13th International Conference on Membrane Computing - MTA Sztaki

B. Aman, G. Ciobanu
Definition 3. A non-hierarchical Colored Petri Net is a nine tuple
CP N = (P, T, A, Σ, X, C, G, E, I), where
1. P is a finite set of places;
2. T is a finite set of transitions such that P ∩ T = ∅;
3. A ⊆ (P × T ) ∪ (T × P ) is a set of directed arcs;
4. Σ is a finite set of non-empty color set;
5. X is a finite set of typed variables such that T ype[x] ∈ Σ for all x ∈ X;
6. C : P → Σ is a color set function that assigns a color set to each place;
7. G : T → EXP R X is a guard function that assigns a guard to each transition
t such that T ype[G(t)] = Bool;
8. E : A → EXP R X is an arc expression function that assigns a guard to
each arc a such that T ype[E(a)] = C(p) MS , where p is the place connected
to the arc a;
9. I : P → EXP R ∅ is an initialization function that assigns an initialization
expression to each place p such that T ype[I(p)] = C(p) MS .
A distribution of tokens over the places of a net is called a marking. Given two
markings m and m ′ , we say that m leads to m ′ by a set U of transitions, and
denote this by m[U〉m ′ .
5 Mobile Membranes as Colored Petri Nets
We denote by Π = (M 0 , R) a system of mobile membranes with a set R of
rules having an initial membrane configuration M 0 = (w 0 1, . . . , w 0 n, µ), where w 0 i
denotes the initial multisets of objects placed on membrane i, and µ the initial
membrane structure. We consider that a well-defined system has at any
point of evolution at most k > 2 membranes. Given such a system of mobile
membranes, the corresponding colored Petri net is denoted by CP N Π =
(P, T, A, Σ, X, C, G, E, I), where the components are defined as follows:
◦ P = {1, . . . , k} ∪ {structure}, where structure is a place that contains the
structure of the corresponding membrane system, namely the pairs (i, j).
◦ T =
⋃
t k , where each t k represents a distinct transition for a rule of R;
1 ≤k≤s
since the rules over mobile membranes contains no explicit label for membranes,
it means that:
• a pino rule can be instantiated at most k times in each step;
k!
• a phago rule can be instantiated at most
times in each step;
2!(k − 2)!
• an exo rule can be instantiated at most
k!
times in each step;
2!(k − 2)!
2 represents the number of membranes from the left hand side of an exo
rule, and
k!
2!(k − 2)!
represents all the possible combinations of membranes.
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