Soft ComputingSoft Computing

Fuzzy Logic

FUZZY LOGIC

Motivation

• Modeling of imprecise conceptsE.g. age, weight, height,…• Modeling of imprecise dependencies (e.g. rules), e.g. if

Temperature is high and Oil is cheap then I will turn-on the generator

• Origin of information- Modeling of expert knowledge-Representation of information extracted from

inherentedly imprecise data

FUZZY LOGIC

To quantify and reason about fuzzy or vague terms of natural language

Example: hot, cold temperaturesmall, medium, tall heightcreeping, slow, fast speed

Fuzzy Variable

A concept that usually has vague (or fuzzy) valuesExample: age, temperature, height, speed

FUZZY LOGIC

Universe of Discourse

Range of possible values of a fuzzy variableExample: Speed: 0 to 100 mph

FUZZY LOGIC

Fuzzy Set (Value)

Let X be a universe of discourse of a fuzzy variable and x be its elements

One or more fuzzy sets (or values) Ai can be defined over X

Example: Fuzzy variable: AgeUniverse of discourse: 0 – 120 yearsFuzzy values: Child, Young, Old

A fuzzy set A is characterized by a membership function µA(x) that associates each element x with a degree of membership value in A

The value of membership is between 0 and 1 and it represents the degree to which an element x belongs to the fuzzy set A

FUZZY LOGIC

Fuzzy Set (Value)

In traditional set theory, an object is either in a set or not in a set (0 or 1), and there are no partial memberships

Such sets are called “crisp sets”

FUZZY LOGIC

Fuzzy Set Representation

Fuzzy Set A = (a1, a2, … an)

ai = µA(xi)

xi = an element of XX = universe of discourse

For clearer representationA = (a1/x1, a2/x2, …, an/xn)

Example: Tall = (0/5’, 0.25/5.5’, 0.9/5.75’, 1/6’, 1/7’, …)

FUZZY LOGIC

Fuzzy Set Representation

For a continuous set of elements, we need some function to map the elements to their membership values

Typical functions: sigmoid, gaussian

FUZZY LOGIC

Formation of Fuzzy Sets

• Opinion of a single person• Average of opinion of a set of persons• Other methods (e.g. function approximation from data

by neural networks)• Modification of existing fuzzy sets

- Hedges- Application of Fuzzy set operators

FUZZY LOGIC

Formation of Fuzzy Sets

Hedges: Modification of existing fuzzy sets to account for some added adverbs

Types:

Concentration (very)Square of memberships Conc(µA(x)) = [µA(x)]2

reduces small memberships values0.1 changes to 0.01 (10 times reduction)0.9 changes to 0.81 (0.1 times reduction)

Example: very tall

FUZZY LOGIC

Formation of Fuzzy Sets

Dilation (somewhat)Square root of memberships

Dil(µA(x)) = [µA(x)]1/2

increases small memberships values0.09 changes to 0.30.81 changes to 0.9

Example: somewhat tall

FUZZY LOGIC

Fuzzy Sets Operations

Intersection (A B)

In classical set theory the intersection of two sets contains those elements that are common to both

In fuzzy set theory, the value of those elements in the intersection:

µA B(x) = min [µA(x), µB(x)]

e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75, 0/6)Tall Short = (0/5, 0.1/5.25, 0.5/5.5, 0.1/5.75, 0/6)

= Medium

FUZZY LOGIC

Fuzzy Sets Operations

Union (A B)

In classical set theory the union of two sets contains those elements that are in any one of the two sets

In fuzzy set theory, the value of those elements in the union: µA B(x) = max [µA(x), µB(x)]

e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75)Tall Short = (1/5, 0.8/5.25, 0.5/5.5, 0.8/5.75, 1/6)

= not Medium

FUZZY LOGIC

Fuzzy Sets Operations

Complement (A)

In fuzzy set theory, the value of complement of A is: µ A(x) = 1 - µA(x)

e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Tall = (1/5, 0.9/5.25, 0.5/5.5, 0.2/5.75, 0/6)

FUZZY RULES

Fuzzy Rules

Relates two or more fuzzy propositions

If X is A then Y is B

e.g. if height is tall then weight is heavy

X and Y are fuzzy variablesA and B are fuzzy sets

FUZZY LOGIC

Fuzzy RelationsClassical relation between two universes U = {1, 2} and V = {a, b, c} is defined as:

a b c

R = U x V = 1 1 1 12 1 1 1

Example: U = Weight (normal, over) V = Height (short, med, tall)

FUZZY LOGIC

Fuzzy Relations

Fuzzy relation between two universes U and V is defined as:

µR (u, v) = µAxB (u, v) = min [µA (u), µB (v)]

i.e. we take the minimum of the memberships of the two elements which are to be related

FUZZY LOGIC

Fuzzy Relations

Example:

Determine fuzzy relation between A1 and A2

A1 = 0.2/x1 + 0.9/x2

A2 = 0.3/y1 + 0.5/y2 + 1/y3

The fuzzy relation R is

R = A1 x A2 = 0.2 x 0.3 0.5 1 0.9

FUZZY LOGIC

Fuzzy Relations

Example:

R = A1 x A2 = 0.2 x 0.3 0.5 1 0.9

= min(0.2, 0.3) min(0.2, 0.5) min(0.2, 1)min(0.9, 0.3) min(0.9, 0.5) min(0.9, 1)

= 0.2 0.2 0.20.3 0.5 0.9

FUZZY LOGIC

Fuzzy Relations

R = R(A1, A2)

= 0.2 0.2 0.20.3 0.5 0.9

A1

A2

a11(0.2)

a12(0.9)

a22(0.5)

a21(0.3) 0.2 0.3

0.2 0.5

a23(1.0)

0.20.9

FUZZY RULES

Fuzzy Associative Matrix So for the fuzzy rule:

If X is A then Y is B

We can define a fuzzy matrix M(nxp) which relates A to B

M = A x B

It maps fuzzy set A to fuzzy set B and is used in the fuzzy inference process

FUZZY RULES

Fuzzy Associative Matrix Concept behind M

a1 b1 a1 b2 …a2 b1 …...

If a1 is true then b1 is true; and so on

FUZZY RULES

Approximate Reasoning Example: Let there be a fuzzy associative matrix M for the rule: if A then B

e.g. If Temperature is normal then Speed is medium

Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]

FUZZY RULES

Approximate Reasoning: Max-Min Inference Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]

B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]then

M = (0, 0) (0, 0.6) . . .(0.5, 0) . . .. . .

= 0 0 0 0 00 0.5 0.5 0.5 0 by taking the minimum0 0.6 1 0.6 0 of each pair0 0.5 0.5 0.5 00 0 0 0 0