xoEPC - Jan Mendling

3.4. EPC Semantics 89
• m ∗ → +1 m ′ if and only if ∃τ : m τ → +1 m ′ .
Since the transition relation covers several marking changes that are not interesting
for an observer of the process, we define the reachability graph RG of an EPC in the
following section. It includes only transitions of the positive state propagation phase.
Calculating the Reachability Graph for EPCs
In this section, we define the reachability graph of an EPC and present an algorithm to
calculate it. First we formalize the concept of reachability related to an EPC.
Definition 3.21 (Reachability related to an EPC). Let EP C = (E, F, C, l, A) be a re-
laxed syntactically correct EPC, N = E ∪ F ∪ C its set of nodes, and MEP C its marking
space. Then, a marking m ′ ∈ MEP C is called reachable from another marking m if
and only if ∃n ∈ N ∧ m1, m2, m3 ∈ MEP C : maxd(m) = m1 ∧ maxw(m1) =
n
m2 ∧ max−1(m2) = m3 ∧ m3 → m
+1 ′ . Furthermore, we define the following notations:
• m n → m ′ if and only if m ′ is reachable from m.
• m → m ′ ⇔ ∃n ∈ N : m n → m ′ .
• m τ → m ′ if and only if ∃n1,...,nq,m1,...,mq+1 : τ = n1n2...nq ∈ N ∗ ∧
m1 = m ∧ mq+1 = m ′ ∧ m1
• m1
∗
τ
→ mq ⇔ ∃τ : m1 → mq.
n1
→ m2, m2
n2
→ ... nq
→ mq+1.
Definition 3.22 (Reachability Graph of an EPC). Let EP C = (E, F, C, l, A) be a relaxed
syntactically correct EPC, N = E ∪ F ∪ C its set of nodes, and MEP C its marking space.
Then, the reachability graph RG ⊆ MEP C ×N ×MEP C of an EPC contains the following
nodes and transitions:
(i) ∀m ∈ IEP C : m ∈ RG.
(ii) (m, n, m ′ ) ∈ RG if and only if m n → m ′ .
The calculation of RG requires an EP C as input and a set of initial markings I ⊆
IEP C. For several EPCs from practice, such a set of initial markings will not be available.