13th International Conference on Membrane Computing - MTA Sztaki

B. Nagy
the SAT and n-SAT languages are not regular; they are much more complex
languages/problems.
5 Conclusions, Further Remarks
The most of new computational paradigms, as membrane computing systems,
solve the SAT in effective ways. Usually the alphabet depends on the particular
problem, i.e., on the number of the variables. Due to page limitations we could
not recall all the details of the mentioned solutions to SAT (it could give a nice
survey). We have shown that the SAT and n-SAT languages are regular over any
(fixed) finite set of variables, and therefore it seems that a set of (much) easier
problems is solved (in a uniform way). Actually, our finite automata check all
Boolean combinations and, therefore, they need exponential number of states.
In membrane systems the evaluation process go in a parallel manner in an exponential
space that can be obtained in a linear time, hence the initial system do
not need to be exponential on any parameter of the input. Our automata check
also the syntax of the input expressions (words), while in membrane systems it
is usually assumed that the input is in a correct form and therefore the computation
checks only the satisfiability of the input formulae. The regular languages
can be recognized in linear time. If there is a correct upper limit to the number
of variables for a given formula, then using the DFA respecting this limit, it is
linear time decidable whether the formula is a satisfiable. This is a very nice
result about an NP-complete problem, isn’t it? Unfortunately, as we discussed
in Subsection 3.3 our result is more theoretical and mathematical than practical.
Our result shows that in SAT the length of the formulae are not so important
factor. It is interesting, because in complexity theory the measure uses the inputlength
as a parameter. In SAT the number of variables of the formula plays a
more essential role.
Let us consider those problems of discrete mathematics (including some NPcomplete
problems) in which there are only finitely many possible answers. They
define regular languages over a finite alphabet (with a finite number of variables,
a finite number of nodes in the graph, etc.) if the syntactically correct description
of them is regular. The problem has only finitely many possible states, and we
can use the same method as we presented here to make a finite automaton, which
accepts the solutions, even if the language (set) of the solutions of the problem is
infinite. It can be a useful tool when an upper bound of the used variables/nodes
etc. is given. We may have a fast algorithm (real time) to solve these problems,
even if the arbitrary case is more difficult.
There are several algorithms to solve SAT by various P-systems. With this
paper we wanted to reopen this particular field. We are looking for new ideas,
collaborations to solve SAT by a method with fixed alphabet. Note here that
there is another new computational paradigm, the so-called interval-valued computation
(introduced in [24, 25] and further developed in [26, 27]). It offers also
a linear solution to SAT (moreover to q-SAT, also). This ‘uniform’ algorithm
gives the answer for every Boolean formula, independently of its length and of
336