894 research outputs found
Large Aperiodic Semigroups
The syntactic complexity of a regular language is the size of its syntactic
semigroup. This semigroup is isomorphic to the transition semigroup of the
minimal deterministic finite automaton accepting the language, that is, to the
semigroup generated by transformations induced by non-empty words on the set of
states of the automaton. In this paper we search for the largest syntactic
semigroup of a star-free language having left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
states.
We introduce two new aperiodic transition semigroups. The first is generated
by transformations that change only one state; we call such transformations and
resulting semigroups unitary. In particular, we study complete unitary
semigroups which have a special structure, and we show that each maximal
unitary semigroup is complete. For there exists a complete unitary
semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by
transformations that map a non-empty subset of states to a single state; we
call such transformations and semigroups semiconstant. In particular, we
examine semiconstant tree semigroups which have a structure based on full
binary trees. The semiconstant tree semigroups are at present the best
candidates for largest aperiodic semigroups.
We also prove that is an upper bound on the state complexity of
reversal of star-free languages, and resolve an open problem about a special
case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table
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
It Is NL-complete to Decide Whether a Hairpin Completion of Regular Languages Is Regular
The hairpin completion is an operation on formal languages which is inspired
by the hairpin formation in biochemistry. Hairpin formations occur naturally
within DNA-computing. It has been known that the hairpin completion of a
regular language is linear context-free, but not regular, in general. However,
for some time it is was open whether the regularity of the hairpin completion
of a regular language is is decidable. In 2009 this decidability problem has
been solved positively by providing a polynomial time algorithm. In this paper
we improve the complexity bound by showing that the decision problem is
actually NL-complete. This complexity bound holds for both, the one-sided and
the two-sided hairpin completions
Learning Moore Machines from Input-Output Traces
The problem of learning automata from example traces (but no equivalence or
membership queries) is fundamental in automata learning theory and practice. In
this paper we study this problem for finite state machines with inputs and
outputs, and in particular for Moore machines. We develop three algorithms for
solving this problem: (1) the PTAP algorithm, which transforms a set of
input-output traces into an incomplete Moore machine and then completes the
machine with self-loops; (2) the PRPNI algorithm, which uses the well-known
RPNI algorithm for automata learning to learn a product of automata encoding a
Moore machine; and (3) the MooreMI algorithm, which directly learns a Moore
machine using PTAP extended with state merging. We prove that MooreMI has the
fundamental identification in the limit property. We also compare the
algorithms experimentally in terms of the size of the learned machine and
several notions of accuracy, introduced in this paper. Finally, we compare with
OSTIA, an algorithm that learns a more general class of transducers, and find
that OSTIA generally does not learn a Moore machine, even when fed with a
characteristic sample
A Characterization of Completely Reachable Automata
A complete deterministic finite automaton in which every non-empty subset of
the state set occurs as the image of the whole state set under the action of a
suitable input word is called completely reachable. We characterize completely
reachable automata in terms of certain directed graphs.Comment: 12 pages, 3 figures, submitted to DLT 201
More Structural Characterizations of Some Subregular Language Families by Biautomata
We study structural restrictions on biautomata such as, e.g., acyclicity,
permutation-freeness, strongly permutation-freeness, and orderability, to
mention a few. We compare the obtained language families with those induced by
deterministic finite automata with the same property. In some cases, it is
shown that there is no difference in characterization between deterministic
finite automata and biautomata as for the permutation-freeness, but there are
also other cases, where it makes a big difference whether one considers
deterministic finite automata or biautomata. This is, for instance, the case
when comparing strongly permutation-freeness, which results in the family of
definite language for deterministic finite automata, while biautomata induce
the family of finite and co-finite languages. The obtained results nicely fall
into the known landscape on classical language families.Comment: In Proceedings AFL 2014, arXiv:1405.527
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