Fuzzy Sets and Logic Sarah Spence Adams Discrete Mathematics

Rethinking Regular Sets Universe U = {a, b, c, d, e, f, g}, set S = {a, c, g} We can define S by mapping each element of the universe to 0 or 1. If we map it to 1, it goes in S. Else, not in S. Here, f(a) =1, f(b) = 0, f(c)=1, f(d) = 0, f(e) = 0, f(f) = 0, f(g) = 1.

Fuzzy Sets Every element in a fuzzy set S has a degree of membership Map each element of U to a value within the interval [0, 1] S = { 0.6 a, 0.3 c, 0.9 g} c has 0.3 degree of membership in S

Fuzzy Logic This business of having “degrees” of membership rather than “in or out” Truth values are between true and false Introduced in 1965 to model uncertainty in natural language: tall, fair, nice, large, hot

Why use fuzzy logic? PROS: Used to solve highly complex problems where math modeling is too difficult/impossible Tolerant of imprecise data Approximation: can model arbitrary nonlinear functions Intuitive, based on linguistic terms Convenient way to express expert and common sense knowledge

Why use fuzzy logic? Cons: Not a cure for all Crisp/precise models can be more efficient and even convenient Other approaches might be formally verified to work

Apply to Computer Games Can have different characteristics of players Strength: strong, medium, weak Aggressiveness: meek, medium, nasty If meek and attacked, run away fast If medium and attacked, run away slowly If nasty and strong and attacked, attack back Control of a vehicle Should slow down when close to car in front Should speed up when far behind car in front Provides smoother transitions, no sharp boundary

Other Applications compensation against vibrations in camcorders home appliances (washing machines, dish washers, rice cookers, etc.) recognition of handwriting, objects, voice image processing flight aid for helicopters simulation for legal proceedings improvement of fuel-consumption for automobiles early recognition of earthquakes “and in almost any other field you can think of”

Sound interesting? It’s never to early to start thinking about course project topics!