the paltry number of insights I may have acquired over the years. But if it turns out I have not learned much of enduring value about fuzzy thinking, the efforts I have made may have been worth the candle - because people can learn by studying other people's errors. &&&     The dynamic evidence page
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Lotfi Zadeh Wins Major Award

LOTFI A. ZADEH

The BBVA Foundation Frontiers of Knowledge Award in the Information and Communication Technologies (ICT) category has been granted in this fifth edition to the electrical engineer Lotfi A. Zadeh, “for the invention and development of fuzzy logic.” This “revolutionary” breakthrough, affirms the jury in its citation, has enabled machines to work with imprecise concepts, in the same way humans do, and thus secure more efficient results more aligned with reality. In the last fifty years, this methodology has generated over 50,000 patents in Japan and the U.S. alone. 

On hearing of the award, Zadeh remarked that it meant a lot to him for several reasons: “First, because fuzzy logic has been somewhat controversial. Some people have greeted it with enthusiasm but others have been skeptical. It also has a special significance for me because I am a great admirer of Spain and the Spanish people. I’d therefore like to take this opportunity to express my deep appreciation to all those who were involved in my receiving this award, particularly Luis Magdalena and Enric Trillas of the European Centre for Soft Computing in Mieres, who were among those putting forward my nomination."

Classical logic, based on class membership, imposes that an element should strictly belong ir not belong to a clearly demarcated set, like for instance the set of even numbers. But reality is a lot more complex. Hence we have groups, classes and sets whose boundaries are blurred, like that of “good basketball players.” To belong to this set, a basketball player must “be tall” and “shoot well”, but these concepts are imprecise. A binary system would specify, for example, that “be tall” equates to “measure more than 185 cm” and discard all players below this height, regardless of their shooting prowess. But fuzzy logic, like a human coach, would find room in the set of good players for one who measures 184 cm but is an excellent shooter. In this sense, what fuzzy logic does is bridge the gap between classical logic and the real world.

This indeed is what Zadeh was seeking when the began the research that led him to fuzzy logic: “As an engineer, I was always convinced that mathematics held the answers to almost any problem, but I also realized that classical mathematics was constrained by its inability to tolerate imprecision.” To get over this shortcoming, Zadeh turned to the human model: “We humans have a remarkable capability to reason and make decisions in an environment of uncertainty and incompleteness of information (…). The principal objective of fuzzy logic is the formalization of this capability.”

Human beings intuitively apply fuzzy logic to their decisions, juggling imprecise data and weighing up each relevant element. Zadeh’s contribution was to apply such logic to the decision-making processes of systems and computers, so they cease to operate as mere calculating machines and become capable of evaluating degrees and shades of reality and deciding accordingly in an autonomous or semi-autonomous fashion (with little or no human intervention). 

According to the jury, the contributions of Lotfi A. Zadeh (Baku, Azerbaijan, 1921) have been “enthusiastically adopted by industry, where thousands of engineers have designed a whole plethora of complex and intelligent systems (…)."

But Zadeh’s work has also changed the face of numerous industrial processes, where it has simplified design, providing more efficient products that are easier to use and more tractable to change, while bringing down production costs. 

A seminal paper

En 1965, Lotfi Zadeh articulated fuzzy sets for the first time in a paper that would come to be among the most cited of the 20th century, with over 35,000 mentions. And the next step from there was the development of fuzzy logic, a brilliant contribution to extending the frontiers of knowledge. Indeed Zadeh is defined in the jury’s citation as the founder of “a new field of research which has proved powerful in many application domains.”

The controversy around fuzzy logic began with the name: “The word fuzzy has a pejorative connotation in English, and this turned out to be a handicap when it came to gaining the acceptance of the scientific community. But it was the word that came closest to what I had in mind. In Asia, however, they don’t have problems with the word fuzzy, so they were more receptive to my work. They also have a culture that accepts shades of grey, as opposed to the western – Cartesian – tradition where everything is either black or white.”

This was perhaps the reason, he speculates, that one of the earliest applications of his concept was the automated subway system in the Japanese city of Sendai. 

