96 research outputs found
Universal Witnesses for State Complexity of Basic Operations Combined with Reversal
We study the state complexity of boolean operations, concatenation and star
with one or two of the argument languages reversed. We derive tight upper
bounds for the symmetric differences and differences of such languages. We
prove that the previously discovered bounds for union, intersection,
concatenation and star of such languages can all be met by the recently
introduced universal witnesses and their variants.Comment: 18 pages, 8 figures. LNCS forma
Maximally Atomic Languages
The atoms of a regular language are non-empty intersections of complemented
and uncomplemented quotients of the language. Tight upper bounds on the number
of atoms of a language and on the quotient complexities of atoms are known. We
introduce a new class of regular languages, called the maximally atomic
languages, consisting of all languages meeting these bounds. We prove the
following result: If L is a regular language of quotient complexity n and G is
the subgroup of permutations in the transition semigroup T of the minimal DFA
of L, then L is maximally atomic if and only if G is transitive on k-subsets of
1,...,n for 0 <= k <= n and T contains a transformation of rank n-1.Comment: In Proceedings AFL 2014, arXiv:1405.527
Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages
We study the state complexity of binary operations on regular languages over
different alphabets. It is known that if and are languages of
state complexities and , respectively, and restricted to the same
alphabet, the state complexity of any binary boolean operation on and
is , and that of product (concatenation) is . In
contrast to this, we show that if and are over different
alphabets, the state complexity of union and symmetric difference is
, that of difference is , that of intersection is , and
that of product is . We also study unrestricted complexity of
binary operations in the classes of regular right, left, and two-sided ideals,
and derive tight upper bounds. The bounds for product of the unrestricted cases
(with the bounds for the restricted cases in parentheses) are as follows: right
ideals (); left ideals ();
two-sided ideals (). The state complexities of boolean operations
on all three types of ideals are the same as those of arbitrary regular
languages, whereas that is not the case if the alphabets of the arguments are
the same. Finally, we update the known results about most complex regular,
right-ideal, left-ideal, and two-sided-ideal languages to include the
unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of
the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3.
The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59,
2017, the issue of selected papers from DCFS 2016. This version corrects the
proof of distinguishability of states in the difference operation on p. 12 in
arXiv:1609.04439v
Theory of Atomata
We show that every regular language defines a unique nondeterministic finite
automaton (NFA), which we call "\'atomaton", whose states are the "atoms" of
the language, that is, non-empty intersections of complemented or
uncomplemented left quotients of the language. We describe methods of
constructing the \'atomaton, and prove that it is isomorphic to the reverse
automaton of the minimal deterministic finite automaton (DFA) of the reverse
language. We study "atomic" NFAs in which the right language of every state is
a union of atoms. We generalize Brzozowski's double-reversal method for
minimizing a deterministic finite automaton (DFA), showing that the result of
applying the subset construction to an NFA is a minimal DFA if and only if the
reverse of the NFA is atomic. We prove that Sengoku's claim that his method
always finds a minimal NFA is false.Comment: 29 pages, 2 figures, 28 table
Quotient Complexities of Atoms in Regular Ideal Languages
A (left) quotient of a language by a word is the language
. The quotient complexity of a regular language
is the number of quotients of ; it is equal to the state complexity of ,
which is the number of states in a minimal deterministic finite automaton
accepting . An atom of is an equivalence class of the relation in which
two words are equivalent if for each quotient, they either are both in the
quotient or both not in it; hence it is a non-empty intersection of
complemented and uncomplemented quotients of . A right (respectively, left
and two-sided) ideal is a language over an alphabet that satisfies
(respectively, and ). We
compute the maximal number of atoms and the maximal quotient complexities of
atoms of right, left and two-sided regular ideals.Comment: 17 pages, 4 figures, two table
Syntactic Complexity of R- and J-Trivial Regular Languages
The syntactic complexity of a regular language is the cardinality of its
syntactic semigroup. The syntactic complexity of a subclass of the class of
regular languages is the maximal syntactic complexity of languages in that
class, taken as a function of the state complexity n of these languages. We
study the syntactic complexity of R- and J-trivial regular languages, and prove
that n! and floor of [e(n-1)!] are tight upper bounds for these languages,
respectively. We also prove that 2^{n-1} is the tight upper bound on the state
complexity of reversal of J-trivial regular languages.Comment: 17 pages, 5 figures, 1 tabl
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