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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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B. Nagy<br />

the SAT and n-SAT languages are not regular; they are much more complex<br />

languages/problems.<br />

5 C<strong>on</strong>clusi<strong>on</strong>s, Further Remarks<br />

The most of new computati<strong>on</strong>al paradigms, as membrane computing systems,<br />

solve the SAT in effective ways. Usually the alphabet depends <strong>on</strong> the particular<br />

problem, i.e., <strong>on</strong> the number of the variables. Due to page limitati<strong>on</strong>s we could<br />

not recall all the details of the menti<strong>on</strong>ed soluti<strong>on</strong>s to SAT (it could give a nice<br />

survey). We have shown that the SAT and n-SAT languages are regular over any<br />

(fixed) finite set of variables, and therefore it seems that a set of (much) easier<br />

problems is solved (in a uniform way). Actually, our finite automata check all<br />

Boolean combinati<strong>on</strong>s and, therefore, they need exp<strong>on</strong>ential number of states.<br />

In membrane systems the evaluati<strong>on</strong> process go in a parallel manner in an exp<strong>on</strong>ential<br />

space that can be obtained in a linear time, hence the initial system do<br />

not need to be exp<strong>on</strong>ential <strong>on</strong> any parameter of the input. Our automata check<br />

also the syntax of the input expressi<strong>on</strong>s (words), while in membrane systems it<br />

is usually assumed that the input is in a correct form and therefore the computati<strong>on</strong><br />

checks <strong>on</strong>ly the satisfiability of the input formulae. The regular languages<br />

can be recognized in linear time. If there is a correct upper limit to the number<br />

of variables for a given formula, then using the DFA respecting this limit, it is<br />

linear time decidable whether the formula is a satisfiable. This is a very nice<br />

result about an NP-complete problem, isn’t it? Unfortunately, as we discussed<br />

in Subsecti<strong>on</strong> 3.3 our result is more theoretical and mathematical than practical.<br />

Our result shows that in SAT the length of the formulae are not so important<br />

factor. It is interesting, because in complexity theory the measure uses the inputlength<br />

as a parameter. In SAT the number of variables of the formula plays a<br />

more essential role.<br />

Let us c<strong>on</strong>sider those problems of discrete mathematics (including some NPcomplete<br />

problems) in which there are <strong>on</strong>ly finitely many possible answers. They<br />

define regular languages over a finite alphabet (with a finite number of variables,<br />

a finite number of nodes in the graph, etc.) if the syntactically correct descripti<strong>on</strong><br />

of them is regular. The problem has <strong>on</strong>ly finitely many possible states, and we<br />

can use the same method as we presented here to make a finite automat<strong>on</strong>, which<br />

accepts the soluti<strong>on</strong>s, even if the language (set) of the soluti<strong>on</strong>s of the problem is<br />

infinite. It can be a useful tool when an upper bound of the used variables/nodes<br />

etc. is given. We may have a fast algorithm (real time) to solve these problems,<br />

even if the arbitrary case is more difficult.<br />

There are several algorithms to solve SAT by various P-systems. With this<br />

paper we wanted to reopen this particular field. We are looking for new ideas,<br />

collaborati<strong>on</strong>s to solve SAT by a method with fixed alphabet. Note here that<br />

there is another new computati<strong>on</strong>al paradigm, the so-called interval-valued computati<strong>on</strong><br />

(introduced in [24, 25] and further developed in [26, 27]). It offers also<br />

a linear soluti<strong>on</strong> to SAT (moreover to q-SAT, also). This ‘uniform’ algorithm<br />

gives the answer for every Boolean formula, independently of its length and of<br />

336

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