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Our variables are now functions, and since relations between functions
are usually referred to as “functionals,” the essential features of a calculus
of recursive functionals will be briefly sketched.
Consider a system like the one suggested in Fig. 3a, with the only differ-
ence that it operates on a finite set of functions of two kinds, {
f
yi
} and {
f
zj
}.
These functions, in turn, operate on their appropriate set of states {
y
i
} and
{
z
j
}. The rules of operation for such a finite function machine are modeled
exactly according to the rules of finite state machines. Hence:
=
[
]
f
Fxf
,
(45a)
y
y
z
¢=
[
]
f
Fxf
,
(45b)
z
z
z
where
F
y
and
F
z
are the functionals which generate the driving functions
f
y
and the subsequent internal function
f
z
¢ from the present internal function
f
z
and an input
x
. One should note, however, that the input here is still a
state. This indicates an important feature of this formalism, namely, the pro-
vision of a link between the domain of states with the entirely different
domain of functions. In other words, this formalism takes notice of the dis-
tinction between entities and their representations and establishes a rela-
tion between these two domains.
Following a procedure similar to that carried out in Eqs. (10) through
(14), the functions of type
f
z
can be eliminated by expressing the present
driving function as result of earlier states of affairs. However, due to some
properties that distinguish functionals from functions, these earlier states
of affairs include both input states as well as output functions. We have for
n
recursive steps:
[
]
()
n
()
n
()
n
f
=
F
x x
,
*,
x
**,
x
***,...,
x
*; *,
f
f
**,...,
f
*
(46)
y
y
y
y
y
Comparing this expression with its analog for finite state machines [Eq.
(14)], it is clear that here the reference to past events is not only to those
events that were the system's history of inputs {
x
(
i
)
*
}, but also to its history
of potential actions {
f
y
(
i
)
*}. Moreover, when this recursive functional is
solved explicitly for time (
t
=
k
D;
k
= 0,1,2,3,...;) [compare with Eq. (16)],
it is again the history of inputs that is “integrated out”; however, the history
of potential actions remains intact, because of a set of
n
“eigenfunctions”
which satisfy Eq. (46). We have explicitly for
k
D), and for the
i
th eigen-
function:
()
=
(
◊
(
[
)
+
(
)
]
i
()
i
*
( )
n
fk
D
Kk
D
p
f
Gxx x
,
*,
**,...,
x
*
(47)
y
i
i
y
i
i
=
123
, , ,...,
n
with
K
i
and
G
i
being functions of (
k
D), the latter one giving a value that
depends on a tail of values in
x
(
j
)
* which is
n
steps long. p
i
is again a func-
tional, representing the output function
f
y
of
i
steps in the past in terms of
another function.
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