13th International Conference on Membrane Computing - MTA Sztaki

Zs. Gazdag, G. Kolonits
• In the last three steps of the system, the symbol yes goes out to the
environment, and the computation halts.
It is not difficult to see that Π(n) works correctly. Indeed, Π(n) sends in
every computation to the environment either the symbol no or the symbol yes.
The symbol no can be introduced only in Case 1 above, but in this case ϕ ′
should contain every complete clause in C n , and it follows from Proposition 1
that in this case ϕ is not satisfiable. On the other hand, yes can be introduced
only in Case 2, but in this case there is a complete clause in C n that does not
occur in ϕ ′ , which, again by Proposition 1 means that ϕ is satisfiable. Moreover,
for every formula ϕ in CNF over X n , it is easy to see that Π(n) halts in n + 5
steps. Thus we have the following theorem.
Theorem 1. The SAT can be solved by a uniform family of a polarizationless
deterministic P systems with active membranes using separation rules in weak
linear time, where the size of an input formula is described by the number of
variables occurring in the formula.
4 Conclusions
We proposed a new approach for solving SAT by P systems with active membranes.
Although our family of P systems can not be constructed in polynomial
time, once a P system Π(n) for a given n ∈ N is constructed, for every formula
ϕ in CNF over the variables in X n , Π(n) can decide whether ϕ is satisfiable
or not in linear steps in n. Our P systems use the standard rules of active
membrane systems and, in addition, separation rules that can change membrane
labels. Moreover, our P systems are polarizationless. It is an interesting question,
whether we could somehow get rid of membrane label changing in our P
systems. However, this seems to be difficult without using other types of rules
(for example, in Section 5 of [2], rules of type [[ ] i [ ] j ] k → [[ ] i ] k [[ ] j ] k , where
i, j, and k are labels, were used to get a linear time solution of SAT without
polarizations and membrane label changing).
It should be also mentioned that the rules in (a), (c)-(f) in the definition
of Π(n) have constant size. Moreover, it is not difficult to see that during the
evolution of Π(n), membranes with label i (3 ≤ i ≤ n + 3) have no more objects
than the number m of the clauses in the input formula. Thus the separation
rules in (b) always should act on membranes with no more than m objects.
Our P system Π(n) has exponential size in n, thus it is a reasonable question
whether a constant time solution of SAT exists based on our P systems. Since our
solution is uniform, i.e., the size of Π(n) depends only on n, and the encoding
of the input formula is computable in linear time, it seems that such a constant
time solution can not be easily given. On the other hand, one can see that slightly
modifying Π(n), a P system Π ′ (n) could be easily given such that, for a formula
ϕ over X n , Π ′ (n) can create the complete clauses of ϕ ′ only in one step. However,
it is not clear how could we ensure Π ′ (n) to send out to the environment the
correct symbol yes or no using only constant number of steps.
218