13th International Conference on Membrane Computing - MTA Sztaki

Time-varying sequential P systems
Let G mM = (G m , M, =⇒ GmM ) be a matrix grammar without appearance
checking and G m = (N, T, w, P, =⇒ Gm ) the underlying multiset grammar. We
now construct a P system with regular control Π C = (Π, H, L) with
Π = (G m , [ 1 ] 1 , P, {(1, w)} , 1)
generating L (G mM ) as follows: Let M = {m i | 1 ≤ i ≤ k} and m i =
(m i,1 , · · · , m i,ki ), m i,j ∈ P , 1 ≤ j ≤ k i , 1 ≤ i ≤ k. A matrix m i
can be simulated by Π C by having the sequence of labels of singleton sets
H C ({m i,1 }) · · · H C ({m i,ki }) in L, i.e., we just take
L = {H C ({m i,1 }) · · · H C ({m i,ki }) | 1 ≤ i ≤ k} ∗ .
This basic result with a one-star control language containing words of arbitrary
length can be improved to a one-star control language containing words of length
two only when starting with a matrix grammar in binary normal form, i.e., N is
divided into two disjoint alphabets N 1 and N 2 , the axiom w is of the form X 0 S
with X 0 ∈ N 1 and S ∈ N 2 , and all the matrices are of the special (binary) form
(X → Y, A → w) with X ∈ N 1 , Y ∈ N 1 ∪ {λ}, A ∈ N 2 , and w ∈ (N 2 ∪ T ) ∗ .
In the case of allowing cooperative rules, the two rules in the binary matrix
(X → Y, A → w) can be put together into the single rule (XA → Y w), i.e., for
this new set of cooperative multiset rules
P ′ = {XA → Y w | (X → Y, A → w) ∈ M}
and the corresponding labeling function H C ′ we can take the control language
and, equivalently,
L ′ = {H ′ C ({XA → Y w}) | (X → Y, A → w) ∈ M} ∗
L ′′ = {H ′ C ({XA → Y w | (X → Y, A → w) ∈ M})} ∗ ,
which proves the assertion for time-varying P systems with cooperative rules,
i.e.,
L (mCF -MAT ) ⊆ L (coo-T V (1) OP 1 ) .
In fact, we have proved even more, as the multiset grammar
G ′ m = ( N 1 ∪ N 2 , T, X 0 S, P ′ , =⇒ G ′ m
)
generates the same multiset language as the original matrix grammar G mM ,
which shows that
L (mCF -MAT ) ⊆ L (mARB) .
As the binary normal form for matrix grammars is not restricted to context-free
multiset rules, we immediately infer that we even have
L (mARB-MAT ) ⊆ L (coo-T V (1) OP 1 ) .
In sum, all the families of languages considered in the statement of the theorem
coincide with L (mARB-MAT ).
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