Information Technology Reference
In-Depth Information
This definition considers that a scenario is plausible as soon as the lowest cor-
relation weight of an action is somewhat plausible. The higher is the correlation
weight, the more plausible is the scenario.
This definition is however too restrictive. Assume that we have two scenarios
S
=(
A
1
,A
2
, ..., A
n
,O
) and
S
=(
B
1
,B
2
, ..., B
n
,O
). Concerning the scenario
S
assume that
1
,n,ω
S
(
A
i
)=
α
and
ω
S
(
O
)=
α
. Assume also that for
the scenario
S
there exists
j
such that
ω
S
(
B
j
)=
α
and that
∀
i
∈
1
,n
,i
=
j, ω
S
(
B
i
)
>α,ω
S
(
O
)
>α
. In such a case, one would clearly prefer scenario
S
to
S
since
S
contains stronger correlated actions. But they are considered
equally plausible when compared with the minimum operator since we only
consider the worst correlated action.
A possible refinement of the minimum operator is to use a so-called “leximin
operator”, well-known in social-choice theory [5]. The leximin ordering makes
sense only when comparing two equally sized vectors. Hence when comparing
two scenarios having a different number of actions, we have to duplicate the
lowest weight in the shortest scenario to obtain two equally sized vectors of
weights. The following describes the leximin ordering:
Definition 12:
Let
v
=(
v
1
, ..., v
n
) and
v
=(
v
1
, ..., v
n
) be two vectors of weights
ranked increasingly, namely
v
1
>...>v
n
and
v
1
>...>v
n
. Then
v
is said to
be leximin prefered to
v
if
∃i
such that
v
i
>v
i
and
∀j<i
,
v
j
=
v
j
.
∀
i
∈
In order to apply this definition to rank-order scenarios, we view the corre-
lation weights associated to each scenario
S
as a vector of weights
v
S
.By
−−−→
v
σ
(
S
)
we denote the vector obtained from
v
S
by ranking the weights increasingly.
Then, the selection of plausible scenarios is given by the following definition:
Definition 13:
A scenario
S
is prefered to
S
, denoted by
S>S
,if
−−−→
v
σ
(
S
)
>
leximin
−−−→
v
σ
(
S
)
. A scenario
S
is among the most plausible scenarios if there is no scenario
S
such that
S
>S
.
Let us go back to our illegal file access example.
As we said in the section
possible scenarios
, according to definition 3 we can
build the following seven scenarios:
-
scenario 1:
S
1
=(
A, B, C, D, E, F, G, H, O
)
-
scenario 2:
S
2
=(
A, C, G, H, O
)
-
scenario 3:
S
3
=(
A, D, E, G, H, O
)
-
scenario 4:
S
4
=(
B, F, G, H, O
)
-
scenario 5:
S
5
=(
A, C, D, E, G, H, O
)
-
scenario 6:
S
6
=(
A, B, D, E, F, G, H, O
)
-
scenario 7:
S
7
=(
A, B, C, F, G, H, O
)
According to definition 10, their corresponding vectors of weights are:
-
scenario 1:
−
v
S
1
=(1
,
1
,
1
,
1
,
1
,
2
,
1
,
2
,
2
) and
−−−→
v
σ
(
S
1
)
=(
2
,
2
,
2
,
1
,
1
,
1
,
1
,
1
,
1)
-
scenario 2:
−
v
S
2
=(1
,
1
,
3
,
2
,
2
) and
−−−→
v
σ
(
S
2
)
=(
3
,
2
,
2
,
1
,
1)
