# Random Subgroups of Rationals

## TL;DR

This paper explores the algorithmic randomness of subgroups of rationals, analyzing their logical properties and learnability, revealing decidability results and limitations in syntactic identification of their finitely generated subgroups.

## Contribution

It introduces a notion of algorithmic randomness for rational subgroups and studies their model-theoretic, recursion-theoretic, and learnability properties, connecting generator complexity with subgroup learnability.

## Key findings

- The theory of the subgroup coincides with that of integers and is decidable.
- There exists a generating sequence with a co-recursively enumerable word problem.
- Finitely generated subgroups are learnable in the limit but not syntactically identifiable.

## Abstract

This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of $(G,+)$; second, what learnability properties can one extract from $G$ and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of $(G,+)$ coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for $G$ with respect to any generating sequence for $G$ is not even semi-decidable, one can build a generating sequence $\beta$ such that the word problem for $G$ with respect to $\beta$ is co-recursively enumerable (assuming that the set of generators of $G$ is limit-recursive). In regard to the second question, it is proven that there is a generating sequence $\beta$ for $G$ such that every non-trivial finitely generated subgroup of $G$ is recursively enumerable and the class of all such subgroups of $G$ is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of $G$ is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of $G$ cannot be syntactically identified in the limit with respect to any generating sequence for $G$. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.