Abstract 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.

Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega $ , $\zeta $ , and $\eta $ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $\omega $ . If $\mathcal {L}$ is a computable copy of $\omega $ that is computably isomorphic to the usual presentation of $\omega $ , then every cohesive power of $\mathcal {L}$ has order-type $\omega + \zeta \eta $ . However, there are computable copies of $\omega $ , necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $\omega + \zeta \eta $ . For example, we show that there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \eta $ . Our most general result is that if $X \subseteq \mathbb {N} \setminus \{0\}$ is a Boolean combination of $\Sigma _2$ sets, thought of as a set of finite order-types, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ , where $\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ denotes the shuffle of the order-types inXand the order-type $\omega + \zeta \eta + \omega ^*$ . Furthermore, ifXis finite and non-empty, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X)$ .

Abstract There is a Turing computable embedding $\Phi $ of directed graphs $\mathcal {A}$ in undirected graphs (see [15]). Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs $\mathcal {A}$ , these formulas interpret $\mathcal {A}$ in $\Phi (\mathcal {A})$ . It follows that $\mathcal {A}$ is Medvedev reducible to $\Phi (\mathcal {A})$ uniformly; i.e., $\mathcal {A}\leq _s\Phi (\mathcal {A})$ with a fixed Turing operator that serves for all $\mathcal {A}$ . We observe that there is a graph G that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $\Sigma _2$ formulas. Any graph can be interpreted in a linear ordering using computable $\Sigma _3$ formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{\omega _1\omega }$ -formulas that, for all G, interpret the input graph G in the output linear ordering $L(G)$ . Harrison-Trainor and Montalbán [7] have also shown this, by a quite different proof.

We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion} provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some $\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some $\mathcal{B}\cong\mathcal{A}$. Jockusch and Soare \cite{JS} showed that there are low linear orderings without computable copies, but Downey and Jockusch \cite{DJ} showed that every Boolean algebra admits strong jump inversion. More recently, D.\ Marker and R.\ Miller \cite{MM} have shown that all countable models of $DCF_0$ (the theory of differentially closed fields of characteristic $0$) admit strong jump inversion. We establish a general result with sufficient conditions for a structure $\mathcal{A}$ to admit strong jump inversion. Our conditions involve an enumeration of $B_1$-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no $1$-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of $DCF_0$. This also yields the fact that the saturated model of $DCF_0$ has a decidable copy.