On von Neumann regularity of cellular automata

TL;DR

This paper characterizes when one-dimensional cellular automata are von Neumann regular, providing decidability results, conditions for non-regularity, and probabilistic insights into typical automata.

Contribution

It establishes a precise algebraic criterion for von Neumann regularity in cellular automata and applies it to elementary CA, including optimal inverse radii and probabilistic non-regularity results.

Findings

Von Neumann regularity characterized by split epicness onto images.

Decidability of regularity for all elementary CA.

High probability that a random cellular automaton is non-regular.

Abstract

We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and T\"orm\"a that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.