13th International Conference on Membrane Computing - MTA Sztaki

<strong>13th</strong> <strong>International</strong> <strong>Conference</strong> on Membrane Computing, CMC13,
Budapest, Hungary, August 28 - 31, 2012. Proceedings, pages 323 - 339.
On Efficient Algorithms for SAT
Benedek Nagy
Department of Computer Science, Faculty of Informatics
University of Debrecen, Egyetem tér 1., 4032 Debrecen, Hungary
nbenedek@inf.unideb.hu
Abstract. There are several papers in which SAT is solved in linear time
by various new computing paradigms, and specially by various membrane
computing systems. In these approaches the used alphabet depends on
the number of variables. In this paper we show that the set of valid
SAT-formulae and n-SAT-formulae (for any fixed n) over finite sets of
variables are regular languages. We show a construction of deterministic
finite automata which accept the SAT and n-SAT languages in conjunctive
normal form checking both their syntax and satisfiable evaluations.
Theoretically the words of the SAT languages can be accepted by linear
time on their lengths by a traditional computer.
Keywords: SAT-problem, membrane computing, efficiency, new computing
paradigms, P-NP, regular languages, finite automata
1 Introduction
Computer science deals with problems that can be solved by algorithms. Some
problems can be solved by very effective algorithms, some of them seem not to
be. In complexity theory there are several classes of problems depending on the
complexity of the possible solving algorithms. A problem is in P if polynomial
deterministic algorithm solves it (on Turing machine). A problem is in NP if
non-deterministic polynomial algorithm solves it (on Turing machine). One of
the most challenging problems is to prove or disprove that the classes P=NP.
Most scientists think that NP is strictly includes P.
The SAT problem is the most basic NP-complete problem [14, 29]. It has
several forms. The first is the satisfiability of arbitrary Boolean formulae. A
restricted, and widely used version uses only formulae in conjunctive normal
forms. In conjunctive normal form a formula is build up from clauses. The clauses
are connected by conjunction. Each clause is build up form literals, and they are
connected by disjunction. A literal is nothing else, but a Boolean variable or its
negative (i.e., negated) form. The most restricted version we deal with is the
so-called 3-SAT. It is still NP-complete; and it has a huge literature. In 3-SAT
every clause has exactly 3 literals. It is very interesting fact, that SAT connects
some of the most important fields of theoretical computer science, such as logic,
formal languages, theory of algorithms and complexity theory.
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