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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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<strong>Membrane</strong>s with local envir<strong>on</strong>ments<br />

The structure of the rest of the paper is the following. At first, in Secti<strong>on</strong><br />

2, we present the general theory of multiset approximati<strong>on</strong>s. Having given the<br />

basic noti<strong>on</strong>s of multisets, we define the general partial multiset approximati<strong>on</strong><br />

space and discuss its fundamental properties. Secti<strong>on</strong> 3 c<strong>on</strong>nects partial multiset<br />

approximati<strong>on</strong> spaces with membrane systems. Using the approximati<strong>on</strong> technique,<br />

we can specify the closeness to membranes, even from inside and outside,<br />

via border z<strong>on</strong>es. In our membrane system model, at least for now, <strong>on</strong>ly communicati<strong>on</strong><br />

rules are defined. Their executi<strong>on</strong>s are restricted to membrane borders.<br />

In the present paper we focus <strong>on</strong> hierarchical P systems.<br />

2 Multiset Approximati<strong>on</strong>s<br />

This secti<strong>on</strong> presents a general theory of multiset approximati<strong>on</strong>s of multisets.<br />

There are (at least) two readings of different versi<strong>on</strong>s of rough set theory. The<br />

first <strong>on</strong>e is about vagueness based <strong>on</strong> indiscernibility, whereas the sec<strong>on</strong>d <strong>on</strong>e<br />

is about possible approximati<strong>on</strong>s of sets. In the present paper we focus <strong>on</strong> the<br />

sec<strong>on</strong>d reading, and we ask how to treat multisets in a very general approximati<strong>on</strong><br />

framework. The answer to this questi<strong>on</strong> is a minimal c<strong>on</strong>diti<strong>on</strong> for applying<br />

multiset approximati<strong>on</strong>s of multisets in membrane computing.<br />

2.1 Fundamental Noti<strong>on</strong>s of Multiset Theory<br />

A multiset is a well–known generalizati<strong>on</strong> of a set. We can say that an object can<br />

have more than <strong>on</strong>e occurrences in a multiset. The use of multisets in mathematics<br />

has a l<strong>on</strong>g history. For instance, Richard Dedekind used the term multiset in<br />

a paper published in 1888. Nowadays multisets are used not <strong>on</strong>ly in mathematics<br />

but informatics.<br />

Definiti<strong>on</strong> 1. Let U be a finite n<strong>on</strong>empty set. A multiset M, or mset M for<br />

short, over U is a mapping M : U → N ∪ {∞}, where N is the set of natural<br />

numbers.<br />

1. Multiplicity relati<strong>on</strong> for an mset M over U is:<br />

a ∈ M (a ∈ U), if M(a) ≥ 1;<br />

2. n–times multiplicity relati<strong>on</strong> for an mset M over U is:<br />

a ∈ n M (a ∈ U), if M(a) = n;<br />

3. an mset M is said to be an empty mset (in notati<strong>on</strong> M = ∅) if M(a) = 0<br />

for all a ∈ U;<br />

4. MS(U) is the set of msets over U;<br />

5. MS n (U) (n ∈ N) is the set of msets over U such that if M ∈ MS n (U),<br />

then M(a) ≤ n for all a ∈ U;<br />

6. MS

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