Fuzzy Sets, Fuzzy Logic, Fuzzy Inference

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Boolean Logic<br>Boolean LogicFuzzy logic<br>degrees of membership<br>degrees of truth<br>set of mathematical principles of knowledge representation based on degrees of membership rather than on crisp membership of classic binary logic<br>Multivalued Logic<br>Multivalued Logic<br>Fuzzy logic adds a range of logical values to Boolean logic<br>principle of dichotomy = Classical set theory imposes 0 and 1<br>crisp theory<br>fuzzy theorycharacteristic function of A → crisp set<br>fA(x): X → {0, 1}

membership function of set A → fuzzy set<br>fA(x) = {<br>1, if x ∈ A<br>0, if x ∉ A<br>} continuum of possible choices

sigmoid/ gaussian/ foi<br>functions can increase computation. Hence, linear fit functions are used.<br>At the root of fuzzy set theory → linguistic variables<br>Linguistic variables → Fuzzy variable<br>John is tall implies var(john) takes val(tall)<br>Hedges → fuzzy set qualifiers<br>↳ acts as operations<br>Fuzzy rules → conditional statements in the form<br>↳ relates to Fuzzy sets<br>where x, y → linguistic variables<br>and A, B are linguistic values<br>Fuzzy Reasoning<br>evaluating antecedent → IF part (Antecedent)<br>applying result to consequent → THEN part (Consequent)<br>Antecedent vs Consequent:<br>In classical rule based; IF antecedent is True then consequent is also True.<br>In Fuzzy rule systems, all rules fire to some extent; Antecedent true to some degree of membership then consequent also true to same degree.<br>monotonic selectionmonotonic selection<br>value of output/ truth membership grade of consequent can be estimated directly from corresponding truth membership grade in antecedent<br>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 →<br>process of mapping<br>from given input to an output<br>using theory of fuzzy sets.<br>Mamdani Style Inference → most common fuzzy inference<br>4-step process<br>input variables (fuzzification)<br>rule evaluation<br>aggregation of rule outputs<br>defuzzification

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Blog by Sparsh