Emerging Approach to Computing Techniques.pptx

EMERGING APPROACH TO COMPUTING TECHNIQUES
(Email id: pksharma@davjalandhar.com)
Live Webinar on Google Meet
Link: https://meet.google.com/tgn-teeo-xab
At
R.R. BAWA D.A.V. COLLEGE FOR GIRLS, BATALA
(24th March, 2022)
By
Dr. P. K. Sharma
Associate Professor in Mathematics
D.A.V. College, Jalandhar.

What is an “Emerging approach to Computing”?
The emerging approach to computing is refer to Soft computing which is placed parallel
to the remarkable ability of the human mind to reason and learn in a environment of
uncertainty and imprecision.
Some of it’s principle components includes:
Neural Network(NN)
Genetic Algorithm(GA)
Machine Learning (ML)
Probabilistic Reasoning(PR)
Fuzzy Logic(FL)
These methodologies (techniques) form the core of soft computing.

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.

SOFT COMPUTING -
DEVELOPMENT HISTORY
Soft = Evolutionary + Neural + Fuzzy
Computing Computing Network Logic
Zadeh Rechenberg McCulloch Zadeh
1981 1960 1943 1965
Evolutionary = Genetic + Evolution + Evolutionary + Genetic
Computing Programming Strategies programming Algorithms
Rechenberg Koza Rechenberg Fogel Holland
1960 1992 1965 1962 1970

NEURAL NETWORKS(NN)
Neural Network is a network of artificial neurons, inspired by biological network
of neurons, that uses mathematical models as information processing units to
discover patterns in data which is too complex to notice by human.
An NN, in general, is a highly interconnected network of a large number of processing elements called neurons in an architecture
inspired by the brain.
NN Characteristics are:-
 Mapping Capabilities / Pattern Association
 Generalisation
 Robustness
 Fault Tolerance
 Parallel and High speed information processing

6
Neuron
Biological neuron
Model of a neuron

BIOLOGICAL BACKGROUND
All living organism consist of cell. In each cell, there is a set of chromosomes
which are strings of DNA and serves as a model of the organism. A
chromosomes consist of genes of blocks of DNA. Each gene encodes a particular
pattern. Basically, it can be said that each gene encodes a traits.
Fig.
Genome
consisting
Of
chromosomes.
A
T
G
C
T
A
G
C
A
G
T
A
C

GENETIC ALGORITHM (GA)
Genetic Algorithms initiated and developed in the early 1970’s by
John Holland are unorthodox search and optimization algorithms,
which mimic some of the process of natural evolution.
Gas perform directed random search through a given set of
alternative with the aim of finding the best alternative with
respect to the given criteria of goodness.
These criteria are required to be expressed in terms of an object
function which is usually referred to as a fitness function.

BENEFITS OF GENETIC ALGORITHM
Easy to understand.
We always get an answer and the answer gets better with time.
Good for noisy environment.
Flexible in forming building blocks for hybrid application.
Has substantial history and range of use.
Supports multi-objective optimization.
Modular, separate from application.

FUZZY LOGIC(FL)
Fuzzy set theory proposed in 1965 by L. A. Zadeh is a
generalization of classical set theory.
In classical set theory, an element either belong to or does not
belong to a set and hence, such set are termed as crisp set. But
in fuzzy set, many degrees of membership (between o/1) are
allowed.

FUZZY VERSES CRISP
FUZZY CRISP
IS R AM HONEST ?
IS WATER COLORLESS ?
FUZZY CRISP
Extremely
Honest(1)
Very
Honest(0.8)
Honest at
Times(0.4)
Extremely
Dishonest(0)
YES!(1)
NO!(0)

OPERTIONS
CRISP FUZZY
1.Union
2.Intersection
3.Complement
4.Difference
1.Union
2.Intersection
3.Complement
4.Equality
5.Difference
6.Disjunctive Sum

PROPERTIES
CRISP FUZZY
Commutativity
Associativity
Distributivity
Idempotence
Identity
Law Of Absorption
Transitivity
Involution
De Morgan’s Law
Law Of the Excluded Middle
Law Of Contradiction
Commutativity
Associativity
Distributivity
Idempotence
Identity
Law Of Absorption
Transitivity
Involution
De Morgan’s Law

ENCODING
There are many ways of representing individual genes.
Binary Encoding
Octal Encoding
Hexadecimal Encoding
Permutation Encoding
Value Encoding
Tree Encoding.

