13th International Conference on Membrane Computing - MTA Sztaki

F.G.C. Cabarle, H.N. Adorna
a spike. Similarly, given an SNP system Π that performs sequential routing of a
spike, there exists a Petri net N Π simulating Π that performs sequential routing
of a token.
Proof. (An illustration of the proof can be seen in Fig. 3.) Given a Petri net N
with places p, q and a transition t, we have p ∈ •t and t ∈ •q. Given a marking
M(p) over p, N can be simulated by an SNP system Π N having neurons σ x , σ y
where R x = {a + /a → a}, α x = M(p), and (x, y) ∈ syn such that t is fired
iff σ x applies rule a + /a → a. M(p) serves as the number of spikes in σ x . Rule
R x in σ x is of the form a + /a → a: R x consumes one spike whenever α x ≥ 1,
and produces one spike. Note that variable overloading is performed because of
the use of R x to mean the set of rules in σ x and to mean the only rule in σ x .
Transition t is fired if there is at least one token in p. The firing of transition t
sends one token to output place q. Similarly, rule R x is applied if neuron σ t has
at least one spike. The neuron sends a spike to its output neuron σ y after R x is
applied. For N, if M 0 = (1, 0) (i.e. only p is marked) and the final configuration
is (0, 1), Π N similarly has C 0 = 〈1, 0〉 and a final configuration of 〈0, 1〉.
The reverse can be shown in a similar manner i.e. given an SNP system Π
that routes a spike sequentially, there exists a Petri net N Π that can simulate
Π: a forgetting rule is a transition with an outgoing arc weight of zero so no
token is ever produced, and the environment is a sink place. For spiking rules,
the regular expression E is an additional condition before a transition t in a
Petri net is fired: if place p ∈ •t, then t is enabled iff a M(p) ∈ L(E) i.e. when
rule R x is applied then transition t is also fired.
⊓⊔
Fig. 3. The “basic” transformation idea from a Petri net performing sequential
routing to an SNP system (and back).
Lemma 2. Given Petri net N that performs AND-split (AND-join) routing of
a token, there exists an SNP system Π N simulating N that performs AND-split
(AND-join) routing of a spike.
Proof. (An illustration of the proof can be seen in Fig. 4.) The proof follows
from Lemma 1 and the following constructions: Given an AND-split Petri net N
with places i, j, k, transition t, such that i ∈ •t and j, k ∈ t•, the AND-split SNP
system Π N that simulates N has neurons σ x , σ y , σ z where R x = {a + /a → a},
with (t, j), (t, k) ∈ syn. For N we have M 0 = (1, 0, 0) i.e. only i is marked, with
a final marking of (0, 1, 1) after the firing of t. N performs an AND-split, sending
one token each to j and k. For Π N we have C 0 = 〈1, 0, 0〉 and the firing of σ t
sends one spike each to σ j and σ k . The final configuration is 〈0, 1, 1〉, thus Π N
performs an AND-split.
150