13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, R. Freund, H. Heikenwälder, M. Oswald, Yu. Rogozhin, S. Verlan
A configuration C of the P system Π can be represented as a set
{(h, w h ) | h ∈ H}, where w h is the current contents of objects contained in the
membrane labeled by h. In the sequential transition mode, one rule from R is
applied to the objects in the current configuration in order to obtain the next configuration
in one transition. A sequence of transitions between configurations of
Π, starting from the initial configuration A, is called a computation of Π. A halting
computation is a computation ending with a configuration {(h, w h ) | h ∈ H}
such that no from R can be applied to the objects w h , h ∈ H, anymore, and
the object w from (f, w) then is called the result of this halting computation if
w ∈ O T . L (Π), the language generated by Π, consists of all terminal objects
obtained as results of a halting computation in Π. By L (X-OP ) (L (X-OP n ))
we denote the family of languages generated by P systems (with at most n
membranes) of type X.
In a similar way as for grammars themselves, we are able to consider various
control mechanisms as defined in the previous section for P systems, too, e.g.,
using a control graph. In this paper, we are going to investigate the power of
regular control.
A (sequential) P system of type X with n membranes and regular control
is a construct Π C = (Π, H C , L, F ) where Π = (G, µ, R, A, f) is a (sequential)
P system of type X, L is a regular language over H C , where H C is the set of
labels identifying the subsets of productions from R in a one-to-one manner,
and F ⊆ H C . The language generated by Π C consists of all terminal objects
z obtained in membrane region f as results of a halting computation in Π.
Observe that as in the case of normal grammars, the sequence of computation
steps must correspond to a string H C (R 1 ) · · · H C (R m ) ∈ L with R 1 , · · · , R m
being subsets of R. The corresponding families of languages generated by P
systems with regular control Π C (with at most n membranes) are denoted by
L
(X-αC (REG) β
OP n
)
, α ∈ {λ, w}, β ∈ {λ, ac, ut}.
Yet in contrast to the previous case, appearance checking and unconditional
transfer have a special effect, as we cannot make a derivation step without applying
a rule, but the derivation thus will halt immediately. In order to cope
with this problem specific for P systems, we allow the system to be inactive
for a bounded number of steps before it really “dies”, i.e., halts. We call this
specific way of terminating a computation halting with delay d, i.e., a computation
halts if for a whole sequence of length d of production sets in a control
word no rule has become applicable. In that way we obtain the language classes
(
)
L X-αC (REG) β
OP n , d , α ∈ {λ, w}, β ∈ {λ, ac, ut}; if any of the numbers n
or d may be arbitrarily large, we replace it by ∗. The case k = 0 describes the
situation with normal halting, i.e., by definition
(
)
)
L X-αC (REG) β
OP n , 0 = L
(X-αC (REG) β
OP n .
In the P systems area we often deal with multisets, i.e., the underlying grammar
is a multiset grammar. In the following, we first restrict ourselves to noncooperative
rules, the corresponding type is abbreviated by ncoo.
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