Complexity of Categorical Theories with Computable Models

M. Lerman and J. Scmerl specified some sufficient conditions for computable models of countably categorical arithmetical theories to exist. More precisely, it was shown that if $T$ is a countably categorical arithmetical theory, and the set of its sentences beginning with an existential quantifier and having at most $n+1$ alternations of quantifiers is $\Sigma_{n+1}^0$ for any $n$, then $T$ has a computable model. J. Night improved this result by allowing certain uniformity and omitting the requirement that $T$ is arithmetical. However, all of the known examples of theories of $\aleph_0$-categorical computable models had low level of algorithmic complexity, and whether there are theories that would satisfy the above conditions for sufficiently large $n$ was unknown. This paper will include such examples.