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. (shrink)

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs (...) where possible, followed by a survey of recent advances. (shrink)

This paper is devoted to fill the gap in studying logics for biprobability structures. We introduce the logic equation image with two conditional expectation operators and prove the completeness theorem. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The aim of the paper is to prove tha analytic completeness theorem for a logic LAs with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.