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Chap3 Mathematical basics of fuzzy logic

Abstract

  • Fuzzy information and fuzzy sets
  • Operations with fuzzy sets
  • Fuzzy relations
  • Linguistic variables

Basics of fuzzy logic

Fuzzy information and Fuzzy sets

Given μ is a fuzzy set in G, then {x| x∈G,μ(x)>0} is called support of fuzzy set μ {x| x∈G,μ(x)=1} is called kernel of fuzzy set μ

fuzzy-similar: have the same kernal strongly fuzzy-similar: have the same kernal and the same support

And link(intersection)

μ1∩μ2:G→[0,1],with(μ1∩μ2)(x)=MIN{μ1(x)+μ2(x)} \mu_1 \cap \mu_2:G \rightarrow[0,1], with (\mu_1 \cap \mu_2)(x) = MIN\{\mu_1 (x) + \mu_2(x)\}

Or link(union)

μ1∪μ2:G→[0,1]mit(μ1∩μ2)(x)=MAX{μ1(x),μ2(x)} \mu_1 \cup \mu_2 : G\rightarrow[0,1] mit (\mu_1 \cap \mu_2)(x) = MAX \{\mu_1 (x), \mu_2(x)\}

Special operators: Modifiers - Concentration operator CON - Dilatation operator DIL

fuzzy relations R:G1×G2→[0,1] R: G_1 \times G_2 \rightarrow [0,1]

(μ1×μ2)(x,y)=MIN{μ1(x),μ2(y)},with(x,y)∈G1×G2 (\mu_1 \times \mu_2)(x,y)= MIN\{ \mu_1(x), \mu_2(y)\}, with (x,y) \in G_1 \times G_2
μ1×μ2=μ1T∘μ2 \mu_1 \times \mu_2 =\mu_1^T \circ \mu_2

product calculation is replaced by the MIN opertor and the addition is replaced by the MAX operatior

μR∘S(x,z)=MAXy∈G2MIN{μR(x,y),μS(y,z)},with(x,z)∈G1×G3 \mu_{R \circ S}(x, z)= MAX_{y \in G_2} {MIN \{\mu_R(x,y), \mu_S(y,z)}\}, with (x,z) \in G_1 \times G_3

fuzzy inference

μ2(y)=MAXx∈G1{MINμ1(x),μR(x,y)}=μ1∘R,y∈G2 \mu_2(y) = MAX_{x \in G_1} \{ MIN{\mu_1 (x), \mu_R(x,y)}\} = \mu_1 \circ R, y\in G_2

inference with several rules

  1. Construct implication relation for each rule i
μRi(x,y)=I(μAi(x),μBi(y)) \mu_{R_i}(x,y) = I(\mu_{A_i}(x), \mu_{B_i}(y))
  1. Aggregate relations RiR_i into one
μR(x,y)=aggr(μRi(x,y)) \mu_R(x,y) = aggr(\mu_{R_i}(x,y))

The aggr operator is the minimum for implications and the maximum for junctions

  1. Use relational composition to derive B' from A'
B′=A′∘R B^\prime = A^\prime \circ R

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