  Cellular automata are topological dynamical systems. We consider the problem
of deciding whether two cellular automata are conjugate or not. We also
consider deciding strong conjugacy, that is, conjugacy by a map that commutes
with the shift maps. We show that the following two sets of pairs of
one-dimensional one-sided cellular automata are recursively inseparable: (i)
pairs where the first cellular automaton has strictly higher entropy than the
second one, and (ii) pairs that are strongly conjugate and both have zero
topological entropies.
  This implies that the following decision problems are undecidable: Given two
one-dimensional one-sided cellular automata $F$ and $G$: 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).
  We also prove the same results for reversible two-dimensional cellular
automata.