Difference between Soft Computing and Hard Computing:
Soft Computing
1. Soft Computing is liberal of
inexactness, uncertainty, partial
truth and approximation.
2. Soft Computing relies on formal
logic and probabilistic reasoning.
3. Soft computing has the features
of approximation and
dispositionality.
4. Soft computing is stochastic in
nature.
5. Soft computing works on
ambiguous and noisy data.
Hard Computing
1. Hard computing needs a
exactly state analytic model.
2. Hard computing relies on
binary logic and crisp system.
3. Hard computing has the
features of exactitude
(precision) and categoricity.
4. Hard computing is
deterministic in nature.
5. Hard computing works on
exact data.

Difference between Soft Computing and Hard Computing(Cont.)
Soft Computing
6. Soft computing can perform
parallel computations.
7. Soft computing produces
approximate results.
8. Soft computing will emerge its
own programs.
9. Soft computing incorporates
randomness.
10. Soft computing will use multi-
valued logic.
Hard Computing
6. Hard computing performs
sequential computations.
7. Hard computing produces
precise results.
8. Hard computing requires
programs to be written.
9. Hard computing is settled.
10. Hard computing uses two-
valued logic.

GEORGE CANTOR
George Cantor, in 1870’s, gave the notion of a Set
and developed Set theory which is of great
importance in mathematics.
SET THEORY
Set theory developed by George cantor now a days is known as classical set theory and the sets are also called crisp sets
Set: A set is a collection of well defined
and distinct objects

Basic Principal Underlying Set Theory:
The basic principal under lying the set theory is that an element can (exclusively) either
belong to set or not belong to a set, but cannot be both.
This means that the set A of a universal set X can be characterized by a characteristic
function defined by
If X = { a, b , e , i , o, u, z} be the universal set, then the set A = { a, e, i, o , u} of X
can also be expressed as
: {0,1
}
A X
 
1
( )
0
A
if x A
x
if x A



 


{(a,1), (b, 0), (e,1), (i,1), (o,1), (u,1), (z,1)}
A 
. ., the set can be written as : .
A
i e A A x x x X

 
{( , ( )) }

21
Fuzzy Thinking
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 nigh impossible to represent much of human discourse.
How is one to represent notions like:
large profit
high pressure
tall man
moderate temperature.

FUZZY SET THEORY
 Fuzzy Set Theory was formalized by Professor
Lotfi Asker Zadeh at the University of
California in 1965 to generalize classical set
theory. Zadeh was almost single handedly
responsible for the early develop-ment in this
field.
LOTFI ZADEH
REFERENCES:
Zadeh L.A.(1965)Fuzzy sets. Information and Control, 8(1965), 338-353.
Zadeh L.A.(1978)Fuzzy Sets as the Basis for a Theory of Possibility, Fuzzy Sets and Systems
100 Supplement (1999) 9-34.

• Formal definition:
A fuzzy set A in X is expressed as a set of ordered pairs:
A
A x x x X

 
{( , ( )) | }
Fuzzy set Membership
function
(MF)
Universe or
universe of discourse
A fuzzy set is totally characterized by a
membership function (MF).
Fuzzy Sets
Remark: The Characteristic function is replaced by a membership
function
A

A


Definition of Crisp Set and Fuzzy Sets
A ‘crisp’ set, A, can be defined as a set which
consists of elements with either full or no
membership at all in the set.
 It is characterized by a characteristic function
A “fuzzy set” is defined as a class of objects with a
continuum of grades of membership.
It is characterized by a “membership function that
assigns to each member of the fuzzy set a degree of
membership in the unit interval [0,1].

25
One can define the crisp set “circles” as:
The fuzzy set “circles can be defined as:

Example (Boolean Logic V/S Fuzzy Logic)

Membership Functions (MF)
 One of the key issues in all fuzzy sets is how to determine membership
functions.
 The membership function fully defines the fuzzy set.
 A membership function provides a measure of the degree of similarity of an
element to a fuzzy set.
 Membership functions can take any form, but there are some common examples
that appear in real applications.

