13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, Yu. Rogozhin
applied, followed by rule 19, forcing a non-ending computation. Yet, if rule 4
is applied again immediately after rule 5, then rule 5 is no longer applicable,
forcing rule 6 and a non-ending computation. If decrement is attempted on a
counter with a zero value, then rule 17 is applied instead of rule 16, forcing an
infinite computation.
The conflicting counter semantics is ensured by rule 8.
Notice that once the simulation of M ′ arrives to q f , symbols x 1 , x 3 , x 5 , q f
stay in the skin, while symbols from Q \ {q f } stay in the environment. Hence,
all the rules moving objects p 1 , p 3 or A i in, i.e., rules 7,10,12,15,18 could not be
applied even if objects from O ′ are sent out, which is carried out by rules 20,21.
Rules 13,19 temporary become applicable, but lead to an infinite computation;
they are no longer applicable once all object from O ′ are taken out, and q f
stays in the skin. The computation halts with the skin containing the result (the
desired number of copies of a 1 ) as well as 6 extra objects: x 1 , x 3 , x 5 , d, q f , b. □
3.4 Symport of Weight at Most 4
If we allow up to 4 objects to participate in symport rules, then the number of
extra objects can be decreased to 2.
Theorem 2. NOP 1 (sym 4 ) ⊇ N 2 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 counter machines is modified such that whenever
counters i and ī are non-zero, both are instantly decremented.
Then, all zero-test instructions for any counter i are performed by incrementing
a conflicting counter ī, and then decrementing it (nothing changes except the
states if counter i has value zero). Otherwise, both counters are decremented,
and then the decrement of counter ī fails. Therefore, we have transformed M
into a counter automaton M ′ = (Q, q 0 , q f , P, C), which is equivalent under the
conflicting counter semantics.
We construct a P system simulating a counter automaton M ′ :
Π = (O, E, [ ] 1
, w, R, 1), where
O = E ∪ O ′ ∪ {T, N},
E = {a i | i ∈ C} ∪ Q ∪ {p 2 | p ∈ P },
O ′ = {p i | i ∈ {1, 3, 4}, p ∈ P },
w = q 0 T s c |O′ |−1 N, where s represents O ′ ,
R = {1 : (qp 1 T, out), 2 : (p 1 q ′ a i T, in) | p : (q → q ′ , i+) ∈ P }
∪ {3 : (qp 1 T, out), 4 : (p 1 p 2 T, in), 5 : (p 2 p 3 a i T, out), 6 : (p 2 p 4 T, out),
7 : (p 2 p 4 T, in), 8 : (p 3 q ′ T, in) | p : (q → q ′ , i−) ∈ P }
∪ {9 : (a i a ī , out) | i ∈ C}
∪ {10 : (q f bx, out | x ∈ O ′ )} ∪ {11 : (Nc, out), 12 : (q f bN, in).
The simulation of a transition in A is performed as follows:
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