13th International Conference on Membrane Computing - MTA Sztaki

Mobile membranes with objects on surface as colored Petri nets
k!
Thus, s = s 1 ∗ k + s 2 ∗
2!(k − 2)! + s k!
3 ∗
2!(k − 2)! , where s 1, s 2 and s 3
represent the numbers of pino, phago and exo rules from R.
◦ A contains input arcs (P × T ) and output arcs (T × P ); for a rule r and its
associated transition t, we build the arcs as follows:
• the input arcs are both from the places that represent the membranes
appearing in the left hand side of the evolution rule r and from the place
structure to the transition t;
• the output arcs are from the transition t to both the places that represent
the membranes appearing in the right hand side of the evolution rule r
and to the place structure.
◦ Σ = U ∪ L, where U represents tokens (color) set containing all the objects
from O, and L = {1, . . . , k} × {1, . . . , k} is a color set containing the
membrane structure.
◦ X = {x, y, z, . . .} is a set of variables used when modifying the content of
place structure.
{
U, if p ∈ {1, . . . , k}
◦ C(p) =
L, if p = structure.
⎧
[x = y], if t is a transition corresponding to a phago rule; it checks if
⎪⎨
both membranes from the left hand side of a phago rule
◦ G(t) =
have the same parent;
⎪⎩
true, otherwise.
◦ For a rule r and its associated transition t, we build E as follows:
• we place the multiset of objects u on an input arc from a place that
represents a membrane appearing in the left hand side of the evolution
rule r (being marked with a multiset of objects u) to the transition t;
• we place all the pairs (i, j) describing the membrane structure appearing
in the left hand side rule r on the input arc from the place structure to
the transition t;
• we place the multiset of objects v on an output arc from a transition t
to a place that represents a membrane appearing in the right hand side
of the evolution rule r (being marked with a multiset of objects v);
• we place all the pairs (i, j) describing the membrane structure appearing
in the right hand side rule r on the output arc from the transition t to
the place structure.
{
w 0
◦ I(p) =
p, if p ∈ {1, . . . , k}
{(i, j) | i, j ∈ {1, . . . , k}, (i, j) ∈ µ}.
We prove formally the relationship between the dynamics of the mobile membrane
Π and that of the corresponding colored Petri net CP N Π .
Theorem 1. If M and M ′ are two membrane configurations of Π, then
where
M ⇒ R′
M ′ if and only if φ(M) [ψ(R ′ )〉 φ(M ′ ),
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