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However,
y
* is given recursively through Eq. (13)
()
1
(
)
yFxxy
*
=
*,
**,
**
(13*)
y
and inserting this into Eq. (13) we have
(
)
()
2
yF xxx y
y
=
,*,**,**
and for
n
recursive steps
(
)
()
n
()
n
()
n
yF xxx x
=
,
*,
**,
***...
x
*,
y
*
(14)
y
This expression suggests that in a nontrivial machine the output is not
merely a function of its present input, but may be dependent on the par-
ticular sequence of inputs reaching into the remote past, and an output state
at this remote past. While this is only to a certain extent true—the “remote-
ness” is carried only over
Z
recursive steps and, moreover, Eq. (14) does
not uniquely determine the properties of the machine—this dependence of
the machine's behavior on its past history should not tempt one to project
into this system a capacity for memory, for at best it may look upon its
present internal state which may well serve as
token
for the past, but
without the powers to recapture for the system all that which has gone by.
This may be most easily seen when Eq. (13) is rewritten in its full recur-
sive form for a linear machine (with
x
and
y
now real numbers)
(
)
-
()
=
()
yt
+
D
ayt
bxt
(15a)
or in its differential analog expanding
y
(
t
+D) =
y
(
t
) +D
dy
/
dt
:
dy
dt
-=
()
a
yxt
(15b)
with the corresponding solutions
n
È
Í
˘
˙
Â
()
=
()
+
()
yn
D
a
n
y
0
b
a xi
-
i
D
(16a)
i
=
0
and
[
]
t
Ú
()
=
a
t
()
+
-
at
()
yt
e
y
0
e
x
tt
d
(16b)
0
From these expressions it is clear that the course of events represented
by
x
(
i
D) (or
x
(t)) is “integrated out,” and is manifest only in an additive
term which, nevertheless, changes as time goes on.
However, the failure of this simple machine to account for memory
should not discourage one from contemplating it as a possible useful
element in a system that remembers.
While in these examples the internal states
z
provided the machine with
an appreciation—however small—of its past history, we shall now give an
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