Fuzzy Sets, Fuzzy Logic, Fuzzy Inference

Sparsh -

Boolean Logic

Boolean Logic

Fuzzy logic

  • degrees of membership
  • degrees of truth

set of mathematical principles of knowledge representation based on degrees of membership rather than on crisp membership of classic binary logic

Multivalued Logic

Multivalued Logic

Fuzzy logic adds a range of logical values to Boolean logic

principle of dichotomy = Classical set theory imposes 0 and 1

crisp theory

fuzzy theory

characteristic function of A → crisp set

fA(x) : X ⟶ {0, 1}

membership function of set A → fuzzy set

fA(x) = {
1, if x ∈ A
0, if x ∉ A
}

continuum of possible choices

Fuzziness in Trapezoidal

sigmoid/ gaussian/ foi
functions can increase computation. Hence, linear fit functions are used.

At the root of fuzzy set theory → linguistic variables

Linguistic variables → Fuzzy variable

John is tall implies var(john) takes val(tall)

Hedges → fuzzy set qualifiers
↳ acts as operations

degree of membership

Fuzzy rules → conditional statements in the form
↳ relates to Fuzzy sets

where x, y → linguistic variables
and A, B are linguistic values

Fuzzy Reasoning

  1. evaluating antecedent → IF part (Antecedent)
  2. applying result to consequent → THEN part (Consequent)

Antecedent vs Consequent:

In classical rule based; IF antecedent is True then consequent is also True.

In Fuzzy rule systems, all rules fire to some extent; Antecedent true to some degree of membership then consequent also true to same degree.

Fuzziness Height and Weight Axis
monotonic selection

monotonic selection

value of output/ truth membership grade of consequent can be estimated directly from corresponding truth membership grade in antecedent

Examples of multiple Antecedents

IF   project duration is Long
AND  project staffing is Large
AND  project funding is Inadequate
THEN risk is High
IF   service is excellent
AND  food is delicious
THEN  tip is generous

Examples of multiple Consequents

IF temperature is hot
THEN
hot water is less big
cold water is more

all antecedents are affected equally by consequents

Fuzzy Inference →

  • process of mapping
  • from given input to an output
  • using theory of fuzzy sets.

Mamdani Style Inference → most common fuzzy inference

4-step process

  1. input variables (fuzzification)
  2. rule evaluation
  3. aggregation of rule outputs
  4. defuzzification