13th International Conference on Membrane Computing - MTA Sztaki

On structures and behaviors of spiking neural P systems and Petri nets
If N is an AND-join net such that i, j ∈ •t and k ∈ t•, then Π N has neurons
σ t , σ i , σ j , σ k with synapses (i, t), (j, t), (t, k) and R t = {(a + ) v → a} for v = | • t|
(in the example figure, v = 2). Given M 0 = (1, 1, 0) for N i.e. only k has no
marking, the final marking after the firing of t will be (0, 0, 1) since N is an
AND-split net, combining the tokens from i and j and producing one token
to k. Similarly for Π N there is C 0 = 〈1, 1, 0, 0〉 and the final configuration is
〈0, 0, 0, 1〉. The rule in σ t combines the spikes from σ i and σ j and produces one
spike to σ k . However if M 0 = 〈0, 1, 0〉 then t cannot fire. Similarly σ x will not
spike given C 0 = 〈0, 1, 0, 0〉. Therefore Π N also performs an AND-join. ⊓⊔
Fig. 4. SNP system (a) AND-join, and (b) AND-split.
Observation 1 If N is a nonsafe Petri net that performs an AND-join, using
the construction for Lemma 2, SNP system Π N does not perform an AND-join.
An example of Observation 1 is shown in Fig. 5: the SNP system does not
perform an AND-join since the joining neuron σ t still spikes once accumulating
two spikes from its top input neuron σ j (and from the other spike from σ i ). In
the Petri net however, transition t is never fired (a deadlock) since k is never
marked, and j is a nonsafe place.
Lemma 3. Given a Petri net N that performs an OR-split (OR-join) of a token,
there exists an SNP system Π N that performs an OR-split (OR-join) of a spike
simulating N.
Fig. 5. A nonsafe AND-join Petri net and the “bad” AND-join SNP system
simulating the net.
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