13th International Conference on Membrane Computing - MTA Sztaki

T. Mihálydeák, Z. Csajbók
are able to give a special meaning of ‘to be close enough to a membrane’. But
we have no precise information about the nature of the space of objects or their
positions in general. If we look at the regions of a P system as multisets, then
a very general theory (the theory of approximation for multisets) can help us to
introduce a correct concept of ‘closeness to a membrane’ (or ‘to be close enough
to a membrane’).
Different ways of set approximations go back (at least) to rough set theory
which was originated by Pawlak in the early 1980’s ([7], [8]). In his theory and its
different generalizations lower and upper approximations of a given set appear
which are based on different kinds of indiscernibility relations. An indiscernibility
relation on a given set of objects provides the set of base sets by which any set
can be approximated from lower and upper sides. The general theory of partial
approximation of sets (see [4]) gives a possibility to embed available knowledge
into an approximation space. The lower and upper approximations of a given
set rely on base sets which represent available knowledge. If we have concepts of
lower and upper approximations, the concept of border can be introduced.
From the set–theoretical point of view, regions in membrane computing are
multisets and so, first, we have to generalize the theory of set approximations for
multisets. In the present paper we give a very general theory of multiset approximations
called partial multiset approximations and provide a partial multiset
approximation space. In this space, approximations are based on a beforehand
given set of base multisets. Using the approximation technique, borders of multisets
can be given. Since the set–theoretic representations of regions are multisets,
borders of regions delimited by membranes can be formed. In short, they are also
called borders of membranes or simply membrane borders. Then, we can say that
an object is close enough to a membrane if it is a member of its border. What is
more we can specify inside and outside borders of membranes, thus the closeness
to membranes from inside and outside. Last, it is assumed that the communication
rules in the P system execute only in membrane borders. Thus, a living
cell can be represented more precisely.
Communication rules change the regions by changing the inside and outside
borders of membranes. However, these changes take place within the base multisets.
Consequently, the changes does not modify either the whole borders or
the partial multiset approximation space. The latter can be changed only then
when there is no any communication rule which can be executed in the borders,
i.e. the membrane computation has halted. Just then, daemons are activated.
Daemons are rules which are assigned to the base multisets. Their forms
are similar to communication rules, more precisely to symport rules of the form
〈u, in〉. However, we strictly have to differentiate regions from base multisets and
rules in the membrane computing sense from daemons. In order to set the daemons
to work, a certain event is specified which starts a daemon each time when
that event occurs. Immediately when the membrane computation has halted,
first, the daemons assigned to the base multisets within membrane borders are
activated.
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