Agent Versus Decomposition of an Algorithm - A Perspective on Agent Systems - page 11

=

X 1 × X 1 ×···×

in further considerations the Cartesian product X X n (considered

as a n-tuple x i of variables x j describing characteristic features of the element u i )
[69, 150, 151].
The concept of the Cartesian product may be applied not only to the set U but
also to the algorithm Al g .
Itmeans that through the algorithm Al g = ( , wemay consider decomposition

of this algorithm, applying the concept of the Cartesian product to the algorithm Al g ,
or to the set of states of the algorithm U .

U , F )

2.4.1 The Decomposition of an Algorithm Based
on the Cartesian Product Versus
Problem of Autonomy

Let us consider the Cartesian product X = X 1 × X 2 ×···× X m , where the set X is

associated with the set U . Considering the function F :

u k u k + 1 F ( ) = (2.9)

and using the notation

u k corresponds x k u k + 1 corresponds x k + 1

, ,

(2.10)

x k + 1
1

x k + 1
2

where x k x 1 , x 2 ,..., x m ), x k + 1 x k + 1

m

)
it is noticeable that the partial function F may be replaced with the function (also
partial) f

= ( = ( , ,...,

X whose domain is the set defined on the basis of characteristic
features. The function f is defined by

: X ₒ

x 1 , x 2 ,..., x m ) = ( x k + 1

1

x k + 1
2

x k + 1
m

u k u k + 1 f ( , ,..., ) ⃔ F ( ) = . (2.11)

In further considerations without the loss of generality we may limit the decompo-
sition of the set U into only to two sets X 1 and X 2 . The constraint to only two sets does
not limit further considerations and all of the significant problems may be further
successfully analyzed. The set X is the Cartesian product of two sets X

X 2 .
This limitation may be treated as the result of grouping the elements of the Carte-
sian product: X

= X 1 ×

= X 1 × X 2 , where X 1 = X 1 × X 2 ×··· X i , X 2 = X i + 1 × X i + 2 ×··· X m

(Fig. 2.3 ).
The projection of the set X onto the set X 1 and X 2 may be introduced by

Proj 1 : X 1 × X 2 ₒ X 1 ∀ ( x 1 , x 2 ) ∈ X 1 × X 2 Proj 1 ( x 1 , x 2 ) = x 1

(2.12)

Proj 2 : X 1 × X 2 ₒ X 2 ∀ ( x 1 , x 2 ) ∈ X 1 × X 2 Proj 2 ( x 1 , x 2 ) = x 2

The decomposition may now come down to the fact that it is necessary to form two
component algorithms Al g 1 and Al g 2 as in Fig. 2.4 . For this reason, it is necessary to