13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, R. Freund, H. Heikenwälder, M. Oswald, Yu. Rogozhin, S. Verlan
We now turn our attention to the case of time-varying P systems with delay
d > 0. Already allowing halting with delay two, in contrast to the preceding
results, we obtain computational completeness, needing only time-varying P
systems using non-cooperative rules in one membrane region even with unconditional
transfer:
Theorem 3. For all α ∈ {λ, w}, β ∈ {ac, ut}, n ≥ 1, p ≥ 12, and d ≥ 2,
L (ncoo-αT V β OP n (p) , d) = P sRE.
Proof. As appearance checking is at least as powerful as unconditional transfer,
we only have to show that
P sRE ⊆ L (ncoo-T V ut OP 1 (12) , 2) .
The proof is based on a construction used for purely catalytic P systems, see
[7] having in mind that the rules being applied with the (three) catalysts in
parallel there can be applied sequentially when periodically using different sets
of rules. In fact, the first two catalysts were used to guide the simulation of the
instructions applied to the first two registers of a register machine, whereas the
third one was used for all the trapping rules only to be applied in case a nondeterministic
choice for a rule assigned to the other two catalysts was taken in
a wrong way. As the simulation of a SUB-instruction there took four steps with
rules for the first two catalysts, we now need three sequential substeps for each
of these four steps, i.e., in total a period of 12.
Now let us consider a language from P sRE, i.e., there exists a register machine
M = (m, B, 1, f, P ) which uses its first two registers for the necessary
computations; during a computation of M, only these registers 1 and 2 can be
decremented. The remaining registers 3 to m are used to store the results of a
computation. We now construct a time-varying P system Π C = (Π, H, L) where
Π = (G m , [ 1 ] 1 , R, A, 1), G m = (N, T, w, P, =⇒ Gm ) is a multiset grammar and
L is a control language having periodicity 12; Π C halts with bounded delay 2,
i.e., the P system Π C definitely halts if for more than two steps no rule can be
applied anymore.
One basic principle for the construction of the P system Π C is that we
represent the contents of register i by the corresponding number of symbols
o i and variants of the labels of instructions to be simulated lead through the
simulation steps. In the following we give a sketch of how the rule sets P i , 1 ≤
i ≤ 12, are to be constructed, which contain the rules to be applied periodically
in the derivation steps 12k + i, k ≥ 0. We start with the axiom A = p 1 ˜p 1 ; in
fact,when reaching P 1 again, only such a pair p j ˜p j for some label j ∈ B \ {f}
should be present besides the symbols o i , 1 ≤ i ≤ m; the numbers of copies of
these symbols represent the number currently stored in the registers i.
The following table shows which rules have to be taken into the rule sets P i ,
1 ≤ i ≤ 12, to simulate a SUB-instruction j : (SUB (a) , k, l), with j ∈ B \ {l h },
k, l ∈ B, a ∈ {1, 2}; in any case, the rule sets P 3 , P 6 , P 9 , and P 12 contain the
rule # → #, where # is a trap symbol which guarantees that as soon as this
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