13th International Conference on Membrane Computing - MTA Sztaki

On structures and behaviors of spiking neural P systems and Petri nets
Fig. 9. Not well-handled (a) and (b), and well-handled (c) and (d) Petri nets
and SNP systems.
The routing types using SNP systems are shown in Fig. 8. The proofs for
Theorems 2 and 3 below follow from Theorem 1, Lemma 2, and Lemma 3.
Theorem 2. If a Petri net N is free-choice then the SNP system Π N that simulates
N is free-choice. If N is non-free-choice then Π N is non-free-choice. ⊓⊔
Theorem 3. If a Petri net N is well-handled then the SNP system Π N that
simulates N is well-handled. If N is not well-handled then Π N is not well-handled.
⊓⊔
Observation 2 Transforming an SNP system Π using the construction for
Lemma 1 not necessarily produce an ordinary Petri net N.
A neuron in Π with a rule a 2 → a requires and consumes 2 spikes, which in N
means an arc for such a rule must have a weight equal to 2. Since AND-joins and
OR-splits in particular can be quite complex to visualize for SNP systems, we
introduce “shorthand” illustrations seen in Fig. 10. An AND-split neuron simply
has a thicker border or membrane, meaning it will only spike once enough spikes
are sent to the neuron. An OR-split neuron simply has thicker synapses or arcs,
indicating that only one of the output neurons will get to fire a spike. After the
structural properties in Definition 5 we present next some behavioral properties
of SNP systems from Petri nets.
Definition 6 (Live, Bounded, Safe (SNP system)). An SNP system Π
is live iff for every reachable configuration C k from C 0 and every rule r in Π
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