On Symmetry versus Asynchronism: at the Edge of Universality in Automata Networks

TL;DR

This paper investigates how different update schemes in automata networks can overcome limitations of symmetric local interactions, exploring the boundary of universality through complexity analysis and a novel network glueing technique.

Contribution

It introduces a new proof technique based on network glueing and studies the interplay of symmetry, asynchronism, and universality in automata networks.

Findings

Update schemes can compensate for symmetric interaction limitations.

A new glueing operation enables complex orbit construction in large networks.

The study identifies conditions at the edge of universality in automata networks.

Abstract

An automata network (AN) is a finite graph where each node holds a state from a finite alphabet and is equipped with a local map defining the evolution of the state of the node depending on its neighbors. The global dynamics of the network is then induced by an update scheme describing which nodes are updated at each time step. We study how update schemes can compensate the limitations coming from symmetric local interactions. Our approach is based on intrinsic simulations and universality and we study both dynamical and computational complexity. By considering several families of concrete symmetric AN under several different update schemes, we explore the edge of universality in this two-dimensional landscape. On the way, we develop a proof technique based on an operation of glueing of networks, which allows to produce complex orbits in large networks from compatible pseudo-orbits in small networks.