## Tuesday, May 08, 2012

Lotfi A. Zadeh Mon, May 7, 2012 at 8:45 PM

 *********************************************************************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/ ```