mark shelton | merrick cloete saman majrouh | sahithi jadav

this presentation fuzzy set theory graduation granulation fuzzy control strengths limitations applications (to be continued) fuzzy logics

“as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until […] precision and significance become almost mutually exclusive characteristics.” - Zadeh, 1965 fuzzy set theory

fuzzy sets ‘classical’ sets are called crisp sets - membership values of 0 or 1 only a set where each element has a degree of membership a membership function converts values into grades of membership

fuzzy logic is a more ‘human’ approach to computation. it involves two main concepts: graduationgranulation

inputs are then grouped together, e.g. cold, lukewarm, warm, hot inputs are drawn together by similarity, proximity or functionality we don’t know exactly where each object starts and ends

graduation the designer decides what constitutes as ‘cold’, as well as all degrees of it. everything is a matter of degree, e.g. not cold, a bit cold, a lot cold we assign a value between 0 and 1, e.g 0.7 is hot, 0.3 is cold

fuzzy control In a single cycle, the system read all inputs each option is weighted and used to output the result rather than select a single option to evaluate, the system evaluates all options

fuzzy operators conjunction (P AND Q) union (P OR Q) zadeh operator probabilistic operator bounded operator min(P, Q)P x Q max (0, P + Q - 1) max(P, Q) P + Q – P x Q min(1, P + Q) 1 if tv(P) ≤ tv(Q) else 0 min(1, 1 – tv(P) + tv(Q)) max(1- tv(P), min(tv(P), tv(Q))) implication (IF P THEN Q)

example system reads the temperature as 0.9 cold, 0.1 warm, 0.0 hot if cold, set heater to on if warm, set heater to off system sets heater to on 90% of the time and off 10% of the time within a cycle

fuzzy logics Most fuzzy logic systems are variations on t-norm fuzzy logics. A t-norm is a continuous function that satisfies the following properties between 0 and 1: commutativity T(a, b) = T(b, a) monotonicity T(a, b) ≤ T(c, d) if a≤ c and b ≤ d associativity T(a, T(b, c)) = T(T(a,b), c) identity T(a, 1) = a

fuzzy logics Some of the types of fuzzy logics are: monoidalleft continuous t-norms basiccontinuous t-norms productfor strong conjunction: T prod (a, b) = a  b pavelka’s stems from Lukasiewicz, each formula has an evaluation But today we are going to focus on two key types – lukasiewicz and godel

fuzzy logics lukasiewicz logic is similar to a basic t-norm T luk (a, b) = max(0, a+b-1)

fuzzy logics godel is the minimum t-norm and is the standard for weak conjunction. T min (a, b) = min(a,b)

strengths convenient user interface with easy end-user interpretation can model problems with imprecise and incomplete data, and nonlinear functions of arbitrary complexity corresponds well with human perceptions

limitations requires ad-hoc tuning of membership functions may not scale well to large or complex problems deals with imprecision and vagueness, but not uncertainty

applications coal power plant refuse disposal plant water treatment system ac induction motor fraud detection

conclusion fuzzy reasoning binary computation human experience natural language artificial intelligence biotechnology

mark shelton | merrick cloete saman majrouh | sahithi jadav