Conjugacy of one-dimensional one-sided cellular automata is undecidable

TL;DR

This paper proves that determining conjugacy, factor, or subsystem relationships between one-dimensional one-sided cellular automata is undecidable, highlighting fundamental limits in classifying their dynamical behaviors.

Contribution

It establishes the undecidability of key decision problems for cellular automata, including conjugacy and embedding, by linking them to entropy and conjugacy properties.

Findings

Conjugacy and related problems are undecidable for one-dimensional one-sided cellular automata.

Pairs with different entropy levels cannot be strongly conjugate, and this is undecidable.

The results extend to two-sided cellular automata, showing broad undecidability.

Abstract

Two cellular automata are strongly conjugate if there exists a shift-commuting conjugacy between them. We prove that the following two sets of pairs $(F,G)$ of one-dimensional one-sided cellular automata over a full shift are recursively inseparable: (i) pairs where $F$ has strictly larger topological entropy than $G$, and (ii) pairs that are strongly conjugate and have zero topological entropy. Because there is no factor map from a lower entropy system to a higher entropy one, and there is no embedding of a higher entropy system into a lower entropy system, we also get as corollaries that the following decision problems are undecidable: Given two one-dimensional one-sided cellular automata $F$ and $G$ over a full shift: Are $F$ and $G$ conjugate? Is $F$ a factor of $G$? Is $F$ a subsystem of $G$? All of these are undecidable in both strong and weak variants (whether the homomorphism is required to commute with the shift or not, respectively). It also immediately follows that these results hold for one-dimensional two-sided cellular automata.