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In-Depth Information
If
s
t
,
u
t
,and
y
t
denote
state
,
input
,and
output
of the Turing machine at time
t
,
respectively, and if
f
and
g
are functions on pairs of
s
t
and
u
t
, then the machine-
operation is characterized by the following set of state equations:
s
t
+
1
=
f
(
s
t
,
u
t
)
and
y
t
=
g
(
s
t
,
u
t
)
,
t
=
0
,
1
,
2
,...,
(3.7)
If the system is a differential system instead of a discrete-state system,
state
,
input
,
and
output
of the system are represented by vectors
s
(
t
)
,
y
(
t
)
,and
u
(
t
)
, respectively.
With
s
(
t
)=
d
/
dts
(
t
)
these
state equations
assume the forms
s
(
t
)=
f
((
s
(
t
)
,
u
(
t
))
,
y
(
t
)=
g
(
s
(
t
)
,
u
(
t
))
(3.8)
Some mathematicians and control theorists in the Soviet Union in the 1940s and
1950s - e.g. Pontryagin - used these state equations earlier than western scientists,
and Zadeh took notice of the scientific progress in the Soviet Union after he migrated
in the United States. He referred to the fact that “in the United States, the introduc-
tion of the notion of state and related techniques into the theory of optimization
of linear as well as nonlinear systems is due primarily to Richard Ernest Bellman,
whose invention of dynamic programming has contributed by far the most powerful
tool since the inception of the variational calculus to the solution of a whole gamut
of maximization and minimization problems.” [101, p. 858]
“Among the scientists dealing with animate systems, it was a biologist — Ludwig
von Bertalanffy -- who long ago perceived the essential unity of systems concepts
and techniques in various fields of science and who in writings and lectures sought
to attain recognition for “general systems theory” as a distinct scientific discipline.
It is pertinent to note, however, that the work of Bertalanffy and his school, being
motivated primarily by problems arising in the study of biological systems, is much
more empirical and qualitative in spirit than the work of those system theorists who
received their training in the exact sciences.” [101, p. 857]
8
Then he demanded a new mathematics: “In fact, there is a fairly wide gap be-
tween what might be regarded as 'animate' system theorists and “inanimate” system
theorists at the present time, and it is not at all certain that this gap will be narrowed,
much less closed, in the near future. There are some who feel that this gap reflects
the fundamental inadequacy of the conventional mathematics — the mathematics
of precisely-defined points, functions, sets, probability measures, etc. -- for cop-
ing with the analysis of biological systems, and that to deal effectively with such
systems, which are generally orders of magnitude more complex than man-made
systems, we need a radically different kind of mathematics, the mathematics of
fuzzy or cloudy quantities which are not describable in terms of probability distri-
butions. Indeed, the need for such mathematics is becoming increasingly apparent
even in the realm of inanimate systems, for in most practical cases the a priori data
as well as the criteria by which the performance of a man-made system is judged
are far from being precisely specified or having accurately-known probability dis-
tributions.” [101, p. 857]
8
For more details on
General Systems Theory
and
Cybernetics
in the history of the theory
of Fuzzy Sets and Systesm see chapter III in [77].
