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，矩阵可消亡性问题询问是否存在这些矩阵的一个乘积等于零矩阵。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）的可消亡性问题，该问题在给定一个NFA时，要求找到一个词$w$，使得在$w$作用下每个状态的像均为空集。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，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与NFA及确定性有限自动机（DFA）可消亡性相关的其他开放问题。