The entropy and reversibility of cellular automata on Cayley tree

TL;DR

This paper investigates the entropy and reversibility of linear cellular automata on Cayley trees, providing rule matrix construction, reversibility conditions, and showing infinite measure entropy for these automata.

Contribution

It introduces a method to analyze reversibility and entropy of cellular automata on Cayley trees, including explicit rule matrix construction and entropy computation.

Findings

Reversibility depends on specific parameters a, b, c, d in the field.

The measure-theoretical entropy is infinite for these automata.

Constructed rule matrices facilitate analysis of cellular automata on Cayley trees.

Abstract

In this paper, we study linear cellular automata (CAs) on Cayley tree of order 2 over the field $\mathbb F_p$ (the set of prime numbers modulo $p$). We construct the rule matrix corresponding to finite cellular automata on Cayley tree. Further, we analyze the reversibility problem of this cellular automaton for some given values of $a,b,c,d\in \mathbb{F}_{p}\setminus \{0\}$ and the levels $n$ of Cayley tree. We compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree the measure entropy with respect to uniform Bernoulli measure is infinity.