"Fuzzy logic" is reasoning with fuzzy sets. Bart Kosko refers to the "fuzzy principle" in stating that "everything is a matter of degree." Instead of using the crisp truth values "1" and "0," fuzzy logic uses truth values as fractions from 0 to 1. Thus, the statement "John is tall" can be 66% true, and John would have a membership value of 0.66 in the fuzzy set of tall people. When using these percentages, fuzzy logicians do not imply that probability or chance is involved. It would not make sense to say that John has a 66% chance of being tall or that my lawn has an 89% probability of being green.
To illustrate a fuzzy set further, let us look again at the green lawn example. Few lawns are 100% green. Often, a lawn contains a few brown or yellow patches. Thus, the word "green," in the context of lawns, stands for a fuzzy set of colors that constitute green. "We think in fuzzy sets and we each define our fuzzy boundaries in different ways and with different examples." While the definition of these boundaries may differ from person to person, "the very looseness of the fuzzy set enhances its expressiveness." So, while I may make the statement, "My lawn is green," in reality, my lawn might be 89% green, or may have a membership value of 0.89 in the fuzzy set of green lawns, because of a few yellow and brown spots. Most people round up to 100% as a matter of convenience.
Fuzzy reasoning requires the creation of fuzzy rules in the form of "if-then" statements. The fuzzy "if-then" rules express the relation between fuzzy sets. By combining fuzzy rules, we can create a fuzzy system that automatically converts inputs into outputs. Building a fuzzy system can be done in three steps: first, select the inputs and outputs of the system; second, pick the fuzzy sets; and third, choose the fuzzy rules.
My favorite illustration of a fuzzy system of fuzzy rules is the washing machine example. Suppose we want to construct a machine that ""knows' to wash dirtier clothes for a longer duration than clothes which are relatively clean." The "input is the degree of dirtiness and [the] output is the duration of the wash." The fuzzy inputs would be: almost completely clean, relatively clean, slightly dirty, dirty, and very dirty. The fuzzy outputs would be: rinse, wash lightly, wash, wash thoroughly, and wash vigorously. Finally, we choose the fuzzy rules: (1) if the clothes are almost completely clean, then only rinse them; (2) if the clothes are relatively clean, then they are lightly washed; (3) if the clothes are slightly dirty, then they are washed; (4) if the clothes are dirty, then they are washed thoroughly; (5) if the clothes are very dirty, then they are washed vigorously.
This fuzzy system can be "defuzzified" by attaching specific numbers to the vague concepts. Fuzzy concepts can be defuzzified by averaging or finding the centroid (i.e., center of mass) of the output numbers. Defining dirtiness as a range of particles of dirt from 10 to 100 and duration of the wash from 10 to 100 minutes, we can assign certain values to our fuzzy sets. Thus, the washing machine will literally think for itself and determine how long to wash laundry based on how dirty it is. Such products have been developed in Japan "to raise the machine IQ of camcorders and transmissions and vacuum sweepers and hundreds of other devices and systems."
Saturday, September 13, 2008
Michael Nguyen: Fuzzy Logic in Patent Law
Michael T. Nguyen recently published a very interesting law journal note proposing the use of fuzzy logic in trials of certain kinds of patent cases. See "The Myth of 'Lucky' Patent Verdicts: Improving the Quality of Appellate Review by Incorporating Fuzzy Logic in Jury Verdicts," 59 Hastings L.J. 1257 (2008). Mr. Nguyen provides a nifty summary of how fuzzy logic can control one kind of process (footnotes omitted):