36 research outputs found
A Fast Algorithm Finding the Shortest Reset Words
In this paper we present a new fast algorithm finding minimal reset words for
finite synchronizing automata. The problem is know to be computationally hard,
and our algorithm is exponential. Yet, it is faster than the algorithms used so
far and it works well in practice. The main idea is to use a bidirectional BFS
and radix (Patricia) tries to store and compare resulted subsets. We give both
theoretical and practical arguments showing that the branching factor is
reduced efficiently. As a practical test we perform an experimental study of
the length of the shortest reset word for random automata with states and 2
input letters. We follow Skvorsov and Tipikin, who have performed such a study
using a SAT solver and considering automata up to states. With our
algorithm we are able to consider much larger sample of automata with up to
states. In particular, we obtain a new more precise estimation of the
expected length of the shortest reset word .Comment: COCOON 2013. The final publication is available at
http://link.springer.com/chapter/10.1007%2F978-3-642-38768-5_1
Complexity of checking whether two automata are synchronized by the same language
A deterministic finite automaton is said to be synchronizing if it has a
reset word, i.e. a word that brings all states of the automaton to a particular
one. We prove that it is a PSPACE-complete problem to check whether the
language of reset words for a given automaton coincides with the language of
reset words for some particular automaton.Comment: 12 pages, 4 figure
Checking Whether an Automaton Is Monotonic Is NP-complete
An automaton is monotonic if its states can be arranged in a linear order
that is preserved by the action of every letter. We prove that the problem of
deciding whether a given automaton is monotonic is NP-complete. The same result
is obtained for oriented automata, whose states can be arranged in a cyclic
order. Moreover, both problems remain hard under the restriction to binary
input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at
http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2
Synchronization Problems in Automata without Non-trivial Cycles
We study the computational complexity of various problems related to
synchronization of weakly acyclic automata, a subclass of widely studied
aperiodic automata. We provide upper and lower bounds on the length of a
shortest word synchronizing a weakly acyclic automaton or, more generally, a
subset of its states, and show that the problem of approximating this length is
hard. We investigate the complexity of finding a synchronizing set of states of
maximum size. We also show inapproximability of the problem of computing the
rank of a subset of states in a binary weakly acyclic automaton and prove that
several problems related to recognizing a synchronizing subset of states in
such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889.
Conference version was published at CIAA 2017, LNCS vol. 10329, pages
188-200, 201
On the Number of Synchronizing Colorings of Digraphs
We deal with -out-regular directed multigraphs with loops (called simply
\emph{digraphs}). The edges of such a digraph can be colored by elements of
some fixed -element set in such a way that outgoing edges of every vertex
have different colors. Such a coloring corresponds naturally to an automaton.
The road coloring theorem states that every primitive digraph has a
synchronizing coloring.
In the present paper we study how many synchronizing colorings can exist for
a digraph with vertices. We performed an extensive experimental
investigation of digraphs with small number of vertices. This was done by using
our dedicated algorithm exhaustively enumerating all small digraphs. We also
present a series of digraphs whose fraction of synchronizing colorings is equal
to , for every and the number of vertices large enough.
On the basis of our results we state several conjectures and open problems.
In particular, we conjecture that is the smallest possible fraction of
synchronizing colorings, except for a single exceptional example on 6 vertices
for .Comment: CIAA 2015. The final publication is available at
http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1
Image reducing words and subgroups of free groups
AbstractA word w over a finite alphabet Σ is said to be n-collapsing if for an arbitrary finite automaton A=〈Q,Σ−·−〉, the inequality |Q·w|⩽|Q|−n holds provided that |Q·u|⩽|Q|−n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a finite family of finitely generated subgroups in finitely generated free groups and prove that the property of being 2-collapsing reflects in the property that each of these subgroups has index at most 2 in the corresponding free group. We also find a similar characterization for the closely related class of so-called 2-synchronizing words
Strong inapproximability of the shortest reset word
The \v{C}ern\'y conjecture states that every -state synchronizing
automaton has a reset word of length at most . We study the hardness
of finding short reset words. It is known that the exact version of the
problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and
complete for the DP class, and that approximating the length of the shortest
reset word within a factor of is NP-hard [Gerbush and Heeringa,
CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly
improve on these results by showing that, for every , it is NP-hard
to approximate the length of the shortest reset word within a factor of
. This is essentially tight since a simple -approximation
algorithm exists.Comment: extended abstract to appear in MFCS 201
Synchronizing Automata on Quasi Eulerian Digraph
In 1964 \v{C}ern\'{y} conjectured that each -state synchronizing automaton
posesses a reset word of length at most . From the other side the best
known upper bound on the reset length (minimum length of reset words) is cubic
in . Thus the main problem here is to prove quadratic (in ) upper bounds.
Since 1964, this problem has been solved for few special classes of \sa. One of
this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In
this paper we introduce a new approach to prove quadratic upper bounds and
explain it in terms of Markov chains and Perron-Frobenius theories. Using this
approach we obtain a quadratic upper bound for a generalization of Eulerian
automata.Comment: 8 pages, 1 figur
Using Sat solvers for synchronization issues in partial deterministic automata
We approach the task of computing a carefully synchronizing word of minimum
length for a given partial deterministic automaton, encoding the problem as an
instance of SAT and invoking a SAT solver. Our experimental results demonstrate
that this approach gives satisfactory results for automata with up to 100
states even if very modest computational resources are used.Comment: 15 pages, 3 figure