Decision algorithms for reversibility of one-dimensional non-linear cellular automata under null boundary conditions

TL;DR

This paper develops algorithms to determine the reversibility of one-dimensional cellular automata under null boundary conditions, extending analysis from linear to non-linear rules and confirming the periodic nature of reversibility functions.

Contribution

It introduces new decision algorithms for reversibility of 1D CA under null boundary conditions, applicable to both linear and non-linear rules, and explores the periodicity of reversibility functions.

Findings

Algorithms successfully decide reversibility for linear and non-linear CA.

Reversibility functions exhibit periodicity related to bucket chain periodicity.

Experimental results validate theoretical predictions and algorithm effectiveness.

Abstract

The property of reversibility is quite meaningful for the classic theoretical computer science model, cellular automata. For the reversibility problem for a CA under null boundary conditions, while linear rules have been studied a lot, the non-linear rules remain unexplored at present. The paper investigates the reversibility problem of general one-dimensional CA on a finite field $\mathbb{Z}_p$, and proposes an approach to optimize the Amoroso's infinite CA surjectivity detection algorithm. This paper proposes algorithms for deciding the reversibility of one-dimensional CA under null boundary conditions. We propose a method to decide the strict reversibility of one-dimensional CA under null boundary conditions. We also provide a bucket chain based algorithm for calculating the reversibility function of one-dimensional CA under null boundary conditions. These decision algorithms work for not only linear rules but also non-linear rules. In addition, it has been confirmed that the reversibility function always has a period, and its periodicity is related to the periodicity of the corresponding bucket chain. Some of our experiment results of reversible CA are presented in the paper, complementing and validating the theoretical aspects, and thereby further supporting the research conclusions of this paper.