13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, Yu. Rogozhin
With cooperation of up to 3 objects, a single membrane suffices. The regions
are called the skin and the environment, the latter contains an unbounded supply
of some objects, while the contents of the former is always finite. With antiport-
2/1 alone (i.e., exchanging 1 object against 2), the computational completeness
is obtained with a single superfluous object. With symport-3 (i.e., symport rules
only, of weight up to 3), one proved in [6] that 13 extra objects suffice for computational
completeness. This result has been improved in [1] to 7 superfluous
symbols. In the same paper it was shown that without any superfluous symbols
such systems only generate finite sets. Although one-membrane pure symport
systems are, in principle, universal, their exact characterization remains open,
and narrowing the gap between 7 objects and 1 object presents an interesting
combinatorics-style problem. While this line of research may seem like a corner
case study, it is precisely the case of exact generative power in one-membrane
systems that demonstrates the difference between symport rules and antiport
rules, and reveals certain intricacies of the former.
The computation consists of multiple, sometimes simultaneous, actions of two
types: move objects from the skin to the environment, and move objects from the
environment into the skin. It is obvious that trying to move all objects out in the
environment will activate the rules of the second type. Since, clearly, rules of the
first type alone cannot generate more than finite sets, it immediately follows that
the “garbage” is unavoidable. Recently, in [3] one obtained some partial results
on the power of one-membrane systems with symport-3, concerning intermediate
number of extra objects. This paper tries to further improve the currently best
bounds on how much “garbage” is sufficient. For instance, we claim that 6 extra
objects are enough for symport of weight at most 3, and 2 objects are enough
for symport of weight at most 4.
2 Definitions
Throughout the paper, by “number” we will mean a non-negative integer. We
write SEG 1 to denote finite consecutive numeric segments and SEG 2 to denote
all elements of SEG 1 with the same parity:
SEG 1 = {{j | m ≤ j ≤ n} | m, n ∈ N},
SEG 2 = {{i + 2j | 0 ≤ j ≤ m} | i, m ∈ N}.
We write N j F IN k to denote the family of all sets of numbers each not smaller
than j, of cardinality k. By N j REG k we denote the family of all sets {x + j |
|u| = x, u ∈ L} for all languages L accepted by some finite automata with k
states.
We assume the reader to be familiar with the basics of the formal language
theory, and we recall that for a finite set V , the set of words over V is denoted
by V ∗ , the set of non-empty words is denoted by V + , and a multiset may be
represented by a string, representing the multiplicity of each symbol by the
number of its occurrences in the string, their order not being important.
116