13th International Conference on Membrane Computing - MTA Sztaki

Limits of the power of tissue P systems with cell division
Definition 2. A recognizer tissue P system with cell division of degree q ≥ 1 is
a tuple
Π = (Γ, Σ, E, M 1 , . . . , M q , R, i in , i out )
where:
1. (Γ, E, M 1 , . . . , M q , R, i out ) is a tissue P system with cell division of degree
q ≥ 1 (as defined in the previous section).
2. The working alphabet Γ has two distinguished objects yes and no being, at
least, one copy of them present in some initial multisets M 1 , . . . , M q , but
none of them are present in E.
3. Σ is an (input) alphabet strictly contained in Γ , and E ⊆ Γ \ Σ.
4. M 1 , . . . , M q are strings over Γ \ Σ;
5. i in ∈ {1, . . . , q} is the input cell.
6. The output region i out is the environment.
7. All computations halt.
8. If C is a computation of Π, then either object yes or object no (but not both)
must have been released into the environment, and only at the last step of
the computation.
For each w ∈ Σ ∗ , the computation of the system Π with input w ∈ Σ ∗ starts
from the configuration of the form (M 1 , M 2 , . . . , M iin + w, . . . , M q ; ∅), that
is, the input multiset w has been added to the contents of the input cell i in .
Therefore, we have an initial configuration associated with each input multiset
w (over the input alphabet Σ) in this kind of systems.
Given a recognizer tissue P system with cell division, we say that a computation
C is an accepting computation (respectively, rejecting computation) if object
yes (respectively, object no) appears in the environment associated with the
corresponding halting configuration of C, and neither object yes nor no appears
in the environment associated with any non-halting configuration of C.
For each natural number k ≥ 1, we denote by TDC(k) the class of recognizer
tissue P systems with cell division and communication rules of length at most k.
We denote by TDC the class of recognizer tissue P systems with cell division and
without restriction on the length of communication rules. Obviously, TDC(k) ⊆
TDC for all k ≥ 1.
2.3 Polynomial Complexity Classes of Tissue P Systems
Next, we define what means solving a decision problem in the framework of
tissue P systems efficiently and in a uniform way. Bearing in mind that they
provide devices with a finite description, a numerable family of tissue P systems
will be necessary in order to solve a decision problem.
Definition 3. We say that a decision problem X = (I X , θ X ) is solvable in a
uniform way and polynomial time by a family Π = {Π(n) | n ∈ N} of recognizer
tissue P systems (with cell division) if the following holds:
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