Computable classes of constructivizations for models of finite computability type

Computable classes of weak constructivizations are studied for models admitting strong constructivizations. We prove that for strongly constnictivizable models $n$-complete in some finite expansion with constants but $(n+1)$-complete in any expansion with constants, it is possible, given an arbitraty class of constructivizations, to construct effectively a constructivization beyond the class which is not an $(n+1)$-constructivization, i.e., whose $(n+1)$-restricted theory is not decidable in any expansion with constants for indices.