Decidability, complexity and index sets of classes of computable models

We will discuss the problems about decidability of models with different properties. The first problem is connected with existence of decidable presentations for models. The second class is problems about autostability and another equivalence relations on computable models. We will consider these problems on the base of approach suggested by S. S. Goncharov and J.Knight via computable numberings and index sets of classes numbers in some computable numberings.
We will give the results about complexity of characterization of autostable relative to strong constructivizations computable models in differen classes of models and slgebraic structures. We will consider the connection of complexity definable relations of computable models with computable numberings in general sense relative to classification of index sets [1], [4], [3], [5].