WHAT IS FUZZY LOGIC?

Is it the paradox of the century?

How can fuzzy thinking be logical? However difficult this may be to accept,Kosko makes it clear that it has come to be a reality and a new successful way to dealwith uncertainty. In particular, we wish to have a computer be capable of reasoning withvague statements that have no probabilistic meaning. This is what we are preparingto illustrate. There is no complete agreement within the fuzzy community on a definitivedefinition for fuzzy logic. However it seems only reasonable to follow a definition accordingto L. Zadeh when he said in a talk he gave at the National Science Foundationthat:

* "In a Narrow sense, fuzzy logic is a logical system that aims at a formalization of approximatereasoning. As such, it is firmly established in multiple valued logic, butits agenda is quite different from that of the traditional Lukasiewicz's logic becausefuzzy logic as logic of approximate reasoning is not part of the traditional multi-valuedlogic. Although originally there was much antagonism, the scientific community hasremarkably changed its attitude. For example, Zadeh has kept a close count, and washappy to announce at a BISC seminar in September 1999, that there were 3240 papers,between 1995 and 1998, containing fuzzy in its title cited in Mathematical Reviews incontrast of only 521 in 1993.

* In a Broad sense, Zadeh often adds, fuzzy logic is almost synonym with fuzzy set theorywhich, as the name suggests, is basically the theory of "classes without boundaries".

Thus, it is highly capable to handle ambiguities and vagueness.

Finally, most of us ingeniously propose that fuzzy logic is an extension of multiple-valuedlogic to the continuum case. With the sharp distinction that in the multiple-valuedcase we are limited to the use of rational numbers within the unit interval, while in thecontinuum case we may use any real number within the unit interval".

Thornber makes a clear case for this claim. In [12], with the introduction of thelinguistic variable, Zadeh makes it even more clear how the tool of fuzzy numbers allowsthe representation and manipulation of the meaning of vague concepts whose value is notprobabilistic, and certainly not stochastic. Why insist that only probability can handle uncertaintywhen we have long recognized different forms of uncertainty. For further explanation,read Novak, see references [11,12]. Different kinds of uncertainty should be treatedand studied accordingly.

EXAMPLE 1. Suppose we say: "Wanted: a cheap house in a good neighborhood with anexcellent school system".

The vagueness, that is absolutely linguistic, of the above statement, an example due tomy good friend Mike Smith, is without a doubt, unsuitable to any attempt at a probabilisticapproach, and keep its original meaning.

EXAMPLE 2. Suppose we say "Matthew is young".

A probabilistic expression of the above would change the statement into somethinglike:

"The probability that Matthew is young is 0.9".

Note the implicit law of the excluded middle: Matthew is either young or he is not.What happens is that we only have a 90 % likelihood of being right in knowing what he isThus, we could have said, using probabilistic language:

"There is only a 90% chance that Matthew is young".

In a fuzzy set context, we would say:

"The membership grade of Matthew within the set of young people is 0.9"or that, by using any of the permissible fuzzy quantifiers:

"Matthew is more or less, [orsomewhat] young".

The semantic is clearly distinct. Again a strikingly distinct semantic that includes quantifiersof a new kind. In other words, the distinction is in the use, in its meaning, thus in thesemantic. We shall often have to return to this distinction that is fundamental to removethe antagonism and the confusion often existing between the idea of probability values anddegrees of membership because most often they end up to be from the unit interval.

In a fuzzy context, we have a continuum of possible valuations which means that mathematicallywe can use anything within the unit interval. If we were able to say that the valueof the truth is either zero or one, then we do not hesitate, and we have a crisp case. It is whenwe are unsure that the truth-value is neither 0 or 1, maybe it is somewhere in between, thenwe may believe that it is a likely event, then we have a probability, when it is not this caseeither then we have a fuzzy case and thus we cannot force a crispness or probability modelingapproach. Thus we must interpolate. Note the frequent mention of the term Interpolativein the quote below. It turns out that these matters lend themselves to the use of fuzzynumbers. Finally, we conclude this introductory section by quoting a concise viewpoint asexpressed by Zadeh at his NSF talk in May 1993. Namely, he said:

"The basic concepts of fuzzy logic are:

* The concept of linguistic variable, that is, a variable whose values are words rather thannumbers;

* The concept of canonical form which represents information as an elastic constrainton a variable;

* The concept of Interpolative reasoning which makes it possible to reason with incompleteinformation.

