A Reversibility Characterization of Locally Finite Groups by Cellular Automata

Abstract: For cellular automata over finite alphabets, bijectivity already implies reversibility. Over infinite alphabets this implication may fail, and the remaining obstruction in the periodic case was recorded by Ceccherini-Silberstein and Coornaert as Open Problem 2 in \emph{Cellular Automata and Groups}. We prove an exact group-theoretic characterization. A group $G$ is locally finite if and only if, over every alphabet, every bijective cellular automaton $A^G\to A^G$ is reversible. Equivalently, if $G$ is not locally finite, then for every infinite alphabet $A$ there exists a bijective cellular automaton $A^G\to A^G$ whose inverse is not a cellular automaton. The counterexample is already obtained on a countable alphabet. Its local rule has a rank track, a direction track and a binary data track; the forward map is triangular along finite directed chains of arbitrary length, so its inverse is defined pointwise but has no uniform finite memory. As a consequence, Open Problem 2 has an affirmative answer, and the periodicity hypothesis is unnecessary for the negative direction.