13th International Conference on Membrane Computing - MTA Sztaki

Membranes with local environments
The structure of the rest of the paper is the following. At first, in Section
2, we present the general theory of multiset approximations. Having given the
basic notions of multisets, we define the general partial multiset approximation
space and discuss its fundamental properties. Section 3 connects partial multiset
approximation spaces with membrane systems. Using the approximation technique,
we can specify the closeness to membranes, even from inside and outside,
via border zones. In our membrane system model, at least for now, only communication
rules are defined. Their executions are restricted to membrane borders.
In the present paper we focus on hierarchical P systems.
2 Multiset Approximations
This section presents a general theory of multiset approximations of multisets.
There are (at least) two readings of different versions of rough set theory. The
first one is about vagueness based on indiscernibility, whereas the second one
is about possible approximations of sets. In the present paper we focus on the
second reading, and we ask how to treat multisets in a very general approximation
framework. The answer to this question is a minimal condition for applying
multiset approximations of multisets in membrane computing.
2.1 Fundamental Notions of Multiset Theory
A multiset is a well–known generalization of a set. We can say that an object can
have more than one occurrences in a multiset. The use of multisets in mathematics
has a long history. For instance, Richard Dedekind used the term multiset in
a paper published in 1888. Nowadays multisets are used not only in mathematics
but informatics.
Definition 1. Let U be a finite nonempty set. A multiset M, or mset M for
short, over U is a mapping M : U → N ∪ {∞}, where N is the set of natural
numbers.
1. Multiplicity relation for an mset M over U is:
a ∈ M (a ∈ U), if M(a) ≥ 1;
2. n–times multiplicity relation for an mset M over U is:
a ∈ n M (a ∈ U), if M(a) = n;
3. an mset M is said to be an empty mset (in notation M = ∅) if M(a) = 0
for all a ∈ U;
4. MS(U) is the set of msets over U;
5. MS n (U) (n ∈ N) is the set of msets over U such that if M ∈ MS n (U),
then M(a) ≤ n for all a ∈ U;
6. MS