* Interpolative reasoning plays a key role in human cognition and lies at the basis ofpattern classification, qualitative reasoning, system identification, system modelingand neural network modeling. In the context of fuzzy logic, a concept which is centralto Interpolative reasoning is that a collection of fuzzy if-then rules which serveto provide an approximate characterization of the input–output relation of partiallyspecified system approximation. To fill the gaps in knowledge, a number of differentarchitectures may be employed, prominent among which are disjunctively combinedfuzzy if-then rules with a defuzzifier and the Takagi-Sugeno-Kang fuzzy if-then ruleswith a convex aggregator. The role-played by Interpolative reasoning in the implicationof fuzzy logic. The importance of fuzzy logic derives from the fact that almostall of human reasoning is approximate in nature. In fact, it is the ability to recognizedistorted speech, decipher sloppy handwriting, and, more generally, make rational decisionsin an environment of uncertainty and imprecision". Note that the membershipfunction is not subject to any restrictions other than to be continuous on the intervalof interest.

* EXAMPLE. To emphasize the distinction between probability and the concept ofmembership function, consider the case in which a patient reports abdominal painsto a doctor, and she is then asked to quantify the severity of the pain on a scale from 1[no pain] to 10 [very high]. There is no probability involved, the pain exists, its severityis the only number that has any meaning.

* Personally, I believe that a good intuitive definition is to say that fuzzy logic is an approximationto classical logic just as numerical analysis provides an approximation toexact solutions.

ABILITY OF FUZZY LOGIC OF HANDLINGFUZZY QUANTIFIERS

This major ability is what sets it drastically apart from any logic that has been studied previously.

In [17, page 13 and following], Yamakawa, a famous Japanese scientist, gives a clearinstance of the use of fuzzy numbers for the modeling of complex systems. He makes aparallel between a mathematical formula and a linguistic rule. This rule would be expressedin a way that is something like the one below:

RULE i: "If x is Ai, then f (x) is Bi".

where A i and B i are to be interpreted as fuzzy sets or linguistic constants. In fact, arelation between a variable x and f (x) can be stated by a set of linguistic rules or by a mathematicalequation. Linguistic constants are generally well-defined languages, namely eithercrisp information or exact numerical values in terms of crisp rules, we have slightly differentif-then clauses:

EXAMPLE 3. For f (x) = [ (x - 3) 2 ÷2; then

RULE 1: if x is around -2 then f (x) is around 25/2.

In the same paper, continuing his summary, he adds that a set of rules forms a knowledgebase. The inference system cannot contain contradictory rules. In [17, page 16], hegives examples of linguistic rules with ill-defined language such as rule 1 above or as:

RULE 2."If the temperature grows higher then the opening of the valve should be reducedsignificantly, and the fuel should be reduced a little".

Note that many cooking, and driving, instructions are quite similar to rule2, yet mostpeople succeed in making sensible interpretations. Can we achieve enough ability to enablea computer to make sense out of these instructions? Knowledge bases constructed fromexperts are rules that are hardly ever precise. The incredible success in using fuzzy logic tosolve control problems has many scientists wonder whether teaching differential equationshas become obsolete. Balancing the inverted pendulum leads to a complex system of differentialequations. Not so, if the control has a fuzzy logic rule base. The ease of such techniqueis clearly appealing, as it will be demonstrated in section 9. Likewise, if we wish to balancea moving platform with a glass of wine and a mouse on it. To conclude, why take our measurementsto be exact when they are so rarely that way? And as a final

EXAMPLE 4:

When we say "Kathy is six feet tall", we do not understand "She is about six feet tall".Rather we understand that she is exactly six feet tall.

Likewise, if we say "Dr. Dara is possibly mid-sixty"

We do not take that to mean that she is exactly 65 years old. We correctly discern thatthe value as an approximation, not as an exact value. Thus, we should treat these values asimprecise rather than exact. A well-known interpretation of fuzziness is that just as numericalanalysis deals with the approximation of problems that are so complex that we cannotfind the solution in a closed form, so fuzzy sets and fuzzy logic respectively deal with an approximationof crisp sets and of truth-values. Keep in mind the parallel between numericalanalysis enabling us to find solutions to difficult problems in an approximate fashion andfuzzy logic, giving us approximation methods to find solution to communication problems.Why always cling to an illusion of exactness when it is not often needed and cannot be usedeither.

DIFFERENT KINDS OF NUMBERS.

RossT. [AP26, page165]

Interval arithmetic can be thought the following way. When we add, multiplyany two crisp real numbers, the result is still a single crisp real number,or a singleton. If we combine an infinite number of crisp singletons from aninterval, we expect a crisp singleton. Thus, an interval is the result from combininginterval.

