13th International Conference on Membrane Computing - MTA Sztaki

B. Nagy
3. If this clause contain a contradictory literal, then increase the value of the
variable i. If there is an i-th clause, then go back to the previous step, else the
formula is unsatisfiable.
A version of the above algorithm solves the problem to decide if a Boolean
formula in CNF is tautology or not. Therefore one may ask, what is the “problem”
when a formula is given in other form. There is a straightforward way to
translate a formula from DNF to CNF and vice-versa based on the logical equivalences
called distributive laws. To dissolve the problem, i.e., this contradiction,
it is enough to know that the size of the formula grows exponentially in this
translation...
After this short logical bypass let us get back to computing, especially to
formal languages.
In the next definition we use the well-known regular operators, such as union,
concatenation and Kleene-star (iteration); we use the usual notation +, ·, ∗ for
these operators respectively.
Definition 2. The finite expressions are regular expressions using the letters of
the alphabet and symbols +, ·, ∗ in the following way.
The letters of the alphabet with the empty word (λ) and the empty set (∅) are
regular expressions. They refer for the singleton languages containing only a
1-letter long word, and for the languages {λ}, {}, respectively.
If r, q are regular expressions, then (r + q), (r · q) and (r ∗ ) are regular expressions,
as well. These complex expressions refer for the languages obtained by
the union, concatenation and Kleene-iteration of the languages referred by the
subexpressions r and q, respectively.
Note that some of the brackets can be eliminated by the usual precedence
relation among the operations and by associativity of union and concatenation.
Usually the sign of the concatenation (·) is also omitted. We will use the abbreviation
r n , denoting the regular expression in which the regular expression r is
concatenated by itself with (a fixed) n (non-overlapping) occurrences.
A language is regular if there is a regular expression which describes it.
Now we recall the definition of finite automaton.
Definition 3. The ordered quintuple A = (K, T, M, σ 0 , H) is called a deterministic
finite automaton (DFA), where K is the finite, non-empty set of states, T
is the finite alphabet of input symbols, M is the transition function, mapping
from K × T to K, σ 0 ∈ K is the initial state, and H ⊆ K is the set of accepting
states.
The well-known Kleene’s theorem states that each regular language can be
accepted by a DFA and each language accepted by a DFA is regular.
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