13th International Conference on Membrane Computing - MTA Sztaki

On efficient algorithms for SAT
include it here. All the problem solutions in [4] use alphabets depending on the
size of the input: they are solved in a semi-uniform way with larger size alphabet
than 6n, where n is the cardinality of the input set of numbers (weights).
2.11 Asynchronous P Systems
In [38] fully asynchronous parallelism in membrane computing and an asynchronous
P systems for the SAT problem is considered. The proposed P system
computes SAT in approximately mn2 n sequential steps or in approximately mn
parallel steps using approximately mn kinds of objects.
2.12 Solving SAT by Pre-computed Resources
In [33] one of the fastest algorithms for SAT uses a pre-computation technique.
It is assumed that the initial membrane structure is given “for free”; the precomputation
(without any costs) give a system that is large enough for the
input formula. (If a larger formula is given, then we need to shift to a larger
pre-computed system.) In this way a membrane structure that one can obtain,
for instance, by membrane divisions, is assumed to ready to use at the beginning
of the process. However the size of the used alphabet is exponential on k.
In some models the cardinality of the alphabet is cubic or exponential with
the number of the variables. Common fact of these approaches that the alphabet
depends on the problem, i.e., it has at least linear size on the number of variables.
3 Solving SAT in Linear Way by Traditional Computing
In the next part of this paper we analyse the SAT in a similar form as the new
computing paradigms solve it (theoretically) in effective ways allowing linear size
alphabet with the number of variables (see also the previous section).
We will prove an interesting and surprising (at least for first sight) result in
a constructive way. The construction goes in two steps. In the next subsection,
the first step, the syntactically correct (CNF) formulae will be described.
3.1 The Syntactic Forms of the SAT Languages
In this part we present the syntax of some of the SAT problems.
How should a word w look like, if the following question is answerable? Is w
is satisfiable, i.e. is w ∈ SAT ?
We describe the syntactically correct CNF formulae. We use the signs [, ]
for the real brackets. (Our alphabet is {a, [, ] , ¬, ∧, ∨} to allow to use the curly
brackets ‘(’ and ‘)’ to show the order of the regular operations of the expression.)
For the (n-)SAT languages we need the CNF forms:
331