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In-Depth Information
∀
u ∈M
u
0
, u
1
, ···
u
0
=
u
: the sequence
starts with
v
)as
u
i
=
(
−−→
u
n
=
v
generated by
a
(
·
a
u
i−
1
)
such as lim
n→∞
(1)
v
v
The family of functions
A
represents a set of functions
{a
v
}
that converge to
v
given values of
v
for all the elements
u ∈M
.
In order to be complete, it is necessary to define how the limit of the sequence
is computed (eq. 1). As
M
is a normed vector space according to the properties
presented in Section 3.3:
u
n
=
v
n→∞
−
u
n
v
=0
1
lim
n→∞
=
⇒
lim
(2)
The creation of the Attenuated Mood Space is necessary to create the emotional
dynamics required at
R4
.
3.6 Emotional Agent System in a Mood Vector Space
It is possible to define an Emotional Agent System
A
as an algebraic structure
A
=(
M, A, E, m
0
), where
M
is a Mood Vector Space,
A
a finite set of agents
A
=
{A
0
,A
1
, ··· ,A
n
}
,and
E
is a set of elements defined as
E
=
{
(
a, t, v, α
)
/a ∈
A, t ∈
N
,v∈M,α∈
R
}
, named the emotion set, which represents the emotions
elicited from all the agents, together with its intensity at a given point in time.
Finally,
m
0
is a function that represents the default mood state for the agents
(the mood state in absence of any emotion, thus the initial state),
m
0
:
A −−→
−−−−−→
u
m
0
0
i
The state of an Emotional Agent System, can be represented as the mood
state of all its agents, and it is denoted as
M
:
A
i
{
u
0
,
u
1
, ··· ,
u
t
n
}
M
(
A,t
)=
,being
t ∈
N
. This state can be defined as follows:
=0
u
0
i
∀i ∈
,n
t
m
0
(
A
i
)
[0
] :if
=
Initial mood state
(3)
u
i
=
u
t−
1
if
t>
0
i
⊕ E
(
A
i
,t
)
(4)
α
0
v
0
⊕···⊕α
m
−
v
m
∀
where
E
(
A
i
,t
)
=
(
A
i
,t,v
j
,α
j
)
∈ E
(5)
E
(
A
i
,t
)
represents the aggregation of the emotions elicited from the agent
A
i
at
time
t
scaled according to the intensities
α
j
and combined by means of the
⊕
operator.
If we want to include a mechanism that ensures that the mood state of the
agents returns to their default state along time (and in absence of new emotions),
we must assure that
M
is an Attenuated Mood Space and redefine the equation 4
as:
u
i
(
u
t−
1
if
t>
0
=
a
i
⊕ E
(
A
i
,t
)
)
(6)
v
where
v
=
m
0
(
A
i
) is the default (initial) mood state and
a
(
·
) is the function,
v
from the family of function
in the Attenuated Mood Space, that defines an
infinite sequence that converges to this default (initial) mood state
A
A
i
).
1
The
operator represents the addition operation
⊕
using the inverse element.
m
0
(
