13th International Conference on Membrane Computing - MTA Sztaki

B. Nagy
One of the aim of many people in computer science to solve the SAT (or the
3-SAT) in an efficient way. There are several attempts by usual algorithms on
traditional computers for both the original and some more restricted versions
[5, 7, 17]. There is an annual conference series: <strong>International</strong> <strong>Conference</strong> on Theory
and Applications of Satisfiability Testing. Scientists also have competitions
whose program solves the given instances of the problem faster.
One of the main motivations of new computational paradigms to solve intractable
problems (SAT and n-SAT are frequently used candidates) by fast
methods. Membrane computing offers various ways for polynomial solutions to
SAT, by trading exponential space for time [33]. For a small collection of these
methods, see [21]. In these new computing paradigms the preparation of the
solution depends on the formula as we recall in Section 2. In this paper we assume
that the reader is familiar with most of the terms of membrane computing
and therefore we do not spend a large number of pages for full descriptions of
the recalled systems. We list several approaches and we point out the fact that
the size of the used alphabet depends on the problem instance (or some of its
parameters).
In Section 3, first we examine the syntactical part of the logical formulae
in conjunctive normal form. To obtain the syntactically correct formulae we
present regular expressions. After this we show how we can recognize the satisfiable
formulae by a deterministic finite automaton. It is well-known that regular
languages can be recognized in linear time, moreover deciding if a word belongs
to the language or not can be done in “real time” by the deterministic finite automaton
for the given language. Therefore the fact that the languages of (n-)SAT
are regular can help us to solve these problems in a very fast way, even if they
are NP-complete problems. We will discuss this possibility in the Subsection 3.3.
Finally we see the original problem and discuss its hardness.
Due to the numerous number of published solutions one may think that to
solve the SAT by membrane computing is not a challenging task any more.
With this paper we want to reopen this research field asking for new solution
algorithms that requires only a fixed number of object types independently of
the input. These new algorithms could play the same role as the traditional
algorithms play in classical computing defining a “more uniform” approach in
membrane computing.
1.1 Basic Definitions and Preliminaries
We recall some basic definitions, such as normal forms, CNF and SAT expressions
and regular expressions. We will deal with SAT only containing formulae in
conjunctive normal form.
Definition 1. The Boolean variables and their negations are literals. A logical
formula is called an elementary conjunction (clause), if it is a conjunction of
literals. The disjunction of elementary conjunctions is a disjunctive normal form
(DNF). If all clauses contain the same number (let us say, n) of literals, then we
call the form n-ary disjunctive form. Similarly, an elementary disjunction is a
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