13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, R. Freund, H. Heikenwälder, M. Oswald, Yu. Rogozhin, S. Verlan
[18] following the work on time-varying automata [17]. This notion was also considered
in the area of Lindenmayer systems, corresponding to controlled tabled
Lindenmayer systems, with the tables being used periodically (see [12]). We can
also interpret these systems as counterparts of cooperating distributed grammar
systems ([2]) with the order of enabling the components controlled by a graph
having the shape of a ring. In the field of DNA computing several models using
the variation in time of the set of available rules were considered. The first
model in this area – time-varying distributed H systems – was introduced in [15]
and using the splicing operation. A similar model having some differences in the
operation application was considered in [10]. In [20] the time-varying mechanism
was used in conjunction with splicing test tube systems; there no direct action
on the splicing rules was considered, yet instead a time-varying dependency in
the communication step.
2 Preliminaries
After some preliminaries from formal language theory, we define the main concept
of P systems with control languages considered in this paper.
The set of integers is denoted by Z, the set of non-negative integers by N.
An alphabet V is a finite non-empty set of abstract symbols. Given V , the free
monoid generated by V under the operation of concatenation is denoted by
V ∗ ; the elements of V ∗ are called strings, and the empty string is denoted by
λ; V ∗ \ {λ} is denoted by V + . Let {a 1 , · · · , a n } be an arbitrary alphabet; the
number of occurrences of a symbol a i in a string x is denoted by |x| ai
; the
Parikh vector associated with x with respect to a 1 , · · · , a n is ( )
|x| a1
, · · · , |x| an .
The Parikh image of a language L over {a 1 , · · · , a n } is the set of all Parikh
vectors of strings in L, and we denote it by P s (L). For a family of languages
F L, the family of Parikh images of languages in F L is denoted by P sF L.
A (finite) multiset over the (finite) alphabet V , V = {a 1 , · · · , a n }, is a mapping
f : V −→ N and represented by 〈f (a 1 ) , a 1 〉 · · · 〈f (a n ) , a n 〉 or by any string
x the Parikh vector of which with respect to a 1 , · · · , a n is (f (a 1 ) , · · · , f (a n )).
In the following we will not distinguish between a vector (m 1 , · · · , m n ) , its representation
by a multiset 〈m 1 , a 1 〉 · · · 〈m n , a n 〉 or its representation by a string x
having the Parikh vector ( )
|x| a1
, · · · , |x| an = (m1 , · · · , m n ). Fixing the sequence
of symbols a 1 , · · · , a n in the alphabet V in advance, the representation of the
multiset 〈m 1 , a 1 〉 · · · 〈m n , a n 〉 by the string a m1
1 · · · a mn
n is unique. The set of all
finite multisets over an alphabet V is denoted by V ◦ .
The family of regular and recursively enumerable string languages is denoted
by REG and RE, respectively. For more details of formal language theory the
reader is referred to the monographs and handbooks in this area as [3] and [16].
2.1 Register Machines
For our main result establishing computational completeness for time-varying
P systems, we will need to simulate register machines. A register machine is a
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