Fuzzy Logic

Fuzzy logic is a many-valued logic in which the truth of a statement is not a sharp “true or false” but a degree that can take any value between 0 (completely false) and 1 (completely true). Its purpose is to model reasoning with vague, imprecise concepts, such as “tall,” “hot” or “fast,” which binary logic does not capture well: at exactly which centimeter does someone go from “not tall” to “tall”?

Where it comes from

It was devised by Lotfi Zadeh, an engineer at the University of California, Berkeley, who in 1965 introduced fuzzy sets in a paper called, precisely, “Fuzzy Sets.” Fuzzy logic derived from it shortly afterward.

Sets and degrees of membership

The basic piece is the fuzzy set. In a classical set, an element either belongs or does not; in a fuzzy one, it belongs to a degree. That is formalized by a membership function, which assigns each element a value in the interval [0, 1]: a person of a certain height may belong to the set “tall” with a degree of 0.7. The logical operators are redefined accordingly—“and” as the minimum, “or” as the maximum, negation as one minus the value.

Fuzzy inference systems

On this basis, fuzzy inference systems are built, which reason with rules of the type “IF the temperature is high THEN cool a lot.” The process has three phases: fuzzify the inputs (translate them into degrees), apply the rules and defuzzify the result (convert it back into a concrete number). The two classic models are Mamdani's and Takagi-Sugeno's.

What it is for

Fuzzy logic shone in control. It is behind household appliances that adjust their operation smoothly—washing machines, cameras, air conditioners—and more ambitious systems, such as the acceleration and braking control of the subway in the Japanese city of Sendai, running since 1987. It is considered one of the techniques of soft computing, alongside neural networks and evolutionary algorithms, with which it is sometimes combined in neuro-fuzzy systems.