Abelian networks II. Halting on all inputs

TL;DR

This paper characterizes when finite irreducible abelian networks halt on all inputs by analyzing the eigenvalues of their production matrix, providing a clear mathematical criterion for halting behavior.

Contribution

It establishes a necessary and sufficient condition for halting of finite irreducible abelian networks based on eigenvalues of the production matrix.

Findings

Halting occurs iff all eigenvalues of the production matrix are within the open unit disk.

Provides a spectral criterion for analyzing abelian networks.

Connects network dynamics to linear algebra properties.

Abstract

Abelian networks are systems of communicating automata satisfying a local commutativity condition. We show that a finite irreducible abelian network halts on all inputs if and only if all eigenvalues of its production matrix lie in the open unit disk.