39 research outputs found
Primitive digraphs with large exponents and slowly synchronizing automata
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.
All these automata are tightly related to primitive digraphs with large
exponent.Comment: 23 pages, 11 figures, 3 tables. This is a translation (with a
slightly updated bibliography) of the authors' paper published in Russian in:
Zapiski Nauchnyh Seminarov POMI [Kombinatorika i Teorija Grafov. IV], Vol.
402, 9-39 (2012), see ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v402/p009.pdf
Version 2: a few typos are correcte
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
Reset thresholds of automata with two cycle lengths
We present several series of synchronizing automata with multiple parameters,
generalizing previously known results. Let p and q be two arbitrary co-prime
positive integers, q > p. We describe reset thresholds of the colorings of
primitive digraphs with exactly one cycle of length p and one cycle of length
q. Also, we study reset thresholds of the colorings of primitive digraphs with
exactly one cycle of length q and two cycles of length p.Comment: 11 pages, 5 figures, submitted to CIAA 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
Attainable Values of Reset Thresholds
An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and
[(p-1)(q-1), p(q-2)+n-q+1], where 2 n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n).
This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers.
We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case.
Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n)
Lower bounds for the length of reset words in eulerian automata
For each odd n ≥ 5 we present a synchronizing Eulerian automaton with n states for which the minimum length of reset words is equal to n 2-3n+4/2. We also discuss various connections between the reset threshold of a synchronizing automaton and a sequence of reachability properties in its underlying graph. © 2013 World Scientific Publishing Company
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