Membership functions can
- either be chosen by the user arbitrarily, based on the user’s
experience (MF chosen by two users could be different depending
upon their experiences, perspectives, etc.)
- Or be designed using machine learning methods (e.g., artificial
neural networks, genetic algorithms, etc.)
 There are different shapes of membership functions; triangular,
trapezoidal, piecewise-linear, Gaussian, bell-shaped, etc.

 
0 x<
( ) / ( )
, , , , = 1
( ) / ( )
0
for
x for x
X for x
x for x
for x

    
     
    


    


 

    





 
0 x< a
( ) / ( ) a
, a, b, c =
( ) / ( ) b
0
for
x a b a for x b
T X
c x c b for x c
for x c

    


   

 

Membership Functions
Trapezoidal Membership Function
Triangular Membership Function

• Gaussian membership function
Where c – centre , s - width and m - fuzzification factor
1
( , , , ) exp
2
m
A
x c
x c s m
s

 

 
 
 
 
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
µA(x)
c=5
s=2
m=2

0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c=5
s=0.5
m=2
c=5
s=5
m=2
0 1 2 3 4 5 6 7 8 9 10
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c=5
s=2
m=0.2
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c=5
s=5
m=5

The fuzzy set operations of union, intersection and complementation are defined in terms of
membership functions as follows:
•Union:
A∪B(x) = max{A(x), B(x)}
•Intersection:
A∩B(x) = min{A(x), B(x)}
•Complement:
not A(x) = 1- A(x)
The other fuzzy set theory constructs that are essential are:
•Fuzzy Set Inclusion:
A ⊂ B if and only if ∀x (for all x) A(x) ≤ B(x)
•Fuzzy Set Equality:
A= B if and only if ∀x (for all x) A(x) = B(x).
.
FUZZY SET OPERATIONS

Representation of Union of two crisp sets and fuzzy sets

Representation of Intersection of two crisp sets and fuzzy sets

Representation of Complement of a crisp set and a Fuzzy set

Emerging Approach to Computing Techniques.pptx

Law of Excluded Middle in Fuzzy set Theory
Min-Max fuzzy logic fails: The Law of Excluded Middle.
A A
  
Since min{ A(x) , 1-A(x)}  0
Thus, (the set of numbers close to 2) AND (the set of numbers not close
to 2)  null set

Law of Contradiction in Fuzzy set theory
Min-Max fuzzy logic fails: The Law of Contradiction.
A A X
 
Since max {A(x) , 1-A(x)}  1
Thus, (the set of numbers close to 2) OR (the set of numbers not close
to 2)  universal set

40
Example of Fuzzy relation (Approximate Equal)
 
 
A fuzzy relation R is a 2D membership
,
( , )
funct
( (
ion
,
, )
) |
R x y
R X Y
x y x y

  
{1, 2, 3, 4, 5}
Let X Y
 
1 1 0.8 0.3 0 0
2 0.8 1 0.8 0.3 0
3 0.3 0.8 1 0.8 0.3
1
4 0 0.3 0.8 1 0.8
5 0 0 0.3 0.8 1
2 3 4 5
R
M
 
 
 
 

 
 
 
 
 














otherwise
y
x
y
x
y
x
y
x
R
0
2
3
.
0
1
8
.
0
0
1
,

Fuzzy Relations and example
Note that a fuzzy relation R is a fuzzy set on XxY.
Matrix representation of relation

41
Example of Fuzzy Relations
( , ) ( , ) min{ ( ), ( )}
R A B A B
x y x y x y
   

 
For example, let:
1 2 3 1 2
{ (x ,0.2), (x ,0.5), (x ,1)} and B ={ (y ,0.3), (y , 0.9)}
A 
1 2
1
2
3
0.2 0.2
, M = 0.3 0.5
0.3 0.9
y y
R
x
A B R x
x
 
 
   
 
 
