On shortest products for nonnegative matrix mortality

TL;DR

This paper investigates the minimal length of words that lead to the zero matrix in nonnegative matrix products, linking it to the NFA mortality problem and providing exponential lower bounds based on alphabet size.

Contribution

It establishes new exponential lower bounds on the shortest words causing mortality in NFAs with nonnegative matrices, depending on alphabet size.

Findings

Shortest such words can be exponentially long in the number of states.

Lower bounds vary with alphabet size, reaching up to 2^n - 1 for size n.

Open problems remain in NFA and DFA mortality complexities.

Abstract

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.