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F uzzy L ogic : Background The concept of a set and set theory are powerful concepts in mathematics. However, the principal notion underlying set theory, that an element can (exclusively) either belong to set or not belong to a set, makes it well high impossible to represent much of human discourse. How is one to represent notions like: large profit high pressure tall man wealthy woman moderate temperature
B ackground & D efinitions “ Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise.” Fuzzy set theory , originally introduced by Lotfi Zadeh in the 1960's, resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit.
F uzzy S ets & F uzzy L ogic A fuzzy set is a collection of objects that might belong to the set to a degree, varying from 1 for full belongingness to 0 for full non-belongingness , through all intermediate values. " Fuzzy logic is a generalization of standard logic, in which a concept can possess a degree of truth anywhere between 0.0 and 1.0. Standard logic applies only to concepts that are completely true (having degree of truth 1.0) or completely false (having degree of truth 0.0). Fuzzy logic is supposed to be used for reasoning about inherently vague concepts, such as 'tallness.' For example, we might say that ‘Michael Jordan is tall,' with degree of truth of 0.9
Fuzzy Logic Example: What is Tall?
Height Voted for
Jack is 6 feet tall
Probability theory - cumulative probability
There is a 75 percent chance that Jack is tall
Membership Functions in Fuzzy Sets Membership Short Medium Tall Height in inches (1 inch = 2.54 cm) 0.5 1.0 64 69 74
Fuzzy logic - Jack's degree of membership within the set of tall people is 0.75
We are not completely sure whether he is tall or not.
Fuzzy logic - We agree that Jack is more or less tall.
Membership Function < Jack, 0.75 Tall >
Knowledge-based system approach: Jack is tall (CF = .75)
Can use fuzzy logic in rule-based systems (belief functions)
F uzzy L ogic & F uzzy S ystems
The term fuzzy logic is used in two senses:
Narrow sense : Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge. Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory.
Broad Sense: F uzzy logic synonymously with fuzzy set theory.
A fuzzy system consists of:
Fuzzy (linguistic) variables
Example: Fuzzy variables Linguistic variables/
Example: Fuzzy rules
A fuzzy rule is a linguistic expression of causal dependencies between linguistic variables in form of if-then statements.
General form: IF <antecedent> then <consequence>
If temperature is cold and oil price is cheap
Then heating is high
Linguistic variables Linguistic values
Example: Fuzzy inference
Inputs to a fuzzy system can be:
fuzzy, e.g. (Score = Moderate), defined by membership functions;
exact, e.g.: (Score = 190); defined by crisp values
Outputs from a fuzzy system can be:
fuzzy, i.e. a whole membership function.
exact, i.e. a single value is produced .
Fuzzy system applications
Pattern recognition and classification
Image and speech processing
Fuzzy systems for prediction
F uzzy systems http:// www.austinlinkscom /Fuzzy http://www.industry.siemens.de/water/en/solutions/sector_fuzzy-logic.htm http://www.mathworks.com/access/helpdesk/help/toolbox/fuzzy/index.html The MathWorks
Fuzzy Logic Advantages
Allows for observation
Shortens system development time
Increases the system's maintainability
Handles control or decision-making problems not easily defined by mathematical models