Nilpotency and periodic points in non-uniform cellular automata

Supreeti Kamilya,Jarkko Kari

doi:10.1007/s00236-020-00390-7

Abstract

Nilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

Highlights

A one-dimensional cellular automaton (CA) consists of an infinite line of cells that evolve in discrete time following some local update rule
We prove that it is undecidable to establish whether a given nonuniform CA (NUCA) has any temporally periodic configurations
The undecidability holds for NUCAs with very simple rule distributions so it is not due to non-recursiveness hidden in the distribution

Summary

Introduction

A one-dimensional cellular automaton (CA) consists of an infinite line of cells that evolve in discrete time following some local update rule. It is known that for any finite set of local rules the distributions that yield a surjective For reversible uniform cellular automata the property is known to hold since spatially periodic configurations are dense in the configuration space and they are automatically temporally periodic. In the non-uniform setting the situation is very different: we show that there are reversible equicontinuous NUCAs without any temporally periodic configurations. A non-uniform cellular automaton (NUCA) is specified by a distribution b ∈ RZ where bi identifies the local rule used at cell i. Note that AZ is a compact metric space so the notions of uniform equicontinuity and pointwise equicontinuity coincide.) Clearly every nilpotent NUCA is equicontinuous since traces may depend on the initial configuration only until the NUCA reaches its fixed point.

Wang tiles and cellular automata

Temporal periodicity in non-uniform CAs

Nilpotency in non-uniform CAs

Conclusions