## Tuesday, November 30, 2010

### Fuzzy Concepts. Hedges

Another important feature of fuzzy systems is the ability to define "hedges," or

modifier of fuzzy values. These operations are provided in an effort to maintain

close ties to natural language, and to allow for the generation of fuzzy statements

through mathematical calculations. As such, the initial definition of hedges and

operations upon them will be quite a subjective process and may vary from one

project to another. Nonetheless, the system ultimately derived operates with the

same formality as classic logic. The simplest example is in which one transforms

the statement "Jane is old" to "Jane is very old." The hedge "very" is usually

defined as follows:

m"very"A(x) = mA(x)^2 Thus, if mOLD(Jane) = 0.8, then mVERYOLD(Jane) = 0.64.

Other common hedges are "more or less" [typically SQRT(mA(x))], "somewhat,"

"rather," "sort of," and so on. Again, their definition is entirely subjective, but

their operation is consistent: they serve to transform membership/truth values in a

systematic manner according to standard mathematical functions.

A more involved approach to hedges is best shown through the work of Wenstop

in his attempt to model organizational behavior. For his study, he constructed

arrays of values for various terms, either as vectors or matrices. Each term and

hedge was represented as a 7-element vector or 7x7 matrix. He ten intuitively

assigned each element of every vector and matrix a value between 0.0 and 1.0,

inclusive, in what he hoped was intuitively a consistent manner. For example, the

term "high" was assigned the vector 0.0 0.0 0.1 0.3 0.7 1.0 1.0

and "low" was set equal to the reverse of "high," or 1.0 1.0 0.7 0.3 0.1 0.0 0.0

Wenstop was then able to combine groupings of fuzzy statements to create new

fuzzy statements, using the APL function of Max-Min matrix multiplication.

These values were then translated back into natural language statements, so as to

allow fuzzy statements as both input to and output from his simulator. For

example, when the program was asked to generate a label "lower than sortof

low," it returned "very low;" "(slightly higher) than low" yielded "rather low,"

etc. The point of this example is to note that algorithmic procedures can be

devised  which translate "fuzzy" terminology into numeric values, perform

reliable operations upon those values, and then return natural language statements

in a  reliable manner.