Mikhail V. Berlinkov

Approximating the Minimum Length of Synchronizing Words Is Hard

We prove that, unless P=NP, no polynomial-time algorithm can approximate the minimum length of reset words for a given synchronizing automaton within a constant factor.

On Two Algorithmic Problems about Synchronizing Automata

Under the assumption P≠NP, we prove that two natural problems from the theory of synchronizing automata cannot be solved in polynomial time. The first problem is to decide whether a given reachable partial automaton is synchronizing. The second one is, given an n-state binary complete synchronizing automaton, to compute its reset threshold within performance ratio less than d ln (n) for a specific constant d > 0.

On the Probability of Being Synchronizable

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

Algebraic Synchronization Criterion and Computing Reset Words

1-\varTheta \frac{1}{n}

Synchronizing Random Almost-Group Automata.

as conjectured by Cameroni¾ź[4] for the most interesting binary alphabet case.

Synchronizing automata on quasi-eulerian digraph

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 713 potential modulators targeting 199 cellular and two viral proteins. Anti-influenza activity for 48 of them has been reported previously, whereas the antiviral efficacy of the 665 remains unknown. Studying anti-influenza efficacy and immuno/neuro-modulating properties of these compounds and their combinations as well as potential viral and host resistance to them may lead to the discovery of novel modulators of IAV–host interactions, which might be more effective than the currently available anti-influenza therapeutics.

On the probability to be synchronizable

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.

Extending word problems in deterministic finite automata.

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 particular, we improve the upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(nlog3n). In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary n-state decoder is at most O(nlogźn). We prove the źerný conjecture for n-state automata with a letter of rank ź 6 n - 6 3 . In another corollary, we show that the probability that the źerný conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and that the expected value of the reset threshold is at most n 3 / 2 + o ( 1 ) .Moreover, all of our bounds are constructible. We present suitable polynomial algorithms for the task of finding a reset word of length within our bounds.