Solvability of the problem of completeness of automaton basis depending on its boolean part

We consider the problem of completeness of systems of automaton functions with operations of superposition and feedback of the form $\Phi\cup\nu,$ where $\Phi\subseteq P_2$, $\nu$ is finite. The solution of this problem leads to separation of the lattice of closed Post classes into strong ones (whose presence in the studied system guarantees the solvability of the completeness problem of finite bases) and weak ones (whose presence in the studied system does not guarantee this). It turns out that the classifications of bases by the properties of completeness and A-completeness coincide. The paper describes this classification.