Given a finite set of matrices with integer entries, the matrix mortality problem asks if there exists a product of these matrices equal to the zero matrix. We consider a special case of this problem where all entries of the matrices are nonnegative. This case is equivalent to the NFA mortality problem, which, given an NFA, asks for a word $w$ such that the image of every state under $w$ is the empty set. The size of the alphabet of the NFA is then equal to the number of matrices in the set. We study the length of shortest such words depending on the size of the alphabet. We show that this length for an NFA with $n$ states can be at least $2^n - 1$, $2^{(n - 4)/2}$ and $2^{(n - 2)/3}$ if the size of the alphabet is, respectively, equal to $n$, three and two.

翻译：给定一个具有整数元素的有限矩阵集合，矩阵消亡性问题询问是否存在这些矩阵的乘积等于零矩阵。我们考虑该问题的一个特殊情况，即矩阵的所有元素均为非负。此情况等价于非确定性有限自动机（NFA）消亡性问题，该问题给定一个 NFA，要求找到一个词 $w$，使得每个状态在 $w$ 下的像均为空集。此时，NFA 字母表的大小等于集合中矩阵的数量。我们研究了此类最短词的长度与字母表大小的关系。我们证明，对于一个具有 $n$ 个状态的 NFA，若其字母表大小分别等于 $n$、三和二，则最短词的长度至少分别为 $2^n - 1$、$2^{(n - 4)/2}$ 和 $2^{(n - 2)/3}$。