On Finite $1$-Dimensional Cellular Automata: Reversibility and Semi-reversibility

TL;DR

This paper investigates the reversibility of finite one-dimensional cellular automata, introducing semi-reversibility, and classifies CAs into reversible, semi-reversible, and irreversible types using reachability trees.

Contribution

It introduces the concept of semi-reversibility and classifies finite CAs based on reversibility, linking finite and infinite CA behaviors.

Findings

Reversibility depends on lattice size for finite CAs.

A new classification into reversible, semi-reversible, and irreversible CAs.

Reachability trees effectively determine the CA's reversibility class.

Abstract

Reversibility of a one-dimensional finite cellular automaton (CA) is dependent on lattice size. A finite CA can be reversible for a set of lattice sizes. On the other hand, reversibility of an infinite CA, which is decided by exploring the rule only, is different in its kind from that of finite CA. Can we, however, link the reversibility of finite CA to that of infinite CA? In order to address this issue, we introduce a new notion, named semi-reversibility. We classify the CAs into three types with respect to reversibility property -- reversible, semi-reversible and strictly irreversible. A tool, reachability tree, has been used to decide the reversibility class of any CA. Finally, relation among the existing cases of reversibility is established.