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briefly sketched to save those who may be unfamiliar with this formalism
from having to consult other sources (Ashby, 1956; Ashby, 1962; Gill, 1962).
B. Finite State Machines
1. Deterministic Machines
Essentially, the theory of finite state machines is that of computation. It pos-
tulates two finite sets of external states called “input states” and “output
states,” one finite set of “internal states,” and two explicitly defined opera-
tions (computations) which determine the instantaneous and temporal rela-
tions between these states.*
Let
X
i
(
i
= 1,2,...,
n
x
) be the
n
x
receptacles for inputs
x
i
each of which
can assume a finite number,
v
i
> 0, of different values. The number of dis-
tinguishable input states is then
n
x
'
1
Xv
i
i
=
(3)
=
A particular input state
x
(
t
) at time
t
(or
x
for short) is then the identifica-
tion of the values
x
i
on all
n
x
input receptacles
X
i
at that “moment”:
()
∫=
{}
xt
x
x
i
(4)
t
Similarly, let
Y
j
(
j
= 1,2,...,
n
y
) be the
n
y
outlets for outputs
y
j
, each of
which can assume a finite number,
v
j
> 0, of different values. The number
of distinguishable output states is then
n
y
'
1
Yv
j
j
=
(5)
=
A particular output state
y
(
t
) at time
t
(or
y
for short) is then the identifi-
cation of the values
y
i
on all
n
y
outlets
Y
j
at that “moment”:
()
∫=
{}
yt
y
y
i
(6)
Finally, let
Z
be the number of internal states
z
which, for this discussion
(unless specified otherwise), may be considered as being not further ana-
* Although the interpretation of states and operations with regard to observables
is left completely open, some caution is advisable at this point if these are to serve
as mathematical models, say, for the behavior of a living organism. A specific phys-
ical spatiotemporal configuration which is identifiable by the experimenter who
wishes that this configuration be appreciated by the organism as a “stimulus” cannot
sui modo
be taken as “input state” for the machine. Such a stimulus may be a stim-
ulant for the experimenter, but be ignored by the organism. An input state, on the
other hand, cannot be ignored by the machine, except when explicitly instructed to
do so. More appropriately, the distribution of the activity of the afferent fibers has
to be taken as an input, and similarly, the distribution of activity of efferent fibers
may be taken as the output of the system.
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