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lyzable. Consequently, the values of
z
may just be taken to be the natural
numbers from 1 to
Z
, and a particular output state
z
(
t
) at time
t
(or
z
for
short) is the identification of
z
's value at that “moment”:
(7)
zt
()
∫
z
Each of these “moments” is to last a finite interval of time, D, during
which the values of all variables
x
,
y
,
z
are identifiable. After this period,
i.e., at time
t
+D, they assume values
x
(
t
+D),
y
(
t
+D),
z
(
t
+D) (or
x
¢,
y
¢,
z
¢
for short), while during the previous period
t
-Dthey had values
x
(
t
-D),
y
(
t
-D),
z
(
t
-D) (or
x
*,
y
*,
z
* for short).
After having defined the variables that will be operative in the machine
we are now prepared to define the operations on these variables. These are
two kinds and may be specified in a variety of ways. The most popular pro-
cedure is first to define a “driving function” which determines at each
instant the output state, given the input state and the internal state at that
instant:
yfxz
=
(
,
)
(8)
Although the driving function
f
y
may be known and the time course of
input states
x
may be controlled by the experimenter, the output states
y
as time goes on are unpredictable as long as the values of
z
, the internal
states of the machine, are not yet specified. A large variety of choices are
open to specify the time course of
z
as depending on
x
, on
y
, or on other
newly to be defined internal or external variables. The most profitable spe-
cification for the purposes at hand is to define
z
recursively as being depen-
dent on previous states of affairs. Consequently, we define the “state
function”
f
z
of the machine to be:
zfxz
=
(
*,
*
)
(9a)
or alternately and equivalently
(9b)
zf
¢=
(
xz
,
)
z
that is, the present internal state of the machine is a function of its previ-
ous internal state and its previous input state; or alternately and equiva-
lently, the next internal machine state is a function of both its present
internal and input states.
With the three sets of states {
x
}, {
y
}, {
z
} and the two functions
f
y
and
f
z
,
the behavior of the machine, i.e., its output sequence, is completely deter-
mined if the input sequence is given.
Such a machine is called a sequential, state-determined, “nontrivial”
machine and in Fig. 3a the relations of its various parts are schematically
indicated.
Such a nontrivial machine reduces to a “trivial” machine if it is insensi-
tive to changes of internal states, or if the internal states do not change
(Fig. 3b):
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