On shortest products for nonnegative matrix mortality

2024-05-30

Reachability Problems

AI Summary

This paper investigates the “matrix mortality” problem for sets of nonnegative integer matrices—i.e., deciding whether some finite product equals the zero matrix—and establishes its equivalence to the “dead-word” problem for nondeterministic finite automata (NFAs): finding the shortest input word that maps all states to the empty set. For NFAs with $n$ states and alphabet sizes $|Sigma| = 2, 3, n$, we derive the first tight exponential lower bounds on the length of the shortest dead word: $2^{(n-2)/3}$, $2^{(n-4)/2}$, and $2^{n-1}$, respectively. Our approach integrates combinatorial automata theory, structural analysis of nonnegative matrix products, modeling of state reachability, and inductive construction techniques. These results precisely characterize the exponential computational complexity of the problem in terms of both state count and alphabet size, substantially advancing the understanding of the intrinsic computational hardness of zero-product problems for nonnegative matrices.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Studies shortest product of nonnegative matrices yielding zero matrix
• Examines NFA mortality problem equivalence with matrix mortality
• Determines bounds on shortest word length for NFA mortality

Innovation

• Nonnegative matrix mortality problem solution
• Equivalence to NFA mortality problem
• Bounds on shortest word lengths