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n
21
-
Â
0
P
=
2
Z
D
2
2
n
-
1
.
(59)
Z
0
The terms
K
P
are called “Schroeder's constituents”. Consequently, there are
2
2
n
different conjunctions possible with these constituents. Each of these
conjunctions represents uniquely one logical function.
Expanding these conjunctions by virtue of the distributivity of (·) and (v)
into a disjunction of conjunctions
(60)
CCC
1
v
v
...,
2
3
one arrives, after cancellation of contradictory terms (
X
i
· ), at Hilbert's
disjunctive normal form which, derived this way, is again a unique repre-
sentation of one of the 2
2
n
X
i
logical functions. This form will be used in the
synthesis of networks.
As an example of this procedure take the two-variable logical function
expressing the equivalence of
A
and
B
. Let
X
1
and
X
2
be
A
and
B
respec-
tively. The four Schroeder constituents are with
Z
from 0 to 3:
=
(
)
DAB
DAB
DAB
DAB
v
v
v
v
,
,
,
.
o
=
(
)
1
=
(
)
2
=
(
)
3
Since
∫
ABDD
∫◊
1
2
,
we have
P
=◊ +◊ +◊ +◊ =
02
o
12
1
12
2
02
3
6
and
)
◊
(
)
∫
(
KDD ABAB
6
∫◊
v
v
.
1
2
Expanding the right hand side gives
(
)
(
)
AA BA
◊
v
◊
v
AB BB
◊
v
◊
,
which after cancellation of contradictions yields the desired expression in
Hilbert's disjunctive normal form:
∫
.
(
)
(
)
∫
BA
◊
v
AB
◊
A
B
The second step considers the synaptic delay D
t
at each McCulloch element.
Let
N
i
(
t
) denote the action performed by the
i
th element at time
t
, or for
short
()
.
(61)
NNt
i
∫
i
In order to facilitate expressions that consider
n
synaptic delays earlier, a
recursive operator
S
is introduced and defined as
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