Graphics Programs Reference
In-Depth Information
t
p
1
p
2
t
1
t
2
p
3
p
4
Figure 2.10: Modelling concurrency
A case of particular interest is free-choice conflicts: two transitions are in a
free-choice conflict if they are in effective conflict, they are always enabled
together and their effective conflict is symmetric.
Concurrency — Contrarily to conflict, concurrency is characterized by
the parallelism of activities; therefore two transitions t
l
and t
m
are said to
be concurrent in marking M if they are both enabled in M, and they are
not in conflict. In formulae:
Definition 2.3.7 (Concurrency) For any Petri net system, transitions t
l
and t
m
are concurrent in marking M iff
t
m
,t
l
∈
E(M)
⇒
not(t
l
EC(M)t
m
) and not(t
m
EC(M)t
l
)
the depicted marking (2p
1
+ p
2
+ p
4
), since t
1
∈
E(M),t
2
∈
E(M),t
1
is not
in effective conflict with t
2
and t
2
is not in effective conflict with t
1
.
Confusion — An intriguing situation arises if concurrency and conflict
are mixed, thus invalidating the naive interpretation that concurrency stands
for “independency”. Observe the net in Fig.
2.11:
if we consider the marking
M in which only places p
1
and p
2
are marked, then t
1
and t
2
are concurrent.
However, if t
1
fires first, then t
2
is in conflict with t
3
, whereas if t
2
fires
first, no conflict is generated. What actually happens is that, although t
1
and t
2
are concurrent, their firing order is not irrelevant from the point
of view of conflict resolution, as one ordering may implicitly resolve the
conflict ahead of time. In particular, if we start from marking p
1
+ p
2
and
we end up in marking p
4
+ p
3
, then we are not able to tell whether it was
necessary to solve a conflict, because this depends on the firing order of
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