690 research outputs found
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
A New Technique for Reachability of States in Concatenation Automata
We present a new technique for demonstrating the reachability of states in
deterministic finite automata representing the concatenation of two languages.
Such demonstrations are a necessary step in establishing the state complexity
of the concatenation of two languages, and thus in establishing the state
complexity of concatenation as an operation. Typically, ad-hoc induction
arguments are used to show particular states are reachable in concatenation
automata. We prove some results that seem to capture the essence of many of
these induction arguments. Using these results, reachability proofs in
concatenation automata can often be done more simply and without using
induction directly.Comment: 23 pages, 1 table. Added missing affiliation/funding informatio
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
Operations on Automata with All States Final
We study the complexity of basic regular operations on languages represented
by incomplete deterministic or nondeterministic automata, in which all states
are final. Such languages are known to be prefix-closed. We get tight bounds on
both incomplete and nondeterministic state complexity of complement,
intersection, union, concatenation, star, and reversal on prefix-closed
languages.Comment: In Proceedings AFL 2014, arXiv:1405.527
Operational State Complexity of Deterministic Unranked Tree Automata
We consider the state complexity of basic operations on tree languages
recognized by deterministic unranked tree automata. For the operations of union
and intersection the upper and lower bounds of both weakly and strongly
deterministic tree automata are obtained. For tree concatenation we establish a
tight upper bound that is of a different order than the known state complexity
of concatenation of regular string languages. We show that (n+1) (
(m+1)2^n-2^(n-1) )-1 vertical states are sufficient, and necessary in the worst
case, to recognize the concatenation of tree languages recognized by (strongly
or weakly) deterministic automata with, respectively, m and n vertical states.Comment: In Proceedings DCFS 2010, arXiv:1008.127
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