13th International Conference on Membrane Computing - MTA Sztaki

On structures and behaviors of spiking neural P systems and Petri nets
purpose of the previous step was to create 2 spikes for this step) so σ i ′ accumulates
2 spikes giving C 2 = 〈0, 0, 2, 0, 0, 0〉. Due to the construction of Π N (and
thus the rules in σ i ′) and due to nondeterminism, the final configuration could
either be 〈0, 0, 0, 0, 1, 0〉 (only σ s fires) or 〈0, 0, 0, 0, 0, 2〉 (only σ t fires). Therefore
N and Π N both perform an OR-split.
If N performs an OR-join having places i, j, k, transitions s, t, such that i ∈ •t, j ∈
•s, t, s ∈ •k, the OR-join SNP system Π N that simulates N has neurons σ s , σ t , σ k
with synapses (k, s), (k, t). An initial marking of M 0 = (1, 1, 0) for N results in
a final marking of (0, 0, 2) after t and s fire. For Π N we have C 0 = 〈1, 1, 0〉 and
a final configuration of 〈0, 0, 2〉 after σ t and σ s fire and each send one spike to
σ k . An OR-join is therefore performed by N and Π N .
⊓⊔
Lemma 2 assumes a safe Petri net so that no place is marked by more than one
token. SNP systems by nature split spikes in an AND-split manner. Referring to
Lemma 3, an OR-split is a nondeterministic decision point where a token in place
i is routed only to a single transition t, among the other transitions in i•, and is
eventually deposited to a place j. Recall that in SNP systems, nondeterminism
is in the form of rule selection. To capture this nondeterministic target selection,
neuron σ i uses its only rule a + /a → a to create k spikes, where k = |i • |.
Each output neuron of σ i (in this case, σ m and σ n ) receives one spike each,
which they then use by applying their rule a → a. The k neurons each send a
spike to their output neuron σ i ′. This intermediary stage involving the first k
parallel neurons is responsible for the creation of k spikes needed to make the
nondeterministic rule selection in the second stage. The second stage (involving
the next k parallel neurons) is an AND-join SNP system involving neuron σ i ′
which nondeterministically chooses one rule to apply among its k rules. At this
point σ i ′ has received k spikes from its k input neurons in the intermediary stage.
For the left hand side of the k number of rules of σ i ′, all rules have a regular
expression E equal to a k and all consume k spikes (now that α i ′ = k, σ i ′ has
to nondeterministically choose which rule to apply). For the right hand side of
the rules, a rule r j in σ i ′ produces one less spike than the succeeding rule r j+1 ,
1 ≤ j ≤ k. This increase in produced spikes per rule in σ i ′ permits the routing
of spikes to a unique output neuron σ u , (i ′ , u) ∈ syn, among the other output
neurons of σ i ′, because each unique a v on the right-hand side of every r j is
“mapped” to exactly one σ u . By mapping we mean that for every r j of σ i ′, only
one σ u is able to use the spikes produced by σ i ′, while the remaining k−1 output
neurons forget the spikes they received. Therefore, an OR-split is performed and
a spike is routed nondeterministically.
In order to return the resulting OR-split SNP system Π N back to the original
OR-split Petri net N, two reduction techniques (see e.g. in [4] and [15]) are used in
the following order: (i) The fusion of parallel places, resulting from transforming
the k neurons in the second stage to k parallel places (Fig. 7(c)) ; (ii) the
reduction of sequential places, resulting from the fused sequential places in (i)
(Fig. 7(d)). An OR-join net is where two or more transitions depositing a token
to a common output place i. In an SNP system an OR-join is simply two or more
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