Slices and levels of extensions of the minimal logic

We consider two classifications of extensions of Johansson's minimal logic J. Logics and then calculi are divided into levels and slices with numbers from 0 to $\omega$. We prove that the first classification is strongly decidable over J, i.e., from any finite list $Rul$ of axiom schemes and inference rules, we can effectively compute the level number of the calculus $(J+Rul)$. We prove the strong decidability of each slice with finite number: for each $n$ and arbitrary finite $Rul$, we can effectively check whether the calculus $(J+Rul)$ belongs to the nth slice.