Reversibility of additive CA as function of cylinder size

TL;DR

This paper investigates the reversibility of additive cellular automata on cylinders, providing criteria and solutions for specific rule classes, and exploring how cylinder size affects reversibility.

Contribution

It introduces a criterion for reversibility of additive CA rules and completely solves the problem for block collections, with reductions for exponential collections and conjectures for mixed classes.

Findings

Reversibility depends on the class of rule collections and cylinder size.

Complete solution provided for block collections of rules.

Reductions achieved for exponential rule collections.

Abstract

Additive CA on a cylinder of size $n$ can be represented by 01-string $V$ of length $n$ which is its rule. We study a problem: a class $S$ of rules given, for any $V\in S$ describe all sizes $n', n'>n,$ of cylinders such that extension of $V$ by zeros to length $n'$ represents reversible additive CA on a cylinder of size $n'$. Since all extensions of $V$ have the same collection of positions of units, it is convenient to say about classes of collections of positions instead of classes of rules. A criterion of reversibility is proven. The problem is completely solved for infinite class of "block collections", i.e. $\{(0,1,\dots,h)|h\in\mathbb Z^+\}$. Results obtained for "exponential collections" $\{(1,2,4,\dots,2^h)|h\in\mathbb Z^+\}$ essentially reduce the complexity of the problem for this class. Ways to transfer the results on other classes of rules/collections are described. A conjecture is formulated for class $\{(0,1,2^m)|m\in\mathbb Z^+\}$.