Basic Concepts of Fuzzy Logic Apparatus of fuzzy logic is built on: Fuzzy sets: describe the value of variables Linguistic variables: qualitatively and quantitatively described by fuzzy sets Possibility distributions: constraints on the value of a linguistic variable Fuzzy if-then rules: a knowledge *Fuzzy Logic: Intelligence, Control, and Information - J. Yen and R. Langari, Prentice Hall 1999

Linguistic variables A fuzzy set can be used to describe the value of a variable. - Temperature is high. - The load is heavy.

Linguistic variables The variable Temperature (x) is characterized both by a symbolic variable (“High”) and a numeric variable expressing its membership in the fuzzy set “High”.

Linguistic variables A linguistic variable is “a variable whose values are words or sentences in a natural or artificial language”. Each linguistic variable may be assigned one or more linguistic values, which are in turn connected to a numeric value through the mechanism of membership functions.

Linguistic variables An example of a fuzzy linguistic variable and membership functions

Possibility Distribution Assigning a fuzzy set to a linguistic variable constrains the value of the variable. Possible vs. Impossible values of the variable are a matter of degree.

Possibility Distribution Example: a suspect is a male between 20 and 30 years old. A crisp set defines the age of this suspect as [20,30]. A 19-years old male would not be a suspect, as this age is an impossible value for this set.

Possibility Distribution A fuzzy set defines the age of this suspect as  (age) that may have a smooth boundary.

Possibility Distribution A possibility that the suspect is 19 years old is 0.75

Possibility Distribution In general, when we assign a fuzzy set A to a variable X, the assignment results in a possibility distribution of X, which is defined by A’s membership function.  X (x)=  A (x)

If-Then rules “If temperature is hot then AC_setting is high” Provide fuzzy inference. Can be viewed as: - Interpolation scheme - Multi-expert panel - Generalization of logic inference

If-Then rules “If temperature is hot then AC_setting is high” Provide fuzzy inference. Can be viewed as: - Multi-expert panel - Interpolation scheme - Generalization of logic inference

If-Then rules Multi-expert panel: A kingdom with 3 mathematicians. 1. Can sqrt numbers between 0 and Can sqrt numbers between 1001 and Can sqrt numbers between 2001 and 5000 The task: What is the sqrt of 1156.

If-Then rules M1: 31.6How sure? – 0 M2: 34 How sure? – 1 M3: 44.73How sure? =0 The answer – 34.

If-Then rules Interpolation:

If-Then rules Inference: Rule: if a person’s income is more than 100K then the person is rich Fact: Jack’s income is 101K Consequence: Jack is rich

If-Then rules Structure of fuzzy rules: IF THEN Example: IF a person’s income is high THEN the person is rich

If-Then rules An antecedent may combine multiple conditions using logic connectives (AND, OR, NOT): IF a person’s income is high AND the income figure is true THEN the person is rich

If-Then rules Consequent 1. Crisp: IF … THEN y=nonfuzzy_value 2. Fuzzy: IF …THEN y is A_fuzzy_set 3. Functional: IF x 1 is A 1 AND x 2 is A 2 … AND x n is A n THEN