1 2 3 1 2
{ , , } , Y = { y , y }
X x x x

1 1 1 2 2 1
2 2 3 1 3 2
(( , ), 0.2), (( , ), 0.2), (( , ), 0.3),
R
(( , ), 0.5), (( , ), 0.3), (( , ), 0.9)
x y x y x y
x y x y x y
 
  
 

Composition of Fuzzy Relations
Fuzzy composition can be defined just as it is for crisp relations. Suppose R be a
fuzzy relation that relates elements from set X to set Y, and let S be a fuzzy
relation that relates elements from set Y to set Z and T is a fuzzy relation that
relates the same elements in set X that R contains to the same elements in set Z
that S contains. Then the fuzzy max-min composition is defined as:
R S
X Y Z
S
R
T 

S
R
T 

 
( , ) ( , ) ( , )
T R S
y Y
x z x y y z
  

  

0.1 0.2 0.0 1.0
0.9 0.2 0.8 0.4
min
0.1 0.2 0.0 0.4
max
Example
1 0.4 0.2 0.3
2 0.3 0.3 0.3
3 0.8 0.9 0.8
R S   
 
( , ) max min ( , z), ( , )
R S Z R S
x y x z y
  
  
 
1 0.1 0.2 0.0 1.0
2 0.3 0.3 0.0 0.2
3 0.8 0.9 1.0 0.4
R a b c d
0.9 0.0 0.3
0.2 1.0 0.8
0.8 0.0 0.7
0.4 0.2 0.3
S
a
b
c
d
  
((1
, ), 0.4), ((1
, ), 0.2), ((1
, ), 0.3),
RoS ((2, ), 0.3), ((2, ), 0.3), ((2, ), 0.3),
((3, ), 0.8), ((3, ), 0.9), ((3, ), 0.8)
  
  
  
 
 
  
 
 

Classical concepts Versus Fuzzy Concepts
Classical Concepts
1)Boolean Logic.
2) No Partial Membership.
3) Sharp Boundaries of
membership function.
4) No uncertainties
allowed.
5) Discrete or continuous
variables.
6) Limited applications to
real world because of its
idealism.
Fuzzy Concepts
1) Fuzzy Logic.
2) Partial Membership allowed.
3) Smooth Boundaries of membership
function in the range [0,1].
4) uncertainties allowed.
5) Linguistic variables.
6) Countless applications to real
world. It can deal with non-
linear and ill-understood
problems.

Fuzzification is to transform crisp inputs into fuzzy subsets.
Defuzzification is to map fuzzy subsets of real numbers into real numbers.

Simple example of Fuzzy Logic
Controlling a fan:
 Conventional model –
if temperature > X, run fan , else stop fan
 Fuzzy System -
if temperature = hot, run fan at full speed
if temperature = warm, run fan at moderate speed
if temperature = comfortable, maintain fan speed
if temperature = cool, slow fan
if temperature = cold, stop fan

47
Fuzzy Applications
Theory of fuzzy sets and fuzzy logic has been applied to problems in a variety of
fields:
 pattern recognition, decision support, data mining & information retrieval, medicine, law,
taxonomy, topology, linguistics, automata theory, game theory, etc.
And more recently fuzzy machines have been developed including:
 automatic train control, tunnel digging machinery,
home appliances: washing machines, air conditioners, etc.

APPLICATION OF SOFT COMPUTING
Stock market prediction in Business.
Computer aided diagnosis in medical.
Handwriting recognition in fraud detection.
Image retrieval.
Biological application in image processing.
Consumer appliance like AC, Refrigerators, Heaters, Washing
machine.
Robotics like Emotional Pet robots.
Food preparation appliances like Rice cookers and Microwave.
Game playing like Poker, checker etc.

FUTURE SCOPE
Soft Computing can be extended to include bio- informatics aspects.
Fuzzy system can be applied to the construction of more advanced
intelligent industrial systems.
Soft computing is very effective when it’s applied to real world problems
that are not able to solved by traditional hard computing.
Soft computing enables industrial to be innovative due to the
characteristics of soft computing: tractability, low cost and high machine
intelligent quotient.

END OF THE SLIDE
THANK YOU VERY MUCH

Emerging Approach to Computing Techniques.pptx

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