13th International Conference on Membrane Computing - MTA Sztaki

R. Freund
Yet in contrast to the results proved in the preceding section for the 1-
restricted minimally transition mode, now with noncooperative rules we only
obtain semilinear sets when using the k-restricted maximally parallel transition
mode:
Theorem 8. For every ρ ∈ {N, T } and every k ∈ N as well as any possible
partitioning Θ of the rule sets in the P systems, i.e., for all p ∈ N, and for any
halting condition γ ∈ {H, F },
NREG = NO ∗ C ∗ (max k (p), γ, ρ) [ncoo] .
Again, with decaying objects, the conditions of Lemma 1 are fulfilled, hence,
we get the following results:
Theorem 9. For all n, d ≥ 1, each Y ∈ {N, P s}, as well as for any halting
condition γ ∈ {H, h, A, F },
Y REG = Y O [d]
∗ C n (max k , γ, E) [coo] .
Theorem 10. For all n, d ≥ 1, each Y ∈ {N, P s}, as well as for any halting
condition γ ∈ {H, h, F }, for any ρ ∈ {N, T } ∪ {−l | l ∈ N},
Y F IN = Y O [d]
∗ C n (max k , γ, ρ) [coo] .
4 Computational Completeness Results for P Systems
with Decaying Objects
In this section we prove computational completeness for catalytic P systems as
well as for P systems using cooperative rules with decaying objects. Moreover,
we only consider P systems with one membrane/cell.
Catalytic P systems can be seen as a specific variant of P systems using
cooperative rules, hence, we first establish the computational completeness result
for P systems using cooperative rules; when using arbitrary cooperative rules,
additional ingredients such as context conditions can be avoided, yet only when
using the transition mode maxobj instead of max as well as with adult halting
or halting with final state:
Theorem 11. For all n ≥ 1 and all d ≥ 2 as well as any γ ∈ {A, F }, any
ρ ∈ {N, T } ∪ {−l | l ∈ N}, and each Y ∈ {N, P s},
Y RE = Y O [d]
∗ C n (maxobj, γ, ρ) [coo] .
Proof (sketch). We only show P sRE ⊆ P sO [2]
∗ C 1 (maxobj, γ, ρ) [coo]. The instructions
of a register machine M = (m, B, l 0 , l h , P ) can be simulated by a P
system Π = (1, V, T, l 0 , R, 1) with decaying objects of decay d = 2 using cooperative
rules in the transition mode maxobj. As usual, the contents of a register j
is represented by the corresponding number of copies of the symbol a j ; T consists
of the symbols a j , 3 ≤ j ≤ m. For keeping the objects a j , 1 ≤ j ≤ m, alive,
we use the rules a j → a j .
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