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In-Depth Information
We may consider this decomposition as follows:
Al
g
=
(
,
)
U
F
has been
decomposed into the “main” algorithm
Al
g
1
=
(
U
1
,
F
1
)
and the algorithm
Al
g
2
=
(
, which represents the subprogram or the service (depending on the
perspective).
However, the function
F
1
(
U
2
,
F
2
)
u
l
1
)
represents the operation of calling the subprogram
u
2
)
and the function
F
2
(
represents the return from the subprogram to the main
program (an algorithm). The result of the decomposition is the cooperation of the
algorithms
Al
g
1
i
Al
g
2
) as the call through the main (calling) algorithm
Al
g
1
of
the called algorithm
Al
g
2
. Summing up, this idea leads us to decomposition of an
algorithm or rather decomposition of a program into subprograms—the procedures
and functions known and used in programming.
2.4 Decomposition Inspired by the Concept
of the Cartesian Product
In order to define the elements u of the certain set
U
(
u
U
) wemay use characteristic
features of a given element
u
. It is connected with the fact that in practice it is easier to
describe an element of a set with the use of a given set of features which take values
from this defined set (e.g., the set of natural numbers, real numbers, etc.) These
features may now be considered as the variables taking values from the defined sets
(Fig.
2.2
).
Then, we assume that the elements of the set
U
are associated with the elements
of a given set
X
, that represent the sets of features characteristic of these elements
x
(
x
∈
∈
X
), so they have the form of n-tuples:
u
i
x
i
x
i
1
,
x
i
2
,...,
x
i
n
),
is equivalent to
=
(
(2.8)
where
u
i
∈
U
,
x
i
X
n
.
Each element
x
j
defines the characteristic feature (or attribute)
j
of the element ui
(or corresponding to its element xi). In consequence, instead of the set
U
we may use
∈
X
=
X
1
×
X
2
×···×
x
1
1
x
2
1
x
n
1
x
1
m
x
2
m
x
n
m
x
1
i
x
2
i
x
n
i
u
1
u
m
u
i
Fig. 2.2
Schema of the Cartesian product application—in the form of features of the elements of
the set
U
(
(
x
i
1
,
x
i
2
, ...
x
i
n
)
—the set of features describing the element
u
i
)
