13th International Conference on Membrane Computing - MTA Sztaki

On structures and behaviors of spiking neural P systems and Petri nets
reach the end goal, corresponding to (iv) or a proper termination of a workflow
process.
Fig. 1. Routing types: (a) sequential, (b) conditional, (c) parallel, (d) iteration.
As mentioned earlier, the routing of tokens are fundamental to WF-nets, and
[21] identifies four types: sequential, parallel, conditional, and iteration (See Fig.
1). In order to perform these routing types, building blocks in Petri net semantics
are used. These building blocks are (again referring to Fig. 1): AND-split is the
sending of a token by transition D (from p5) to two or more output places of t
in parallel, in this case p6 and p7. An AND-join is the removal, in parallel, of a
token from every input place of E (in this case p6 and p7) in order to fire E. An
OR-split is a nondeterministic routing of a token in p3 to only one among many
output transitions of p3 (in this case firing either B or C). An OR-join is the
sending of a token from B (or C) to p4, among several input transitions of p4.
From [19] we have the following: For Petri net N, a path H from node x 0
to node x k is a sequence 〈x 0 , x 1 , . . . , x k−1 , x k 〉 where (x i , x i+1 ) ∈ A, for 1 ≤
i ≤ k − 1. The alphabet of H denoted as alph(H) = {x 0 , x 1 , ..., x k−1 , x k }. H
is elementary if for any nodes x i and x k in H, i ≠ k implies x i ≠ x k . An
elementary path H implies that H must have unique nodes in path. Using paths
and alphabets we have the following definition.
Definition 3 (Well-handled, Free-Choice (Petri nets) [19]). A Petri net
N is well-handled iff for any pair of nodes x and y such that one of the nodes
is a place and the other a transition, and for any pair of elementary paths H 1
and H 2 leading from x to y, alph(H 1 ) ∩ alph(H 2 ) = {x, y} implies H 1 = H 2 .
N is free-choice iff for every two transitions t 1 and t 2 , •t 1 ∩ •t 2 ≠ ∅ implies
•t 1 = •t 2 .
Definition 3 provides structural properties of Petri nets. The well-handled property
makes sure that a token that is split using parallel routing (AND-split) is
synchronized or terminated with an AND-join. The property also assures that
a conditionally routed token (OR-split) is synchronized with an OR-join. If an
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