13th International Conference on Membrane Computing - MTA Sztaki

(Tissue) P systems with decaying objects
2.3 Networks of Cells
In [10], a formal framework for (tissue) P systems capturing the formal features
of various transition modes was developed, based on a general model of membrane
systems as a collection of interacting cells containing multisets of objects,
which can be compared with the models of networks of cells as discussed in [2]
and networks of language processors as considered in [4]. Continuing the formal
approach started in [10], k-restricted variants of the minimally and the maximally
parallel transition modes were considered in [11], i.e., we considered a
partitioning of the whole set of rules and allowed only multisets of rules to be
applied in parallel which could not be extended by adding a rule from a partition
from which no rule had already been taken into this multiset of rules, but
only at most k rules could be taken from each partition. Most of the following
definitions are taken from [7] and [11].
Definition 1. A network of cells with checking sets of degree n ≥ 1 is a construct
where
Π = (n, V, T, w, R, i 0 )
1. n is the number of cells;
2. V is a finite alphabet;
3. T ⊆ V is the terminal alphabet;
4. w = (w 1 , . . . , w n ) where w i ∈ 〈V, N〉, for each i with 1 ≤ i ≤ n, is the multiset
initially associated to cell i;
5. R is a finite set of rules of the form
(E : X → Y )
where E is a recursive condition for configurations of Π (see definition
below), while X = (x 1 , . . . , x n ), Y = (y 1 , . . . , y n ), with x i , y i ∈ 〈V, N〉,
1 ≤ i ≤ n, are vectors of multisets over V . We will also use the notation
(E : (x 1 , 1) . . . (x n , n) → (y 1 , 1) . . . (y n , n))
for a rule (E : X → Y ); moreover, the multisets x i and y i may be split into
several parts or be omitted in case they equal the empty multiset;
6. i 0 is the output cell.
A network of cells (in the following also simply called P system) consists
of n cells, numbered from 1 to n and containing multisets of objects over V ;
initially cell i contains w i . A configuration C of Π is an n-tuple of multisets
over V (u 1 , . . . , u n ); the initial configuration of Π, C 0 , is described by w, i.e.,
C 0 = w = (w 1 , . . . , w n ). Cells can interact with each other by means of the rules
in R. A rule
(E : (x 1 , 1) . . . (x n , n) → (y 1 , 1) . . . (y n , n))
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