AI - a Guide to Intelligent Systems.pdf - Member of EEPIS

192
ARTIFICIAL NEURAL NETWORKS
In our example, we may assume all thresholds to be zero. Thus,
82
32
3
>< 0 2 2 1
6 76
7
Y 1 ¼ sign 4 2 0 254
1 5
>:
2 2 0 1
2 39
2 3
0 >= 1
6 7
4 0 5
>; ¼ 6 7
4 1 5
0 1
and
82
32
>< 0 2 2
6 76
Y 2 ¼ sign 4 2 0 254
>:
2 2 0
3
1
7
1 5
1
2 39
2 3
0 >= 1
6 7
4 0 5
>; ¼ 6
4 7
1 5
0 1
As we see, Y 1 ¼ X 1 and Y 2 ¼ X 2 . Thus, both states, ð1; 1; 1Þ and ð 1; 1; 1Þ, are
said to be stable.
How about other states?
With three neurons in the network, there are eight possible states. The remaining
six states are all unstable. However, stable states (also called fundamental
memories) are capable of attracting states that are close to them. As shown in
Table 6.5, the fundamental memory ð1; 1; 1Þ attracts unstable states ð 1; 1; 1Þ,
ð1; 1; 1Þ and ð1; 1; 1Þ. Each of these unstable states represents a single
error, compared to the fundamental memory ð1; 1; 1Þ. On the other hand, the
Table 6.5
Operation of the three-neuron Hopfield network
Possible
Inputs
Outputs
Fundamental
state Iteration x 1 x 2 x 3 y 1 y 2 y 3 memory
1 1 1 0 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1