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 and imprecise concepts such as ‘tall’, ‘hot’ or ‘fast’, which binary logic does not capture well: at what exact centimeter does someone change from ‘not tall’ to ‘tall’?

In Zadeh's original formulation for fuzzy sets, ‘and’ is represented by the minimum, ‘or’ by the maximum and negation by one minus the value; these are classic operators, not the only possible choice across all fuzzy logics.

Fuzzy logic is a family of many-valued logics that models reasoning with vague statements through degrees of truth, commonly between 0 and 1, rather than restricting truth to a binary choice.

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. Logical connectives are interpreted as functions over those degrees; minimum, maximum and one minus the value are a common choice, not a universal definition of every fuzzy logic.

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. Documentation: scholarly fuzzy-logic overview.

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. Documentation: scholarly fuzzy-logic overview.