13th International Conference on Membrane Computing - MTA Sztaki

Simplifying Event-B models of P systems using functions
3 Simplified Model of a Tissue P System with
Active Membranes Solving the 3-colouring
Problem
We consider now the 3-colouring problem: given an undirected graph G =
(V,E), decide if it is possible to colour its nodes using 3 colors such that, for
every edge (u, v) ∈ E, the colours of u and v are different. This problem can be
solved in linear time by Π(n) a family of recognizer tissue P systems with cell
division of degree 2 [2]. In the following we will consider only a restricted set of
objects, V = {A i ,T i ,R i ,G i ,B i :1≤ i ≤ n}, and the division rules associated
with the second membrane of the P systems given in [2]:
• r 1i :[A i ] 2 → [R i ] 2 [T i ] 2 , for 1 ≤ i ≤ n
• r 2i :[T i ] 2 → [B i ] 2 [G i ] 2 , for 1 ≤ i ≤ n
Here, A i and T i are nonterminal symbols, A i encodes the i−th vertex of the
graph and R i , B i , G i are terminal symbols corresponding to the three colors
red, blue and green.
As we presented in [5] the Event-B model of a P system has two components.
The first one is a context that contains the set of symbols (denoted SYMBOLS)
and the constant n with a fixed nonnegative value.
The second component is a machine with the following variables:
- a partial function used to represent the structure of the P system, except the
skin: cell ∈ N → (SY MBOLS → ({1,...,n}→N)) with the following signification:
cell(x)(a)(i) represents the number of objects a with the index i in the
membrane x;
- mark : N →{0, 1} a partial function used “to mark” cells produced in a division
rule or involved in a communication rule between two steps of maximal
parallelism; such cells cannot be the subject of another division or communication
rule in the same computation step.
Using the modelling approach that we propose, we associate to the first set
of rules the following event:
Rule 1i
any
x, y, z, i
where
then
grd1 : x ∈ dom(cell)&mark(x) =0&i ∈ 1 ...n& cell(x)(a)(i) ≥ 1
grd2 : y ∈ N \ dom(cell)&z ∈ N \ dom(cell)&y ≠ z
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