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Example 2 The Fig.4 (a) represents two arbitrary graphs generated in Phase I
for n =4 and r max =1 . Patterns to be learnt P 1 = 0000 and P 2 = 1111 are
mapped onto the nodes of the attractor cycle of length 3 and 1 respectively. The
sets 0001 , 0010 , 0100 , 1000 and 1110 , 1101 , 1011 , 0111 are the noisy patterns
with noise of 1 bit added to P i (i =1 , 2 , ···) respectively. These are mapped in
two attractor basins as shown in Fig.4 (a) .
Phase II - From all the graphs selected in Phase I , we generate a state tran-
sition table. The Fig.4 (b) represents state transition tables derived from two
directed graphs shown in Fig.4 (a) . Total number entries in the state transition
table is k · p , where k is the number of patterns to be learnt and p is the number
of states in the graph generated as per the Equation 1 . For the example, graphs
of Fig.4 (a) , the number of patterns to be learnt ( k )is2( P 1 and P 2 ) and the
number of states in each basin ( p ) is 5. So the total number entries in the state
transition table of Fig.4 (b) is 10.
Phase III - Design the rule vector of the GMACA cells from the state tran-
sition table.
Consider i th cell whose rule is to be identified. We concentrate on 3 columns
-( i − 1) th , i th and ( i +1) th columns of all the k · p number of patterns of the
state transition table of Fig.4 (b) . Suppose, for a present state configuration of
the i th cell, the next state is '0' for n 0 times and '1' for n 1 times, then state
'0' and '1' collides with each other to become the next state of the cell for that
configuration.
The Fig.4 (c) represents the neighborhood configurations along with the next
state of 2 nd and 3 rd cells of the patterns of two basins noted in Fig.4 (b) . For 2 nd
cell, there is no collision between next state '0' and '1' of the cell for 8 possible
configurations. Whereas for '000' neighborhood configuration, the next state of
3 rd row of Fig.4 (b) ) for 1 time and '1' (4 th row of Fig.4 (b) ) for 1 time. That
is, n 0 = 1 and n 1 = 1. So, for '000' configuration, the next state of 3 rd cell may
be '0' or '1' generating an instance of collision.
In order to resolve this conflict we introduce the following heuristics.
3.2 Resolution of Collision
- If n 0 n 1 , the collision between state '0' and '1' is high. In that case, we
randomly decide the next state of a cell.
- If n 0 >> n 1 or n 0 << n 1 , the collision is minimum. In that case, the next
state of a cell is '0' if n 0 >n 1 , otherwise '1'.
Design of a synthesis algorithm to arrive at an effective GMACA ,weob-
served, is a hard problem . So, we fall back on the Simulated Annealing ( SA )
program to solve this problem.
3.3 Simulated Annealing Program to Evolve Effective GMACA
Simulated annealing is a generalization of a Monte Carlo method for examining
the equations of state and frozen states of n-body systems. The GMACA synthe-
sis scheme can be elegantly mapped to this Monte Carlo approach. The current
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