13th International Conference on Membrane Computing - MTA Sztaki

F.G.C. Cabarle, H.N. Adorna
Fig. 10. Shorthand illustrations for an AND-join (a) and an OR-split (b) neuron.
Fig. 11. (a) A safe, nonlive Petri net N 1 (from [15]), (b) A nonlive, k = 2
bounded (nonsafe) SNP system Π N1 , simulating N 1 .
there exists a configuration C j reachable from C k wherein rule r is applied. Π is
bounded iff for every configuration each neuron has at most n spikes, where n
is a finite positive integer. If n = 1 then we say Π is safe.
In [14], properties such as liveness, boundedness, deadlock-free, and terminating
properties were introduced. A similar presentation with [14] is an earlier
work on P systems and Petri nets in [22]. In our work the definition for liveness
and boundedness are similar to those in [14], although our liveness definition
is identified by rule application and not by configurations. From the previous
results and the properties in Definition 6, we have the following corollary.
Corollary 2. If a safe Petri net N is is simulated by an SNP system Π N , the
bound k for Π N is given by the AND-join transition t in N such that k = | • t|
is maximum in N.
As seen in Fig. 11 and using the shorthand illustrations from Fig. 10, Π N1
is 2-bounded, even though N 1 is a safe net, since transitions | • t3| = | • t4| = 2.
4 Final Remarks
In this work we have added additional relationships between certain classes (e.g.
safe, ordinary) of Petri nets to SNP systems without delays. In particular we
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