13th International Conference on Membrane Computing - MTA Sztaki

(Tissue) P systems with decaying objects
On the other hand, when using the asynchronous, the sequential or even the
maximally parallel transition mode, we only obtain regular sets (see [11]):
Theorem 5. For each Y ∈ {N, P s}, for any ϑ ∈ {asyn, sequ, max}, any γ ∈
{H, h, A, F }, and any ρ ∈ {N, T } ∪ {−l | l ∈ N},
Y REG = Y O ∗ C ∗ (ϑ, γ, ρ) [ncoo] .
Combining the results of Theorem 5 with those from Theorem 1, we immediately
obtain the following corollary for the sequential transition mode:
Corollary 2. For any halting condition γ ∈ {H, h, A, F }, any ρ ∈ {N, T } ∪
{−l | l ∈ N}, and each Y ∈ {N, P s},
for all d ≥ 1.
Y REG = Y O [d]
∗ C ∗ (sequ, γ, E) [ncoo] = Y O ∗ C ∗ (sequ, γ, ρ) [ncoo] ,
For purely catalytic P systems with decaying objects, even in the maximally
parallel transition mode the conditions of Lemma 1 are fulfilled, hence, we get
the following results:
Theorem 6. For all n, d, k ≥ 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, γ, E) [pcat k ] .
Theorem 7. For all n, d, k ≥ 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, γ, −k) [pcat k ]
= Y O [d]
∗ C n+1 (max, γ, ρ) [pcat k ] .
In all these systems with decaying objects, the catalysts are assumed to only
have life time d, too.
3.4 The k-Restricted Maximally Parallel Transition Mode
In this subsection, we investigate the k-restricted maximally parallel transition
mode. With cooperative rules, we again easily obtain computational completeness
when using the k-restricted maximally parallel transition mode, a result
which immediately follows from the results proved in the preceding section, i.e.,
from Theorem 4 and Corollary 1 (see [11]):
Corollary 3. For all n ≥ 1 and k ≥ 3, as well as for any halting condition
γ ∈ {H, h, F },
NRE = NO ∗ C n (max k , −k) [coo] = NO ∗ C n (max k , −k) [pcat k ] .
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