13th International Conference on Membrane Computing - MTA Sztaki

Membranes with local environments
If we have a membrane system Π, and the generated membrane approximation
space PASM(Π), then we can define the border of a region as an mset. To
introduce constrains on symport/antiport rules we need not only the border of
a region but ‘inside’ and ‘outside’ borders as well.
Definition 8. Let Π be a P system, V = {a 1 , a 2 , . . . , a n } and PASM(Π) be a
membrane approximation space generated by the P system Π as in Definition 7.
Let the sets Γi u, Γ i l of indices (for all w0 i ) be as follows:
– u(w 0 i ) = ⊔ {B γ | B γ ∈ B γ ∈ Γ u i },
– l(w 0 i ) = ⊔ {B γ | B γ ∈ B γ ∈ Γ l i }.
Then
1. border(w 0 i ) = ⊔ {B γ | γ ∈ Γ u i \ Γ l i };
2. border out (w 0 i ) = ⊔ {B γ ⊖ w 0 i | γ ∈ Γ u i \ Γ l i };
3. border in (w 0 i ) = ⊔ {B γ ⊖ (B γ ⊖ w 0 i ) | γ ∈ Γ u i \ Γ l i }.
Remark 9. The mset border(wi 0 ) is definable in the generated membrane approximation
space PASM(Π) for all i.
Using the borders of regions, the following constraint for rule executions can
be prescribed: a given rule r ∈ R i of a membrane i has to work only in the border
of its region. In order to be so, let the execution of a rule r ∈ R i (i = 1, 2, . . . , m)
define in the following forms:
– if a symport rule has the form 〈u, in〉, it is executed only in that case when
u ⊑ border out (w 0 i );
– if a symport rule has the form 〈u, out〉, it is executed only in that case when
u ⊑ border in (w 0 i );
– if an antiport rule has the form 〈u, in; v, out〉, it is executed only in that case
when u ⊑ border out (w 0 i ) and v ⊑ borderin (w 0 i ).
The next theorem shows that the membrane computation actually works in
the membrane borders.
Theorem 1. Let Π = 〈V, µ, w1, 0 w2, 0 . . . , wm, 0 R 1 , R 2 , . . . , R m 〉 be a P system
where the communication rules in R i (i = 1, 2, . . . , m) are constrained as above,
and PASM(Π) = 〈MS