13th International Conference on Membrane Computing - MTA Sztaki

On efficient algorithms for SAT
Note that in this paper we do not try to make minimal DFAs to accept the
examined languages. Our result is more theoretical than practical. However we
note here that in practice automata with high number (over some millions) of
states are used. Sometimes the so-called lazy evaluation mode is used [10], especially,
in natural language processing. From our point of view it means the
following. For a particular problem one do not need all the states of the automaton,
only the reached states are used. Therefore, in some cases only those
states are stored which are already reached from the initial state. Using this
method large memory can be saved. Moreover processing a word at most as
many states are needed as the length of the word (plus 1). The disadvantage of
this method that the computation must compute the automaton as well, establishing
the possible new states which can take much time. This hard computation
was used as a pre-computation at the construction of our DFAs (similarly to the
pre-computation is used in [31]).
4 The SAT over Unbounded Set of Variables
In [8] seven circumstances are given when the power of context-free languages is
not enough to describe some phenomena of the world. One of them is a logical
example: the language of tautologies is not context free, as it is shown in [30].
The complexity of the decision whether a Boolean formula is tautology is
closely connected to the complexity of SAT as we already described.
Let a Boolean formula be given. It is a tautology if and only if its negation
is unsatisfiable. The formula is satisfiable if and only if its negation is not a
tautology.
In [8, 30] the authors use the tautologies over arbitrarily many variables (coding
their names by finite letters) and using the connectives negation, conjunction,
disjunction and implication. It is easy to show that the languages of tautologies
in arbitrary form cannot be regular, because we use brackets. (Using a homomorphism
to substitute all symbols except the brackets by the empty word, we
get the DYCK language, which is not regular, but it is context-free. Since the
regular languages are closed under homomorphism, the original language cannot
be regular.) Note that we can avoid using the brackets by prefix notation of
formulae, but the language cannot be regular, because a counter is needed to
know how many operands have to follow the operators.
In [23] it is proved that the language of Boolean tautologies over an infinite
alphabet (using coding to a finite alphabet) is not regular and not context-free,
but it is a context-sensitive language, even if only formulae in DNF is used.
It is not a surprising fact, since the membership problem of context-sensitive
languages is a P-SPACE complete problem ([12, 15]), while the word problem
for context-free languages is in P. Therefore, this language can be accepted by
a linear bounded Turing-machine. The dual problems of the SAT and n-SAT
are hard with unbounded number of variables. Knowing that the dual problems
have similarly large complexities, we can say that over arbitrary many variables
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