xoEPC - Jan Mendling

82 3. Event-driven Process Chains (EPC)
• m max
→ d m ′ if and only if ∃τ : m τ → d m ′ ∧ �∃ m ′′ �=m ′ : m ′ → d m ′′ .
• maxd : MEP C → MEP C such that maxd(m) = m ′ if and only if m max
→ d m ′ . The
existence of a unique maxd(m) is the subject of Theorem 3.1 below.
Theorem 3.1 (Dead Context Propagation terminates). For an EPC and a given marking
m, there exists a unique maxd(m) which is determined in a finite number of propagation
steps.
Proof. Regarding uniqueness, by contradiction: Consider an original marking m0 ∈
MEP C and two markings mmax1, mmax2 ∈ MEP C such that m0
max
→d mmax1, m0
mmax2, and mmax1 �= mmax2. Since both mmax1 and mmax2 can be produced from m0
they share at least those arcs with dead context that were already dead in m0. Further-
more, following from the inequality, there must be an arc a that has a wait context in
one marking, but not in the other. Let us assume that this marking is mmax1. But if
τ
∃τ : m0 → mmax2 such that κmmax2(a) = dead, then there must also ∃τ
d ′ τ
: mmax1
′
→ m
d ′
such that κm ′(a) = dead because mmax2 is produced applying the propagation rules
without ever changing a dead context to a wait context. Accordingly, there are further
max
propagation rules that can be applied on mmax1 and the assumption m0 →d mmax1 is
wrong. Therefore, if there are two mmax1 and mmax2, they must have the same set of arcs
with dead context, and therefore also the same set of arcs with wait context. Since both
their states are equal to the state of m0 they are equivalent, i.e., maxd(m) is unique.
Regarding finiteness: Following Definition 3.11 on page 50, the number of nodes of an
EP C is finite, and therefore the set of arcs is also finite. Since the number of dead con-
text arcs is increased in each propagation step, no new propagation rule can be applied,
at the latest after each arc has a dead context. Accordingly, dead context propagation
terminates at the latest after |A| steps.
Phase 2: Transition Relation for Wait Context Propagation
For the wait context propagation, we also distinguish two cases based on the different
transition relations of Figure 3.13. The first case covers (a) function, (b) intermediate
event, (c) split, (d) and-join nodes. If the node belongs to this group and all input arcs
max
→d