AI TECHNIQUES

Fuzzy Logic

(Fuzzy System)

Fuzzy Logic : An Idea

Fuzzy Logic: Background

The concept of a set and set theory are powerful concepts in mathematics. However, the principal notion underlying set theory, that an element can (exclusively) either belong to set or not belong to a set, makes it well high impossible to represent much of human discourse. How is one to represent notions like: 

large profit

high pressure

tall man

wealthy woman

moderate temperature

Background & Definitions

“Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise.”

Fuzzy set theory, originally introduced by Lotfi Zadeh in the 1960's, resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit.

Fuzzy Sets & Fuzzy Logic

A fuzzy set is a collection of objects that might belong to the set to a degree, varying from 1 for full belongingness to 0 for full non-belongingness, through all intermediate values.

"Fuzzy logic is a generalization of standard logic, in which a concept can possess a degree of truth anywhere between 0.0 and 1.0. Standard logic applies only to concepts that are completely true (having degree of truth 1.0) or completely false (having degree of truth 0.0). Fuzzy logic is supposed to be used for reasoning about inherently vague concepts, such as 'tallness.' For example, we might say that ‘Michael Jordan is tall,' with degree of truth of 0.9

Fuzzy Logic Example:What is Tall?

In-Class ExerciseProportion

Height Voted for5’10” 0.055’11” 0.106’ 0.606’1” 0.156’2” 0.10

Jack is 6 feet tall Probability theory - cumulative probability There is a 75 percent chance that Jack is tall

Membership Functions in Fuzzy Sets

Membership

Short Medium Tall

Height in inches (1 inch = 2.54 cm)

0.5

1.0

64 69 74

Fuzzy logic - Jack's degree of membership within the set of tall people is 0.75

We are not completely sure whether he is tall or not. Fuzzy logic - We agree that Jack is more or less tall. Membership Function

< Jack, 0.75 Tall > Knowledge-based system approach: Jack is tall

(CF = .75) Can use fuzzy logic in rule-based systems (belief

functions)

Fuzzy Logic & Fuzzy Systems

The term fuzzy logic is used in two senses:

Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals (as logical systems do) with the representation and inference from knowledge. Fuzzy logic, unlike other logical systems, deals with imprecise or uncertain knowledge. In this narrow, and perhaps correct sense, fuzzy logic is just one of the branches of fuzzy set theory.

Broad Sense: Fuzzy logic synonymously with fuzzy set theory.

Fuzzy systems

A fuzzy system consists of: Fuzzy (linguistic) variables Fuzzy rules Fuzzy inference

Example: Fuzzy variables

Fuzzy label Numerical interval Typical value

Always [1.00, 1.00] 1.0Very strong [0.95, 0.99] 0.99Strong [0.80, 0.94] 0.9More or less strong [0.65, 0.79] 0.7Medium [0.45, 0.64] 0.5More or less weak [0.30, 0.44] 0.3Weak [0.10, 0.29] 0.2Very weak [0.01, 0.09] 0.05No [0.00, 0.00] 0.0

Linguistic variables/

Example: Fuzzy rules

A fuzzy rule is a linguistic expression of causal dependencies between linguistic variables in form of if-then statements.

General form: IF <antecedent> then <consequence> Example:

If temperature is cold and oil price is cheap Then heating is high

Linguistic variables Linguistic values

Example: Fuzzy inference

Inputs to a fuzzy system can be: fuzzy, e.g. (Score = Moderate), defined by

membership functions; exact, e.g.: (Score = 190); defined by crisp

values Outputs from a fuzzy system can be:

fuzzy, i.e. a whole membership function. exact, i.e. a single value is produced .

Fuzzy system applications Pattern recognition and classification Fuzzy clustering Image and speech processing Fuzzy systems for prediction Fuzzy control Monitoring Diagnosis

Speech processing

µ

dB30 50

Low Medium High

(a)

IF 0-1000 is Medium AND1000-2000 is Medium AND2000-3000 is Low THEN

the note is Middle C

IF 0-1000 is High AND1000-2000 is Medium AND2000-3000 is Low THEN

the note is D above Middle C

IF 0-1000 is High AND1000-2000 is Low AND2000-3000 is Low THEN

the note is E above Middle C

IF 0-1000 is Medium AND1000-2000 is Medium AND2000-3000 is Medium THEN

the note is F above Middle C

IF 0-1000 is Low AND100-2000 is Medium AND2000-3000 is Medium AND3000-4000 is Low THEN

the note is G above Middle C

IF 0-1000 is Low AND1000-2000 is Medium AND2000-3000 is Medium AND3000-4000 is Medium THEN

the note is A above Middle C

IF 0-1000 is Low AND1000-2000 is Medium AND2000-3000 is Low THEN

the note is B above Middle C

IF 0-1000 is High AND1000-2000 is High AND2000-3000 is High THEN

the note is C above Middle C

(b)

i

i

i

I

Iu

u

u

New Zealand EnglishGeneral Australian EnglishR.P. English

1000 1500 2000 2500

200

300

400

500

600

800

900

0 500

F1

freq

uenc

y(H

ertz

)

F2 frequency (Hertz)

3

3

3

Monitoring

1062

Quick Normal Slow

Brakes' response (seconds) Cooling system (tº)

50

Normal

Underheating Overheating

90 120 150

Gauge sensitivity (levels)

1 2 3 4 5

Damaged OK

0

µ

Temperature (ºC)

50

Normal

90 120 150

Low High

Fuzzy systems

http://www.austinlinkscom/Fuzzy

http://www.industry.siemens.de/water/en/solutions/sector_fuzzy-logic.htm

http://www.mathworks.com/access/helpdesk/help/toolbox/fuzzy/index.html

The MathWorks

Fuzzy Logic Advantages

Provides flexibility Allows for observation Shortens system development time Increases the system's maintainability Handles control or decision-making problems not

easily defined by mathematical models

Intelligence Density Dimension

Accuracy Response speed Flexibility Tolerance for complexity