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13th International Conference on Membrane Computing - MTA Sztaki

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On structures and behaviors of spiking neural P systems and Petri nets<br />

focused <strong>on</strong> some structural properties of Petri nets that is fundamental to routing<br />

tokens: the AND- and OR-splits and joins. As it turns out, even the relatively<br />

simple mechanism of c<strong>on</strong>diti<strong>on</strong>al routing in Petri nets, the OR-split, can be quite<br />

complex in terms of SNP systems (an additi<strong>on</strong>al 2k + 1 neur<strong>on</strong>s to route a spike<br />

am<strong>on</strong>g k output neur<strong>on</strong>s). It seems that, at least for “standard” SNP systems (as<br />

defined in this work) without delays, the routing of spikes to specific or targeted<br />

neur<strong>on</strong>s is quite “unnatural” (again recall that splits in SNP systems are by<br />

nature AND-splits). Perhaps a similarly complex structure is required in order<br />

to perform AND-joins for n<strong>on</strong>safe and n<strong>on</strong>ordinary (i.e. generalized) Petri nets.<br />

If SNP systems are to be used for modeling processes and phenomena (aside<br />

from the more usual computability results) then results <strong>on</strong> structural and behavioral<br />

properties are certainly desirable. Since Petri nets enjoy a rich theory<br />

<strong>on</strong> both kinds of properties, it seems reas<strong>on</strong>able to further link Petri nets to SNP<br />

systems, as several previous works have already d<strong>on</strong>e. For our part, this work<br />

can be seen as a precursor to using SNP systems to be used in modeling processes.<br />

Even biological processes perhaps, after further theoretical developments<br />

are pursued, as menti<strong>on</strong>ed by Păun at the beginning of [17]. Some of these processes<br />

or phenomena might include multi or distributed processors and workflow<br />

processes just to name a few. Many of these have been modeled and analyzed<br />

using Petri nets such as in [15] and [21], am<strong>on</strong>g others.<br />

Additi<strong>on</strong>ally, other classes of Petri nets such as colored and stochastic nets<br />

(see for example [15]) just to name a few, could be simulated by SNP systems.<br />

Other variants of SNP systems such as those with neur<strong>on</strong> budding and divisi<strong>on</strong><br />

as in [16] can also be transformed and simulated by Petri nets. Such investigati<strong>on</strong>s<br />

will most likely yield interesting and useful theoretical and even applicative<br />

results for both models.<br />

Lastly, Petri nets and their behaviors can be represented as matrix equati<strong>on</strong>s,<br />

and using these equati<strong>on</strong>s several tools have been produced for Petri nets (see<br />

[15] and [21]). Similarly, the behavior of SNP systems have been represented as<br />

matrices in [23] which was used in the creati<strong>on</strong> of an SNP system simulator in<br />

[1] and [2]. One desirable feature of the various Petri net tools is their utility for<br />

analyses and modeling of processes. It is also the hope of further realizing and<br />

opening up c<strong>on</strong>necti<strong>on</strong>s between Petri nets and SNP systems that motivates this<br />

work.<br />

Acknowledgments<br />

F.G.C. Cabarle is supported by the DOST-ERDT program. H.N. Adorna is<br />

funded by a DOST-ERDT research grant and the Alexan professorial chair of<br />

the UP Diliman Department of Computer Science. The authors would also like<br />

to acknowledge the comments and suggesti<strong>on</strong>s of the an<strong>on</strong>ymous reviewers that<br />

helped improve this work. Both authors would like to acknowledge the help of<br />

R.A.B. Juay<strong>on</strong>g and J.B. Clemente.<br />

157

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