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such internal states of algorithms as
x
1
and
x
2
that the realization of algorithms
x
0
,
x
1
))
x
0
,
x
2
))
(
Al
g
1
,(
(
Al
g
2
,(
Exec
and
Exec
exists in the following form:
x
k
0
,
x
k
1
)))
x
0
,
x
1
))
=
((
x
0
,
x
1
), (
x
0
,
x
1
), . . . , (
Exec
(
Al
g
1
,(
x
0
,
x
k
0
,
x
k
2
)))
x
0
,
x
2
))
=
((
x
0
,
x
2
), (
x
2
),...,(
Exec
(
Al
g
2
,(
If additionally it occurs that
x
k
0
x
k
0
, then these algorithms are equivalent in
a given problem solving (internal states of the algorithms
x
k
1
and
x
k
2
are of no
importance here, and only the final state of the environment is essential). If a solu-
tion to a given problem
x
0
through the realization of the first algorithm
Al
g
1
is
equivalent to the solution to this problem through the realization of the algorithm
Al
g
2
, then we may replace the algorithm
Al
g
1
with the equivalent algorithm
Al
g
2
and choose the one which is, e.g., faster or makes better use of the resources.
=
•
The algorithms
Al
g
1
=
(
X
,
f
1
)
and
Al
g
2
=
(
X
,
f
2
)
cooperate through the
environment but
X
X
2
. It means that certain changes in the environ-
ment, those necessary for problem solving, are realized by the former algorithm,
and the other ones by the latter (Fig.
2.10
), but simultaneously the algorithm
Al
g
1
affects the changes of the state
x
2
of the algorithm
Al
g
2
and vice versa. It may
be noted that the algorithm
Al
g
1
(or
Al
g
2
) is non-autonomous, then it is neces-
sary to broaden the definition of autonomy to algorithms cooperating through the
environment.
=
X
0
×
X
1
×
The concept of the environment and the cooperation of algorithms through the envi-
ronment, under the definition of autonomy presented above, is that the cooperating
algorithms are not autonomous but dependent through the environment. However,
apart from the states of the environment there are internal states of individual algo-
rithms, and therefore we may broaden definitions of autonomy introduced earlier.
On the other hand, the concept of the environment allows consideration of the phe-
nomenon of communication through the environment. For example, the algorithm
Al
g
1
may make such changes in the environment that can be “read” by the algorithm
(
Al
g
2
) as changes connected with certain established information, which is a kind of
sending a message by the algorithm
Al
g
1
to the algorithm
Al
g
2
. This way of commu-
nication allows for more complex forms of cooperation such as negotiations, planning
and forming groups of cooperating algorithms, or realization of cooperation as such.
2.5.1.3 The Autonomy of Cooperating Algorithms
Based on the Environment
The notion of
autonomy
has been defined and discussed in Sect.
2.4.2
, but after the
introduction of the environment concept and the possibility of cooperation between
the algorithms based on these environments it needs certain modification.
