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M
is a general part of
-
M
;
≤
=
-
≤
|E
.
In Fig. 6, the messages
m
1and
m
2 form a safe part into the bSDs
M
1,
M
2and
M
3. The order of the events of a safe part is the same as the order of the events
of the initial bSD restricted to the events of the safe part (
≤
=
≤
|E
). That is
why the messages
m
1and
m
2 do not form a safe part into
M
4, because with
only the messages
m
1and
m
2, the receiving of the message
m
1 and the sending
of the message
m
2 are not ordered whereas in the bSD
M
4, these two events are
ordered (by transitivity) because of the message
5.
Finally, an enclosed part defines a strict sequence of messages but this se-
quence can be “surrounded” by others messages. More formally:
m
Definition 6 (Enclosed Part).
Let
M
=(
I,E,≤,A,α,φ,≺
)
be a bSD. We
will say that
M
=(
I
,E
, ≤
,A
,α
,φ
, ≺
)
is an enclosed part of
M
if:
• M
is a safe part of
M
;
E
)
E
)=
E
.
• pred
≤,E
(
∩ succ
≤,E
(
In Figure 6, the messages
m
1and
m
2 form an enclosed part into the bSDs
M
1
and
2. Since an enclosed part is a part where no event can be present between
the events forming the enclosed part, the message
M
m
1and
m
2donotforman
enclosed part into
M
3.
E
)
E
), which represents the intersection be-
The set
pred
≤,E
(
∩ succ
≤,E
(
tween the set of predecessors of
E
and the set of successors of
E
,
5
, indi-
cates the presence of events “between” the events of
E
.Indeed,ifanevent
e∈ E
come between two events
e
and
e
of
M
(
e
≤ e ≤ e
and
φ
(
e
)=
e
)), then
E
)andto
E
). Therefore
φ
(
e
)=
φ
(
e
belongs to
pred
≤,E
(
succ
≤,E
(
pred
≤,E
E
∩ succ
≤,E
(
E
Let us note that for the four proposed definitions of part of a bSD, the definitions
are based on the semantics of the language of scenarios used, since we take account
of the message names, but also of the partial order induced by the pointcut.
E
)
=
3.2
Join Point
Roughly speaking, a join point is defined as a part of the base bSD such that this
part corresponds to the pointcut. Since we have defined four notions for parts of
a bSD, we have four corresponding strategies for detecting join points. It remains
to define the notion of correspondence between the pointcut and the part. To do
so, we introduce the notions of morphisms and isomorphisms between bSDs.
M
=
Definition 7 (bSD Morphism).
Let
M
=(
I,E,≤,A,α,φ,≺
)
and
I
,E
, ≤
,A
,α
,φ
, ≺
)
be two bSDs. A bSD morphism from
M
to
M
is a triple
(
μ
<μ
0
,μ
1
,μ
2
>
of morphisms, where
μ
0
:
I → I
,
μ
1
:
E → E
,
μ
2
:
A → A
=
and:
5
Let us note that
E
is necessarily inclued in
pred
≤,E
(
E
)andin
succ
≤,E
(
E
)because
each event of
E
is its own predecessor and its own successor (
e ≤ e
,
≤
being reflexive
by definition).
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