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13th International Conference on Membrane Computing - MTA Sztaki

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B. Nagy<br />

Due to the deterministic finite automata accepting these languages, <strong>on</strong>e can<br />

decide if a word is in the language in at most as many steps as the length of the<br />

word. So, as an immediate c<strong>on</strong>sequence of the previous theorem we state about<br />

the classical computing paradigm the following<br />

Corollary 1. The SAT and n-SAT problems (over any finite sets of variables)<br />

can be solved by deterministic linear time sequential algorithms.<br />

We note here that our soluti<strong>on</strong> is a uniform soluti<strong>on</strong>. However the size of<br />

our DFA is not necessarily polynomial <strong>on</strong> the size of the input. Actually, if it<br />

was polynomial, then it would prove that P=NP since the structure of the DFA<br />

cannot change during the computati<strong>on</strong>. Opposite to this fact the structure of<br />

the membrane system can grow (exp<strong>on</strong>entially) during the computati<strong>on</strong>, and<br />

therefore in uniform soluti<strong>on</strong> it is usually required that the initial size of the<br />

membrane structure is polynomial <strong>on</strong> the length of the input. At Boolean circuits<br />

(see [29]) the uniform method is also frequently used, but with a fixed set of gates<br />

(alphabet).<br />

3.3 Complexity Issues<br />

Automata accepting the languages of satisfiable formulae in (n-ary) CNF were<br />

given in the previous subsecti<strong>on</strong>. Now we are going to make some short notes <strong>on</strong><br />

complexity.<br />

It is an important property of regular languages that they can be recognized<br />

in linear, moreover in “real”-time. With a large enough memory (i.e., number of<br />

possible states) we know the answer immediately after reading the formula. If<br />

there is a correct upper limit to the number of variables for a given CNF/SATformula,<br />

then using the DFA respecting this limit, it is linear time decidable<br />

whether the formula is satisfiable or not.<br />

Due to our c<strong>on</strong>structi<strong>on</strong> we can say that the language of (n-)SAT over a<br />

finite set of variables is not <strong>on</strong>ly an NP, but an L (linear time decidable by a<br />

deterministic Turing Machine) problem. This fact seem to be a very surprising<br />

because there is a big difference: for NP-complete problems there are not any<br />

methods known to compute them in deterministic polynomial time by traditi<strong>on</strong>al<br />

sequential machine, while problems in L are the most simplest <strong>on</strong>es.<br />

Let us say something about the complexity of the c<strong>on</strong>structed automata.<br />

Look the part C of the states, which is the most complex part of our automata.<br />

This element in this DFAs c<strong>on</strong>tains elements in the number of the powerset of the<br />

powerset of the variables. For a k element set of variables it means 2 (2k) states<br />

of C. For k = 10 it is 2 1024 which is about 10 308 . This number is incredibly huge,<br />

it is much more than the number of atoms in the known Universe. The statecomplexity<br />

of our automat<strong>on</strong> (depending <strong>on</strong> k) is EXP (EXP (k)), therefore<br />

there is no way to make such automat<strong>on</strong> which is useful in practice. (For small<br />

values of k there are some efficient programs which can decide the SAT-problem<br />

in reas<strong>on</strong>able time [5, 37].)<br />

334

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