Graphics Programs Reference
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follow the net arcs backwards starting from transition t k until we find a new
transition t i such that π i π j . All the transitions on the path (including
t i ) with priority greater than or equal to π j , are in indirect structural con-
flict relation with t j . Indeed, any transition on the path can trigger a firing
sequence of transitions with priority higher than π j , that may eventually
enable transition t k whose firing would decrease the enabling degree of tran-
sition t j . Notice that the transitions on the path are in causal connection
relation.
A formal recursive definition for the ISC relation follows:
Definition 4.4.1 Given a priority PN model, two transitions t i and t j are
in indirect structural conflict relation (denoted t i ISC t j ) iff
π i π j ;
•∃ t k : (π k > π j ) (t i SCCt k ) ((t k SCt j ) (t k ISCt j ))
where SCC is the structural causal connection relation (see Definition 2.5.8) .
Let us consider the model in Fig. 4.2; since transition t 3 is in SC relation
with transition t 2 , π 3 > π 2 and t 1 SCCt 3 , it is possible to conclude that
t 1 ISCt 2 .
Given the definition of structural conflict, it is possible to introduce the
notion of conflict set and extended conflict set. The motivation for
the introduction of these concepts will become apparent in the following
chapters.
We define the symmetric structural conflict as follows:
Definition 4.4.2 Transition t i is in symmetric structural conflict
with t j
(denoted t i SSC t j ) iff
π i = π j and
t i SCt j t j SCt i t i ISCt j t j ISCt i .
The conflict set associated with a given transition t i is the set of transitions
that might be in conflict with t i in some marking.
Definition 4.4.3 The conflict set associated with a given transition t i is
defined as
CS(t i ) = { t j : (t i SSCt j ) }
The transitive closure of the SSC relation is an equivalence relation that
allows the partition of the set T into equivalence classes called extended
conflict sets.
Definition 4.4.4 ECS(t i ) = { t j : t i SSC t j ∧∼ (t i SME t j ) } .
 
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