Let L be a computable vocabulary, let XL be the space of L-structures with universe ω and let $f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all $x,y \in {2^\omega }$, if $x{ \equiv _h}y$ then $f\left( x \right) \cong f\left( y \right)$. One of the following two properties must hold. (1) The Scott rank of f (0) is $\omega _1^{CK} + 1$. (2) For all $x \in {2^\omega },f\left( x \right) \cong f\left( 0 \right)$.

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure ${\cal S}$, there exists a countable field ${\cal F}$ of arbitrary characteristic with the same essential computable-model-theoretic properties as ${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle α ∈ 2ω, we can ask the same question relativized to α. A negative answer for every α implies Vaught's Conjecture for Lω1ω.

For a countable structure , the spectrum is the set of Turing degrees of isomorphic copies of . For a complete elementary first order theory T, the spectrum is the set of Turing degrees of models of T. We answer a question from [1] by showing that there is an atomic theory T whose spectrum does not match the spectrum of any structure.