Graphics Programs Reference
In-Depth Information
p
1
t
1
p
3
t
3
p
5
p
2
t
2
p
4
Figure 2.11: Confusion
the two concurrent transitions t
1
and t
2
. This situation is called confusion,
and it may be particularly annoying if we are modelling a system where
conflicts are not resolved non-deterministically, but with the intervention
of an external oracle (or through some metrics): the firing order of two
nonconflicting transitions as t
1
and t
2
is thus going to decide whether the
oracle is called or not (the metrics are used or not) to resolve the potential
conflict between t
2
and t
3
. As we shall see in the next chapters, the solution
of a conflict due to the firing order can cause very subtle problems in the
stochastic definition of GSPNs.
Causal connection — Another possible relation among transitions is
causality. Two transitions t
l
and t
m
are in causal relation in M if t
l
is
enabled in M, t
m
has enabling degree k in M (possibly k = 0, that is to say
t
m
is not enabled in M), and the firing of t
l
generates a marking where the
enabling degree of t
m
is strictly greater than k. In formulae:
Definition 2.3.8 (Causal connection) For any PN system
S
, transitions
t
l
and t
m
are in causal relation in marking M, and we write t
l
CC(M)t
m
iff
M[t
l
i
M
0
⇒
ED(t
m
,M
0
) > ED(t
m
,M)
(2.7)
An example of causal connection can be seen in Fig.
2.11.
In marking
M = p
1
+ 2p
2
+ p
3
, t
1
CC(M) t
3
since the enabling degree of t
3
in marking
M is 1, while after the firing of t
1
the enabling degree of t
3
is 2. Observe
that in marking M = p
1
+ p
2
+ p
3
transition t
1
is not in causal connection
with t
3
since the enabling degree of t
3
is bounded to one by the marking of
the input place p
2
.
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