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Fig. 2.3
Schema of the use
of grouping to the Cartesian
product in order to constrain
the dimension of the problem
Alg
1
Alg
2
x
1
x
2
(x
1
, x
2
,... x
i
, x
i+1
, ... x
n
)
Fig. 2.4
Schema of the
Cartesian product application
to the decomposition of an
algorithm
define the sets of states of these algorithms as well as their (partial) functions of the
transition.
The following problems and questions appear:
•
The component algorithms
Al
g
1
and
Al
g
2
resulting from the decomposition of the
algorithm
Al
g
should have defined sets of states
X
1
and
X
2
, specified with the use
of the set of states of the algorithm
Al
g
. Decomposition of the sets of states
X
based on the concept of the Cartesian product (Fig.
2.3
) may be the starting point
of realizing such decomposition.
•
Another problem is that the algorithms should have the transition partial functions
f
1
and
f
2
, which should be formed on the basis of the transition function
f
,sothe
function
f
should be decomposed into two functions
f
1
and
f
2
. That will allow the
algorithms
Al
g
1
=
(
X
1
,
f
1
)
and
Al
g
2
=
(
X
2
,
f
2
)
to create, with the appropriate
transition functions (Fig.
2.4
).
•
It should also be analyzed whether the algorithms
Al
g
1
and
Al
g
2
are related to
each other through mutual interactions. In particular, the mutual relationship of
the algorithms should be analyzed. We need to define what the notion
autonomous
means (or should mean) and how we should really understand it. In other words,
how the notion
autonomous
should be defined to make it clear that we deal with
the
autonomous algorithm
.
•
Further, it should be considered whether it is possible (and to what extent) to make
algorithms
Al
g
1
and
Al
g
2
independent so that they can be autonomous algorithms.
•
The question arises as to whether the decomposition of the algorithm
Al
g
into the
algorithms
Al
g
1
and
Al
g
2
ensures that they may solve the problems that are solved
with the use of the algorithm
Al
g
=
(
. The term of equivalence of algorithms,
which is considered in Sect.
2.5.1
, seems to be useful here.
X
,
f
)
