Synchronizing Automata on Quasi Eulerian Digraph

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 solved for few special classes of \sa. One of this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In this paper we introduce a new approach to prove quadratic upper bounds and explain it in terms of Markov chains and Perron-Frobenius theories. Using this approach we obtain a quadratic upper bound for a generalization of Eulerian automata.Comment: 8 pages, 1 figur

Algebraic synchronization criterion and computing reset words

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 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 best general 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 $O(n \log^3 n)$. In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary $n$-state decoder is at most $O(n \log n)$. We also show that for any non-unary alphabet there exist decoders whose reset threshold is in $\varTheta(n)$. We prove the \v{C}ern\'{y} conjecture for $n$-state automata with a letter of rank at most $\sqrt[3]{6n-6}$. In another corollary, based on the recent results of Nicaud, we show that the probability that the \v{C}ern\'y conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and also that the expected value of the reset threshold of an $n$-state random synchronizing binary automaton is at most $n^{3/2+o(1)}$. Moreover, reset words of lengths within all of our bounds are computable in polynomial time. We present suitable algorithms for this task for various classes of automata, such as (quasi-)one-cluster and (quasi-)Eulerian automata, for which our results can be applied.Comment: 18 pages, 2 figure

Multi-Omics Studies towards Novel Modulators of Influenza A Virus-Host Interaction

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

Words of Minimum Rank in Deterministic Finite Automata

International audienceThe rank of a word in a deterministic finite automaton is the size of the image of the whole state set under the mapping defined by this word. We study the length of shortest words of minimum rank in several classes of complete deterministic finite automata, namely, strongly connected and Eulerian automata. A conjecture bounding this length is known as the Rank Conjecture, a generalization of the well known Černý Conjecture. We prove upper bounds on the length of shortest words of minimum rank in automata from the mentioned classes, and provide several families of automata with long words of minimum rank. Some results in this direction are also obtained for automata with rank equal to period (the greatest common divisor of lengths of all cycles) and for circular automata