13th International Conference on Membrane Computing - MTA Sztaki

F.G.C. Cabarle, H.N. Adorna
Fig. 8. Routing types using SNP systems: (a) sequential, (b) conditional, (c)
parallel, (d) iteration.
neurons sending a spike to a common output neuron. Joins in SNP systems are
by nature OR-joins. The following corollary follows from Lemma 3.
Corollary 1. Given a k-way OR-split net where the deciding (origin) place is p
(i.e. |p • | = k), then the simulating OR-split SNP system has additional 2k + 1
neurons.
Corollary 1 is evident from Fig. 7. Before moving on, we provide definitions
of free-choice and well-handled SNP systems as follows.
Definition 5 (Well-handled, free-choice (SNP system)). Given an SNP
system Π, Π is well-handled if a spike that is split with an AND-split (ORsplit)
is synchronized or terminated with an AND-join (OR-join). Let in Π exist
neurons σ x , σ y and σ w . Π is free-choice whenever (w, x), (w, y) ∈ syn, this
implies every neuron σ z such that (z, x) ∈ syn then (z, y) ∈ syn as well.
The definition of the well-handled and free-choice properties in Definition 5
follow the idea of the same properties for Petri nets (Definition 3). From the
previous definitions and lemmas we have the following theorems.
Theorem 1. Given a safe ordinary Petri net N that performs one or a combination
of the following routing types: sequential, parallel, conditional, and iterative,
then there exists an SNP system Π N that can simulate N.
Proof. Proof for sequential routing follows from Lemma 1, from Lemma 3 for
conditional, and Lemma 2 for parallel routing. For iterative routing, this is simply
a synapse from one neuron to another, different neuron in Π N .
⊓⊔
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