Semantics, Verification, and Implementation of Workflows ... - YAWL

Chapter 2. Formal foundations
The reachability graph is a directed labelled graph where the elements of the
reachability set form the nodes and the tuple consisting of a source marking that
enables a transition, the transition and the target marking that is reached by
firing the transition form the edges. The graph can be used to determine the
possible states of a reset net from an initial marking.
Definition 2.13 (Reachability graph) Let N = (P, T, F, R) be a reset net and
M ∈ IM(N). The directed labelled graph G = (V, E) with label set L = T is the
reachability graph of the marked net (N, M) iff
1. V = N[M〉 and
t
2. for any transition t ∈ T and markings M 1 , M 2 ∈ IM(N) : M 1 → M 2 ⇔
(M 1 , t, M 2 ) ∈ E.
p1
p2 t2
t2
t1
t1 p4 p1 p2+p3
t3
p3+p4
p4
t3
2p4
p3
t3
Figure 2.3: A reset net with an initial marking and its reachability graph
Example 4: In Figure 2.3, a reset net with an initial marking M = p1 is shown
together with its reachability graph. The reachability set N[M〉 = {p1, p2 +
p3, p3 + p4, p4, 2p4}. There are two occurrence sequences t1, t2, t3 and t1, t2.
From marking p2 + p3, it is possible to fire either t2 or t3. If t2 is fired at p2 + p3,
another marking p3 + p4 is reached, which still enables t3. Firing t3 at p3 + p4
results in 2p4. Note that there are no tokens in p2 and therefore, the reset arc
of p3 does not have an effect. On the other hand, if t3 is fired at p2 + p3, a
marking p4 is reached as t3 will remove all tokens from p2. As a result, transition
t2 cannot be enabled.
Liveness, boundedness and safeness are defined as in previous work [Pet81,
Mur89]. Liveness, boundedness and safeness can be determined from the reachability
graph.
Definition 2.14 (Liveness, boundedness, safeness [Pet81, Mur89]) A transition
is live if it can be enabled from every reachable marking. A place is safe if
it never contains more than one token at the same time. A place is k-bounded if
it will never contain more than k tokens. A place is bounded if it is k-bounded
for some k.
If all places in a reset net are bounded, the reset net is also bounded and
hence, it is possible to generate a finite reachability set. If a place is unbounded,
the reachability set contains an infinite number of states (an infinite state space).
In such cases, reachability of a marking cannot be determined but coverability
can be determined. Coverability is a relaxed notion that can handle unbounded
behaviour. A marking M 2 is said to be coverable from another marking M 1 in a
PhD Thesis – c○ 2006 M.T.K Wynn – Page 15