Fuzzy logic opened the door to machine understanding of such imprecise instructions as “brake smoothly” or “refrigerate until the air is cool,” which would be instantly understood by any human being acquainted with the system, but are utterly impenetrable for a conventional computer program. The conceptual shift was so abrupt that Zadeh initially had to face the skepticism of many scientist colleagues, until the success of the practical applications of his theory dissipated all such doubts. 

Zadeh’s work has enabled us to communicate with machines through an increasingly natural, human language. 

The laureate, still working at the age of 91, sees this as the most promising research avenue in the fuzzy logic field, and hopes to author some further advance that will connect computers and systems more closely with natural language.   

International jury

The jury in this category was chaired by George Gottlob, Professor of Computer Science at the University of Oxford (United Kingdom), with Ramón López de Mántaras, Director of the Artificial Intelligence Research Institute of the Spanish National Research Council (CSIC) acting as secretary. Remaining members were Oussama Khatib, Professor in the Artificial Intelligence Laboratory in the Computer Sciences Department of Stanford University (United States), Rudolf Kruse, Head of the Department of Knowledge Processing and Language Engineering at Otto-von-Guerike-Universität Magdeburg (Germany), Mateo Varelo, Director of the Barcelona Supercomputing Center (Spain) and Joos Vandewalle, Head of the SDC Division in the Department of Electrical Engineering at the Katholieke Universiteit Leuven (Belgium).

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Is Fuzzy Logic Passé?

About ten years ago, I mentioned to a colleague that I was wrestling, in a fuzzy way, with some question about fuzzy logic. He replied (whimsically?). "Isn't fuzzy logic passé?"
Well, I now have an answer for him. Professor Lotfi Zadeh today issued the following report to subscribers to his discussion list BISC (to join BISC send message to sympa@lists.EECS.Berkeley.EDU with the following command in the body of his/her email message: subscribe bisc-group; or from another account, subscribe bisc-group your_email_address): Report on the Impact of Fuzzy Logic   PATENTS  Number of fuzzy-logic-related patents applied for or issued in Japan: 22,541 (not  updated) Number of fuzzy-logic-related patents (containing the word “fuzzy”) applied for or issued in the US: 33,022
JOURNALS Fuzzy in title
Fuzzy Sets and Systems  IEEE Transactions on Fuzzy Systems  Fuzzy Optimization and Decision Making  Journal of Intelligent & Fuzzy Systems  Fuzzy Economic Review  International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Journal of Japan Society for Fuzzy Theory and Systems  International Journal of Fuzzy Systems International Review of Fuzzy Mathematics  Fuzzy Systems and Soft Computing Turkish Journal of Fuzzy Systems Annals of Fuzzy Sets, Fuzzy Logic and Fuzzy Systems Iranian Journal of Fuzzy Systems Fuzzy Information and Engineering Advances in Fuzzy Systems  International Journal of Fuzzy System Applications  Advances in Fuzzy Sets and Systems  International Journal of Fuzzy Systems and Rough Systems  International Journal of Fuzzy Logic Systems  Journal of Biomedical Fuzzy Systems Association  Advances in Fuzzy Mathematics Journal of Fuzzy Mathematics Journal of Advanced Research in Fuzzy and Uncertain Fuzzy Systems & AI—Reports & Letters Neural and Fuzzy Modeling Technology in Economics Soft  Computing in title 1.     Soft Computing2.     Applied Soft Computing3.     Mathware & Soft Computing4.     Journal of Multiple-Valued Logic and Soft Computing5.     Applied Computational Intelligence and Soft Computing6.     Autosoft Journal. Intelligent Automation & Soft Computing7.     International Journal of Advances in Soft Computing and Its Applications8.     International Journal of Artificial Intelligence and Soft Computing9.     International Journal of Soft Computing Applications10.  International Journal on Soft Computing11.  International Journal of Soft Computing12.  International Journal of Mathematics and Soft Computing13.  International Journal of Soft Computing Simulation and Software Engineering14.  International Journal of Soft Computing and Bioinformatics15.  Journal of Artificial Intelligence and Soft Computing Research16.  International Journal of Soft Computing and Engineering17.  Fuzzy Systems and Soft Computing18.  International Journal of Research and Reviews in Soft and Intelligent Computing19.  International Journal of Factory Automation, Robotics and Soft Computing20.  International Journal of Biomedical Soft Computing and Human Sciences  
COUNT of PUBLICATIONS   Count of publications containing the word “fuzzy” in the title, as cited in INSPEC and MATH.SCI.NET          databases. Compiled on August 13, 2012. INSPEC Database
1970-1979:   567 1980-1989:   2,375 1990-1999:   21,555 2000-2009: 44,615 2010-present:  16,247 Total:   85,359   MathSciNet Database   1970-1979:   446 1980-1989:   2,474 1990-1999:   5,526 2000-2009: 10,281 2010-present: 2895 Total:   21,622   Total number of papers with “fuzzy” in title (Google Scholar): 281,000 Number of citations/results of papers by L. Zadeh (Google Scholar): 101,802 Number of citations of L. Zadeh’s paper “Fuzzy sets,” Information and Control, 1965 (Google Scholar): 36,933 Number of members of the BISC Group (subscribers on BISC mailing list) worldwide: 1010 --  Lotfi A. Zadeh  Professor Emeritus Director, Berkeley Initiative in Soft Computing (BISC)  Address:  729 Soda Hall #1776 Computer Science Division Department of Electrical Engineering and Computer Sciences University of California  Berkeley, CA 94720-1776  zadeh@eecs.berkeley.edu  Tel.(office): (510) 642-4959  Fax (office): (510) 642-1712  Tel.(home): (510) 526-2569  Fax (home): (510) 526-2433  URL: http://www.cs.berkeley.edu/~zadeh/ BISC Homepage URLs URL: http://zadeh.cs.berkeley.edu/ &&& The dynamic evidence page