The topic of interval arithmetic has found its proper place in modern mathematics. Its applicationto imprecise sentences is quite natural. Everyone has the correct understanding ofa sentence such as:

"They are a pair of thirtysomething people".

Nobody expects that to mean other than their age is something between t=30 and t =40,namely the value of t belongs to the interval [30, 40]. In other words, the interval becomesa representation of the imprecision of the sentence. What we do now is to introduce othermethods to establish a correspondence between an imprecise sentence and a mathematicalexpression.

What is our goal? In this discussion we accept realistically that often we need to usenumbers that approximate closely our evaluation of an item or even of a situation whenwe cannot do so exactly. Everyone agrees that to assume or make use of those approximationsas if they were exact is conducive to nonsense. This is quite natural, and in physics weare taught early on that every instrument has an error associated with the measurement ityields, thus, we must be aware of it and include it in our calculations. To accommodate thisline of thinking there have been attempts to method probabilistic reasoning, see references[U 1, 11, 19], none with great success, check and read the various works in the section ofthe references on modeling under uncertainty, [U], page47. Thus, it is desirable to conserveclearly the character of imprecision of the evaluations for example by creating in a suitablefashion numbers that reflect the imprecision and then manipulate these numbers withmuch savvy. In the next chapter, for example, mid-sixty can be represented by a trianglewith vertices (60, 0), (65, 1), (70, 0).

Note that 65 has been associated with a value one because it is truly mid-sixty, whilesixty and seventy are associated to the value zero because they are clearly not mid-sixty.The latter representation is referred to as a Triangular Fuzzy Numbers. Incidentally, thereexists a well formulated fuzzy arithmetic that some believe it will soon included in handcalculators. Other fuzzy numbers, very popular in applications, have the shape of atrapezoid, or exponential.

Trapezoidal Numbers are also very popular in the representation of imprecise expressions.Buckley uses them successfully to generate a computational scheme to analyze financialmatters. Their construction is similar to that of triangular numbers as it will beillustrated in the later example, next chapter, example 6, page 24. Arithmetic that handleseither numbers is available and it will be included in later sections, see Buckley's paper,where he proposes a nifty technique, or a recent text by Kauffman and Gupta, and others,e.g.. As a matter of fact, some are already predicting the inclusion of fuzzy numbersin hand held calculators.

NOTATION. The trapezoid with vertices (a,0), (b,1), (c,1), (d,0) is denoted concisely by T[a, b, c, d].

Where a, b, c, d are any real numbers of the horizontal axis, and these vertices are joinedby segments of a line in the order they are given. See example 6 in the next section.

WHAT IS A LINGUISTIC VARIABLE?

We quote an accepted definition [19, 20].

DEFINTION 1. We say that X is a Linguistic Variable if its values which are called interpretations,are natural languages expressions [AP19]. As an additional comparison, in classicallogic there are two quantifiers, the existential and the universal. Not so in fuzzy logic,some of which are:

{almost, rather, often, approximately, close to,}........ (3)

{near, few, later, several}........ (3a)

It is no wonder that mathematics must become fairly sophisticated. Zadeh appliesthe concept to approximate reasoning that is at the core of the effort of making machinesachieve some form of reasoning, much like humans do. In fact, the ability of a human tounderstand slur speech and fill in details for some missing information is amazing. Thevaluation of a classical logic implication is rather straightforward, not so for the fuzzy logiccase. Many solutions have been proposed over the past two decades with none becomingstandardized. With the final selection of a method being primarily due to personal preference.Some texts do a splendid job in examining many options, see Ross, Klir's,and Asai&Terano &Sugeno.

One of the most interesting issues still strongly debated within the fuzzy communityis how to compute the validity of an implication. A comparative review here is beyond thepurpose of this concise presentation, thus we refer the reader to consult the authors thatwere mentioned above. Finally, it is time to consider, see [2,18].

For the example in section 9, the stabilization of the inverted pendulum, the neededlinguistic variables that describe the state of each variable are fuzzy numbers in a triangularshape, see here figure5 are:

{AZ= Approximately Zero, NS=Negative Small, NL= Negative Large,}............... (4)

{PS=Positive Small, PM= Positive Medium, PL=Positive Large}.................. (4a)

And their interaction appears in the rules matrix in figure4, page46, given by their firstletter, where each is a fuzzy set in the shape of a triangle shown in figure5.