Information Technology Reference
In-Depth Information
This section describes the
GA
based solution to evolve the
MACA
with the
desired functionality. The three major functions of
GA
- Random Generation
of initial population, Crossover and Mutation, as developed in the current
GA
formulation, are next discussed.
3.1
Random Generation of Initial Population with Desired Format
To form the population, it must be ensured that each solution randomly gener-
ated is an
n
bit
MACA
with 2
m
number of attractors where
m
can assume any
value from 1 to
n
. From
Result II
of
Section 2.1
, the elementary divisor form of
MACA
is (1 +
x
)(1 +
x
)
··m times x
d
1
x
d
2
··x
d
p
&
d
1
+
d
2
···
+
d
p
=(
n − m
),
where the number of (1 +
x
) determines the number of attractors.
As per [3], the elementary divisors, if arranged in different order, produces
MACA
with different attractor basins. For synthesis, the elementary divisors
are first randomly distributed among itself forming a sequence.
Fig 4
shows one
such sequence for the characteristic polynomial
x
7
(1 +
x
)
3
:
x
2
· x
2
·
(1 +
x
)
· x ·
(1 +
x
)
·
(1 +
x
)
· x
2
. This is referred to as
pseudo-chromosome format
has been
detailed in
Section 3.2
.
Each elementary divisor (
φ
i
(
x
)) can be converted to a
CA
[1]. A tri-diagonal
T
matrix with characteristic polynomial
φ
i
(
x
) is accordingly synthesized. Let
T
i
matrices correspond to elementary divisors
x
d
i
(it will be a
d
i
× d
i
matrix) and
T
j
matrices correspond to each elementary divisors (
x
+ 1). If
T
i
s and
T
j
s are
randomly arranged in block diagonal form, the characteristic polynomial of the
resultant
T
is
x
n−m
·
(1 +
x
)
m
and the minimal polynomial is
x
d
p
·
(1 +
x
) and
it generates the
MACA
[1].
2
0
2
0
-1
1
-1
-1
2
0
x
2
x
2
2
1+x
x
1+x
1+x
x
Characteristics Polynomial - x (1+x)
3
7
2
Minimal Polynomial - x (1+x)
Fig. 4.
MACA in Diagonal Representation Form
3.2
Pseudo-Chromosome Format
It is a method of representing an
MACA
with respect to the sequence in which
its
x
d
i
s
and (1 +
x
)
s
are arranged. It gives a semblance of the chromosome
and hence termed as
pseudo-chromosome format
. It a string of
n
bits where (a)
