13th International Conference on Membrane Computing - MTA Sztaki

D. Sburlan
represents a 0L system. The language generated by H is
L(H) = {x ∈ ∆ ∗ | ω =⇒ Pj1 w 1 =⇒ Pj2 . . . =⇒ Pjm w m = x,
m ≥ 0, 1 ≤ j i ≤ k, 1 ≤ i ≤ m}.
It is known that CF ⊂ ET 0L ⊂ CS and that NCF ⊂ NET 0L ⊂ NCS.
The set {2 n | n ≥ 0} ∈ NET 0L \ NCF .
The following result represents a normal form for the ET0L systems.
Lemma 1. For each L ∈ ET 0L there is an extended tabled Lindenmayer system
H = (V, T, ω, ∆) with two tables (T = {T 1 , T 2 }) generating L, such that for each
a ∈ ∆ if a → α ∈ T 1 ∪ T 2 then α = a.
P Systems with Simbol Objects and Multiset Rewriting Rules
A P system with symbol objects and multiset rewriting rules of degree m ≥ 1 is
a tuple
Π = (O, C, µ, w 1 , . . . , w m , R 1 , . . . , R m , i 0 ) where
• O is a finite set of objects;
• C ⊆ O is the set of catalysts;
• µ is a tree structure of m uniquely labeled membranes which delimit the regions
of Π; the set of labels is {1, . . . , m};
• w i , 1 ≤ i ≤ m, is the multiset of objects, initially present in the region i of Π;
• R i , 1 ≤ i ≤ m, is a finite set of multiset rewriting rules associated with the
region i; the rules are of type ca → cv or a → v, where c ∈ C, a ∈ O \ C, and
v ∈ ((O \ C) × {here, out, in}) ∗ .
The initial configuration of Π is C 0 = (µ, w 1 , . . . , w m ). A transition between
configurations means to apply in parallel a maximal multiset of evolution rules
(the rules are nondeterministically chosen and they compete for the available
objects), in all the regions of Π. The application of a rule u → v in a region
containing the multiset w consists of subtracting from w the multiset u and
then adding the objects composing v in the regions indicated by the targets
in, out, and here (we usually omit the target here). The P system iteratively
takes parallel steps until there remain no applicable rules in any region Π; then,
the system halts. The number of objects in the region i 0 of Π in the halting
configuration represents the result of the underlaying computation of Π. By
collecting the results of all possible computations of Π one gets the set of natural
numbers N(Π) generated by Π. The families of all sets of numbers generated
by P systems with symbol objects, multiset rewriting rules, with at most m
membranes, and with at most k catalysts (i.e., card(C) = k) is denoted by
NOP m (cat k ).
The following results regard the computational power of the P system model
defined above.
Proposition 1. NOP m (cat k ) = NOP 1 (cat k ), for any k ≥ 0
Theorem 2. NREG = NOP 1 (cat 0 ) ⊂ NOP 1 (cat 2 ) = NRE.
The exact characterization of the computational power of catalytic P systems
with only one catalyst remains an open problem.
410