13th International Conference on Membrane Computing - MTA Sztaki

On structures and behaviors of spiking neural P systems and Petri nets
only one rule will be nondeterministically chosen and applied. The parallelism
is global for SNP systems, since neurons operate in parallel.
Given a neuron ordering of 1, . . . , m we can define an initial system configuration
as a vector C 0 = 〈α 10 , α 20 , . . . , α m0 〉. A computation is a sequence of
transitions from an initial configuration. A computation may halt (no more rules
can be applied for a given configuration) or not. One way to obtain a result is
to take the time difference between the first spike of the output neuron to the
environment and the output neuron’s second spike e.g. if σ out first spikes at
time t and spikes for the second time at time t + k then we say the number
(t + k) − t = k is “computed” by the system.Another way to obtain results is
to take the time difference between t and every other successive spiking time of
σ out .
3 Main Results
Now we move to routing in SNP systems i.e. routing of spikes. First, we provide
our results in order to simulate routing in Petri nets using SNP systems. The
simulation as mentioned earlier is relation from a set of configurations of a simulated
system and a set of configurations of a simulating system as in [7]. The
simulated and simulating systems in this work can either be Petri nets or SNP
systems i.e. our results allow the simulation of routing (either tokens or spikes)
between Petri nets and SNP systems. In contrast to [12], our results include
Petri nets with transitions having more than one incoming arc, and without using
synchronizing places as was done in [14]. On one hand, simulations of SNP
systems to Petri nets seem to be relatively straightforward (e.g. initially in [12]
and [13] with some modifications in [14]). On the other hand, simulations of
even ordinary Petri nets to SNP systems seem to be straightforward, although
we show in this section it is not quite so (at the least for certain routing types).
In this work we focus on ordinary Petri nets (as defined in Definition 1) for
the following reasons: (i) numerous analysis tools and techniques developed for
ordinary Petri nets since Petri nets were introduced, including linear algebraic
methods, structural and behavioral properties, etc. (ii) ordinary Petri nets have
been used extensively in literature to model processes and phenomena, (iii) ordinary
Petri nets are sufficient in order to model workflows (WF-nets) See for
example [21], [19], [4], and (especially) [15] just to name a few sources. For our
following results we refer to ordinary Petri nets unless otherwise stated. We
introduce similar routing blocks to SNP systems as was done with Petri nets:
parallel (AND-joins and splits) and conditional (OR-joins and splits). Sequential
and iteration routing also follow. The functioning of the blocks should be the
same for Petri nets and SNP systems i.e. if an AND-join Petri net combines
tokens from one or more input places in parallel, then an AND-join SNP system
should combine spikes from two or more input neurons, and so on. First, we
perform (easy) sequential routing.
Lemma 1. Given a Petri net N that performs sequential routing of a token,
there exists an SNP system Π N simulating N that performs sequential routing of
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