13th International Conference on Membrane Computing - MTA Sztaki

F.G.C. Cabarle, H.N. Adorna
weight of arc (p, t) is equal to 1. Petri nets where the arc weight is always 1 are
known as ordinary Petri nets. Note that ordinary and nonordinary Petri nets
(i.e. arc weights greater than or equal to 1) have the same modeling power [15].
We use the notation •p to denote the set of input transitions of place p, and the
notation •t as the set of input places of transition t. Similarly we use p• and t•
to denote the sets of output transitions and places of p and t respectively. Tokens
are indistinct, and a place without an input transition is called a source place
(•p = ∅) while a place without an output transition is a sink place (p• = ∅).
We assume that there is no place p or transition t such that •p = p• = ∅ or
•t = t• = ∅. A marking of a place p is denoted as M(p). A marking of a place is
always a non-negative integer.
A transition t is enabled iff for every p ∈ •t, p has at least one token. An
enabled transition t is fired when t removes one token from every p, and deposits
one token to every place p ′ ∈ t•. When |p • | ≥ 1 and there exists another
transition t ′ ∈ p•, then p has to nondeterministically choose (also known as a
decision point, choice, or conflict in literature e.g. [4,15]) which among t and t ′
will be enabled. If p ′ , p ′′ ∈ t• then if t fires, t deposits one token each to p ′ and p ′′ .
Parallelism in Petri nets comes from the fact that both p ′ and p ′′ receive tokens
at the same time after t is fired, thus allowing the firing and marking of possibly
more succeeding transitions and places, respectively. A marking M n is reachable
from a marking M if there is a sequence of enabled transitions 〈t 1 t 2 . . . t n 〉 that
leads from M to M n . The set of all reachable markings from M 0 given a net N
is denoted as R(N, M 0 ) or simply R(M 0 ) assuming there is no confusion on the
referred net. Now we provide some properties of Petri nets as well additional
classes from literature.
Definition 2 (Liveness, Boundedness, Safeness (Petri nets) [15,21]). A
Petri net (N, M 0 ) is live iff for every M ′ ∈ R(M 0 ) and every t ∈ T , there exists
a state M ′′ reachable from M ′ which enables t. (N, M 0 ) is bounded iff for each
p ∈ P there exists a positive integer k, such that for each M ∈ R(M 0 ), M(p) ≤ k
(the net is k-bounded). The net is safe iff for each reachable state M(p) does
not exceed 1.
Definition 2 provides some behavioral properties of Petri nets. A condition known
as a deadlock occurs when a transition t is unable to fire given a certain marking
(see Fig. 2). If N is a live net then it is considered deadlock-free. A class of
Petri nets known as workflow nets or WF-nets were introduced in [19] and were
used to model workflows. A WF-net is a special type of Petri net that has
two special places, the only source place i (•i = ∅) and the only sink place o
(o• = ∅). A net N is a good WF-net, a net that models correct workflow process
definitions, if N satisfied the following conditions: (i) N is a live net, (ii) The
initial configuration is where only M(i) = 1 and all other places are unmarked
(iii) N halts only in a configuration where M(o) = 1 and all other places are
unmarked. These properties as outlined in [21] all make sure that a process
executes correctly i.e. no tasks (modeled by transitions) were left undone and
all resources or states reached (modeled by marked places) have been used to
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