13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, Yu. Rogozhin
Such a system behaves like Π 2 , except it also chooses among different objects
p j to send out symbols a k for k > x j . It halts either with a 1 · · · a y+1 p 1 · · · p m
generating y + m + 1, or with a 1 · · · a xj q j p 1 · · · p m generating x j + m + 1. For
m = 2, 3, 4 this leads to Π 3 generating {x 1 + 3, x 2 + 3, y + 3}, Π 4 generating
{x 1 +4, x 2 +4, x 3 +4, y+4}, and Π 5 generating {x 1 +5, x 2 +5, x 3 +5, x 4 +5, y+5},
i.e., any 3-, 4- or 5-element set with 3, 4 or 5 extra objects, respectively.
3.2 Straightforward Regularity
We now proceed by constructing a P system generating the length set of a
language accepted by a finite automaton A = (Q, Σ, δ, q 0 , F ), where Q = {q j |
0 ≤ j ≤ m}; we assume A satisfies the following property: there is at least one
transition from every non-final state.
Π A = (O = Q ∪ Q ′ ∪ Σ, E = Q ′ ∪ Σ, [ ] 1
, w = q 0 · · · q m q ′ 0, R),
R = {(q j q ′ j, out) | 0 ≤ j ≤ m}
∪ {(q j aq ′ k, in) | q k ∈ δ(q j , a), a ∈ Σ} ∪ {(q ′ j, out) | q j ∈ F }.
Unfortunately, besides the needed number, the skin region at halting also contains
the superfluous symbols, as many as there are states in A. Therefore, we
have obtained all sets N k REG k .
The simplest examples of application of Π A are the set of all positive numbers
and the set of all positive even numbers.
Therefore, NOP 1 (sym 3 ) contains NF IN 1 ∪ ⋃ ∞
k=0 (N kF IN k ∪ N k REG k ).
3.3 Improved Universality
We now revisit the symport-3 construction from [1]. The 7 extra objects were
denoted l h , b, d, x 1 , x 4 , x 5 , x 6 . We present an improvement to this construction,
lowering the number of extra objects to 6 (thus also improving a regularity result
from [3] to a universality result).
Theorem 1. NOP 1 (sym 3 ) ⊇ N 6 RE.
Proof. Consider an arbitrary counter automaton M. We first transform it as follows:
for each counter i, a conflicting counter ī is introduced, initially containing
value zero. The semantics of a counter machine is modified such that whenever
counters i and ī are non-zero, the computation is aborted without producing a
result.
Then, all zero-test instructions for any counter i are performed by incrementing
a conflicting counter ī, and then decrementing it. The counter automaton
M ′ = (Q, q 0 , q f , P, C) under the conflicting counter semantics is equivalent to
the counter automaton M.
120