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Fuzzy Logic, 1965

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Lotfi Zadeh on Misconceptions about Fuzzy Logic

Lotfi A. Zadeh	Mon, May 7, 2012 at 8:45 PM
Reply-To: bisc-group@lists.eecs.berkeley.edu To: bisc-group@lists.eecs.berkeley.edu
********************************************************************* Berkeley Initiative in Soft Computing (BISC) ********************************************************************* Dear members of the BISC [Berkeley Initiative on Soft Computing] Group: Sometimes labels are misleading. This applies to fuzzy logic. There are many kinds of logical systems, some going back to antiquity and some of recent vintage. Among them are:  Aristotelian logic, modal logic, deontic logic, multivalued logic, dynamic logic, probabilistic logic, etc. A common misconception is that fuzzy logic is a member of this list. This is not the case. Fuzzy logic is much broader than a logical system. Basically, fuzzy logic, FL, is a system of  reasoning, modeling and computation. FL has four principal facets. First, the relational facet, FLr. This facet is centered on fuzzy relations, fuzzy if-then rules, fuzzy systems analysis and  fuzzy decision-analysis. Most applications of fuzzy logic relate to this facet. The most visible  application area is fuzzy control. Second, the epistemic facet, FLe. This facet is concerned  with knowledge representation, linguistic variables, possibility theory, search and natural  languages. Third, the fuzzy-set-theoretic facet, FLs. This facet is focused on the theory of  fuzzy sets. Fourth, the logical facet, FLl. In this facet, and only in this facet, fuzzy logic is  viewed as a logical system. To differentiate between FL and FLl, FL and FLl are referred to  as fuzzy logic in a wide sense and fuzzy logic in a narrow sense, respectively. Today, when  we discuss fuzzy logic, it should be understood that we are talking about fuzzy logic in a wide  sense, unless stated to the contrary. Equating FL to its logical facet, FLl, is a common  misconception. This misconception is a source of a great deal of misunderstanding about  what fuzzy logic is and what it has to offer. In science and engineering, precision is respected and imprecision is not. What is widely unrecognized is that in many important applications of fuzzy logic imprecision is deliberate. The underlying rationale is the following. In most real-world problems there is some tolerance for imprecision. Fuzzy logic exploits this tolerance for imprecision through the use of words in place of numbers. Resulting in lower costs and greater simplicity. This is a key idea which underlies Computing with Words (CWW). This idea is one of the most important features of fuzzy logic, and it is unique to fuzzy logic. In CWW, concepts and techniques drawn from the realm of natural languages play an important role. Natural languages are intrinsically imprecise. In CWW, the accent is on problem-solving rather than on axiomatics and precisely defined concepts. There is a rationale for this attitude -- a rationale which is embodied in the Impossibility Principle. Briefly, the Impossibility Principle states that as the complexity of a system increases, a point is reached beyond which precision and relevance become incompatible. An example which I employed in my earlier messages, March 23, 24 and April 4, 2011, is the taxicab problem. The taxicab problem is a convenient platform