Fuzzy Logic - Artificial Intelligence PDF

Fuzzy logic Artificial intelligence Kristóf Karacs PPKE-ITK 1 History and motivation n Lotfi Zadeh, 1965 n Concept: truth values may apply partially ¨ InBoolean logic truth values are binary ¨ Not able to e...

Fuzzy logic Artificial intelligence Kristóf Karacs PPKE-ITK 1 History and motivation n Lotfi Zadeh, 1965 n Concept: truth values may apply partially ¨ InBoolean logic truth values are binary ¨ Not able to express uncertainty in the occurrence of events, rather their truthfulness n Logical statements are derived from natural language statements 2 1  Fuzzy sets n Sets are identified by linguistic identifiers ¨ “tall”, “young”, “bigger” n Grade of membership , ∈ , 0≤ ≤1 n Fuzzy set A ¨ = , = , , , ,…, , 3 Membership function n Age – “young” Y Y A A Crisp Fuzzy 4 2  Types of membership functions n Piecewise linear ¨ Straight lines (increasing, decreasing) ¨ Triangular 5 Linear curves 6 3  Types of membership functions n Smooth curves ¨ s-curve ¨ z-curve ¨ p-curve 7 Smooth curves 8 4  Operations n Given 9 Fuzzy intersection n 10 5  Operations on fuzzy sets 11 Algebraic properties n Commutativity ¨ a or b = b or a ¨ a and b = b and a n Associativity ¨ (a or b) or c = a or (b or c) ¨ (a and b) and c = a and (b and c) n Distributivity ¨ a or (b and c) = (a or b) and (b or c) ¨ a and (b or c) = (a and b) or (b and c) n DeMorgan rules ¨ ¬(a and b) = (¬a) or (¬ b) ¨ ¬(a or b) = (¬a) and (¬b) 12 6  Algebraic properties n Absorption ¨ (a and b) or a = a ¨ (a or b) and a = a n Idempotency ¨a or a = a ¨ a and a = a n Exclusion (not satisfied) ¨a or ¬a ¹ 1 ¨ a and ¬a ¹ ∅ 13 Linguistic modifiers n Approximation of Fuzzy Sets: scalar → fuzzy set, modifying the "base" of a fuzzy set ¨ about, around, near and close to n Restriction of Fuzzy Sets: modifying the shape ¨ below and above n Intensification and Dilution of Fuzzy Sets ¨ intensification: very (n = 2) and extremely (n = 3) ¨ dilution: somewhat (n = 1/2) and greatly (n = 5/7) 14 7  Linguistic modifiers n Graphical representation 15 8