Finiteness of the set of automaton Post bases with a decidable completeness problem

The bases under consideration are those of the form $M=\Phi\cup\nu$, where $\Phi$ is a Post class and $\nu$ is a finite system of automaton functions. It is shown that the sets of Post classes $\Phi$ for which the completeness problem for $\Phi\cup\nu$ is solvable, respectively, the $A$-completeness problem for $\Phi\cup\nu$ is solvable, are finite.