13th International Conference on Membrane Computing - MTA Sztaki

(Tissue) P systems with decaying objects
– if possible – has to be used. For the basic variant as defined in the following,
in each transition step we choose a multiset of rules R ′ from Appl (Π, C, asyn)
that cannot be extended to R ′′ ∈ Appl (Π, C, asyn) with R ′′ R ′ and such that
(R ′′ − R ′ ) ∩ R j ≠ ∅ and R ′ ∩ R j = ∅ for some j, 1 ≤ j ≤ p, i.e., extended by a
rule from a set of rules R j from which no rule has been taken into R ′ .
Definition 5. For the minimally parallel transition mode with partitioning Θ
(min(Θ)),
Appl (Π, C, min(Θ)) = {R ′ | R ′ ∈ Appl (Π, C, asyn) and
there is no R ′′ ∈ Appl (Π, C, asyn)
with R ′′ R ′ , (R ′′ \ R ′ ) ∩ R j ≠ ∅
and R ′ ∩ R j = ∅ for some j, 1 ≤ j ≤ p} .
In the k-restricted minimally parallel transition mode, a multiset of rules
from Appl (Π, C, min(Θ)) can only be applied if it contains at most k rules from
each partition R j , 1 ≤ j ≤ p.
Definition 6. For the k-restricted minimally parallel transition mode with partitioning
Θ (min k (Θ)),
Appl (Π, C, min k (Θ)) = {R ′ | R ′ ∈ Appl (Π, C, min(Θ)) and
|R ′ ∩ R j | ≤ k for all j, 1 ≤ j ≤ p} .
Each multiset of rules obtained by min 1 can be seen as a kind of basic
maximally parallel vector; this interpretation also allows for capturing the understanding
of the minimally parallel transition mode as introduced by Gheorghe
Păun:
Definition 7. For the base vector minimally parallel transition mode with partitioning
Θ (min GP (Θ)),
Appl (Π, C, min GP (Θ)) = {R ′ | R ′ ∈ Appl (Π, C, min(Θ)) and R ′ ⊇ R ′′
for some R ′′ ∈ Appl (Π, C, min 1 (Θ))} .
In the k-restricted maximally parallel transition mode, a multiset of rules
can only be applied if it is maximal but only contains at most k rules from each
partition R j , 1 ≤ j ≤ p.
Definition 8. For the k-restricted maximally parallel transition mode with partitioning
Θ (max k (Θ)),
Appl (Π, C, max k (Θ)) = {R ′ | R ′ ∈ Appl (Π, C, max) and
|R ′ ∩ R j | ≤ k for all j, 1 ≤ j ≤ p} .
Definition 9. For the the k-restricted maximally parallel transition mode with
only one partition, max k ({R}), we also use the notion k-restricted maximally
parallel transition mode (max k ), i.e., we get
Appl (Π, C, max k ) = {R ′ | R ′ ∈ Appl (Π, C, max) and |R ′ | ≤ k}.
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