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with the following values of facilitatory and inhibitory synaptic junctions:
n
=+
3
,
n
=-
1
,
1
2
n
=-
2
,
n
=+
1
.
3
4
Consider for the moment all input fibers in the ON state and the threshold
at zero:
xxxx
1
====
1
,
q
=
0
.
(16)
2
3
4
The internal state Z is, according to eq. (16) given by
Z =◊ () +◊ () +◊ () +◊ () =+ >-
13111211 1e;
hence, according to rule (vii) eq. (16), we have:
y = () =
F 1 ,
so the element “fires”; its output is ON. Raising the threshold one unit,
q=1, still keeps the element in its ON state, because its internal state does
not fall below zero. From this we can conclude that a completely discon-
nected element with zero threshold always has its output in the ON state.
If in fig. 9 the threshold is raised to +2, the simultaneous excitation of all
fibers will not activate the element. Furthermore, it is easily seen that with
threshold +5 this element will never fire, whatever the input configuration;
with threshold -4 it will always fire.
The two-valuedness of all variables involved, as well as the possibility of
negation (inhibition) and affirmation (excitation), make this element an
ideal component for computing logical functions in the calculus of pro-
positions where the ON or OFF state of each input fiber represents the
truth or falsity of a proposition X i , and where the ON or OFF state of the
output fiber Y represents the truth value of the logical function F computed
by the element.
Let us explicate this important representation with a simple example of
an element with two input fibers only ( N = 2), each attached to the element
with only a single facilitatory junction ( n 1 = n 2 =+1). We follow classical
usage and call our input fibers A and B ,—rather than X 1 and X 2 (which pays
off only if many input fibers are involved and one runs out of letters of
the alphabet). Fig. 10 illustrates the situation. First, we tabulate all input
configurations—all “input states” that are possible with two input fibers
when each may be independently ON or OFF. We have four cases: A and
B both ON or both OFF, and A ON and B OFF, and A OFF and B ON, as
indicated in the left double column in fig. 10.
In passing, we may point out that with N input fibers, two choices for
each, we have in general
N in = 2 N
(17)
possible input states.
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