LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Eilenberg P Systems with Symbol-Objects 53
If Π is an EOP system, then Ps(Π) will denote the set of Parikh vectors
of natural numbers computed as result of halting computations. The result of
a halting computation in Π is the Parikh vector ΨV (w) =(|w|a1,...,|w|ah ),
where w is the multiset formed by all the objects sent out of the system during
the computation with |w|ai denoting the number of ai’s occurring in w and
V = {a1,...,an}.
The family of Parikh sets of vectors generated by EOP (EOPP) systems
with at most m membranes, at most s states and using at most p sets of rules
is denoted by PsEOPm,s,p(PsEOPPm,s,p). If one of these parameters is not
bounded, then the corresponding subscript is replaced by ∗. The family of sets
of vectors computed by P systems with non-cooperative rules in the basic model
is denoted by PsOP(ncoo) [13].
In what follows, we need the notion of an ET0L system, which is a construct
G =(V,T,w,P1,...,Pm), m ≥ 1, where V is an alphabet, T ⊆ V , w ∈ V ∗ ,and
Pi, 1≤ i ≤ m, are finite sets of rules (tables) of context-free rules over V of the
form a → x. In a derivation step, all the symbols present in the current sentential
form are rewritten using one table. The language generated by G, denoted by
L(G), consists consists of all strings over T which can be generated in this way,
starting from w. An ET0L system with only one table is called an E0L system.
We denote by E0L and ET0L the families of languages generated by E0L systems
and ET0L systems, respectively. Furthermore, we denote by by PsE0L and
PsET0L the families of Parikh sets of vectors associated with languages in ET0L
and E0L, respectively. It is know that PsCF ⊂ PsE0L ⊂ PsET0L ⊂ PsCS.
Details can be found in [14].
3 Computational Power of EP Systems and EPP Systems
We start by presenting some preliminary results concerning the hierarchy on the
number of membranes and on the number of states.
Lemma 1. (i) PsEOP1,1,1 = PsOP1(ncoo) =PsCF,
(ii) PsEOP∗,∗,∗ = PsEOP1,∗,∗,
(iii) PsEOP1,∗,∗ = PsEOP1,2,∗.
Proof. (i) EP systems with one membrane, one state and one set of rules are
equivalent to P systems with non-cooperative rules in the basic model.
(ii) The hierarchy on the number of membranes collapses at level 1. The inclusion
PsEOP1,∗,∗ ⊆ PsEOP∗,∗,∗ is obvious. For the opposite inclusion,
the construction is nearly the same as those provided in [13] for the basic
model of P systems. We associate to each symbol an index that represent the
membrane where this object is placed and, when we move an object from a
membrane to another one, we just change the corresponding index.
(iii) The hierarchy on the number of states collapses at level 2. The inclusion
PsEOP1,2,∗ ⊆ PsEOP∗,∗,∗ is obvious. On the other hand, consider an EP
systems Π, withPs(Π) ∈ PsEOP1,s,p for some s ≥ 3, p ≥ 1 (yet again, the
cases s =1ors = 2 are not interesting at all), such that: