13th International Conference on Membrane Computing - MTA Sztaki

A new approach for solving SAT by P systems with active membranes
Clearly, the main issue with Π(n) is that it can not be constructed in polynomial
time in n. As we have mentioned, the reason of this is that Π(n) creates
complete clauses from the clauses of the input formula, and the number of these
complete clauses can be exponential in n. On the other hand, one can note that
the answer of Π(n) depends only on whether or not every membrane with label
n + 3 contains at least one object regardless of whether the set of these objects
contains every complete clause or not. Thus, one way to turn Π(n) into
a polynomially (semi-)uniform solution of SAT might be to modify Π(n) such
that it reuses somehow the original clauses of the input formula in every step of
the computation. We demonstrate our idea that might be such a solution using
the P system Π(3) from Example 1. Let Π ′ (3) be a slight modification of Π(3)
working as follows. Assume that Π ′ (3) is started with the same input symbols
xyz, ¯x, ȳ, ¯z as in Example 1. In the first step, Π ′ (3) does not create new clauses
from ȳ and ¯z, it only makes two copies of them, one marked with a prime, and
another one marked with a double prime, i.e., Π(3) creates ȳ ′ and ȳ ′′ from ȳ,
and ¯z ′ and ¯z ′′ from ¯z. Then Π ′ (3) separates the new copies into the two new
membranes according to that whether they are marked with a prime or with a
double prime. The remaining clauses are separated in the same way as it is done
in Π(3). Thus, after the first step the membrane with label 3 looks as follows:
[[xyz, ȳ ′ , ¯z ′ ] 4 , [¯x, ȳ ′′ , ¯z ′′ ] 4 ] 3 . Then, in the next step, Π ′ (3) creates the clauses ¯z ′
and ¯z ′′ from ¯z ′ , ¯x ′ and ¯x ′′ from ¯x, ¯z ′ and ¯z ′′ from ¯z ′′ (of course, here Π ′ (3)
requires such rules also that can rewrite ¯z ′ and ¯z ′′ similarly as ¯z was rewritten
before). Moreover, both ȳ ′ and ȳ ′′ are rewritten to ȳ by two corresponding rules.
Then the symbols are separated into the new membranes as follows:
[[[xyz, ¯z ′ ] 5 , [ȳ, ¯z ′′ ] 5 ] 4 , [[¯x ′ , ¯z ′ ] 5 , [¯x ′′ , ȳ, ¯z ′′ ] 5 ] 4 ] 3 .
Finally, after the third step of Π ′ (3), the membrane with label 3 looks as follows:
[[[[xyz] 6 , [¯z] 6 ] 5 , [[ȳ ′ ] 6 , [ȳ ′′ , ¯z] 6 ] 5 ] 4 , [[[¯x ′ ] 6 , [¯x ′′ , ¯z] 6 ] 5 , [[¯x ′ , ȳ ′ ] 6 , [¯x ′′ , ȳ ′′ , ¯z] 6 ] 5 ] 4 ] 3 .
Now, since every membrane with label 6 contains at least one object, Π ′ (3) can
continue the computation and send out the symbol no in the same way as Π(3)
does it.
If we generalise the above idea to Π ′ (n) (n ∈ N), one can see that the
number of objects used in Π ′ (n) is linear in n + m, where m is the number of
clauses of the input formula. Thus, it seems that we can construct a polynomially
semi-uniform family of P systems with active membranes that is based on the
P systems presented in this paper and solves SAT in linear time in n. Our
future plan is the exact definition of this family of P systems and showing its
correctness. Moreover, we are planning to implement these P systems on certain
systems using parallel hardware since we would like to see whether our new
approach can be utilized in practice as well.
Acknowledgements. The authors are grateful to the reviewers for their valuable
comments that improved the manuscript.
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