13th International Conference on Membrane Computing - MTA Sztaki

On efficient algorithms for SAT
2 Solving SAT by Membrane Computing
In this section we recall a very important class of unconventional computing
techniques and analyse the proposed solutions to SAT in that frameworks.
Over the last decade, molecular computing has been a very active field of
research. The great promise of performing computations at a molecular level
is that the small size of the computational units potentially allows for massive
parallelism in the computations. Thus, computations that seem to be intractable
in sequential modes of computation can be performed (at least in theory) in
polynomial or even linear time.
In this section we are dealing with Membrane Computing. It is a branch of
biocomputing, and developing very rapidly. New algorithmic ways are used to
solve hard (intractable) problems. This field was born by the paper [31]. An
early textbook presenting several variations of these systems is [33]. There are
various ways for trading space for time, i.e., by parallelism exponential space
can be obtained in linear time, for instance, by active membranes. The SAT
is solved by various models in effective ways. In the next subsections we recall
some of these methods (dealing only with some aspects that are important
for our point of view, without further details due to the page limit; for further
details we recommend to check the literature at the P-system home page
at http://psystems.disco.unimib.it/ and http://ppage.psystems.eu; we
assume that the reader is familiar with the various concepts of these systems or
she/he can look for them in the cited literature). Since SAT is one of the most
known and most important NP-complete problems there are several attempts
by older and newer methods.
We use the terms uniform and semi-uniform by the definition of [35, 36]: in
semi-uniform algorithm a specific membrane system is created for a given instance
of the problem, while uniform algorithm can solve all instances having
same parameters (e.g., number of clauses, number of variables). We use the parameters:
m clauses and k variables of the solvable CNF formula. Note that in
[22] it is pointed out that at problems connected to complexity classes P, NP
and P-SPACE the choice of uniformity or semi-uniformity leads to the characterizations
of the same complexity classes.
2.1 Membrane Creation
Using membrane creation one can use an exponential growth (exponential space
can easily been obtained) during the computation.
We briefly describe how membrane creating can be used to solve SAT in linear
time. First we have an initial membrane with only one object. Applying the only
applicable rule for this object we introduce two new objects corresponding to
the possible values of the first variable, and a technical object to continue the
process. Then the new objects of the logical variable create new membranes (and
copy some symbols to the new membranes). Now for each new membrane two
new objects are introduced corresponding to the next variable, these new objects
create new ones again, etc. Finally the membrane structure forms a complete
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