13th International Conference on Membrane Computing - MTA Sztaki

Time-varying sequential P systems
Theorem 1. For all α ∈ {λ, w}, β ∈ {λ, ac, ut}, and n ≥ 1,
)
L
(ncoo-αC (REG) β
OP n ⊆ P sL (CF -MAT ) .
Proof. According to the arguments given above, we only have to consider the
case of regular control languages without appearance checking, i.e., we only have
to show that
L (ncoo-αC (REG) OP n ) ⊆ P sL (CF -MAT ) .
So let Π C = (Π 0 , H 0 , L 0 ) be a P system of degree n with regular control
(and without appearance checking) where Π 0 = (G 0 , µ, R 0 , A 0 , f) and
G 0 = (N 0 , T 0 , w 0 , P 0 , =⇒ G0 ) is a multiset grammar. As we are dealing with static
P systems not communicating with the environment, it is clear that we can use
the well-known flattening procedure reducing it to an equivalent system P system
Π 1 = (Π, H, L) where Π = (G m , [ 1 ] 1 , R, A, 1), G m = (N, T, w, P 1 , =⇒ Gm )
is a multiset grammar and L is a regular control set over H, i.e., H is the set of
labels for the subsets of R; Π 1 uses non-cooperative rules in only one membrane
region, i.e., we may consider this P system as a multiset rewriting device where
a symbol b from membrane region i in the original P system Π C is represented
as [i, b]; it is easy to see that the control language L 0 can be changed accordingly
to obtain the regular control set L for Π 1 . We also observe that the terminal
objects b ∈ T 0 in the output region f of the original system Π C in Π 1 now are
represented as objects [f, b] .
Let M = (Q, H, δ, q 0 , Q f ) be the deterministic finite automaton accepting L
where Q is the set of states, δ is the transition function, q 0 is the initial state,
Q f is the set of final states. The simulation then works in several steps:
– We first construct a matrix grammar with context-free rules
where
G M = (G, M, =⇒ GM )
G = ( N ∪ T ∪ Q ∪ ¯Q, N ′ ∪ T ′ ∪ Q ′ , q 0 A, P, =⇒ G
)
.
For any non-cooperative rule a → u ∈ R, we take the matrix (p → q, a → u)
into M if and only if a → u is in the set of rules R p labeled by p and
(p, R p , q) ∈ δ. At the end of a computation, arriving at some q, with q ∈ F
for α = λ or q ∈ Q for α = w, we may prime every remaining symbol to make
it a terminal one by using the matrices (q → ¯q) as well as (¯q → ¯q, a → a ′ ) for
all a ∈ N ∪ T and finally ending up with the matrix (¯q → q ′ ). In that way
we can simulate the computations in Π 1 , but
– it remains to check that we have arrived at a configuration to which no rule
is applicable anymore. This can be achieved by intersecting the language
L (G M ) generated by G M with a regular set L r that cuts out all elements
of L (G M ) representing a configuration containing a primed version of a
symbol which would allow for the application of a rule from the set of rules
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