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TABLE 1. Computing Z logical function
F
z
(
x
) on inputs
x
z
1
1
1
...1
2
2
2
...2
...
ZZZ
...
Z
x
a
bc
...
Xa
bc
...
X
...
a
bc
...
X
y
g
ab...dag
b ...Œ
...
bŒg
...d
interpretation of the internal states
z
as being a selector for a specific func-
tion in a set of multivalued logical functions. This is most easily seen when
writing the driving function
f
y
in form of a table.
Let
a
,
b
,
c
...
X
be the input values
x
; a, b, g ...
Y
be the output values
y
; and 1, 2, 3...
Z
be the values of the internal states. A particular driving
function
f
y
is defined if to all pairs {
xz
} an appropriate value of
y
is associ-
ated. This is suggested in Table 1.
Clearly, under
z
= 1 a particular logical function,
y
=
F
1
(
x
), relating
y
with
x
is defined; under
z
= 2 another logical function,
y
=
F
2
(
x
), is defined; and,
in general, under each
z
a certain logical function
y
=
F
z
(
x
) is defined.
Hence, the driving function
f
y
can be rewritten to read
(17)
yFx
=
()
,
which means that this machine computes another logical function
F
z
¢
on its
inputs
x
, whenever its internal state
z
changes according to the state func-
tion
z
¢=
f
z
(
x
,
z
).
Or, in other words, whenever
z
changes, the machine becomes a
differ-
ent
trivial machine.
While this observation may be significant in grasping the fundamental
difference between nontrivial and trivial machines, and in appreciating the
significance of this difference in a theory of behavior, it permits us to cal-
culate the number of internal states that can be effective in changing the
modus operandi
of this machine.
Following the paradigm of calculating the number
N
of logical functions
as the number of states of the dependent variable raised to the power of
the number of states of the independent variables
)
(
no. of states of indep. variables
)
N
=
(
no. of states of dep. variables
(18)
we have for the number of possible trivial machines which connect
y
with
x
N
T
=
Y
X
(19)
This, however, is the largest number of internal states which can effec-
tively produce a change in the function
F
z
(
x
), for any additional state has
to be paired up with a function to which a state has been already assigned,
hence such additional internal states are redundant or at least indistin-
guishable. Consequently
Z
£
Y
X
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