INTRODUCTION SOFT

COMPUTINGChapter 1

Soft computing is an emerging approach to computing which parallel the remarkable ability of the human mind to reason and learn in a environment of uncertainty and imprecision.

SOFT COMPUTING Soft computing differs from

conventional (hard) computing in that, unlike hard computing

It is tolerant of imprecision Uncertainty partial truth and approximation In effect, the role model for soft

computing is the human mind.

COMPONENTS Fuzzy Systems Neural Networks Evolutionary Computation/Genetic

Algorithm(GA) Machine Learning Probabilistic Reasoning

GOALS OF SOFT COMPUTING The main goal of soft computing is to

develop intelligent machines to provide solutions to real world problems, which are not modeled, or too difficult to model mathematically.

It’s aim is to exploit the tolerance for Approximation, Uncertainty, Imprecision, and Partial Truth in order to achieve close resemblance with human like decision making.

DIFFERENCE BETWEEN HC & SC

Hard Computing Soft Computing

conventional computing

Binary logic, crisp systems, numerical analysis and crisp software 

Tolerant to imprecision, uncertainty, partial truth, and approximation

fuzzy logic, neural nets and probabilistic reasoning.

DIFFERENCE BETWEEN HC & SC

Hard Computing Soft Computing

 requires programs to be written

 two-valued logic

Deterministic requires exact input

data

strictly sequential precise answers

can evolve its own programs

multivalue or fuzzy logic

Stochastic Can deal with

ambiguous and noisy data

parallel computations approximate answers

CLASSICAL SETS & FUZZY SETS

CLASSİCAL SETS AND FUZZY SETS

A classical set is defined by crisp boundaries

A fuzzy set is prescribed by vague or ambiguous properties; hence its boundaries are ambiguously specified

X (Universe of discourse)

Classical Set•A set is defined as a collection of objects, which share certain characteristics•A classical set is a collection of distinct objects -- set of negative integers, set of persons with height<6 ft, days of the week etc

• Each individual entity in a set is called a member or an element of the set.

• The Classical set is defined in such a way that the Universe of Discourse is split into 2 groups: members and Nonmembers

Classical Set

Shoe Polish

Tuesday

Wednesday

Saturday

Liberty

Butter

Days of the week

Classical Set : Law of the Excluded Middle

•X Must either be in set A or in set not-A, ie., Of any subject, one thing must be either asserted or denied

Aristotle

Defining a SetThere are several ways of defining a set •A = {2,4,6,8,10}

•A= {x│x is a prime number <20 }

• A= {xi +1 = (xi +1 )/5, i=1 to 10, where x1=1 }

•A= {x│x is an element belonging to P AND Q }

•µ A(x) = 1 if x A∈ = 0 if x A∈Here µ A(x) is membership function for set A

•Φ is a null or Empty Set i.e., with no elements

•Set consisting of all possible subsets of a given set A is called a Power Set P(A)= {x│x A }⊆

•For crisp set A and B containing some elements in universe X, the notations used are x A ∈ ⇒ x does belong to A x A ∉ ⇒ x does not belong to A x X ∈ ⇒ x does belong to universe X

•For classical sets A and B on X we also have A B ⊂ ⇒ A is completely contained in B (i.e., if x A then x B ) ∈ ∈

A B⊆ ⇒ A is contained in or equivalent to B

A=B ⇒ A B and B A ⊂ ⊂

Operations on Classical Sets

Union : A B = {x│x A or x B }∪ ∈ ∈

Intersection : A ∩ B = {x│x A and x B }∈ ∈ Complement : Ā = {x│x ∉ A , x ∈ X }

Difference: A-B = A │ B = {x│x A and x B }∉ ∉ = A- (A ∩ B ) i.e., All elements in universe that belong to A but do not belong to B

Properties of Classical Sets Commutivity : A B = B A ; ∪ ∪ A ∩ B = B ∩ A

Associatively : A ( B C) = ( A B ) C∪ ∪ ∪ ∪ A ∩ ( B ∩ C) = ( A ∩ B) ∩C

Distributivity: A ( B ∩ C) = ( A B ) ∩ ( A C ) ∪ ∪ ∪ A ∩ ( B C) = ( A ∩ B) ( A ∩ C ) ∪ ∪

Properties of Classical Sets

Idem potency : A A = A ; ∪ A ∩ A = A

Transitivity : If A B⊆ C, then A C⊆ ⊆

Identity: A ∪ Φ = A ; A ∩ Φ = Φ A ∪ X = X ; A ∩ X = X

Properties of Classical SetsLaw of Excluded middle : A Ā = X; ∪

DeMorgan’s Law

Law of contradiction : A ∩ Ā = Φ;