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To model the states of a VO and their transitions, let
VO
be the universe of all the pos-
sible organizations
O
. A multi-agent system based on virtual organizations is a structure
MAS
=(
VO
,
D
)
where, for every time
t
T
,
D
t
∈
⊆
×
VO
defines transitions between
different states of the system. In every state of the organization
O
VO
∈
VO
, in a given time
T
, there is a set of possible next states of the organization, denoted by
D
t
(
O
)
t
∈
⊆
VO
.
D
t
(
O
t
)
, so an organization will only change
to another state when it is allowed to reach from the initial state.
Since the organization is composed by three elements (
OS
,
OE
and
T
, it holds that
O
t
+
1
Thus, for every
t
∈
∈
), before exe-
cuting a change of state it is necessary to check that these elements are able to change
from the initial state to the possible destination state. Formally:
φ
((
OS
t
+
1
,
OE
t
+
1
,
t
+
1
)
D
t
(
OS
t
,
OE
t
,
t
))
φ
∈
φ
↔
((
OS
t
+
1
D
t
OS
(
OS
t
))
(
OE
t
+
1
D
t
OE
(
OE
t
))
t
+
1
D
t
t
)))
∈
∧
∈
∧
(
∈
(
φ
φ
(15)
φ
However, in order to swap from one state to another, it is not necessary to produce a
change in all three elements that compose the VO. A change ranges from a very small
variation in one or few of the elements building the organization to a big amount of
changes in a large amount of entities from the VO. Formally:
O
t
+
1
)
(
O
t
((
OS
t
OS
t
)
(
OE
t
OE
t
)
t
t
)))
∧
→
(
¬
∧
∧
∧
∧
(
φ
∧
φ
(16)
The above formula (that uses LTL [18]) helps us to formalize how
D
t
is build.
D
t
=
(
O
t
,
O
t
+
1
)
O
t
+
1
=
O
t
|
(17)
D
t
is composed by the set of all possible transitions of the MAS, where each new
transition is generated every time that in
OS
,
OE
or
φ
(or in a combination of these
elements) an atomic change is produced.
5
Discussion
In section 3 an analysis of the most relevant formalization proposals was presented. The
Virtual Organization Formalization
(VOF) takes inspiration from features taken from
some of the analyzed proposals. In this section, we depict a comparison between VOF
and these background proposals.
Firstly, the organizational temporal evolution proposed by VOF is mainly based on
PopOrg, which models the dynamics of the
population
(similar to our
OE
)andthe
organization
(similar to our
OS
).
Regarding the structure of an organization, OperA offers relations between roles that
are similar to those included in VOF. The supervision relation of VOF is similar to the
combination of the power and authorization relations of OperA, expressing that an agent
is able to delegate its objectives to a subordinated agent, like the power relation does
(the authorization relation expresses the power relation, but as a temporal situation).
Also, in OperA, the objective that a subordinated agent can take from a superior agent
