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In-Depth Information
Theorem 1
[1] In an n cell MACA with k
=2
m
attractors, there exists
m-bit
positions at which the attractors give pseudo-exhaustive
2
m
patterns.
Theorem 2
[1] The
modulo-2
sum of two states is the non-zero predecessor of
0-state
(pattern with all 0's) if and only if the two states lie in the same MACA
basin.
Example 1
The example MACA of
Fig.1
is used to illustrate the above results.
-
It is a
5
-cell MACA having
4
number of attractors and the depth of the
MACA is 3.
-
Result I:
The characteristic polynomial is x
3
·
(1+
x
)
2
. Therefore, m=2. This
is consistent with the result in the
Fig.1
where attractor(k)is4.
-
Result II:
The characteristic polynomial in elementary divisor form is x
3
·
(1 +
x
)
·
(1 +
x
)
.
-
Result III:
The minimal polynomial is x
3
·
(1 +
x
)
.
-
Result of Theorem 1:
In
Fig.1
, two least significant bit positions constitute
the PEF.
-
Result of Theorem 2:
We take an attractor
00011
and any two states
11111,
11101
of the attractor basin. The
modulo-2
sum of these two patterns is
00010
which is a state in
0
− basin. By contrast, if we take two states
00001
and
11000
belonging to two different attractor basins
00001
and
11000
respectively,
their
modulo-2
sum is
11011
which is a state in a non-zero attractor
(00101)
basin.
2.2
MACA
- As a Classifier
An
n
-bit
MACA
with
k
-attractors can be viewed as a natural classifier (
Fig.2
).
It classifies a given set of patterns into
k
-distinct classes, each class containing
the set of states in the attractor basin.
11
10
01
00
Class I
Class II
MEMORY
Fig. 2.
MACA
based Classification Strategy
For an ideal classifier, to distinguish between two classes, we would need one
bit. The classifier of
Fig 2
requires two bits to distinguish between two classes