for introduction of two basic concepts--the concepts of p-validity (provable validity) and f-validity (fuzzy validity). I hail a taxicab and ask the driver to take me from address A, where I am, to address B. There  are two versions: (a) I ask the driver to take me to B the shortest way; and (b) I ask the driver  to take me to B the fastest way. Abstractly, the street map is assumed to be a graph, and the  problem is to move from node A to node B. Each link (block) is assumed to be associated with  a constant, l, the length of the link, and a random variable, t, the traversal time. The traversal  time, t, is assumed to depend on the time at which the taxicab enters the link. Version (a) has a p-valid solution. The route that the driver chooses is an f-valid solution.  Version (b) has an f-valid solution which is the route that the driver takes. Version (b) does not  have a p-valid solution because we have no way of minimizing the sum of not-well-defined  random variables. In summary, Version (a) is a tractable problem whereas Version (b) is an  intractable problem. An analogy is helpful. Assume that I want to reach the peak of a mountain. I start by driving a  car toward the mountain. At some point, I cannot proceed further because of rough terrain. To  proceed further, I use a mule. Eventually, I reach a point beyond which I have to proceed on  foot. Using a car in the first leg of my trip is analogous to the use of tools which are provided by  traditional bivalent-logic-based mathematics. Classes are assumed to be crisp, that is, have  sharp boundaries. Let us refer to the tools which I use as Modality 1. The second leg is  analogous to the use of tools based on fuzzy logic. Classes are assumed to have unsharp  boundaries which are precisely defined via membership functions. Broadly speaking, we  employ what may be labeled fuzzy mathematics. Call it Modality 2. In the third leg, the  machineries of traditional mathematics and fuzzy mathematics cease to be effective. Classes  have unsharp boundaries which are not precisely defined. This is the world of everyday  reasoning. What we employ may be viewed as quasi-mathematics--a kind of mathematics  which I describe very briefly in my 2009 note on "Toward Extended Fuzzy Logic--A First Step,"  Fuzzy Sets and Systems 160, 3175-3181. Call this Modality 3. The taxicab problem, Version  (b), falls within Modality 3. Given a real-world problem, P, what modality does it fall into? The answer depends on how  P is modeled. Idealization of an intractable problem may make it a tractable problem. This is  common practice when we are faced with an intractable problem which we want to solve  through the use of traditional mathematics.     What I said above carries an important message. You should not assume that every problem  that we are faced with falls into Modality 1, that is, can be solved through the use of traditional  mathematics. Rigor and precision carry a price.     Regards,     Lotfi -- Lotfi A. Zadeh Professor Emeritus Director, Berkeley Initiative in Soft Computing (BISC) Address: 729 Soda Hall #1776 Computer Science Division Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720-1776 zadeh@eecs.berkeley.edu Tel.(office): (510) 642-4959 Fax (office): (510) 642-1712 Tel.(home): (510) 526-2569 Fax (home): (510) 526-2433 URL: http://www.cs.berkeley.edu/~zadeh/ BISC Homepage URLs URL: http://zadeh.cs.berkeley.edu/

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