LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

256 Lila Kari, Carlos Martín-Vide, and Andrei Păun
Recently, a rather unexpected result was reported in [2]: in order to get
the universality, minimal symport and antiport rules, that is of the forms
(a, in), (a, out), (a, out; b, in), where a, b are objects, are sufficient. The price was
to use nine membranes, much more than in the results from [3] and [4], for example.
The problem whether or not the number of membranes can be decreased
was formulated as an open problem in [2]. We contribute here to this question,
by improving the result from [2]: six membranes suffice. The proof uses the same
techniques as the proofs from [2], [4]: simulating a counter automaton by a P
system with minimal symport/antiport rules.
It is highly probable that our result is not optimal, but we conjecture that
it cannot be significantly improved; we believe that at most one membrane can
be saved.
2 Counter Automata
In this section we briefly recall the concept of counter automata, useful in the
proof of our main theorem. We follow here the style of [2] and [4]. Informally
speaking, a counter automaton is a finite state machine that has a finite number
of counters able to store values represented by natural numbers; the machine
runs a program consisting of instructions which can increase or decrease by one
the contents of registers, changing at the same time the state of the automaton;
starting with each counter empty, the machine performs a computation; if it
reaches a terminal state, then the number stored in a specified counter is said
to be generated during this computation. It is known (see, e.g., [5]) that counter
automata (of various types) are computationally universal, they can generate
exactly all Turing computable sets of natural numbers.
More formally, a counter automaton is a construct M =(Q, F, p0,C,cout,S),
where:
– Q is the set of the possible states,
– F ⊆ Q is the set of the final states,
– p0 ∈ Q is the start state,
– C is the set of the counters,
– cout ∈ C is the output counter,
– S is a finite set of instructions of the following forms:
(p → q, +c), with p, q ∈ Q, c ∈ C: add 1 to the value of the counter c
and move from state p into state q;
(p → q, −c), with p, q ∈ Q, c ∈ C: if the current value of the counter c is
not zero, then subtract 1 from the value of the counter c and move from
state p into state q; otherwise the computation is blocked in state p;
(p → q, c = 0), with p, q ∈ Q, c ∈ C: if the current value of the counter c
is zero, then move from state p into state q; otherwise the computation
isblockedinstatep.