600 research outputs found
Synchronizing automata preserving a chain of partial orders
AbstractWe present a new class of automata which strictly contains the class of aperiodic automata and shares with the latter certain synchronization properties. In particular, every strongly connected automaton in this new class is synchronizing and has a synchronizing word of length ⌊n(n+1)6⌋ where n is the number of states of the automaton
Synchronizing Automata Preserving a Chain of Partial Orders
We present a new class of automata which strictly contains the class of aperiodic automata and shares with the latter certain synchronization properties. In particular, every strongly connected automaton in this new class is synchronizing and has a reset word of length [(n(n+1)/6] where n is the number of states of the automaton. © Springer-Verlag Berlin Heidelberg 2007.Supported by the Russian Foundation for Basic Research, grant 05-01-00540. The paper has been completed during the author’s stay at the University of Turku under the Finnish Mathematical Society International Visitors Program 2006–2007 ‘‘Algorithmic and Discrete Mathematics’’
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
Synchronizing automata with random inputs
We study the problem of synchronization of automata with random inputs. We
present a series of automata such that the expected number of steps until
synchronization is exponential in the number of states. At the same time, we
show that the expected number of letters to synchronize any pair of the famous
Cerny automata is at most cubic in the number of states
Synchronizing non-deterministic finite automata
In this paper, we show that every D3-directing CNFA can be mapped uniquely to
a DFA with the same synchronizing word length. This implies that \v{C}ern\'y's
conjecture generalizes to CNFAs and that the general upper bound for the length
of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. As a
second consequence, for several classes of CNFAs sharper bounds are
established. Finally, our results allow us to detect all critical CNFAs on at
most 6 states. It turns out that only very few critical CNFAs exist.Comment: 21 page
A linear bound on the k-rendezvous time for primitive sets of NZ matrices
A set of nonnegative matrices is called primitive if there exists a product
of these matrices that is entrywise positive. Motivated by recent results
relating synchronizing automata and primitive sets, we study the length of the
shortest product of a primitive set having a column or a row with k positive
entries, called its k-rendezvous time (k-RT}), in the case of sets of matrices
having no zero rows and no zero columns. We prove that the k-RT is at most
linear w.r.t. the matrix size n for small k, while the problem is still open
for synchronizing automata. We provide two upper bounds on the k-RT: the second
is an improvement of the first one, although the latter can be written in
closed form. We then report numerical results comparing our upper bounds on the
k-RT with heuristic approximation methods.Comment: 27 pages, 10 figur
Slowly synchronizing automata and digraphs
We present several infinite series of synchronizing automata for which the
minimum length of reset words is close to the square of the number of states.
These automata are closely related to primitive digraphs with large exponent.Comment: 13 pages, 5 figure
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