13th International Conference on Membrane Computing - MTA Sztaki

B. Nagy
Due to the deterministic finite automata accepting these languages, one can
decide if a word is in the language in at most as many steps as the length of the
word. So, as an immediate consequence of the previous theorem we state about
the classical computing paradigm the following
Corollary 1. The SAT and n-SAT problems (over any finite sets of variables)
can be solved by deterministic linear time sequential algorithms.
We note here that our solution is a uniform solution. However the size of
our DFA is not necessarily polynomial on the size of the input. Actually, if it
was polynomial, then it would prove that P=NP since the structure of the DFA
cannot change during the computation. Opposite to this fact the structure of
the membrane system can grow (exponentially) during the computation, and
therefore in uniform solution it is usually required that the initial size of the
membrane structure is polynomial on the length of the input. At Boolean circuits
(see [29]) the uniform method is also frequently used, but with a fixed set of gates
(alphabet).
3.3 Complexity Issues
Automata accepting the languages of satisfiable formulae in (n-ary) CNF were
given in the previous subsection. Now we are going to make some short notes on
complexity.
It is an important property of regular languages that they can be recognized
in linear, moreover in “real”-time. With a large enough memory (i.e., number of
possible states) we know the answer immediately after reading the formula. If
there is a correct upper limit to the number of variables for a given CNF/SATformula,
then using the DFA respecting this limit, it is linear time decidable
whether the formula is satisfiable or not.
Due to our construction we can say that the language of (n-)SAT over a
finite set of variables is not only an NP, but an L (linear time decidable by a
deterministic Turing Machine) problem. This fact seem to be a very surprising
because there is a big difference: for NP-complete problems there are not any
methods known to compute them in deterministic polynomial time by traditional
sequential machine, while problems in L are the most simplest ones.
Let us say something about the complexity of the constructed automata.
Look the part C of the states, which is the most complex part of our automata.
This element in this DFAs contains elements in the number of the powerset of the
powerset of the variables. For a k element set of variables it means 2 (2k) states
of C. For k = 10 it is 2 1024 which is about 10 308 . This number is incredibly huge,
it is much more than the number of atoms in the known Universe. The statecomplexity
of our automaton (depending on k) is EXP (EXP (k)), therefore
there is no way to make such automaton which is useful in practice. (For small
values of k there are some efficient programs which can decide the SAT-problem
in reasonable time [5, 37].)
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