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What in the world is a fuzzy function? For that matter, what is an ordinary function?  Ordinary functions are a bit easier to deal with, so it's worth looking to see if they help understand the fuzzy kind.

Speaking in modern lingo, an ordinary function is a mapping from the elements, x, of a number set, A, to the elements, y, of a number set, B. This is a function of a single variable (x) -- which means that only the elements of the set, A, are associated with elements of the set, B. To each element in A we hook an element from B. (Nothing says A and B can't be the same set.) The function defines the rule for making the connection.

We normally think of this type of function as single-valued, meaning that only one member of B can be associated with any given member of A. That's the rule. Stated another way, the rule says that a given number in A can only go to a single number in B. If x₁ is in A, it maps uniquely to y₁. (This doesn't mean, however, that different numbers in A can't go to the same number in B.) Such a function is normally expressed as:

You've probably already figured out that fuzzy functions have something to do with fuzzy sets, which indeed they do. These fuzzy sets involve fuzzy variables. Strangely enough, the fuzzy variables could be fuzzy numbers or even linguistic variables, which are inherently fuzzy.

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Yikes! Out of the frying pan into the fire! What in the world are fuzzy variables? 

Spikes! That's how Bart Kosko views traditional, ordinary, what-you-see-is-what-you-get numbers. In fuzzy set terms, an ordinary number is a (limiting case) fuzzy number that is 100% in the fuzzy set of that number. Like 0 in the fuzzy set ZERO. Or 1 in the fuzzy set ONE. And so on. What's more, no other number is in the limiting set. This is like a boundary for fuzzy number sets -- a limiting condition.

For most fuzzy numbers, though, close clusters are involved. For instance, numbers close to 0 can be included in the fuzzy set ZERO, but they would have different degrees of membership. We commonly say, for instance, that "there is essentially no money left in the bank -- a few bucks, maybe, but that's about it." In this case, as Kosko says:

The graph for ZERO could take any shape, but in any case zero is 100% in ZERO, whereas numbers like 1 and -1 are only partially in the set -- their degree of membership is less than one. And in this case numbers like 2 and -2 aren't in the set at all -- their degree of membership is zero. When the graph shrinks so that only 0 is in the set, we get back to a correspondence with the ordinary number 0. The same thing applies to all the numbers. So ordinary arithmetic becomes a special case of fuzzy arithmetic.

The situation really gets bizarre when we start talking about variables in terms of ordinary words of the language. (In Visual Basic you can even have variable objects!) What sense can be made of linguistic variables?

In fact, the idea of a linguistic variable makes good sense and is really very clever, because its underlying utility is to solve long-standing insoluble problems. Using the precise techniques in math and logic, we've made fairly good progress, relatively speaking, in the physical sciences, but the methods have been woefully unsuited to deal with ordinary human interactions -- I might say underwhelming. Even if fuzzy theory can only inch us forward in the social arena, they're still worth considering. The use of linguistic variables is one approach.

Linguistic variables are related to, but different from, fuzzy sets -- they have the relationship of goal to instrument. The relationship is neatly spelled out by Kurt J Schmucker, who says:

We use linguistic variables without realizing it, even though we may not formalize the process. Think about 'pressure', for instance. Language allows us to refer to different pressure states. That is, we have many descriptive terms for pressure. We can say the pressure is very low, or low, or medium low, or moderate, or average, or moderately high, or high, or very high, or explosively high, or anything else you can think of. And we relate the terms to different numerical value ranges of the pressure. Bottom line is, these terms are values of 'pressure.'

As another example, Schmuckerdefines a linguistic variable he calls number. He uses the integers between 1 and 10 for the illustration and identifies the set of values {few, several, many} as the natural language expressions that number can take. The expressions are names for the number sets as follows:

In other words, 'few', for example, can mean 1, or 2, or 3, or 4, but to different degrees.

See Fuzzy Sets for details of degree of membership in sets.

In practical applications it would be natural to design the language expressions to maintain a rich but finite set. And because it is likely to be large, the set would best be specified by rule rather than by listing them, as a roster. You would need to have a way to specify the different (non-numerical) values of the variable.

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Traditionally, functions are rules that define mappings from ordinary numbers to ordinary numbers -- from independent to dependent variable. Fuzzy functions, by contrast, are rules mapping fuzzy variables to fuzzy variables. To see what a fuzzy function might look like, let's develop one that uses the fuzzy variable defined for pressure. (You can find other examples in Kosko, Schmucker, and Pappo.)

To this end, assume we wish to simulate an individual's behavior and would like to model the way that person responds to social pressure. Specifically, let's say we need to model the effect of pressure on the person's motivation in specific situations. To keep it simple we will use only the values low, medium, and high as our linguistic representations for pressure, and we'll use goes down, remains the same, and goes up as our linguistic representations for motivation. We will, further, accept the following rules as the applicable conditionals for the individual:

The rules thus define our function. They provide the connection between the social condition and the person's response and form the basis for a fuzzy inference. Between the input (social pressure) and the output (response). Between the independent variable (pressure) and the dependent variable (response).

By defining numerical fuzzy subsets for each of the linguistic representations of the two fuzzy variables, the rules can be embedded in a traditional function to model a specific connection between numerical values for pressure and numerical values for motivation:

For more details of these relationships, you might consult here.

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