# Mikhail BerlinkovUral Federal University | UrFU · Institute of Mathematics and Computer Science

Mikhail Berlinkov

PhD

## About

32

Publications

1,714

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281

Citations

## Publications

Publications (32)

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.

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...

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:...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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....

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...

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...

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...

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...

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.

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...

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...

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.

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

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...

## Projects

Project (1)

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