Information Technology Reference
In-Depth Information
Pr
(
P
B
(
u
)) = 0. The probabilistic parts of (2) and (3)
are proven by contradiction as well, assuming that there exist nodes with dif-
ferent ratios, considering the pair of nodes with the highest ratio and obtaining
information flow for some property.
The converse is proven by considering basic cylinders for which it is possible
to show that there is no information flow. Then we define measurable subsets
P
n
which are disjoint unions of cylinders and we prove that there is no information
flow for these sets. Taking the limit of these sets we show that the absence of
information flow follows for
P
.
Next, we characterize general information flow, which turns out to be a very
strong property:
|
B
(
u
))
>
0and
Pr
(
P
|
H
ω
which are without general infor-
mation flow are those which have a projection on L reduced to a single trace.
Theorem 2.
The only systems with Tr
⊂
H
ω
this trace
w
is different from
and the finite non-empty low-level words
u
such
that
B
(
u
)
Proof.
Suppose that the projection of
Tr
on
L
is a trace
w
.Since
Tr
⊂
are the finite prefixes of
w
. Moreover for such a trace
u
,wehave
B
(
u
)=
Tr
and in this case, the system is without general information flow.
Conversely, suppose that the projection on
L
of the trace set
Tr
contains two
different traces
w
and
w
,andlet
u
be their longest common prefix. Let
a
=
∅
L
such that
ua
is a prefix of
w
.Let
P
be the property which consists of the infinite
sequences in
Tr
whose projection on
L
is equal to
w
.Wehave
Pr
S
(
P
∈
|
B
(
u
))
>
0
and
Pr
S
(
P
To our knowledge, there is no simple characterization of systems which are
without high-level information flow. It is immediate that any system without
sequential information flow is without high-level information flow, since the de-
finition of the latter has a coarser abstraction function. Also directly from the
definition, it follows that the projection of any nonempty bunch
B
(
u
)onto
H
must be the same, otherwise, for a high-level sequence
α
|
B
(
ua
)) = 0. Therefore
S
has general information flow.
H
∗
distinguishing
∈
between
B
(
u
)and
B
(
v
)wecantake
P
=
αH
ω
and we have
Pr
S
(
P
|
B
(
u
))
=
Prb
s
(
P
|
B
(
v
)), since one is zero and the other one not.
4
Relativized Information Flow
The definitions of the previous section capture information flow, but provide
no specific information about the time moment of the low-level observation and
the events whose occurrence are linked to it. For a more refined and relativized
view, one may wish to introduce the moment of observation in the property under
consideration. For example a question of interest could be: observing some partial
low-level trace at the current moment, what is the probability that the potential
trace satisfies some past or future or more generally some relativized property?
For example, what is the probability that starting from the current time, there
is still one high-level action which will occur? Or, what is the probability that
at current time, an event has occurred in the past, and will never occur in the
future?
