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l
1
l
2
...
l
n
k
1
F
(
a
k
1
,
l
1
)
F
(
a
k
1
,
l
2
)...
F
(
a
k
1
,
l
n
)
k
2
F
(
a
k
2
,
l
1
)
F
(
a
k
2
,
l
2
)...
F
(
a
k
2
,
l
n
)
O
F
(
)
=
.
A
.
k
m
F
(
a
k
m
,
l
1
)
F
(
a
k
m
,
l
2
)...
F
(
a
k
m
,
l
n
)
Hence, we can describe the neural network with the form
a
1
W
1
B
1
=
O
F
((
P
)
⊕
),
a
i
a
i
−
1
W
i
B
i
=
O
F
((
)
⊕
).
Therefore,
a
M
−
1
a
M
−
2
W
M
−
1
B
M
−
1
=
O
F
((
)
⊕
)
W
1
B
1
B
2
B
M
−
2
B
M
−
1
=
O
F
((...
O
F
((
O
F
((
)
⊕
)
⊕
)...
⊕
)
⊕
).
P
A more general case is the following: each layer hat its own transfer function, i.e.,
function
F
i
is associated to the
i
-th layer. Therefore, theNNhas the IM-representation
a
M
−
1
W
1
B
1
B
2
B
M
−
2
B
M
−
1
=
O
F
M
−
1
((...
O
F
2
((
O
F
1
((
P
)
⊕
)
⊕
)...
⊕
)
⊕
).
Below, we will extend the results from [23], using the ideas from Sect.
5.4
.Now,
for each layer we juxtapose an IMFE
a
1
,
1
...
a
1
,
s
i
F
i
=
f
1
,
s
i
,
p
0
f
1
,
1
...
x
where
f
i
,
j
s
i
. Therefore, for the
j
-th node
from
i
-th layer of the multilayered network we juxtapose the function
f
i
,
j
and in a
result, we obtain
∈
F
for 1
≤
i
≤
M
−
1 and 1
≤
j
≤
a
1
W
1
B
1
=
F
1
⊕
((
P
)
⊕
),
a
i
a
i
−
1
W
i
B
i
=
F
i
⊕
((
)
⊕
).
Therefore,
a
M
−
1
a
M
−
2
W
M
−
1
B
M
−
1
=
F
M
−
1
⊕
((
)
⊕
)
W
1
B
1
B
2
B
M
−
2
B
M
−
1
=
F
M
−
1
⊕
((...
F
2
⊕
((
F
1
⊕
((
P
)
⊕
)
⊕
)...
⊕
)
⊕
).
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