We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $-formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $\Sigma_1$-fragment of the theory Th(R) is decidable.

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.

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

A Turing degree d is the degree of categoricity of a computable structure ${\cal S}$ if d is the least degree capable of computing isomorphisms among arbitrary computable copies of ${\cal S}$. A degree d is the strong degree of categoricity of ${\cal S}$ if d is the degree of categoricity of ${\cal S}$, and there are computable copies ${\cal A}$ and ${\cal B}$ of ${\cal S}$ such that every isomorphism from ${\cal A}$ onto ${\cal B}$ computes d. In this paper, we build a c.e. degree d and a computable rigid structure ${\cal M}$ such that d is the degree of categoricity of ${\cal M}$, but d is not the strong degree of categoricity of ${\cal M}$. This solves the open problem of Fokina, Kalimullin, and Miller [13].

For a computable structure ${\cal S}$, we introduce the notion of the spectral dimension of ${\cal S}$, which gives a quantitative characteristic of the degree of categoricity of ${\cal S}$. We prove that for a nonzero natural number N, there is a computable rigid structure ${\cal M}$ such that $0\prime$ is the degree of categoricity of ${\cal M}$, and the spectral dimension of ${\cal M}$ is equal to N.

We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure $\mathcal{S}$ , the SC-autostability spectrum of $\mathcal{S}$ is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of $\mathcal{S}$ . The degree of SC-autostability for $\mathcal{S}$ is the least degree in the spectrum (if such a degree exists).

We prove that for a computable successor ordinal α, every Turing degree c.e. in and above 0 (α) is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above 0 (2β+1) is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative to n-constructivizations.