13th International Conference on Membrane Computing - MTA Sztaki

B. Nagy
k-level binary tree. Each path from the initial to a leaf-membrane represents
a possible truth-assignment. Now, each membrane in the k-th level computes
objects corresponding to satisfied clauses of the analysed formula. (It can be
done easily by a comparison among the literals of the clauses and the given
truth-assignment of the membrane.) Using a cooperative rule a special symbol is
sent out if all clauses are satisfied in a membrane. In the next step the previous
level membranes forward this symbol. Therefore this special symbol moves up
all k levels, and finally leaves the system and terminate the process with answer
‘satisfiable’. More technical details about such a method can be found in [33].
In this process the power of parallelism builds up a complete tree by levels
in linear time. In each membrane in the deepest level there are rules for each
clause, therefore the evaluation of clauses can go in a parallel way.
This approach, using membrane creation to solve SAT uses an alphabet with
cardinality approximately 3k + 2 m in [33]. The algorithm has a linear time complexity:
it solves the problem in 3k + 1 steps.
2.2 Membrane Division
Membrane division is another usual option for active membranes to increase the
number of membranes exponentially in the starting phase of the computation.
The SAT problem can be solved by a P system with active membranes in a
time which is linear on the number of variables and the number of clauses. In
the algorithm found in [32] the size of the alphabet is 5k + m.
In another algorithm ([33]) using membrane division the alphabet (the set
of object-symbols) has cardinality about 4k + 2m. This algorithm solves SAT in
linear time with respect to k + 2m.
In the same book a parallel computer model is also shown in which the ‘parallel
core’ do a massive parallel computation (brute-force) and then a ‘checker’
checks the result and a ‘messenger’ sends out the answer. This framework is used
to solve the SAT with alphabet of size 4k + m + 2.
2.3 Active Membranes with Polarization
In [3] SAT can be deterministically decided in linear time (linear with respect
to k + m) by a uniform family of P systems with active membranes with two
polarizations and global evolution rules, move out and membrane division rules.
The size of the alphabet is approximately mk 2 .
It is also proved that the SAT can be deterministically decided in linear time
with respect to km by a uniform family of P systems with active membranes with
two polarizations and special rules: global split rules, exit only with switching
polarization, yes out rule (for ejecting the result) and global polarizationless
division rules. These systems use an alphabet of size approximately mk 2 .
In [39] about 4k + 2m kinds of object are used in a linear algorithm using
polarities of membranes to solve SAT.
In [28] evolution rules, move out rules and separation rules are used; the
alphabet is larger than mk 2 .
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