Fuzzy Logic: Principles, Applications, and Systems, Slides of Robotics

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computingFuzzy logic is part of soft

Contents

• Combining methods and future research• Our approach to decomposition• Approaches to fuzzy logic decomposition• Practical Examples• Fuzzy Logic Minimization and Synthesis• Fuzzy logic systems applications• Fuzzy logic• Fuzzy sets• Review of classical logic and reasoning systems

Logical arguments

Logical arguments

statements.Any logical argument consists of

or holds false.unambiguously either holds trueA statement is a sentence which

Example:

Example:Today is Sunday

Predicates

Predicates

Example:

Example: Seven is an even number

follows:This example can be written in a mathematical form as

7

∈∈∈∈

{x| x is an even number}

or in a more concise way:

7

∈∈∈∈

{x|P(x)}

property P' and it is known as the predicate.where | is read as such that and P(x) stands for `x has

particularNote that a predicate is not a statement until some

x-value

is specified.

statement whose truth or falsity can be worked out.Once a x value is specified then the predicate becomes a

Universe of Discourse

Universe of Discourse

a statement is evaluated for relevant values.Using the universe of discourse one assures that

The above predicate is then true only for

real

numbers.

anything else.numbers rather than buildings, rivers, ordiscourse is most likely to be the set of naturalSimilarly for the first example the universe of

any logical paradoxes can not arise.Hence, using the concept of the universe of discourse

Existential Quantifier

Existential Quantifier

Another type of quantifier is the

existential quantifier (

The existential quantifier is interpreted

least one element of the set.describes a statement as being true for atas 'there exists' or 'for some' and

For example, (

x)

((river(x)

name(Amazon))

Truth Tables

Truth Tables

A

B

AÙB

AÚB

~

A

True

True

True

True

False

True

False

False

True

False

False

True

False

True

True

False

False

False

False

True

And

Or

Not

Fuzzy Logic

Fuzzy Logic

introduced by L.A Zadeh in a 1965 paper.The concept of fuzzy logic was

present us with certain problemsuseful and applicable for many years theyWhereas Aristotelian concepts have been

• Cannot handle exceptions– Lack of quantifiers – Cannot express ambiguity

Traditional Logic Problems

Traditional Logic Problems

Cannot handle exceptions:

• Another limitation of traditional predicate logic is

true.expressing things that are sometimes, but not always

Fuzzy Set Theory

Fuzzy Set Theory

Fuzzy Sets

Definition

Operations

Identities

Transformations

TRADITIONAL

TRADITIONAL vs

vs. FUZZY SETS

. FUZZY SETS

partially to a set.In fuzzy sets, something may belong

terms.somewhat true, 0.2 or 0.9 in numericalTherefore it might be very true or

mapping is as shown below:sets defined in terms of a functionalThe membership function using fuzzy

TRADITIONAL

TRADITIONAL vs

vs. FUZZY SETS

. FUZZY SETS

Fuzzy logic allows you to violate the

laws of

noncontradiction

since an element can be a

member of more than one set.

More set operations are available

necessarily result to an empty set.intersection of a set with its complement does notThe excluded middle is not applicable, i.e., the

cases mightRule based systems using fuzzy logic in some

increase the amount of computation

classical binary logic. required in comparison with systems using