Abelian networks I. Foundations and examples

TL;DR

This paper establishes foundational principles for abelian networks, demonstrating their independence from processing order and applying them to solve complex integer programming problems with novel examples.

Contribution

It introduces the axiomatic framework for abelian networks, proves a least action principle, and presents new non-unary examples like 'oil and water' and 'abelian mobile agents.'

Findings

Final output independence from processing order

Application to solve linear and nonlinear integer programs

Introduction of non-unary abelian network examples

Abstract

In Deepak Dhar's model of abelian distributed processors, automata occupy the vertices of a graph and communicate via the edges. We show that two simple axioms ensure that the final output does not depend on the order in which the automata process their inputs. A collection of automata obeying these axioms is called an "abelian network". We prove a least action principle for abelian networks. As an application, we show how abelian networks can solve certain linear and nonlinear integer programs asynchronously. In most previously studied abelian networks, the input alphabet of each automaton consists of a single letter; in contrast, we propose two non-unary examples of abelian networks: "oil and water" and "abelian mobile agents".