LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

54 Francesco Bernardini, Marian Gheorghe, and Mike Holcombe
Π =([1 ]1,X),
where X =(V,Q,M1,Φ,F,q0), with Φ = {φ1,...,φp}. We construct an EP
systems Π ′ ,withPs(Π ′ ) ∈ PsEOP1,2,p+2 such that:
where
Π ′ =([1 ]1,X ′ ),
X ′ =((V ∪ Q ∪{#,ɛ}), {q, q ′ },M ′ 1 ,Φ′ ,F ′ ,q),
with q, q ′ , #,ɛ /∈ (V ∪ Q) (i.e., q, q ′ , # are new symbols that do not appear
neither in V nor in Q),
M ′ 1 = M ′ 1 ∪{q0},Φ ′ = {φ ′ 1,...,φ ′ p,φp+1,φp+2}.
For each 1 ≤ j ≤ p, wehave:
φ ′ j =(φj ∪{p → q | F (φj,p)=q}∪{p → # | p ∈ Q}∪{# → #}),
and F ′ (φ ′ p ,q)=q. Moreover, we have:
φp+1 =({p→ ɛ | p ∈ Q }),
φp+2 =({a→ # | a → v ∈ φj, 1 ≤ j ≤ p}∪{# → #}),
and F ′ =(φp+1,q)=q ′ , F ′ =(φp+2,q ′ )=q ′ .
We have placed inside the skin membrane the initial state of the system Π.
In general, we may suppose to have inside membrane 1 an object p that
represent the current state of the state machine associated with the system
Π. Thus, we apply the rules as in the system Π, by using some φ ′ j ,andwe
change the state by using rule p → q, ifF (φj,p)=q. At any moment, if we
choose the wrong set of rules with respect to the current state (i.e., there
does not exists any state q such that F (φj,p)=q), then we are forced to
apply a rule p → #, and, due to the rule # → #, we generate an infinite
computation. In order to finish a computation, we have to trigger on φp+1,
which replaces the current state with ɛ and lead the system to the state q ′ .
Here, if there are rules that can be still applied to the objects present inside,
then an infinite computation is generated, as we can continue to use the rules
inside membrane 1 φp+2 forever.
It follows that Ps(Π ′ )=Ps(Π). ✷
Now, we are able to show the main result concerning the power of EP systems,
which provides a characterization of the family of Parikh sets of vectors
associated with ET0L languages.
Theorem 1. PsEOP1,2,∗ = PsEOP1,2,4 = PsET0L.
Proof. (i) PsET0L ⊆ PsEOP1,2,4. According to Theorem 1.3 in [14], for each
language L ∈ ET0L there is an ET0L system that generates L and contains