Ekaterina Fokina

We study classes of countable graphs where every member does not contain a given finite graph as an induced subgraph -- denoted by $\mathsf{Free}(\mathcal{G})$ for a given finite graph $\mathcal{G}$. Our main results establish a structural dichotomy for such classes: If $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is on top under effective bi-interpretability, implying that the members of $\mathsf{Free}(\mathcal{G})$ exhibit the full range of structural and computational behaviors. In contrast, if $\mathcal{G}$ is an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is structurally simple, as witnessed by the fact that every member satisfies the computable embeddability condition. This dichotomy is mirrored in the finite setting when one considers combinatorial and complexity-theoretic properties. Specifically, it is known that $\mathsf{Free}(\mathcal{G})^{fin}$ is complete for graph isomorphism and not a well-quasi-order under embeddability whenever $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, while in all other cases $\mathsf{Free}(\mathcal{G})^{fin}$ forms a well-quasi-order and the isomorphism problem for $\mathsf{Free}(\mathcal{G})^{fin}$ is solvable in polynomial time.

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.

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $Σ_2$ sentences.

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(τ)$ be the set of countable structures with universe $ω$ in vocabulary $τ$ topologized by the Scott topology. We show that an invariant set $X \subseteq Mod(τ)$ is $Π^0_α$ in the effective Borel hierarchy of this topology if and only if it is definable by a $Π^p_α$ - formula, a positive $Π^0_α$ formula in the infinitary logic $L_{ω_1,ω}$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $K$ be positively computably embeddable in $K'$ by $Φ$, then for every $Π^p_α$ formula $ξ$ in the vocabulary of $K'$ there is a $Π^p_α$ formula $ξ^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models ξ^\star$ if and only if $Φ(A) \models ξ$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is $\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their $\Sigma^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable.