arXiv subjects

Elena V. Pribavkina

Publications and source records attributed to Elena V. Pribavkina.

8 recordsLinked to original sources

arXiv2015

Synchronizing automata and principal eigenvectors of the underlying digraphs

A coloring of a digraph with a fixed out-degree k is a distribution of k labels over the edges resulting in a deterministic finite automaton. An automaton is called synchronizing if there exists a word which sends all states of the automaton to a single state. In the present paper we study connections between spectral and synchronizing properties of digraphs. We show that if a coloring of a digraph is not synchronizing, then the stationary distribution of an associated Markov chain has a partition of coordinates into blocks of equal sum. Moreover, if there exists such a partition, then there exists a non-synchronizing automaton with such stationary distribution. We extend these results to bound the number of non-synchronizing colorings for digraphs with particular eigenvectors. We also demonstrate that the length of the shortest synchronizing word of any coloring is at most $w^2 - 3w + 3$, where $w$ is the sum of the coordinates of the integer principal eigenvector of the digraph.

cs.FL↗

arXiv2014

A Coloring Problem for Infinite Words

In this paper we consider the following question in the spirit of Ramsey theory: Given $x\in A^ω,$ where $A$ is a finite non-empty set, does there exist a finite coloring of the non-empty factors of $x$ with the property that no factorization of $x$ is monochromatic? We prove that this question has a positive answer using two colors for almost all words relative to the standard Bernoulli measure on $A^ω.$ We also show that it has a positive answer for various classes of uniformly recurrent words, including all aperiodic balanced words, and all words $x\in A^ω$ satisfying $λ_x(n+1)-λ_x(n)=1$ for all $n$ sufficiently large, where $ λ_x(n)$ denotes the number of distinct factors of $x$ of length $n.$

math.CO↗

arXiv2014

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.

cs.FL↗

arXiv2013

Finitely generated ideal languages and synchronizing automata

We study representations of ideal languages by means of strongly connected synchronizing automata. For every finitely generated ideal language L we construct such an automaton with at most 2^n states, where n is the maximal length of words in L. Our constructions are based on the De Bruijn graph.

cs.FL↗

arXiv2013

Principal ideal languages and synchronizing automata

We study ideal languages generated by a single word. We provide an algorithm to construct a strongly connected synchronizing automaton for which such a language serves as the language of synchronizing words. Also we present a compact formula to calculate the syntactic complexity of this language.

cs.FL↗

arXiv2013

A Coloring Problem for Sturmian and Episturmian Words

We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists whenever $w$ is a Sturmian word or a standard episturmian word.

math.CO↗

arXiv2011

On Non-Complete Sets and Restivo's Conjecture

A finite set S of words over the alphabet A is called non-complete if Fact(S*) is different from A*. A word w in A* - Fact(S*) is said to be uncompletable. We present a series of non-complete sets S_k whose minimal uncompletable words have length 5k^2 - 17k + 13, where k > 3 is the maximal length of words in S_k. This is an infinite series of counterexamples to Restivo's conjecture, which states that any non-complete set possesses an uncompletable word of length at most 2k^2.

cs.FL↗

arXiv2010

On the Minimal Uncompletable Word Problem

Let S be a finite set of words over an alphabet Sigma. The set S is said to be complete if every word w over the alphabet Sigma is a factor of some element of S*, i.e. w belongs to Fact(S*). Otherwise if S is not complete, we are interested in finding bounds on the minimal length of words in Sigma* which are not elements of Fact(S*) in terms of the maximal length of words in S.

cs.FL↗