arXiv 1602.075562016-02-24cs.FLcs.DM

非负矩阵的本原集与同步自动机

Primitive sets of nonnegative matrices and synchronizing automata

Balázs Gerencsér, Vladimir V. Gusev, Raphaël M. Jungers

备注：: 14 pages

摘要：

一组非负矩阵$\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$被称为本原矩阵，如果存在索引$i_1, i_2, \ldots, i_m$使得$M_{i_1} M_{i_2} \ldots M_{i_m}$为正（即具有其所有条目$>0$）。最短乘积的长度称为$\mathcal{M}$的指数。原始矩阵集的概念出现在控制理论、非齐次马尔可夫链、自动机理论等领域的许多问题中。最近，同步自动机和原始矩阵集之间的联系被建立起来。在本文中，我们显着加强这些联系，提供等价的结果，无论是在组合表征，和计算方面。研究了$n \times n$矩阵的所有本原集之间的最大指数，记为$\exp(n)$ .我们证明了$\lim_{n\rightarrow\infty} \tfrac{\log \exp(n)}{n} = \tfrac{\log 3}{3}$，此外，我们还确定该界可以解决仔细同步自动机的Černý问题。我们还研究了一组矩阵，没有零行和列，表示为$\mathcal{NZ}$，由于其有趣的连接到Černandi猜想和最近推广的Perron Frobenius理论这一类。我们刻画了与$\mathcal{NZ}$矩阵集合的指数有关的不同问题的计算复杂性，并给出了属于一个特殊子类的集合的指数的二次界.也就是说，我们证明了一组矩阵的指数有界的全支持$2n^2 -5n +5$。

英文摘要：

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \ldots M_{i_m}$ is positive (i.e. has all its entries $>0$). The length of the shortest such product is called the exponent of $\mathcal{M}$. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata theory etc. Recently, connections between synchronizing automata and primitive sets of matrices were established. In the present paper, we significantly strengthen these links by providing equivalence results, both in terms of combinatorial characterization, and computational aspects. We study the maximal exponent among all primitive sets of $n \times n$ matrices, which we denote by $\exp(n)$. We prove that $\lim_{n\rightarrow\infty} \tfrac{\log \exp(n)}{n} = \tfrac{\log 3}{3}$, and moreover, we establish that this bound leads to a resolution of the Černý problem for carefully synchronizing automata. We also study the set of matrices with no zero rows and columns, denoted by $\mathcal{NZ}$, due to its intriguing connections to the Černý conjecture and the recent generalization of Perron-Frobenius theory for this class. We characterize computational complexity of different problems related to the exponent of $\mathcal{NZ}$ matrix sets, and present a quadratic bound on the exponents of sets belonging to a special subclass. Namely, we show that the exponent of a set of matrices having total support is bounded by $2n^2 -5n +5$.

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英文摘要：

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