Conjugacy of onedimensional onesided cellular automata is undecidable
Abstract
Two cellular automata are strongly conjugate if there exists a shiftcommuting conjugacy between them. We prove that the following two sets of pairs $(F,G)$ of onedimensional onesided 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 onedimensional onesided 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 onedimensional twosided cellular automata.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.08111
 arXiv:
 arXiv:1710.08111
 Bibcode:
 2017arXiv171008111J
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Formal Languages and Automata Theory;
 Mathematics  Dynamical Systems;
 37B15;
 68Q80;
 20F10
 EPrint:
 12 pages, 2 figures, accepted for SOFSEM 2018