13th International Conference on Membrane Computing - MTA Sztaki

On efficient algorithms for SAT
• if the previous symbol was ¬ (we know it from part y of the given state),
then there are two possibilities: if the corresponding value of the given
variable in d is 1, then let the k + 1-st value of d ′ be 1 (and the other
items can be the same as they were in d); if the corresponding value is
not 1, then let it be 2 in d ′ and all other values are the same as they are
in d.
• if the previous symbol is not ¬, then: if the corresponding value of the
given state is 2 in d, then let the k + 1-st item of d ′ be 1 and all other
values of d ′ can be copied from the corresponding values of d. And if the
corresponding value of d is not 2, then let it be 1 in d ′ and each of the
other values will be the same as the corresponding value of d.
– if t =], then let c ′ = c if the k + 1-st value of d is 1. In other cases let
c ′ = c ∪ N, where N is the set containing all k-tuples in which the value of
those variables which have corresponding values of 1 in d is 0 and the value
of those variables which have corresponding values of 2 in d is 1. And let
y ′ = M A (y, ]), finally d ′ = 0.
Let W be the maximal element of C, i.e., it contains all the 2 k possibilities.
For our automaton let the set of final states be the following: all states (c, y f , 0)
for which c ≠ W , i.e., c does not contain all the possibilities and y f is the final
state of A.
Since the form of the accepted expressions are correct, and the part c does
not contain all possible evaluation of the variables in the final state, the automaton
defined above can recognize exactly the SAT languages, i.e., the satisfiable
Boolean formulae in CNF.
Using a language n-SAT instead of SAT, we modify the automaton above in
the following way. Let a DFA B = (Y B , T, M B , y 0B , {y fB }) which accepts the
syntactically correct n-ary CNF expressions. Let the states of the new automaton
be the elements of the Cartesian product of C, Y B and D. (The set Y B is
used instead of Y .) And all of the transitions are the same according to the
corresponding elements of Y B . Finally, in accepting states we use the final state
of the automaton B instead of the final state of A.
Moreover our automata with their final states (especially with part c ∈ C) tell
us for which values of the variables the formula is true. The formula evaluates to
true if the vectors are not in c. Note here that the SAT languages are infinite even
if the set of variables is finite. We have constructed finite automata accepting
the languages of SAT and n-SAT. Therefore it is proved that:
Theorem 1. The languages of satisfiable Boolean formulae in conjunctive normal
form over any (fixed) finite sets of variables are regular languages. Similarly,
the languages of n-SAT formulae (n ∈ N) over any finite sets of variables are
also regular.
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