13th International Conference on Membrane Computing - MTA Sztaki

F.G.C. Cabarle, H.N. Adorna
Fig. 6. OR-join Petri net and OR-join SNP system that simulates the net.
Fig. 7. A 3-way OR-split Petri net N (a), the 3-way OR-split SNP system Π N
simulating N (b). In (c) and (d) we see how Π N , directly transformed from Π N
using Lemma 1 and 3, can be transformed back to the original net N.
Proof. (An illustration of the proof can be seen in Fig. 6 for OR-join and Fig.
7. (a) and (b) for OR-split) The proof follows from Lemma 1 and the following
construction: Given Petri net N that performs an OR-split with places i, x, y,
transitions s, t, where t, s ∈ i•, t ∈ •x, s ∈ •y, the OR-split SNP system Π N
simulating N has neurons σ i , σ i ′, σ m , σ n , σ s , σ t , with (i, m), (i, n), (m, i ′ ),
(n, i ′ ), (i ′ , s), (i ′ , t) as synapses, R i ′ = {r 1 , . . . , r k } for k = |i • | and each rule
in R i ′ is of the form a k → a j , 1 ≤ j ≤ k. For some ordering O = 〈σ 1 , . . . , σ k 〉
of every neuron σ w so that (i ′ , w) ∈ syn, r uj is the j th rule of R u in σ u , where
1 ≤ u ≤ k, such that r uj is of the form:
a j →
{ a, if j = u
λ, if j ≠ u
If an initial marking for N is M 0 = (1, 0, 0) i.e. only i is marked, due to nondeterminism
a final marking could either be (0, 1, 0) (only x is marked) or (0, 0, 1)
(only y is marked). For Π N we have C 0 = 〈1, 0, 0, 0, 0, 0〉 i.e. only σ i has a spike
initially. An AND-split is performed by σ i which sends one spike each to σ m and
σ n , giving C 1 = 〈0, 1, 1, 0, 0, 0〉. Both σ m and σ n fire one spike each to σ i ′ (the
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