The reversibility of cellular automata on trees with loops

[EN] In this work the notion of linear cellular automata on trees with loops is introduced and the reversibility problem in some particular cases is tackled. The explicit expressions of the inverse cellular automata are computed

On the reversibility and the closed image property of linear cellular automata

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $\tau \colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $\tau \colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $\tau_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $\tau_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology

Reversibility of Symmetric Linear Cellular Automata with Radius r = 3

The aim of this work is to completely solve the reversibility problem for symmetric linear cellular automata with radius r = 3 and null boundary conditions. The main result obtained is the explicit computation of the local transition functions of the inverse cellular automata. This allows introduction of possible and interesting applications in digital image encryption.This research was funded by Ministerio de Ciencia, Innovación y Universidades (MCIU, Spain), Agencia Estatal de Investigación (AEI, Spain), and Fondo Europeo de Desarrollo Regional (FEDER, UE) under project TIN2017-84844-C2-2-R (MAGERAN) and project SA054G18 supported by Consejería de Educación (Junta de Castilla y León, Spain)