Fuzzy Logic

Fuzzy Sets and Fuzzy Fuzzy Sets and Fuzzy Logic Logic

© N. Kasabov Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering, MIT Press, 1996

Sub-topics:

Fuzzy sets Operations with fuzzy sets Geometrical representation of fuzzy sets Fuzziness versus probability Conceptualising in fuzzy terms The extension principle Fuzzy relations and fuzzy implications Fuzzy propositions and fuzzy logic


Fuzzy Sets

Figure 3.1

Membership functions representing three fuzzy sets for the variable "height".

Fuzzy Sets...

Figure 3.2

Representing crisp and fuzzy sets as subsets of a domain (universe) U".

Fuzzy Sets...

Figure 3.3

Support of a fuzzy set A

Fuzzy Sets...

Figure 3.4

I-cut of a fuzzy set

Operations with fuzzy sets

Figure 3.5

Five operations with two fuzzy sets A and B approximately represented in a graphical form

Operations with fuzzy sets...

Figure 3.6

Showing graphically one way to measuring similarity and distance between fuzzy sets A and B. The black area represents quantitatively the measure.

Operations with fuzzy sets...


Figure 3.7

A graphical representation of similarity S (the black area) between two fuzzy sets B and A based on possibility P and necessity N measures (see the formulas in the text).

Geometrical representation of fuzzy sets


Figure 3.8

A geometrical representation of the concept of "heavy smoker" from The Smoker and the Risk of Cancer case example as a point in a (three dimensional) cube and as a (two dimensional) membership function.

Fuzziness versus probability


Figure 3.9

Probability density function for throwing a dice and the membership functions of the concepts "Small" number, "Medium", "Big".

Conceptualising in fuzzy terms

Standard membership functions:

• single-valued, or singleton• triangular• trapezoidal• S-function (sigmoid function):

– S(u) = 0, u<=a – S(u) = 2((u-a)/(c-a))2 , a <u <= b – S(u) = 1 - 2((u-a)/(c -a))2 , b <u <= c – S(u) = 1, u > c.


Conceptualising in fuzzy terms...

more standard membership functions...

• Z function:– Z(u)= 1 - S(u)

• Pi - function: • P(u)=S(u), u<=b; P(u)=Z(u), u>b.

Two parameters must be defined for the quantization procedure:

• the number of the fuzzy labels;• the form of the membership functions for each of

the fuzzy labels.




Figure 3.10

Standard types of membership functions: Z function; n function; S function; trapezoidal function; triangular function; singleton.



Figure 3.12

One representation for the fuzzy number "about 600".



Figure 3.13

Representing truthfulness (certainty) of events as fuzzy sets over the [0,1] domain.

The extension principle


Figure 3.14

An illustration of the extension principle in fuzzy theory: an example of transforming a fuzzy set "x is about 2" into a fuzzy set "f is about f(2)" for f(x)= (x 1)2.

Fuzzy relations and fuzzy implications


Figure 3.15

Several useful fuzzy implications. The following short denotations are used: u instead of n(u); v instead of n(v); "n" is minimum; "n" is maximum; "+" is algebraic summation; " " is algebraic substraction.

Fuzzy relations and fuzzy implications...


Figure 3.16

(a) Membership functions for fuzzy sets for the Smoker

and the Risk of Cancer case example.

(b) The Rc implication relation: "heavy smoker > high risk of cancer" in a matrix form.



Figure 3.17

Max min composition applied over the fuzzy set "moderate smoker" and the Rc implication "heavy smoker >high riskof cancer" from figure 3.16 for the Smoker and the Risk of Cancer case example.



see figure 3.18 (not shown)

for examples of fuzzy compositions

see figure 3.19 (not shown)

for examples of the fuzzy logic connectives AND, OR, NOT illustrated over two fuzzy propositions A and B.

.

Fuzzy propositions and fuzzy logic

Fuzzy propositions are propositions which contain fuzzy variables with their fuzzy values. The truth value of a fuzzy proposition " X is A " is given by the membership function mA.

Fuzzy propositions may include modifiers, or also called - hedges. The general form of a fuzzy proposition with a modifier is X is mA. The negation "not" can be viewed as a modifier. Other modifiers are: very A, denoted as A2 (concentration); more or less A, denoted as A1/2 (dilation)


Fuzzy propositions and fuzzy logic...


Figure 3.5

Five operations with two fuzzy sets A and B approximately represented in a graphical form

Fuzzy propositions and fuzzy logic...

Fuzzy logic

• fuzzy propositions• fuzzy connectives• fuzzy rules (laws) of inference

Fuzzy laws of inference:

• generalised modus ponens• generalised modus tolens• de Morgan

see Figure 3.20

for an illustration of properties of fuzzy implications when used for inference.

Fuzzy Logic

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