314 research outputs found
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
Complexity of Road Coloring with Prescribed Reset Words
By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic
directed multigraph with a constant out-degree can be colored such that the
resulting automaton admits a reset word. There may also be a need for a
particular reset word to be admitted. For certain words it is NP-complete to
decide whether there is a suitable coloring of a given multigraph. We present a
classification of all words over the binary alphabet that separates such words
from those that make the problem solvable in polynomial time. We show that the
classification becomes different if we consider only strongly connected
multigraphs. In this restricted setting the classification remains incomplete.Comment: To be presented at LATA 201
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
A quadratic algorithm for road coloring
The Road Coloring Theorem states that every aperiodic directed graph with
constant out-degree has a synchronized coloring. This theorem had been
conjectured during many years as the Road Coloring Problem before being settled
by A. Trahtman. Trahtman's proof leads to an algorithm that finds a
synchronized labeling with a cubic worst-case time complexity. We show a
variant of his construction with a worst-case complexity which is quadratic in
time and linear in space. We also extend the Road Coloring Theorem to the
periodic case
Groups and Semigroups Defined by Colorings of Synchronizing Automata
In this paper we combine the algebraic properties of Mealy machines
generating self-similar groups and the combinatorial properties of the
corresponding deterministic finite automata (DFA). In particular, we relate
bounded automata to finitely generated synchronizing automata and characterize
finite automata groups in terms of nilpotency of the corresponding DFA.
Moreover, we present a decidable sufficient condition to have free semigroups
in an automaton group. A series of examples and applications is widely
discussed, in particular we show a way to color the De Bruijn automata into
Mealy automata whose associated semigroups are free, and we present some
structural results related to the associated groups
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