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Two cellular automata are strongly conjugate if there exists ashift-commuting conjugacy between them. We prove that the following two sets ofpairs $(F,G)$ of one-dimensional one-sided cellular automata over a full shiftare recursively inseparable: (i) pairs where $F$ has strictly largertopological entropy than $G$, and (ii) pairs that are strongly conjugate andhave zero topological entropy.  Because there is no factor map from a lower entropy system to a higherentropy one, and there is no embedding of a higher entropy system into a lowerentropy system, we also get as corollaries that the following decision problemsare 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 weakvariants (whether the homomorphism is required to commute with the shift ornot, respectively). It also immediately follows that these results hold forone-dimensional two-sided cellular automata.

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