Mikhail Berlinkov

Mikhail Berlinkov
Ural Federal University | UrFU · Institute of Mathematics and Computer Science

PhD

About

32
Publications
1,714
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281
Citations

Publications

Publications (32)
Article
Full-text available
We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability. Moreover, we also prove that the convergence rate is exactly $1-\Theta(\frac{1}{n})$ as conjectured by Cameron~\cite{CamConj} for the most interesting binary alphabet case.
Preprint
Full-text available
We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a reset word) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. While...
Article
In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on [Formula:...
Article
Full-text available
Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states mapped to a state in S by the action of w. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preima...
Chapter
In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on \(n-1\) s...
Preprint
In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on $n-1$ sta...
Article
Full-text available
A word $w$ is \emph{extending} a subset of states $S$ of a deterministic finite automaton, if the set of states mapped to $S$ by $w$ (the preimage of $S$ under the action of $w$) is larger than $S$. This notion together with its variations has particular importance in the field of synchronizing automata, where a number of methods and algorithms rel...
Article
Full-text available
Human influenza A viruses (IAVs) cause global pandemics and epidemics. These viruses evolve rapidly, making current treatment options ineffective. To identify novel modulators of IAV–host interactions, we re-analyzed our recent transcriptomics, metabolomics, proteomics, phosphoproteomics, and genomics/virtual ligand screening data. We identified 71...
Article
We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain new upper bounds for automata with a short word of small rank. The results are applied to make several improvements in the area. In particu...
Conference Paper
We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability. Moreover, we also prove that the convergence rate is exactly $1-\Theta(\frac{1}{n})$ as conjectured by [Cameron, 2011] for the most interesting binary alphabet case. Finally, we describe a deterministic algorithm which decid...
Data
Full-text available
We prove the exact asymptotic $1-\left({\frac{2\pi}{3}-\frac{827}{288\pi}}+o(1)\right)/{\sqrt{n}}$ for the probability that the underlying graph of a random mapping of $n$ elements possesses a unique highest tree. The property of having a unique highest tree turned out to be crucial in the solution of the famous Road Coloring Problem as well as the...
Conference Paper
We refine results about relations between Markov chains and synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset thresholds of automata with a short word of a small rank. The results are applied to make several improvements in the area. We improve the b...
Article
Full-text available
We prove the exact asymptotic $1-\left({\frac{2\pi}{3}-\frac{827}{288\pi}}+o(1)\right)/{\sqrt{n}}$ for the probability that the underlying graph of a random mapping of $n$ elements possesses a unique highest tree. The property of having a unique highest tree turned out to be crucial in the solution of the famous Road Coloring Problem as well as the...
Article
Full-text available
We refine results about relations between Markov chains and synchronizing automata. Then we use it to improve the best known upper bound on the reset threshold for finite prefix codes. Next we present polynomial time algorithms to find reset words with lengths of the proven bounds for finite prefix codes and other classes of automata. Another conse...
Article
Full-text available
It is known, that if an $\epsilon$-machine is exactly synchronizable then the probability of generating non-reset words of length $L$ vanishes exponentially fast as $a^L$ where $a$ is the synchronization rate constant. Hence the synchronization rate constant serves as a natural measure of synchronization for $\epsilon$-machines. In the present work...
Technical Report
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The aim of the presentation is to give an overview of the synchronization theory with emphasis on the open questions. The main questions about synchronization properties are discussed in general, random, modifiable and stochastic settings.
Article
Full-text available
We consider the first problem that appears in any application of synchronizing automata, namely, the problem of deciding whether or not a given $n$-state $k$-letter automaton is synchronizing. First we generalize results from \cite{RandSynch},\cite{On2Problems} for the case of strongly connected partial automata. Specifically we show that an automa...
Article
We describe a new version of the so-called extension method that was used to prove quadratic upper bounds on the minimum length of reset words for various important classes of synchronizing automata. Our approach is formulated in terms of Markov chains; it is in a sense dual to the usual extension method and improves on a recent result by Jungers....
Conference Paper
Full-text available
We consider two basic problems arising in the theory of synchronizing automata: deciding, whether or not a given $n$-state automaton is synchronizing and the problem of approximating the reset threshold for a given synchronizing automata. For the first problem of deciding whether or not a given $n$-state automaton is synchronizing we present an alg...
Data
Full-text available
We consider two basic problems arising in the theory of synchronizing automata: deciding, whether or not a given $n$-state automaton is synchronizing and the problem of approximating the reset threshold for a given synchronizing automata. For the first problem of deciding whether or not a given $n$-state automaton is synchronizing we present an al...
Article
The \v{C}ern\'y conjecture (\v{C}ern\'y, 1964) states that each n-state \san\ possess a \sw\ of length $(n-1)^2$. From the other side the best upper bound for the \rl\ of n-state \sa\ known so far is equal to $\frac{n^3-n}6$ (Pin, 1983) and so is cubic (a slightly better though still cubic upper bound $\frac{n(7n^2+6n-16)}{48}$ has been claimed in...
Conference Paper
Full-text available
In 1964 \v{C}ern\'{y} conjectured that each $n$-state synchronizing automaton posesses a reset word of length at most $(n-1)^2$. From the other side the best known upper bound on the reset length (minimum length of reset words) is cubic in $n$. Thus the main problem here is to prove quadratic (in $n$) upper bounds. Since 1964, this problem has been...
Article
Recently, Carpi and D'Alessandro have formulated a conjecture whose validity would imply an O(n2) upper bound for the minimum length of reset for synchronizing automata with n states. We refute this conjecture as well as a related conjecture by Rystsov and suggest a weaker version that still suffices to achieve a quadratic upper bound.
Article
Full-text available
Černý's conjecture asserts the existence of a synchronizing word of length at most (n - 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for...
Conference Paper
Recently, Carpi and D’Alessandro [3] have formulated a conjecture whose validity would imply an O(n 2) upper bound for the minimum length of reset words for synchronizing automata with n states. We refute this conjecture as well as a related conjecture by Rystsov [13] and suggest a weaker version that still suffices to achieve a quadratic upper bou...
Conference Paper
Full-text available
We prove that, unless P = NP, no polynomial-time algorithm can approximate the minimum length of synchronizing words for a given synchronizing automaton within a constant factor.
Article
Full-text available
We prove that, unless $\mathrm{P}=\mathrm{NP}$, no polynomial algorithm can approximate the minimum length of \sws for a given \san within a constant factor. Comment: 12 pages, 1 figure
Article
Full-text available
The well known open \v{C}ern\'y conjecture states that each \san with $n$ states has a \sw of length at most $(n-1)^2$. On the other hand, the best known upper bound is cubic of $n$. Recently, in the paper \cite{CARPI1} of Alessandro and Carpi, the authors introduced the new notion of strongly transitivity for automata and conjectured that this pro...

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Project (1)
Project
1. Prove exponentially small asymptotic bound on the probability of being non-synchronizable strongly-connected binary automaton