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 for an NFA with $n$ states this length can be at least $2^n - 1$ for an alphabet of size $n$, $2^{(n - 4)/2}$ for an alphabet of size $3$ and $2^{(n - 2)/3}$ for an alphabet of size $2$. We also discuss further open problems related to mortality of NFAs and DFAs.

翻译：给定一个具有整数元素的有限矩阵集合，矩阵可灭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NFA可灭性问题，该问题给定一个NFA，要求寻找一个词$w$，使得每个状态在$w$作用下的像为空集。此时NFA的字母表大小等于集合中矩阵的数量。我们研究此类最短词的长度与字母表大小的关系。我们证明：对于具有$n$个状态的NFA，当字母表大小为$n$时，该长度至少可达$2^n - 1$；当字母表大小为$3$时，至少可达$2^{(n - 4)/2}$；当字母表大小为$2$时，至少可达$2^{(n - 2)/3}$。我们还进一步讨论了有关NFA和DFA可灭性的其他开放问题。