arXiv subjects
Publications and source records attributed to Alexandra Soskova.
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 improve on and generalize a 1960 result of Maltsev. For a field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is a copy of $F$ defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair $(u,v)$ as parameters. We show that $F$ is interpreted in $H(F)$ using computable $Σ_1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of $F$ are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of $F$ represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of $F$ in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.
There is a Turing computable embedding $Φ$ of directed graphs $A$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform interpretation; i.e., for all directed graphs $A$, these formulas interpret $A$ in $Φ(G)$. It follows that A is Medvedev reducible to $Φ(A)$ uniformly; i.e., there is a fixed Turing operator that serves for all $A$. 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 $Σ_2$ formulas. Any graph can be interpreted in a linear ordering using computable $Σ_3$ formulas. Friedman and Stanley gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{ω_1,ω}$ formulas that, for all $G$, interpret the input graph $G$ in the output linear ordering $L(G)$. Harrison-Trainor and Montalbán have also shown this, by a quite different proof.
Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $Π_{C}% \mathcal{L}$ for familiar computable linear orders $\mathcal{L}$. If $% \mathcal{L}$ is isomorphic to the ordered set of natural numbers $\mathbb{N}$ and has a computable successor function, then $Π_{C}\mathcal{L}$ is isomorphic to $\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$ Here, $+$ stands for the sum and $\times $ for the lexicographical product of two orders. We construct computable linear orders $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ isomorphic to $\mathbb{N},$ both with noncomputable successor functions, such that $Π_{C}\mathcal{L}_{1}\mathbb{\ }$is isomorphic to $\mathbb{N}+% \mathbb{Q}\times \mathbb{Z}$, while $Π_{C}\mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $Π_{2}^{0}$ and $Σ_{2}^{0}$ sentences, we provide new examples of $Π_{3}^{0}$ sentences $Φ$ and computable structures $% \mathcal{M}$ such that $\mathcal{M}\vDash Φ$ while $Π_{C}\mathcal{M}% \vDash \urcorner Φ.$